Issuu

Contents 1

Whole numbers 1.01

Addition and subtraction

1.02

Multiplication

1.03

Division

1.04

Order of operations with whole numbers

1.05

Check reasonableness of calculations

Investigation: How long does it take to count to a million?

1.06

Number calculations with technology

1.07

Write mathematical statements

Chapter 1 review

Powers, factors and multiples

2.01

Number operation properties

2.02

Square numbers

2.03

Square roots

2.04

Powers of whole numbers

2.05

Powers of ten

2.06

Factors and multiples

Investigation: The sieve of Eratosthenes

Prime factorisations

Investigation: Celtic knots

HCF and LCM

Chapter 2 review

2.07 2.08

Algorithms and coding

102

3.01

Solve problems using an algorithm

104

3.02

Following algorithms

113

3.03

Branching

118

3.04

Iterations

124

3.05

Flow charts

131

3.06

Algorithms in code

141

3.07

Loops

147

Chapter 3 review

158 Contents mathspace.co

iii

Integers 164 4.01

Integers and the number line

166

4.02

Compare integers

173

Investigation: Life-size number line

178

Add and subtract integers

180

Chapter 4 review

188

4.03

Fractions 190 5.01

Identify, order and compare fractions

192

5.02

Negative fractions

200

5.03

Equivalent and simplified fractions

206

Investigation: Fraction flowers

216

Add and subtract fractions

218

Investigation: Egyptian fractions

226

5.05

Fractions of a quantity

228

5.06

Multiply fractions

234

5.07

Divide fractions

240

5.08

Order of operations with fractions

246

Chapter 5 review

251

5.04

Decimals 254 6.01

Decimals and place value

256

6.02

Add and subtract decimals

262

6.03

Convert between fractions and decimals

268

6.04

Multiply decimals

277

6.05

Divide decimals

283

6.06

Round decimals

288

Chapter 6 review

296

Percentages

298

7.01

Fractions, decimals and percentages

300

7.02

Compare percentages

308

7.03

Express a quantity as a percentage

311

7.04

Percentages of a quantity

316

7.05

Percentages with technology

323

Chapter 7 review

329

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ratios 332 8.01

Identify ratios

334

8.02

Equivalent and simplified ratios

339

8.03

Find part or whole measures

346

Chapter 8 review

354

Algebra 356 9.01

Introduction to variables

358

9.02

Build algebraic expressions

363

9.03

Substitution

369

9.04

Collect like terms

374

Chapter 9 review

380

Equations 382 10.01

Introduction to equations

384

10.02

Write rules for relationships

391

10.03

One-step equations

400

10.04

Two-step equations

405

Chapter 10 review

415

Number plane

418

11.01

The number plane

420

11.02

Directed numbers on the number plane

429

11.03

Graphs of linear relationships

438

Chapter 11 review

448

Angles, lines, and shapes

452

12.01

Measure angles

454

12.02

Parallel lines and transversals

463

12.03

Triangles

475

12.04

Quadrilaterals

485

Investigation: Sort shapes with algorithms

494

Chapter 12 review

501

Contents mathspace.co

Length and area 13.01

Units of length

508

13.02

Perimeter

514

13.03

Perimeter of composite shapes

524

13.04

Area of rectangles and squares

530

13.05

Area of triangles

539

13.06

Area of parallelograms

546

13.07

Area of composite shapes

552

Investigation: Explore circles

562

Circumference of a circle

565

Chapter 13 review

573

13.08

Solids and volume

576

14.01

Nets of solids

578

14.02

Visualise solids

583

14.03

Volume using unit cubes

601

14.04

Units of volume

605

14.05

Volume of right prisms

610

14.06

Attributes in measurement formulas

619

Chapter 14 review

633

Transformations 638 15.01

Transformation of shapes

640

15.02

Translations

646

15.03

Reflections

655

15.04

Rotations

663

15.05

Multiple transformations

673

Investigation: Tessellations

685

Line and rotational symmetry

688

Chapter 15 review

697

15.06

506

Mathspace Victorian Curriculum – Year 7 mathspace.co

Probability 702 16.01

Language of probability

704

16.02

Probability as a number

712

16.03

Theoretical probability

718

16.04

Experimental probability

724

Investigation: Simulations

732

Chapter 16 review

737

Statistics

740

17.01

Types of data

742

17.02

Frequency tables

747

17.03

Summarise data

753

17.04

Outliers

762

17.05

Dot plots and column graphs

769

17.06

Stem-and-leaf plots

783

17.07

Pie charts

791

17.08

Line graphs

799

Investigation: Statistical investigations

809

Chapter 17 review

811

Answers

816

Contents mathspace.co

vii

Big ideas Arithmetic skills, including order of operations, estimation, technology, and symbolic expressions, are essential tools for problem - solving and building a strong foundation for advanced mathematical concepts.

1 Whole numbers

Chapter outline 1.01 Addition and subtraction 1.02 Multiplication 1.03 Division 1.04 Order of operations with whole numbers 1.05 Check reasonableness of calculations Investigation: How long does it take to count to a million? 1.06 Number calculations with technology 1.07 Write mathematical statements Chapter 1 review

4 9 15 19 24 29 32 35 41

1.02

Multiplication After this lesson, you will be able to… • multiply multi-digit numbers. • use different methods to multiply small and large numbers.

Multiplication with area models The expression 5 × 3 results in the same total as repeatedly adding 5 three times: 5 + 5 + 5 = 15. The expression 5 × 3 tells us in a more eﬃcient way that we have 3 groups of 5. We can connect multiplication and area by modelling 5 × 3 in a grid of squares. We can see that we need 15 small squares to cover a rectangle that is 5 units long and 3 units high:

1 1

5 × 3 = 15

11 12 13 14 15 5

Just like with addition, we can use the commutative law to change the order in which numbers appear in the expression, but that doesn’t change the result. Here we have a rectangle that is 3 units long and 5 units high and we can see it uses the same number of small squares to cover it: The answer to a multiplication is called the product. The product of 3 and 5 is 15.

3 × 5 = 15

Interactive exploration

Discover this concept in action online

mathspace.co

We can use an area model to visualise multiplying larger numbers. We can draw a large rectangle that is divided into smaller rectangles using place values, like ones, tens, hundreds etc. Looking at the area model we can also see that 5 × 3 has the same value as 3 × 5. This is called the commutative property, it means that multiplications can be done in any order.

1.02 Multiplication mathspace.co

Example 1 Use the area model to find 76 × 68. a Fill in the areas of each rectangle. ×

60 8

Create a strategy

Apply the idea

Multiply the number at the top of the column by the number on the left of the row.

Area 1 = 60 × 70 = 4200 Area 2 = 60 × 6 = 360 Area 3 = 8 × 70 = 560 Area 4 = 8 × 6 = 48

Multiply 6 tens by 7 tens Evaluate Multiply 6 tens by 6 Evaluate Multiply 8 by 7 tens Evaluate Multiply 8 by 6 Evaluate

Filling in the areas of each rectangle, we have × 60 8

70 4200 560

b What is the total area of all four rectangles?

Create a strategy Add all the areas calculated in part (a).

Apply the idea Total area = 4200 + 560 + 360 + 48 = 5168

Add all the values Evaluate

c So what is 76 × 68?

Create a strategy

Apply the idea

Use the answer calculated on part (b).

76 × 68 = 5168

Mathspace Victorian Curriculum – Year 7 mathspace.co

6 360 48

1.02 Practice questions Understanding

Explorer 1–2

Adventurer

Trailblazer

1–2

Explain using a diagram of a rectangle why 5 × 3 is the same as 3 × 5.

Tom made an area model to calculate 12 × 8. How does Tom find the answer to 12 x 8 from this? 8

Fluency Ex 1

3–4, 7a-h, 8a-d, 9a-d, 10a-d

3, 5, 7i-l, 8e-h, 9e-h, 10c-f

5–6, 7i-l, 8i-l, 9e-l, 10cgh

Let’s calculate 50 × 83 using an area model.

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 50 × 83?

Let’s calculate 6 × 608 using an area model. a

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 6 × 608?

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 83 × 660?

600

Let’s calculate 83 × 660 using an area model. a

80 3

Let’s calculate 87 × 974 using an area model.

900

Find the areas of each of the smaller rectangles.

Using the smaller areas found in part (a), what is 87 × 974?

1.02 Multiplication mathspace.co

80 7

Calculate: a

2×3

7×1

9×4

1×8

5×5

6×2

3×7

3 × 10

2 × 15

11 × 7

12 × 3

5 × 12

Ex 2

Calculate: a

6 × 30

45 × 6

87 × 3

39 × 9

432 × 9

789 × 3

567 × 4

876 × 5

5555 × 3

1234 × 9

1298 × 2

3569 × 3

Calculate: a

4000 × 70

500 × 60

10 000 × 120

700 × 30

234 × 17

678 × 23

842 × 15

741 × 14

1234 × 19

3569 × 13

5873 × 42

9326 × 51

Calculate: a

2×5×9

3×2×8

10 × 7 × 9

12 × 2 × 2

4 × 2 × 15

20 × 10 × 6

12 × 5 × 4

15 × 15 × 5

Reasoning

11–12

11, 12ac

Explain why 5 × 6 is the same as 10 × 3.

Write these expressions as a sum of two products. Do not evaluate the product. a

8 × 75

11 × 92

Problem-solving 13a, 14–17 13

82 × 64

13ab, 14–18

475 × 73

13b-d, 14–20

Find two numbers that match these criteria. a

Product of 6 and sum of 5

Product of 24 and difference of 10

Product of 36 and sum of 13

Product of 1296 and difference of 0

Tristan works 8 hours a day, 6 days a week; and Sheila works 9 hours a day, 5 days a week. a

How many hours does Sheila work each week?

Who works more hours each week?

A rock climber descends from the top of a cliff face to the ground in 5 stages, dropping 106 metres each time. What is the height of the cliff face?

Maximilian builds handmade designer dining tables, each table sells for $4865. He has just received a new order for 6 tables. How much are the total sales if all 6 tables are sold?

A showroom has 155 electric bikes. Each electric bike cost $3500. What is the total cost of all the electric bikes?

A tyre company can produce 2512 tyres a day. How many tyres can they produce in 360 days?

Mathspace Victorian Curriculum – Year 7 mathspace.co

A group of countries have decided to donate $3 508 060 every year towards improving freshwater availability. How much money will the group donate over 16 years?

A carton holds 36 packets of biscuits. Each packet has 12 biscuits. How many biscuits can be packed in 55 cartons?

1.03

Division After this lesson, you will be able to… • divide multi-digit numbers. • use different methods to divide small and large numbers.

Division Division uses the ÷ symbol and is essentially the opposite of multiplication. The expression 15 ÷ 3 means “15 split into 3 equal groups”, which is the same as the number 5.

Interactive exploration

Discover this concept in action online

mathspace.co

A division equation is (almost) always related to a multiplication equation: 20 ÷ 5 = 4 is related to 4 × 5 = 20 which is also related to 20 ÷ 4 = 5. When dividing larger numbers, we will use a procedure called short division. We write the number we are dividing by (the divisor) on the left and the number we are dividing (the dividend) on the right and draw a division bracket between them. We will be writing our result (called the quotient) above the bracket.

With short division, we are still dividing the dividend into equal groups of the divisor. However, we consider each place value of the dividend separately, starting with the largest place values.

1.03 Division mathspace.co

Example 1 Evaluate 1616 ÷ 4.

Create a strategy Use short division.

Apply the idea Form the short division tableau with 1616 as the dividend and 4 as the divisor. We cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16. 16 ÷ 4 is 4, so we put the 4 in the hundreds place.

Again, we cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16. We put a 0 in the tens place.

16 ÷ 4 is 4, so we put the 4 in the ones place. Since there is no remainder at this step, we conclude that 1616 ÷ 4 = 404

Example 2 Find the quotient and remainder of 1890 ÷ 8.

Create a strategy Use short division.

Apply the idea Form the short division tableau with 1890 as the dividend and 8 as the divisor. We cannot divide 1 by 8. so we carry the 1 across to the next number along to form 18.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Fluency Ex 1

Ex 2

3c-i, 4d-i, 5d-i, 6d-i

3f-l, 4g-l, 5g-l, 6g-l

Calculate: a

72 ÷ 8

108 ÷ 6

270 ÷ 9

360 ÷ 4

546 ÷ 6

828 ÷ 3

1300 ÷ 5

2056 ÷ 8

2996 ÷ 4

4786 ÷ 2

6349 ÷ 7

9207 ÷ 9

Calculate: a

480 ÷ 12

620 ÷ 10

888 ÷ 12

1800 ÷ 15

2655 ÷ 59

3872 ÷ 88

7850 ÷ 25

8372 ÷ 28

22 644 ÷ 34

31 512 ÷ 52

36 300 ÷ 60

39 000 ÷ 65

5555 ÷ 6

Find the quotient and the remainder for these divisions: a

3, 4a-ei, 5a-ei, 6a-ei

1242 ÷ 7

3126 ÷ 4

4108 ÷ 5

6125 ÷ 9

7134 ÷ 7

7318 ÷ 8

9312 ÷ 5

12 314 ÷ 4

23 968 ÷ 5

29 876 ÷ 8

31 597 ÷ 3

Find the quotient and the remainder for these divisions: a

435 ÷ 18

539 ÷ 73

728 ÷ 55

819 ÷ 37

3126 ÷ 29

4138 ÷ 39

6346 ÷ 22

9721 ÷ 18

10 998 ÷ 22

15 206 ÷ 31

21 491 ÷ 35

37 311 ÷ 29

Reasoning

7–9

12 ÷ 3 is not equal to 3 ÷ 12. If the two numbers are x and y, when is x ÷ y equal to y ÷ x?

Why is it important to avoid making the divisor equal to zero?

Rewrite 28 ÷ 3 in the form of divisor × quotient + remainder, explain your thinking.

Problem-solving 10–13

10, 12–14

13–17

What is the smallest number greater than 30 that leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 7? Explain your answer.

Sandy baked 320 biscuits and divided them equally into 20 packets. How many biscuits did Sandy put in each packet?

Glenn needs 8 oranges to make a glass of orange juice. If Glenn has 400 oranges, how many glasses of orange juice can he make?

Andrea wants to put her 425 books in boxes. If each box can hold 25 books, how many boxes does she need for her books?

A school is able to gather 2068 canned items from a donation drive. Each of the 47 recipients will receive the same number of canned items. How many items does each recipient receive?

Mathspace Victorian Curriculum – Year 7 mathspace.co

1.04 Practice questions Understanding

Explorer

Adventurer

1–2

Trailblazer

1–2

What is the order of operations when performing arithmetic with whole numbers?

The ﬁrst step when calculating 2 × (7 + 3) is to do 7 + 3. What is the ﬁrst step in each of these problems? a

3×5+1

10 − 4 × 2

9÷3+5×2

7−2+6−1

Fluency Ex 1

3a-h, 4a-gi, 5, 8ac

Ex 3

6+4×2

8 × (8 − 6)

6 + 32 ÷ 4

24 ÷ 6 × 4

22 − 18 ÷ 6

37 − 35 ÷ 7 + 8

36 ÷ 4 × 8 + 6

18 ÷ 2 + 7

8×5−3×2

7+4×3−6

11 + 10 ÷ 5 × 8

Calculate: a

(7 + 9) × 10

7 × (5 + 4) − 50

34 − [29 + (12 ÷ 3)]

(8 − 2) × 8

(7 − 2) × (6 − 2)

(10 + 2) × (7 − 2)

[69 − (15 + 24)] ÷ 3 + 8 × 6

(11 + 5) ÷ (6 − 2)

[(8 + 2) × 6] ÷ 5

{[100 − (5 + 15)] + 8} ÷ 8

[7 × (6 − 3)] × [(8 + 2) ÷ 5]

[1 + (6 × 3 − 9)] ÷ [20 − (100 ÷ 10)]

m 8 × 0 ÷ 8 ÷ 1

12 × 2 ÷ 2 ÷ 3

Given the expression 6 + 19 − 4 × 3: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

6 + (19 − 4) × 3

6 + 19 − (4 × 3)

iii

(6 + 19 − 4) × 3

Given the expression 6 × 15 − 5 + 3: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

3g-n, 4g-n, 7, 8bdfh

Calculate:

m Ex 2

3c-j, 4e-kn, 6, 8aceg

6 × 15 − (5 + 3)

Mathspace Victorian Curriculum – Year 7 mathspace.co

6 × (15 − 5) + 3

iii

(6 × 15) − 5 + 3

Given the expression 35 − 6 × 3 + 2: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

35 − (6 × 3) + 2

35 − (6 × 3 + 2)

iii

(35 − 6) × 3 + 2

35 − 6 × (3 + 2)

Insert a pair of brackets in the appropriate location to make these statements true. a

12 − 6 + 6 ÷ 6 = 10

5 + 10 × 4 − 5 × 4 = 40

4 + 5 × 13 − 5 − 4 = 40

5 × 15 − 4 ÷ 2 + 8 = 65

18 × 4 − 1 + 4 − 18 = 40

23 − 10 × 3 + 8 × 9 − 2 = 109

15 ÷ 3 + 7 × 4 + 2 × 6 = 257

30 − 4 × 5 ÷ 5 − 3 = 20

9, 11

10–12

Reasoning 9

Given the expression 8 + 3 × 6: a

Determine whether these expressions result in the same value as the given expression: i

3×6+8

iii

(8 + 3) × 6

8 + (3 × 6)

How do the brackets in (8 + 3) × 6 change the order of operations from the given expression 8 + 3 × 6?

Determine whether these expressions result in the same value as the given expression: i

7 ÷ (5 + 9)

9+7÷5

iii

5+9÷7

(7 ÷ 5) + 9

How do the brackets in 7 ÷ (5 + 9) change the order of operations from the given expression 7 ÷ 5 + 9?

Given the expression 58 − 7 × 8: a

Determine whether these expressions result in the same value as the given expression: i

6×8+3

Given the expression 7 ÷ 5 + 9: a

9–10

58 − (7 × 8)

(58 − 7) × 8

iii

58 × 5 − 7

How do the brackets in (58 − 7) × 8 change the order of operations from the given expression 58 − 7 × 8?

Given the expression 7 ÷ 4 − 1: a

Determine whether these expressions result in the same value as the given expression: i

4−1÷7

(7 ÷ 4) – 1

iii

7 ÷ (4 − 1)

How do the brackets in 7 ÷ (4 − 1) change the order of operations from the given expression 7 ÷ 4 − 1?

1.04 Order of operations with whole numbers mathspace.co

Problem-solving 13ac, 14–15 13

13ace, 15–16

13bdf, 15–17

Fill in the boxes with the right operations to make these statements true. a c e

7 ⬚ 3 ⬚ 4 − 3 = 14

20 ⬚ 5 ⬚ 4 − 6 = 10

15 ⬚ 3 ⬚ 2 − 4 = 43

12 ⬚ 3 ⬚ 2 × 4 = 288

9 × 2 ⬚ 5 ⬚ 3 − 1 = 32

30 × 6 ⬚ 3 ⬚ 2 = 58

A gardener has 5 bags with 12 apples in each bag. He discovers there are 2 rotten apples in each bag. How many good apples does he have?

Before going back to school a student purchases 4 pens costing $3 each and 5 pencils costing $2 each. Find the total spent by the student.

A teacher has 50 lollies in a bag and wants to give 3 lollies to each of her 16 students. How many lollies would the teacher have left?

Josie has 100 marbles in her collection. After picking out her favourite 12, she shared the remainder amongst her 4 friends. How many marbles does each friend receive?

1.05 Check reasonableness of calculations After this lesson, you will be able to… • estimate and check the reasonableness of an answer.

Check reasonableness To estimate, we can round numbers to make the calculation easier. The more we round, the less accurate the estimate will be. To ﬁnd a reasonable estimate, we can round to the nearest 10, 100, or other place value. 10

When rounding to the nearest 10, we are asking whether this number is closer to 10 or 20. In this case, we can see that 17 is closer to 20. So, 17 rounded to the nearest 10 is 20.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Round these numbers to the nearest 100: a

825

276

913

127

166

659

482

995

Round these numbers to the nearest 1000: a

1234

6890

9425

4765

9938

44 987

19 439

66 902

Consider the product of 37 × 94 and determine if each statement is correct or incorrect. Provide the correct statement if necessary. • The product is less than 4000. • The product is more than 2700. • The product is more than 4000. • The product is less than 2700.

Fluency Ex 1

Ex 2

59 − 27 is approximately equal to 30.

68 − 27 is approximately equal to 60.

2883 − 2831 is approximately equal to 80 − 30.

6871 − 5832 is approximately equal to 70 − 30.

3 × 85 will be more than 240.

6+3+3

23 + 42 + 34

93 + 342 + 567

364 + 292 + 311

9+9−7

45 + 12 − 7

68 × 69

Which of these expressions are approximately equal to 10? 7+2+3

9 + 8 − 12

2 + 8 + 19

Which of these expressions are approximately equal to 100? a

4 × 89 will be less than 320.

Determine whether these sums are approximately equal to 10, 100, or 1000:

a 9

5b-d, 6b, 7cd, 9–11, 14

Are these statements accurate? Explain your choice.

a 8

5a-c, 6b, 7bc, 9, 12–14

Are these statements reasonable? Explain your choice.

a 7

5a-c, 6a, 7ab, 8, 12–14

28 + 32 + 43

99 + 88 − 78

131 + 62 + 11

Which of these expressions are approximately equal to 1000? a

452 + 213 + 334

99 + 188 − 68

948 + 827 + 18

946 + 28 + 14 + 19

Determine whether these products approximately equal to 1200: a

33 × 37

31 × 18

Is 821 ÷ 400 approximately equal to 200?

Is 4 ÷ 1221 approximately equal to 300?

Mathspace Victorian Curriculum – Year 7 mathspace.co

203 × 6

Approximate these calculations to the nearest 100: a

45 × 93

Reasoning 15

63 + 75

314 + 78

both values are rounded up?

both values are rounded down?

one value is rounded up and the other down?

5513 ÷ 11

15ac, 16, 17cd, 18, 19ab, 20, 21ac, 22

198 + 47

987 + 332

If you estimate these differences, will the estimate be larger or smaller than the actual answer? 83 − 41

569 − 114

719 − 87

959 − 674

When estimating differences, is it possible to make a generalisation about the size of the estimation compared to the actual answer when: a

both values are rounded up?

both values are rounded down?

the larger value is rounded up and the smaller down?

the larger value is rounded down and the smaller up?

If you estimate these products, will the estimate be larger or smaller than the actual answer? 132 × 43

714 × 98

15 × 41

178 × 9

When estimating products, is it possible to make a generalisation about the size of the estimation compared to the actual answer when: a

both values are rounded up?

both values are rounded down?

one value is rounded up and the other down?

If you estimate these quotients, will the estimate be larger or smaller than the actual answer? a

1621 ÷ 4

15ac, 17bc, 19ad, 21ab

a 20

15, 17ab, 19cd, 21bd

When estimating sums, is it possible to make a generalisation about the size of the estimation compared to the actual answer when:

a 18

178 × 5

If you estimate these sums, will the estimate be larger or smaller than the actual answer? a

275 ÷ 29

148 ÷ 19

599 ÷ 63

125 ÷ 52

When estimating quotients, is it possible to make a generalisation about the size of the estimation compared to the actual answer when: a

both values are rounded up?

both values are rounded down?

the larger value is rounded up and the smaller down?

the larger value is rounded down and the smaller up?

1.05 Check reasonableness of calculations mathspace.co

1.06 Number calculations with technology After this lesson, you will be able to… • use technology to perform operations with whole numbers and check the reasonableness of your answer.

Operations with whole numbers using technology The most common type of technology used for performing operations is the hand-held calculator, so let’s run through the basics. The buttons we will use at the moment are: • The number buttons: 0 - 9

• The equals sign =

• The operations: + , - , × , and ÷

• Round Brackets ( and )

To perform an operation between two numbers with your calculator, • Press the buttons corresponding to the digits of the ﬁrst number, going from left to right • Press the corresponding operation button • Enter the second number the same way as the ﬁrst • Press the equals sign The answer should now appear on the calculator display. If we are using a modern calculator, it will automatically follow the correct order of operations, but if not, we will have to do that for ourselves. It’s always useful to check the reasonableness of your answer. If the answer doesn’t seem right, there’s a good chance it isn’t. Here are some common calculator errors: • Missing a digit from a number • Swapping the digits • Using the wrong operation • Adding an extra digit

Example 1 Use a calculator to evaluate 214 + 443.

Create a strategy

Apply the idea

Press the calculator buttons in order of the digits and symbols in the expression, followed by the equals sign.

Press the buttons: 2 14 + 4 43 = to get: 214 + 443 = 657

Mathspace Victorian Curriculum – Year 7 mathspace.co

Danielle wants to evaluate the expression (224 − 126) ÷ 14. In which order should Danielle push the buttons on her calculator to get the correct answer? A B

224

−

126

224

−

126

Rosey wants to evaluate the expression (789 − 31) × 24. In which order should Rosey push the buttons on her calculator to get the correct answer? A

Fluency

Ex 1

6 Ex 2

31 =

−

5a-d, 6a-c, 7a-e

24 789

5c-f, 6b-d, 7c-g

5e-h, 6c-e, 7f-j

Evaluate: a

214 + 300

99 735 − 37 926

629 + 543

800 ÷ 10

2939 × 46

3563 ÷ 7

5135 ÷ 65

932 × 28

720 ÷ 7 × 7000

44 − (31 + (8 ÷ 2))

34 + 3294 + 909 + 6204

4 × 6 + 86 − 30

4 × 100 000 + 9 × 10 000 + 2 × 100 + 6 × 10

Evaluate: a

Determine whether these equations are true or false: a

7 × (3 + 3) = 14

8÷4×2=1

4 + (5 × 6) = 34

(9 + 7) × 10 = 79

(11 + 9) × 5 = 56

(10 × 7) × 2 = 140

17 − (2 + 4) = 19

23 − (15 − 9) = 17

28 − (16 ÷ 4) = 3

(100 ÷ 10) ÷ 2 = 20

Reasoning 8

−

789

8–10

9–12

11–14

Xavier used his calculator to evaluate 18 × 75 − 8 and got the answer 1018. Is Xavier’s answer correct? If not, explain what he might have done wrong.

Lucy used her calculator to evaluate (20 − 18) × 3 + 19 and got the answer 25. Is Lucy’s answer correct? If not, explain what she might have done wrong.

Maximilian used his calculator to evaluate 3 × 700 and got the answer 210. Is Maximilian’s answer correct? If not, explain what he might have done wrong.

Vincent used his calculator to evaluate 6 × 40 000 and got the answer 2 400 000. Is Vincent’s answer correct? If not, explain what he might have done wrong.

Michael used his calculator to evaluate (2 + 50) ÷ 77 and got the answer

Is Michael’s answer correct? If not, explain what he might have done wrong. 34

Mathspace Victorian Curriculum – Year 7 mathspace.co

Lucy used her calculator to evaluate 20 − (5 + 14) ÷ 21 and got the answer

Is Lucy’s answer correct? If not, explain what she might have done wrong. 14

Brad used his calculator to evaluate (4 × 11 + 9) ÷ 14 − 20 and got the answer

Is Brad’s answer correct? If not, explain what he might have done wrong.

Problem-solving 15–17 15

15–17

18–19

A charity group for tsunami relief has gathered 588 first aid kits. 21 shelters have been affected. The charity decided to distribute the first aid kits equally between these. How many first aid kits does each shelter receive?

A cargo ship is delayed by a storm at sea for two weeks. When it arrives, the crew finds that 3 of the 81 crates of fruit have spoiled due to the delay. They decide to share the remaining good crates equally among their 13 customers, with any extras sold separately. a

Find the number of fruit crates that each customer will receive.

Each crate contained 32 kg of fruit and each kilogram of fruit could be sold for $4. Calculate the loss incurred by the crew due to the spoilage.

A teacher has ﬁfteen bags of lollies. Each bag contains 18 lollies but the teacher eats three lime lollies from each bag before sharing the remainder amongst her 25 students. How many lollies will each student receive?

Gus wants to give away his 120 marbles to his friends. He is going to give half the marbles to Corey. Corey will get three times as many as Alﬁe and Barry will get twice as many as Alﬁe. How many marbles does each friend receive?

In a class of 10 students, two have 2 siblings, four have 1 sibling and one is an only child. If there are 17 siblings in total, how many students have 3 siblings?

1.07 Write mathematical statements After this lesson, you will be able to… • translate written text into mathematical expressions using the four operations of addition, subtraction, multiplication, and division. • describe the relationship between numbers using symbols such as greater than, less than, greater than or equal to, less than or equal to, equal to, and not equal to.

1.07 Write mathematical statements mathspace.co

Fluency Ex 1

Ex 2

The sum of 2 and 7

The difference between 7 and 6

The product of 3 and 10

The quotient of 12 and 4

Four groups of seven

Thirteen is not equal to eleven

4 is less than 10

8 is greater than 5

Write these word statements using mathematical symbols: Fifteen is equal to seven plus eight

Six is equal to sixteen minus ten

Eighty is equal to eight times ten

Five is equal to ﬁfty divided by ten

Nineteen is greater than seven plus ﬁve

Write these word statements using mathematical symbols: a

The difference of 24 and 8, divided by 4

12 times the result of 10 reduced by 6

1 is equal to 12 divided by 3 minus 3

11 times 2 take away 9 is not equal to 10

15 is less than the quotient of 100 and 5

20 is greater than or equal to 2 groups of the sum of 3 and 6

Which option matches the statement “8 more than the quotient of 9 and 3”? B

(2 × 3) − (10 + 12)

(10 − 12) + (2 × 3)

(10 + 12) − (2 × 3)

14 = 7 × (10 − 8)

14 = 7 × 10 − 8

14 = 7 + (10 − 8)

14 = 7 − (10 − 8)

5 × 4 < 25

5 × 4 > 25

5 × 4 ≤ 25

5 × 4 ≥ 25

Which option matches the statement “10 is greater than or equal to double the difference between 11 and 7”? A

(2 × 3) + (10 + 12)

Which option matches the statement “The product of 5 and 4 is less than 25”? A

Which option matches the statement “14 is equal to 7 groups of the difference between 10 and 8”? A

Which option matches the statement “1 is 5 less than the quotient of 24 and 4”? A

Which option matches the statement “The product of 2 and 3 is subtracted from the sum of 10 and 12”? A

3e-h, 4e, 5aef, 8–12

A 7

3c-f, 4e, 5aef, 6–7, 10–11

Write these word statements using mathematical symbols:

3a-d, 4–8

10 ≤ 2 × (11 − 7)

10 ≥ 2 × 11 − 7

10 ≤ 2 × 11 − 7

10 ≥ 2 × (11 − 7)

Given the statement: “The sum of 2 and the quotient of 80 and 10”: a

Write this statement using mathematical symbols.

Find the ﬁnal value of the statement.

1.07 Write mathematical statements mathspace.co

1 Chapter review 1

Insert a pair of brackets in the appropriate location to make these statements true: a

7 × 3 + 9 − 12 + 2 = 20

5 + 10 × 4 − 5 × 4 = 40

9 + 6 ÷ 3 × 4 − 8 = 12

14 ÷ 2 × 4 + 9 − 3 × 2 = 31

Given the expression 4 + 11 − 3 × 2: a

Evaluate the expression.

These expressions are similar to the given expression, except that a pair of brackets has been added. Evaluate these expressions: i

4 + (11 − 3) × 2

4 + 11 − (3 × 2)

iii

(4 + 11 − 3) × 2

Fill in the boxes with the right operation to make these statements true: a c

7 ⬚ 3 ⬚ 6 − 3 = 22

20 ⬚ 2 ⬚ 5 + 3 = 11

8 ⬚ 3 ⬚ 6 − 2 = 28

9 × 2 ⬚ 5 ⬚ 3 − 1 = 32

A student purchases 5 boxes of crayons costing $7 each and 8 sketch pads costing $5 each. Find the total spent by the student.

Round these numbers to the nearest 1000:

1264

8890

7425

2234

9838

43 976

18 439

69 902

Determine whether these statements are reasonable and explain your choice. a

84 + 29 is approximately equal to 110.

76 + 15 is approximately equal to 100.

9134 + 2678 is approximately equal to 9000 + 3000.

7652 − 1389 is approximately equal to 7500 − 1000.

For the product of 47 × 93, determine which of these statements are correct: a

The product is less than 5000.

The product is more than 3600.

The product is more than 5000.

The product is less than 3600.

Determine whether these sums are approximately equal to 10, 100, or 1000: a

29 + 46 + 32

83 + 442 + 667

394 + 292 + 321

5513 ÷ 11

Which of these expressions are approximately equal to 1000? a

652 + 243 + 334

45 + 188 − 68

648 + 227 + 18

646 + 281 + 14 + 19

Approximate these calculations to the nearest 100: a

6+2+3

45 × 93

178 × 5

1621 ÷ 4

Hiro wants to evaluate the expression (24 + 83) × 56. In which order should Hiro push the buttons on his calculator to get the correct answer? A

+ × 24 83 =

Chapter 1 review mathspace.co

Big ideas • Basic rules govern number manipulation, including associative, commutative, and distributive laws, which support accurate mathematical problem-solving. • Mastery of the concepts of factors, multiples, and prime factorisation enables determination of the highest common factor and least common multiple of numbers. Utilising these concepts is crucial when solving problems that involve shared properties and relationships between numbers.

Curriculum Outcomes VC2M7A02 apply the associative, commutative and distributive laws to aid mental and written computation, and formulate algebraic expressions using constants, variables, operations and brackets VC2M7N01 describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems VC2M7N02 represent natural numbers in expanded notation using powers of 10, and as products of powers of prime numbers using exponent notation

2 Powers, factors and multiples Chapter outline 2.01 Number operation properties 2.02 Square numbers 2.03 Square roots 2.04 Powers of whole numbers 2.05 Powers of ten 2.06 Factors and multiples Investigation: The sieve of Eratosthenes 2.07 Prime factorisations Investigation: Celtic knots 2.08 HCF and LCM Chapter 2 review

46 55 61 66 71 76 81 83 89 91 99

The distributive law This is a method we will explore more later when we learn algebra, but it can also help us to perform trickier multiplication questions. For example, if we want to multiply 9 by 43, we could rewrite the larger number 43 as the sum (40 + 3). The multiplication then becomes 9 × (40 + 3). We can then distribute the multiplication over each number in the brackets, like this: 9 × (40 + 3) = (9 × 40) + (9 × 3)

Using the distributive law, we can now expand this as (9 × 40) + (9 × 3), both of which are a lot easier to calculate. We can use a similar strategy to make division questions easier, like 168 ÷ 14. We know that 168 is 140 + 28, which we know are multiples of 14. So we can do: 168 ÷ 14 = (140 + 28) ÷ 14 = (140 ÷ 14) + (28 ÷ 14) = 10 + 2 = 12

Example 4 Given the expression 168 ÷ 7: a Complete the statement: 168 ÷ 7 is the same as (140 + ⬚) ÷ 7

Create a strategy

Find the number to be added to 140 that will give a sum of 168.

Apply the idea 140 + 28 = 168 168 ÷ 7 is the same as (140 + 28) ÷ 7. b Complete the statement: (140 + 28) ÷ 7 is the same as 140 ÷ 7 + ⬚ ÷ 7

Create a strategy

Consider the ﬁrst expression. Find the other number that is being divided by 7.

Apply the idea In (140 + 28) ÷ 7, both 140 and 28 will be divided by 7. (140 + 28) ÷ 7 is the same as 140 ÷ 7 + 28 ÷ 7.

Mathspace Victorian Curriculum – Year 7 mathspace.co

How does the distributive law help simplify calculations? Provide an example.

Is each expression an example of the associative law, commutative law, reordering or distributive law? a

15 + 5 = 5 + 15

54 ÷ 9 ÷ 3 = 54 ÷ 3 ÷ 9

24 + 6 + 2 = 24 + (6 + 2)

126 ÷ 3 = (120 + 6) ÷ 3

(15 + 6) × 4 = (15 × 4) + (6 × 4)

52 − 12 − 24 = 52 − 24 – 12

48 ÷ 2 ÷ 8 = 48 ÷ 8 ÷ 2

Are these statements true or false? a

4 × (3 + 5) = 4 × 3 + 4 × 5

4 + (3 × 5) = 4 × 3 + 4 × 5

(4 × 3) + 5 = 4 × 5 + 3 × 5

(4 + 3) × 5 = 4 × 5 + 3 × 5

5–12, 14, 16–17, 20–21

5–7, 10, 13, 15, 18–19, 21–22

(67 − 19) × 37

67 − 37 − 19

67 − (37 − 19)

35 + 29 + 11 = 35 + (⬚ + ⬚)

Given the expression 2 × 16 × 5: a

Which pair of numbers will be easiest to multiply together ﬁrst?

Complete the statement and then evaluate it: 2 × 16 × 5 = ⬚ × ⬚ × 16

Evaluate 17 + 39 + 23 using reordering.

67 × 37 − 67 × 19

28 + 46 − 16 = 28 + (⬚ − ⬚)

Fill in the boxes to complete the working out using the associative law. 13 × 4 × 5 = ⬚ × (4 × ⬚)

Fill in the boxes to complete the working out using the associative law. a

Ex 2

5–12, 14, 16–17, 20

Which of these expressions is equal to 67 − 19 − 37? A

24 × 6 = 6 × 24

Fluency Ex 1

Helen rearranged the numbers using reordering to make the addition easier. Complete the statement and then evaluate it: 24 + 38 + 16 = ⬚ + 16 + ⬚ Derek is evaluating the subtraction as shown. He noticed that it would be easier to subtract 27 from 47 ﬁrst and rearrange the numbers using reordering. Complete the statement and then evaluate it: 47 − 18 − 27 = ⬚ − 27 − ⬚ Given the expression 56 − 18 − 26: a

Which of the two numbers will be easier to subtract from 56 ﬁrst?

Complete the statement and then evaluate it: 56 − 18 − 26 = 56 − ⬚ − ⬚

Evaluate 57 − 29 − 17 using reordering.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 3

Given the expression 54 + 26 + 38: a

Which pair of numbers will be easiest to add together ﬁrst?

Fill in the boxes to complete the working out. 54 + 26 + 38 = (⬚ + ⬚) + 38

Evaluate 4 × 17 × 5 using regrouping.

Given the expression 35 × 23: a

c 17

Ex 4

Evaluate the expression.

Complete the statement:

26 × 29 is the same as 26 × (⬚ − 1). 26 × (30 − 1) is the same as 26 × 30 − 26 × ⬚.

Evaluate the expression.

9 × 998

7 × 1002

11 × 124

6 × 412

Fill in the boxes to complete his working out, then evaluate the expression. 132 ÷ 11 ÷ 4 = 132 ÷ ⬚ ÷ 11

Given the expression 192 ÷ 6:

Fill in the box to complete the sentence.

192 ÷ 6 = (180 + ⬚) ÷ 6. (180 + 12) ÷ 6 is the same as 180 ÷ 6 + ⬚ ÷ 6.

Evaluate the expression.

Given the expression 119 ÷ 7: a

c 22

35 × 23 is the same as 35 × (20 + ⬚). 35 × (20 + 3) is the same as 35 × ⬚ + 35 × 3.

Dave is evaluating 132 ÷ 11 ÷ 4. He noticed that it would be easier to divide 132 by 4 ﬁrst and rearrange the numbers using reordering.

Complete the statement:

Evaluate: a

Given the expression 26 × 29: a

Complete the statement:

Fill in the box to complete the sentence.

119 ÷ 7 is the same as (140 − ⬚) ÷ 7. (140 − 21) ÷ 7 is the same as 140 ÷ ⬚ − 21 ÷ ⬚.

Evaluate the expression.

Evaluate using the distributive law: a

1616 ÷ 8

396 ÷ 4

522 ÷ 6

585 ÷ 13

2.01 Number operation properties mathspace.co

Reasoning

24–25

23–24

Using an example, explain why the commutative law does not work for division.

Julietta solved the problem as shown: 1

6 × 2 − 1 + 9 = 12 − 1 + 9

= 12 − (1 + 9)

Associative law

= 12 – 10

Evaluate the addition

Evaluate the subtraction

Evaluate the multiplication

Identify and explain Julietta’s mistake. 25

Describe the most eﬃcient method to evaluate 5 × 34 × 200.

Problem-solving 26 26

26–27

27–28

The area of a rectangle is equal to the length multiplied by width. Consider this rectangle: a

Find the area of Area 1.

Find the area of Area 2.

Find the total area.

Write the area calculation of 8 × 22 using the distributive law.

Area 1

Area 2

Consider this rectangle: a

Find the total area.

Find the non-shaded area.

Find the shaded area.

Write the area calculation of 16 × 20 using the distributive law.

Stefano wants to buy 4 shirts that cost $15 each and 3 pairs of pants that cost $25 each. If the store is offering a 10% discount on all purchases over $100, how much will the discount be? Use the distributive law to simplify the calculation.

Mathspace Victorian Curriculum – Year 7 mathspace.co

2.02 Square numbers After this lesson, you will be able to… • write powers using the base and the index. • identify and calculate perfect squares.

Square numbers A square number is a number that can be obtained by multiplying any number by itself. For example, 4 is a square number because it can be obtained by multiplying 2 by itself: 2×2=4 To indicate that a number has been multiplied by itself, we write the number being multiplied, called the base, and then indicate the number of times it is multiplied by itself by writing this number, called the index (also known as the exponent or power), above and to the right of the base, like so: baseindex For example, if we have 3 × 3, we can see that 3 is being multiplied by itself 2 times, so 3 is the base, and the index is 2: 3 × 3 = 32 This can be read as “three to the power of two” or “three raised to the power of 2.” For now, we will just be looking at numbers raised to the power of two, resulting in perfect squares. In the above example 32 can also be referred to as “three squared” and results in the perfect square of 9. Let’s look at the pattern to see what it means to ‘square’ a number: 12

You can see that each formation of dots forms a perfect square. 12 = 1 × 1 = 1 22 = 2 × 2 = 4 32 = 3 × 3 = 9 42 = 4 × 4 = 16 and so on. 2.02 Square numbers mathspace.co

Interactive exploration

Discover this concept in action online

Example 1 Evaluate 92.

Create a strategy Use the fact that squaring a number means multiplying by itself.

Apply the idea 92 = 9 × 9 = 81

Multiply the base by itself Evaluate

Example 2 Evaluate 42 + 92.

Create a strategy Use the fact that squaring a number means multiplying by itself.

Apply the idea 42 + 92 = 4 × 4 + 9 × 9

Multiply the base by itself

= 16 + 81

Evaluate the multiplication

= 97

Evaluate

Example 3 Evaluate

Create a strategy Identify the order of operations.

Mathspace Victorian Curriculum – Year 7 mathspace.co

mathspace.co

6, 7abeh, 8a-d, 9a-im, 10a-d, 11–12

Fluency

Ex 1

Ex 2

State the ﬁrst ﬁve square numbers.

Are the given numbers square numbers?

144

Evaluate: a

122

152

232

502

1002

32 + 72

Evaluate: a

92 + 22 2

7 −3

12 × 22 2

m 8 ÷2 10

7cdfg, 8e-h, 9dh-p, 10e-h, 12–13

Ex 3

7a-d, 8c-f, 9c-im, 10c-f, 11–12

72 + 92

1002 + 102

9 −8

12 − 6

1002 − 102

32 × 42

22 × 102

10 ÷ 2

1002 × 102

15 ÷ 3

402 ÷ 52

(2 + 3)2

Evaluate: a

32 + 5

82 − 3 × 22

(7 − 3)2 + 10

2(32 + 1)2

What would be the area of a square vegetable garden with a side of 24 m?

Find the next number in the sequence. 1, 4, 9, 16, 25, 36, 49, 64

Ace and his team arrange seating for a concert into a square grid with an equal number of rows and columns. If there are 18 chairs in each row, how many chairs will be needed in total?

Reasoning

14–15, 18

16–17, 19

The number 9 can be drawn as a square with side lengths of 3. What would happen if we tried to draw a non-square number as a square? Explain your answer.

Harry was working out 22 × 52 and thought that he could simplify the expression using 2 × 5 = 10.

Evaluate 22 × 52, by ﬁrst expanding and evaluating each square.

Evaluate 102.

Is Harry correct in thinking that to solve 22 × 52, he could use the fact that 2 × 5 = 10? Justify your answer.

Calculate 42 × 22 using the above method.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Najah was working out 62 ÷ 22 and thought that she could simplify the expression using 6 ÷ 2 = 3. a

Evaluate 62 ÷ 22, by ﬁrst expanding and evaluating each square.

Evaluate 32.

Is Najah correct in thinking that to solve 62 ÷ 22 she could use the fact that 6 ÷ 2 = 3? Justify your answer.

Calculate 122 ÷ 32 using the above method.

Francis was working out 42 + 52 and thought that he could simplify the expression using 4 + 5 = 9. a

Evaluate 42 + 52, by ﬁrst expanding and evaluating each square.

Evaluate 92.

Is 42 + 52 = (4 + 5)2 a true statement? Justify your answer.

Chase was working out 122 − 92 and thought that he could simplify the expression using 12 − 9 = 3. a

Evaluate 122 − 92, by ﬁrst expanding and evaluating each square.

Evaluate 32.

Is 122 − 92 = (12 − 9)2 a true statement? Justify your answer.

Explain why the square of an odd number is always odd.

Problem-solving 23–24, 26

21, 24, 26

20–22, 25–27

A number is multiplied by 8 and then added to 32. The result is a perfect square. What is the smallest positive integer value of this number?

Find the smallest perfect square number that can be expressed as the sum of two smaller perfect square numbers.

What is the smallest number that Jimmy could add to 85 to make it a perfect square?

What would be the area of a square having a perimeter of 36 cm?

Barney was considering the pattern of square numbers and thought he spotted a pattern when calculating the differences between consecutive square numbers. a

Calculate the differences between consecutive square numbers. i iii

12 − 02 32 − 22

ii iv

22 − 12 42 − 32

What pattern did Barney identify when he looked at the answers from part (a)?

Barney isn’t sure the pattern will continue. Explain whether you think it will or will not continue forever.

2.02 Square numbers mathspace.co

Consider the ﬁgure: Number

Square Number

100

First Difference

Second Difference

Complete the ﬁgure by ﬁlling in the boxes.

Describe the pattern that exists in the ﬁrst difference row.

Describe the pattern that exists in the second difference row.

Answer these: a

Complete the table; one column has been done for you. Number Square of number Final digit of square number

7 49

19 361

Complete the second table. Number Square of number Final digit of square number

12 144

State the pattern you observe in the ﬁnal digit row.

Use your observation to predict the ﬁnal digit of the square numbers from 21 to 30.

Will this pattern continue? Please justify your answer.

When asked how old she was, a teacher replied: “My age in years is not prime but odd and when reversed and added to my age you have a perfect square. Or you can reverse and subtract, and again you have a perfect square”. How old is the teacher?

Mathspace Victorian Curriculum – Year 7 mathspace.co

2.03 Square roots After this lesson, you will be able to… • define what a square root is. • understand the relationship between squares and square roots. • use the square root symbol and find the square root of a number. • solve problems involving square roots.

The square root A square root is a way of finding a number that can be multiplied by itself to get a certain number. For example, if we want to find the square root of 16, we are looking for a number that, when multiplied by itself, equals 16. That number is 4, because 4 × 4 equals 16. We write the square root of 16 as . Square roots are like the opposite of squaring a number. When we square a number, we multiply it by itself. When we take the square root of a number, we find a number that can be multiplied by itself to get the original number. We can call these opposite operations inverse operations. Not all numbers have whole number square roots. For example, the square root of 2 is a decimal number that goes on forever without repeating.

Example 1 Evaluate

Create a strategy Find the square root by ﬁnding the number that when multiplied by itself would give 121.

Apply the idea Express 121 as a product of prime factors

Find the square root

Example 2 Evaluate

Create a strategy Perform the operation inside the square root symbol then ﬁnd the square root of the result.

2.03 Square roots mathspace.co

Do these expressions result in whole numbers as their answers? a

How can square roots be used in real-life contexts, such as ﬁnding the dimensions of a square garden bed when given its area?

Fluency

6, 8, 9a-h, 10, 12, 13a-d, 15

Given that we know 5 × 5 = 25, what is

Which has a value between 10 and 11?

Evaluate: a

Ex 2

Evaluate:

A square room has an area of 81 m2. Find the length of one side of the room.

Find the dimensions of a square plot of land of area 15 625 m2.

Consider the number a

Ex 3

7, 8cd, 9e-l, 11–12, 13e-h, 14–15

A Ex 1

6, 8, 9c-hkl, 10, 12, 13c-f, 15

Complete the inequality with two consecutive perfect square numbers that 22 lies between: ⬚ < 22 < ⬚

Complete the inequality with two consecutive whole numbers that ⬚<

lies between:

<⬚

Determine the two consecutive whole numbers between which each of these roots lie. a

State the integer that lies between

and

2.03 Square roots mathspace.co

Find the length of fence required to enclose a square garden of area 36 m2.

Sharon has purchased a square piece of fabric with an area of 144 cm2. Find the perimeter of the fabric.

Find the perimeter of a square kitchen ﬂoor with an area of 160 000 cm2.

William has been creating some art patterns using tiles. Consider the pattern of tiles shown:

n =1

n =2

n =3

n =4

Complete the table. Shape number (n)

Total number of tiles (T ) b

Using the table in part (a), describe using words or symbols, the connection between the shape number (n) and the total number of tiles (T ).

Complete the table. Shape number (n)

Number of light purple border tiles (B) d

Using the table in part (c), describe using words or symbols, the connection between the shape number (n) and the number of light purple border tiles (B).

Using your answer from part (d), which shape would have 24 border tiles?

Using your answers from parts (b) and (d), determine the total number of tiles required if there are 36 border tiles.

Find the number of blue central tiles required when there are 100 tiles in total.

Find the number of border tiles required when there are 49 blue central tiles.

2.03 Square roots mathspace.co

2.04 Powers of whole numbers After this lesson, you will be able to… • define and understand the concept of powers of whole numbers. • write powers of whole numbers using index notation. • simplify and evaluate expressions using powers and indices.

Powers of whole numbers An index, or power, is a number that indicates how many times a base number is multiplied by itself. The base number is usually a whole number (an integer), and the index is written as a small number in the upper right hand corner of the base: Index / Exponent / Power

103 = 10 × 10 × 10 Multiplied 3 times Base

The base number written with the small power to the top right is called index notation. The index tells us how many times the base number is multiplied by itself. When this is written out with all the multiplications shown, this is called expanded notation.

Interactive exploration

Discover this concept in action online

mathspace.co

The index is equal to the number of times the base is multiplied together. Some powers have special names. Squares, we have seen, with a power of 2. Cubes are numbers multiplied by themselves three times with a power of 3. For powers higher than 4, we say “to the power of “.

Power 2 3

Example 52 53

Description 5 squared 5 cubed 5 to the power of 6

Mathspace Victorian Curriculum – Year 7 mathspace.co

Expanded notation 5×5 5×5×5 5×5×5×5×5×5

Complete the table: Words Five to the power of six Six to the power of four Eight cubed Eleven to the power of three Twenty-three to the power of ﬁve Seven squared Fifteen to the power of four

Ex 1

Ex 2

9×9

9×9×9

9×9×9×9

9×9×9×9×9

Write these expressions using index notation: a

5 × 5 × 5

1×1×1×1×1×1

3 × 3 × 3 × 3

2×2×2×2

4 × 4 × 4 × 4 × 4

7×7

12 × 12 × 9 × 9 × 9

11 × 7 × 11 × 7 × 7

12 × 12 × 12 × 15 × 15

2×4×2×4

24 × 24 × 24 × 12

2×2×3×3×3×5×5

Write these expressions in expanded form: a

46 × 92

775 × 132

Write these numbers in expanded form: a

Index notation

Write these powers of 9 in index form: a

Expanded notation

75 × 64

Consider the expression 53. a

Write the expression in expanded form.

Using your answer to part (a), evaluate 53.

Complete the table of powers of 2: 21

2 13

Complete these patterns: a

Powers of 3: 1, 3, ⬚, ⬚, 81, ⬚, ⬚

Powers of 5: 1, ⬚, 25, ⬚, 625

2.04 Powers of whole numbers mathspace.co

Reasoning

14–15

14–16

Explain how to evaluate 45.

Answer these: a

14–16

Complete the table: 11

1 b 16

Looking at the answers in the table, what could you say about any power of 1?

Answer these: a

In the table, what do you notice about the answers as you move from right to left? Now consider if an extra column was added to the start of the table.

What number would complete the table? c

Using a similar method, complete these tables. i

30 31 32 33 34 3

iii

100

125

625

101

102

103

104

100

1000

10 000

State a generalisation about any base that has a power of zero. 17a-f, 18e-h, 19–20

17, 18e-l, 19–20

Complete these statements: a

Problem-solving 17a-d, 18a-d, 19 17

024 = ⬚

3⬚ = 81

2⬚ = 32

5⬚ = 625 118

5⬚ = 125 2⬚ = 512

53 − 32

33 × 32

22 + 43

72 + 34

09 + 18 + 27

33 − 22 × 42

50 ⬚ 15

130 = ⬚

3⬚ = 729

Evaluate: a

23 × 42

122 − 22

53 − 33

State which symbol ( < , = or > ) would make these statements true: a

32 ⬚ 31

118 ⬚ 124

28 ⬚ 35

A number of bacteria triples in size every hour. If the initial amount of bacteria is 3, determine how many bacteria will be present after 3 hours.

Mathspace Victorian Curriculum – Year 7 mathspace.co

2.05 Powers of ten After this lesson, you will be able to… • identify the value of any power of ten. • rewrite numbers using powers of ten.

Powers of ten Recall that an index (or power) tells us the number of times to multiply the base by itself.

Interactive exploration

Discover this concept in action online

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The table shown demonstrates another way to think of some of the powers of ten that we see in the applet. Index form

Meaning

Value

105

10 × 10 × 10 × 10 × 10

100 000

One hundred thousand

104

10 × 10 × 10 × 10

10 000

Ten thousand

10 × 10 × 10

1000

One thousand

102

10 × 10

100

One hundred

Ten

One

10 10

In Words

Example 1 Complete this expression: 4000 = ⬚ × 103

Create a strategy

Apply the idea

Remember that multiplying by 10 is the same as multiplying by 1000. So, this question is asking us to ﬁnd the number that multiplies by 1000 to give 4000.

Think about how many times 1000 ﬁts into 4000. 4000 = 4 × 103

2.05 Powers of ten mathspace.co

2.05 Practice questions Understanding

Adventurer

1–3

Trailblazer

1–3

1–2

How many zeros will 1012 have when written as a whole number?

105 is equal to: A

1 million

10 thousand

10 million

100 thousand

Which is smaller: 105 or 107? 4–5, 6a-d, 7a-d, 9a-f, 10a-c

Fluency

Ex 1

Explorer

4, 6aceg, 7c-f, 8, 9c-g, 10aceg

6aefh, 7e-h, 8, 9e-h, 10bdfh

Complete the table: 101

102

100

103

104

105

106

107

108

Complete the statement: 60 000 = ⬚ × 104

Write these as whole numbers: a

2 × 105

8 × 10

3 × 102

7 × 103

1 × 104

3×1

5 × 107

6 × 10

Express these numbers in expanded notation using powers of ten: a

100

1000

400

900

6000

7000

80 000

30 000

Ex 2

The distance between two planets is approximately 5 × 108 kilometres. Express the distance as a whole number.

Ex 3

Express these numbers in expanded notation using powers of ten:

Ex 4

2200

8300

2480

7020

6260

92 000

33 050

62 480

Write these as whole numbers: a

3 × 104 + 7 × 103

1 × 103 + 4 × 101

4 × 102 + 1 × 101

7 × 103 + 6 × 102 + 4 × 101

6 × 104 + 3 × 102 + 5 × 1

3 × 104 + 2 × 103 + 9 × 101 + 5 × 1

2 × 104 + 2 × 103 + 7 × 102 + 7 × 101 + 6 × 1

3 × 10 + 2 × 10 + 4 × 10 + 2 × 10

Mathspace Victorian Curriculum – Year 7 mathspace.co

Fluency 6

Ex 1

Is 5 a factor of these numbers? a

155

252

Write all the factors of these numbers: 14

231

312

Write all the factors of these numbers: 115

Write down the ﬁrst ﬁve multiples of these numbers: a

Are these numbers multiples of 2? a

Write down the 8th multiple of 4.

Write down the 20th multiple of 4.

Fill in the blank: 42 is the ⬚th multiple of 7.

8g-l, 9cd, 10e-h, 13–15

Factor pairs of 50 (⬚, 50) (⬚, 25) (5, ⬚)

a Ex 2

7, 8e-j, 9bc, 10c-f, 11, 13–15

Complete this table of factor pairs:

6–7, 8a-f, 9ab, 10a-d, 11–12, 14

Lucy earns $30 a week. a

Fill in the table to help her keep track of her money.

How much money does Lucy have after 5 weeks?

Week 1 2 3 4 5 6

Total earnings (in dollars) 30 60

2.06 Factors and multiples mathspace.co

Reasoning

18–20

17–20

16–20

What types of numbers have exactly three factors? Justify your answer.

Can a multiple of a number also be a factor of that number? Explain your answer.

A teacher has 280 pencils that she wants to divide equally among her 35 students with no pencils left over. Determine if this is possible and explain your reasoning.

John has 40 marbles that he wants to divide equally among his 6 friends.

How many marbles should each friend receive?

How many marbles are left over?

Explain why John has marbles left over.

Shirah has 16 ﬂowers to plant in her garden. She wants to arrange them in rows with the same number of ﬂowers in each row. a

State all the possible numbers of ﬂowers she can have in each row.

Explain how you determined your answer to part (a).

Problem-solving 21, 24–26

21–26

22–27

There are 12 eggs in a carton. If Harry buys 9 cartons for his shop, determine how many eggs he will have.

Three multiples of a certain number are 12, 15, and 27. The number is larger than 1. What is the number?

A company needs to pack boxes of toys, and each box must have the same number of toys. If they have 144 toys, determine the largest number of boxes they can pack with no toys left over.

The company Mindt sells 6 deluxe bars of chocolate for $36. Madbury sells 4 bars of chocolate for $12. Which brand of chocolate is more expensive?

Two fishermen go fishing at different times during the week. Wally catches 12 fish per day 3 days a week whereas Callum catches 5 fish per day 4 days a week.

How many ﬁsh does Wally catch a week?

How many ﬁsh does Callum catch a week?

Who catches more ﬁsh a week?

A hotel has 180 rooms that need to be renovated. a

If the hotel wants to renovate the same number of rooms each week for 12 weeks, how many rooms should they renovate each week?

If they can only renovate 10 rooms a week, how many weeks will it take to renovate all the rooms?

Find the dimensions of a rectangular ﬁeld with a perimeter of 80 m and an area of 384 m2.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Here is the grid after the first two numbers are circled and their multiples crossed out.

What is the next number to be circled after 2 and 3?

Investigate Repeat step 2 until you reach the end of the grid. Then, answer these questions. 1.

How many numbers are circled between 1 and 200 by the end of the process?

What is special about the numbers you circled?

What does it mean if a number is crossed out only once? What about more than once?

Discussion

What patterns did you notice when finding prime numbers with the sieve of Eratosthenes?

Do you think this method could be used to find prime numbers beyond 200? Explain your reasoning.

Mathspace Victorian Curriculum – Year 7 mathspace.co

2.07 Prime factorisations After this lesson, you will be able to… • express a given number as a product of its prime factors. • use a factor tree to perform prime factorisation.

Prime factorisation Every whole number greater than 1 is either prime or composite, with primes having only themselves and 1 as factors. 0 and 1 are special in that they are neither prime nor composite. Prime numbers are useful because they can be thought of as the building blocks of all other numbers (except 1). We can write any composite number as a product of prime numbers. One way to rewrite a composite number as a product of prime factors is using a factor tree. It is useful to know that the ﬁrst ﬁve prime numbers are 2, 3, 5, 7 and 11 for constructing factor trees. Start with the composite number we want to investigate, draw a box around it, and draw two lines coming out of it. Here is how we might start with the number 12.

Next, choose two numbers that multiply together to make 12, such as 3 and 4. Write these at the end of each of the lines.

12 4

Circle any prime numbers. Since 3 is a prime number, we circle it. 4 is not a prime number, so we draw a box around it instead.

12 4

We then repeat the process with 4, with 2 × 2.

12 4 2

3 2

Since 2 is prime, we circle both of these numbers. Once all the numbers at the end of each branch are circled, the factor tree is complete.

12 4 2

3 2

Factor trees are not always unique - here is another factor tree for 12.

12 6 2

2 3

Multiplying together the circled numbers at the end of each branch always gives the original number. Here, we see 2 × 2 × 3 = 12.

We could have chosen 2 and 6 at the ﬁrst branch of the tree and still ended up with the same circled numbers 2, 2 and 3. These are called the prime factors.

2.07 Prime factorisations mathspace.co

Even if the numbers chosen at each branch are different, the numbers at the end of the branches will always be the same for any number — they will just be shown in a different order. So, the prime factors for every composite number are unique (there is only one set of possible prime factors). Here is a factor tree for 360. 360 30 15 3

12 2

4 2

We can write: 360 = 3 × 5 × 2 × 2 × 2 × 3

Multiply the prime factors together

360 = 2 × 2 × 2 × 3 × 3 × 5

Rewrite the factors in ascending order

360 = 2 × 3 × 5

Express the multiplications using index notation

Notice that the factor tree for 12 we made earlier is a smaller part of the factor tree for 360. This is because 12 is a factor of 360, and when we write 360 = 2 × 2 × 2 × 3 × 3 × 5 we can recognise the prime factorisation of 12 inside it: 360 = 2 × (2 × 2 × 3) × 3 × 5.

Interactive exploration

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Example 1 A number has this factor tree:

2 5

a What is this number at the top of the tree?

Create a strategy Multiply the prime factors together to obtain the composite number at the top of the factor tree.

Apply the idea Number = 2 × 2 × 5 × 5 × 7 × 11 = 7700

Mathspace Victorian Curriculum – Year 7 mathspace.co

Multiply the numbers at the end of each branch Perform the multiplication

2.07 Practice questions Understanding 1

1–2

Trailblazer

1–3

Prime number

iii

Prime factorisation of a number

A positive whole number that divides the number without leaving a remainder.

A positive whole number greater than 1 that has no positive integer divisors other than 1 and itself.

A unique representation of the number as a product of prime numbers.

Factor of a number

Identify which numbers are prime and which are composite in each list: 3, 10, 12, 4, 5, 6

14, 7, 15, 13, 11

21, 19, 27, 17, 1

How are factors and prime factors the same? How are they different?

Fluency 4

Adventurer

Match the terms to the deﬁnitions:

a 3

Explorer

4–5, 7a-d, 8a-d, 9a-d

4–6, 7c-f, 8c-f, 9c-f

4–6, 7e-h, 8e-h, 9e-h

The given ﬁgure shows a partial factor tree of 36.

Construct the remaining factors to complete the factor tree. 2

A number has this factor tree: What is the number at the top of the tree?

6 Ex 1

A number has this factor tree: a

What is the number at the top of the tree?

Write the prime factorisation of the number in index notation.

3 2

Mathspace Victorian Curriculum – Year 7 mathspace.co

5 3

Ex 2

For each number: i

Construct a prime factor tree.

Write the prime factorisation of the number in index notation.

126

144

200

360

Write these numbers as a product of their prime factors in expanded form: a

132

150

Write these values as a product of their prime factors using index notation: a

200

180

225

208

400

560

504

Reasoning

10–11

10–12

Can two different numbers have the same prime factorisation? Explain your reasoning.

Rae and Josie are working together to ﬁnd the prime factorisation for 30. Rae says they should begin by dividing by 5, but Josie says they should begin by dividing by 2. Who is correct? Explain your reasoning.

Explain how we can use prime factors to determine if 35 is a factor of 420 without using a calculator.

Problem-solving 13ace, 14ace 13

175 = 35 × 5

20 = 2 × 2 × 5

286 = 2 × 7 × 11

27 = 33

189 = 33 + 7

20 = 2 + 5

Do these equations represent the correct prime factorisation? If not, write the correct one using either expanded form or index notation. a

56 = 23 × 3

495 = 22 × 55

200 = 23 × 52

63 = 32 × 7

300 = 22 × 3 + 52

e 15

13bdf, 14bdf, 15–16

Do these equations show the correct prime factorisation in expanded form? If not, write the correct one.

e 14

13bdf, 14bdf, 15

504 = 2 + 3 + 7

What is the largest positive number that can evenly divide 108, 252 and 432 without leaving any remainder?

2.07 Prime factorisations mathspace.co

2.08 HCF and LCM After this lesson, you will be able to… • determine the highest common factor of two or more numbers. • determine the lowest common multiple of two or more numbers.

Highest common factor (HCF) The highest common factor (HCF) is the largest factor shared by two or more given numbers. For example, the HCF of 12 and 18 is 6, which we can see: Factors 1, 2, 3, 4, 6, 12 1, 2, 3, 6, 9, 18

12 18

Comparing each list of factors, we can see that 6 is the largest factor shared by 12 and 18. To find the HCF for two numbers, you can simply look at the lists of factors for each number and find the highest number in common. This will always work, but there is a faster way using factor trees. To find the highest common factor of 126 and 294, we start by drawing their factor trees: Prime factorisation in expanded form: 126 = 2 × 3 × 3 × 7

126 6 2

21 3

Prime factorisation in expanded form: 294 = 2 × 3 × 7 × 7

294 6 2

49 3

We then ﬁnd what factors appear in both factorisations — they each have one 2, one 3, and one 7 in common. The highest common factor is the product of the common prime factors, 2 × 3 × 7 = 42.

2.08 HCF and LCM mathspace.co

To ﬁnd the lowest common multiple of 126 and 294, we look again at their factor trees and jump straight to the prime factorisations in expanded form, since we have found these previously: Prime factorisation in expanded form: 126 = 2 × 3 × 3 × 7

126 6 2

21 3

Prime factorisation in expanded form: 294 = 2 × 3 × 7 × 7

294 6 2

49 3

Find the HCF which would be 2 × 3 × 7 = 42. Next, we identify any factors that would be left over for each number after we remove the factors that make up the HCF. • For 126, after removing the common factors we are left with the factor 3. • For 294, after removing the common factors we are left with the factor 7. To ﬁnd the LCM, we multiply the HCF by these remaining values, so the LCM of 126 and 294 is 42 × 3 × 7 = 882.

Example 2 Consider these prime factorisations: 2200 = 2 × 2 × 2 × 5 × 5 × 11 2750 = 2 × 5 × 5 × 5 × 11 Find the lowest common multiple of 2200 and 2750.

Create a strategy Find the HCF of 2200 and 2750. Multiply the HCF by the prime factors left over from each number after removing the HCF factors.

Apply the idea HCF = 2 × 5 × 5 × 11 Find the HCF by identifying common factors = 550

Evaluate the product

Prime factors of 2200 not in the HCF = 2 × 2 Identify the remaining prime factors Prime factors of 2750 not in the HCF = 5 LCM = 550 × 2 × 2 × 5 Find the HCF by identifying common factors = 11 000

Evaluate the product

2.08 HCF and LCM mathspace.co

Fluency 3

3–6, 7a-d, 9a, 10–11, 12a-d, 15–16

3–5, 7e-h, 8, 9ab, 11, 12c-f, 13–17

6, 7g-j, 8, 9cd, 11, 12e-h, 13–17

For each of these numbers: i

List the factors.

6 and 12

9 and 24

State the highest common factor.

14 and 32

28 and 42

Given these prime factorisations: 180 = 2 × 2 × 3 × 3 × 5 600 = 2 × 2 × 2 × 3 × 5 × 5 Find the highest common factor of 180 and 600.

Given these prime factorisations: 224 = 2 × 2 × 2 × 2 × 2 × 7 196 = 2 × 2 × 7 × 7 Find the highest common factor of 224 and 196.

Given these prime factorisations: 4900 = 2 × 2 × 5 × 5 × 7 × 7 1750 = 2 × 5 × 5 × 5 × 7 Find the highest common factor of 4900 and 1750.

7 Ex 1

Find the highest common factor of these groups of numbers: a

8 and 16

18 and 24

21 and 33

36 and 48

42 and 24

72 and 108

121 and 143

150 and 560

48, 80, and 176

168, 312, and 120

The table shows the prime factorisations of several numbers in index form. a b

What is the highest common factor of 2940 and 14 700? What is the highest common factor of 17 640 and 308 700?

Number

Prime factorisation

2940

22 × 3 × 5 × 72

14 700

22 × 3 × 52 × 72

17 640

23 × 32 × 5 × 72

26 250

2 × 3 × 54 × 7

308 700

22 × 32 × 52 × 73

For each of these numbers: i

List the ﬁrst 8 multiples.

6 and 10

8 and 14

State the lowest common multiple.

24 and 42

Given these prime factorisations: 54 = 2 × 3 × 3 × 3 36 = 2 × 2 × 3 × 3 Find the lowest common multiple of 54 and 36.

Mathspace Victorian Curriculum – Year 7 mathspace.co

15 and 30

Given these prime factorisations:

Ex 2

1260 = 2 × 2 × 3 × 3 × 5 × 7 1890 = 2 × 3 × 3 × 3 × 5 × 7 Find the lowest common multiple of 1260 and 1890.

Find the lowest common multiple of these groups of numbers: a

6 and 14

7 and 18

24 and 84

78 and 26

144 and 156

450 and 650

3, 8, and 18

9, 14, and 27

Consider the numbers 1575 and 1650. a

What is the highest common factor of 1575 and 1650?

What is the lowest common multiple of 1575 and 1650?

The table shows the prime factorisations of several numbers in index form. a b

What is the lowest common multiple of 2940 and 17 640? What is the highest common factor of 14 700 and 26 250?

Number

Prime factorisation

2940

22 × 3 × 5 × 72

14 700

22 × 3 × 52 × 72

17 640

23 × 32 × 5 × 72

26 250

2 × 3 × 54 × 7

308 700

22 × 32 × 52 × 73

Allena has two pieces of cloth. One piece is 68 centimetres wide and the other piece is 32 centimetres wide. She wants to cut both pieces into strips so that all of the strips have the same width. a

How wide should she cut the strips if she wants the strips to be as large as possible?

How many strips will she end up with?

Cicada species spend many years underground, emerging as a single brood every generation. Platypedia areolata emerges every 19 years. They are hunted by the Argiope aurantia spider, which has a 4 year breeding cycle. a

How often do these species start their breeding cycle at the same time?

A genetic mutation changes the life cycle of the cicadas so the mutants emerge every 15 years. How often will the mutant cicada species start their life cycle at the same time as the spiders?

Two airlines, JetFuel and LionAir, each land planes at an airport terminal. During the day a JetFuel plane lands every 18 minutes, and a LionAir plane lands every 16 minutes. A shuttle bus takes passengers from the terminal to the city centre, departing every 21 minutes. a

How often does a JetFuel plane land and a shuttle bus depart at the same time?

Which of these two options happens more often in a single day? A

JetFuel plane lands and the shuttle bus departs at the same time.

LionAir plane lands and the shuttle bus departs at the same time.

2.08 HCF and LCM mathspace.co

Reasoning

19–20

18–20

A student claims that two numbers will always have a common factor and a common multiple. Explain if the student is correct.

Explain how the prime factors of two numbers can be used to ﬁnd their highest common factor.

The HCF of two numbers is 4, and the LCM is 48. a

Sasha thinks the two numbers are 8 and 12. Determine if Sasha is correct. Explain your reasoning.

Suri claims the two numbers will be multiples of 4. Determine if Suri is correct. Explain your reasoning.

What could the two numbers be?

Problem-solving 21, 22

22, 23

21–24

Jane has a piece of cloth that has a size of 84 cm by 112 cm. She wants to cut squares of equal size out for a patchwork quilt without wasting any material. What would be the side length of the largest size square that can be cut?

Cicadas are a type of insect that spend most of their life underground and come above ground when they mature. Two variations of the species, Type A and Type B, emerge above ground as a group every 13 and 17 years respectively. If the last year that both Type A and Type B emerged was 1993, what is the next year that both types will emerge together?

Cheryl and Berting are juggling. They have 4 red balls and 1 green ball each. The time between releasing the green ball and releasing the green ball again is 6 seconds for Cheryl and 9 seconds for Berting. At some point in the performance, Cheryl and Berting each release their green ball at exactly the same time.

How long will it be until they each release their green ball at the same time again?

How many red balls has Berting thrown before they each release their green ball at the same time again?

Two students are training by running laps in an oval. They pace themselves and jog at constant, but different speeds. The faster student completes each lap in 90 seconds, and the slower student completes each lap in 120 seconds. If both students begin at the starting line at the same time, how many minutes will pass before they both pass the start line at the same time again?

Mathspace Victorian Curriculum – Year 7 mathspace.co

2 Chapter review 1

Fill in the boxes to complete the working out using the associative law. 45 + 27 + 18 = 45 + (⬚ + ⬚)

32 + 52 − 24 = 32 + (⬚ − ⬚)

152

182

262

802

1202 + 102

62 − 32

122 − 82

162 − 52

(7 − 3)2 + 16

3 (42 + 1)2

a 2

Evaluate:

Jenny and her crew organise chairs for a school play in a square formation with the same number of chairs in each row and column. If there are 25 chairs in each column, how many chairs are required in total?

A man said to his friend: “My age in years is an even number that is not prime. If you reverse the digits, you get a perfect square. If you divide this perfect square by 4, you also get a perfect square.” How old is the man?

Evaluate:

Consider the number a

Complete the inequality with two consecutive perfect square numbers that 70 lie between: ⬚ < 70 < ⬚

Complete the inequality with two consecutive whole numbers that ⬚<

lies between:

<⬚

Determine the two consecutive whole numbers each of these roots lies between. a

Jason has a square garden with an area of 81 m2. Find the perimeter of the garden.

Chapter 2 review mathspace.co

Write these as whole numbers: a

5 × 102 + 3 × 101

8 × 103 + 6 × 102 + 2 × 101

3 × 104 + 2 × 103 + 7 × 101 + 3 × 1

9 × 104 + 3 × 103 + 8 × 102 + 7 × 101 + 4 × 1

A company sold 2 × 103 + 3 × 102 + 5 × 101 units of Product A and 4 × 102 + 7 × 101 + 6 units of Product B last year. Express the total number of units sold for both products as a whole number.

List all the factors of these numbers: a

Write down the 30th multiple of 2.

Write these values as a product of prime factors in expanded form: a

120

150

Given these prime factorisations: 24 = 2 × 2 × 2 × 3 144 = 2 × 2 × 2 × 2 × 3 × 3 Find the highest common factor of 24 and 144.

Consider the numbers 270 and 3465. a

What is the highest common factor of 270 and 3465?

What is the lowest common multiple of 270 and 3465?

The table shows the prime factorisations of several numbers in index form. a

What is the highest common factor of 1260 and 8820?

What is the highest common factor of 26 250 and 220 500? Number 1260 8 820 26 250 31 500 220 500

Prime factorisation 22 × 32 × 5 × 7 22 × 32 × 5 × 72 2 × 3 × 54 × 7 22 × 32 × 53 × 7 22 × 32 × 53 × 72

Sarah has a piece of fabric that has a size of 90 cm by 150 cm. She wants to cut squares of equal size out for a patchwork quilt without wasting any material. What would be the side length of the largest size square that can be cut?

Chapter 2 review mathspace.co

101

Big ideas • Algorithms are systematic guides for problem-solving, which include step-by-step instructions, decision-making through branching, repetition with loops, and visual representations like flow charts and pseudocodes.

Curriculum Outcomes VC2M7N08 compare, order and solve problems involving addition and subtraction of integers

3 Algorithms and coding

Chapter outline 3.01 Solve problems using an algorithm 3.02 Following algorithms 3.03 Branching 3.04 Iterations 3.05 Flow charts 3.06 Algorithms in code 3.07 Loops Chapter 3 review

104 113 118 124 131 141 147 158

3.01 Solve problems using an algorithm After this lesson, you will be able to… • follow an algorithm to solve mathematical problems such as addition and subtraction. • create simple algorithms.

Solve problems with algorithms Here is an example of an algorithm. Algorithm steps Think of any two numbers from 1 to 9. Make two 2-digit numbers by putting them in either order. Subtract the smaller number from the larger number. Divide the result by 9. Finally, add the smaller starting number.

Example 3 and 5 53 and 35 53 − 35 = 18 18 ÷ 9 = 2 2+3=5

In the example, we chose the numbers 3 and 5 to start with. In the last step of the algorithm, we added the smaller of these numbers, 3, and got a result which was the larger number, 5. Does this always happen? Try it yourself with two different numbers using the algorithm. The key to a good algorithm is that each step is clear, and is made up on just one speciﬁc instruction or question.

Example 1 This function machine is designed to subtract fourteen. ⬚ ⟶ −14 ⟶ ⬚

a If we put 37 into the function machine, will we need to regroup before subtracting the units?

Create a strategy

Apply the idea

Check if the units digit of the input number is smaller than 4.

The units digit of 37 is 7, which is greater than 4. So, the answer is no.

b If we put 33 into the function machine, will we need to regroup before subtracting the units?

Create a strategy

Apply the idea

Check if the units digit of the input number is smaller than 4.

The units digit of 33 is 3, which is smaller than 4.

104

Mathspace Victorian Curriculum – Year 7 mathspace.co

So, the answer is yes.

c Describe which type of numbers will involve regrouping when put into this function machine.

Create a strategy

Apply the idea

Identify the range of unit digits that would require regrouping when subtracting 14.

When subtracting 14, regrouping is necessary if the units digit of the original number is less than 4. So, the answer is numbers with a units digit from 0, 1, 2, or 3.

Example 2 We are going to add these two numbers together, using the addition algorithm: 37 + 46 1. Add the units. 2. Is the result at least 10? • If no - GO TO Step 3. • If yes - regroup into tens and units, then GO TO Step 3. 3. Add the tens. 4. Is the result at least 100? • If no - GO TO Step 5. • If yes - regroup into hundreds and tens, then GO TO Step 5. 5. Write the ﬁnal answer by adding the hundreds, tens and units. a Which two numbers will be added in Step 1?

Create a strategy

Apply the idea

Follow Step 1.

Step 1 tells us to add the units which are the numbers 7 and 6.

b Complete Step 1: Add the units.

Create a strategy Add the units found in part (a).

Apply the idea 7 + 6 = 13

Add the units

c What do we need to do at Step 2?

Create a strategy

Apply the idea

Use the answer from part (b).

Since the result from part (b) is at least 10, 13 > 10, Step 2 tells us to regroup into tens and units, and then GO TO Step 3.

3.01 Solve problems using an algorithm mathspace.co

105

d Regroup 13 into tens and units: 13 = ⬚ + ⬚

Create a strategy

Apply the idea

Write the number in a place value table.

Here is 13 in a place value table: Tens 1

Units 3

We have 1 tens which equals 10, and 3 units which equals 3. 13 = 10 + 3 e Now complete Step 3: Add the tens.

Create a strategy

Apply the idea

Find the tens in each number.

37 has 3 tens which equals 30. 46 has 4 tens which equals 40. 13 has 1 ten which equals 10. 30 + 40 + 10 = 80

Add the tens

What do we need to do at Step 4?

Create a strategy

Apply the idea

Use the answer from part (e).

Since 80 < 100 we need to GO TO Step 5.

g Complete Step 5: Write the ﬁnal answer by adding the hundreds, tens and units.

Create a strategy Add the tens from part (e) and the units digit of the number found in part (b).

Apply the idea From part (b) we have 13 which has a units digit of 3. We found from part (e) that the sum of the tens is 80. This includes the 10 from 13. Final answer = 80 + 3 = 83

106

Add the tens and units Evaluate

Mathspace Victorian Curriculum – Year 7 mathspace.co

Example 3 We are going to subtract these two numbers, using the subtraction algorithm: 91 − 26 1. Is the units digit of the first number less than the units digit of the second number? • If no - GO TO Step 2. • If yes - regroup the tens from the first number to give a ten to the units, then GO TO Step 2. 2. Subtract the units of the second number from the units of the first number. 3. Subtract the tens of the second number from the remaining tens of the first number. 4. Write the final answer by adding the results of Steps 2 and 3. a Fill in the blanks: The units digit of the first number is ⬚.

The units digit of the second number is ⬚.

Create a strategy

Apply the idea

Write the numbers in a place value table.

Here are 91 and 26 in a place value table: Tens 9 2

Units 1 6

The units digit of the ﬁrst number is 1. The units digit of the second number is 6.

b What do we need to do at Step 1?

Create a strategy

Apply the idea

Use the answers from part (a).

Since 1 < 6, we need to regroup the tens from the ﬁrst number, 91, to give a ten to the units, then GO TO Step 2.

c Fill in the blanks: We can regroup the tens part of 91 as 90 = ⬚ + 10, in order to give 10 to the units part, which will become 1 + 10 = ⬚.

Create a strategy

Apply the idea

For the ﬁrst blank, subtract 10 from the tens value of the ﬁrst number.

The tens value of 91 is 90. So, 90 − 10 = 80.

For the second blank, evaluate the number sentence.

We can regroup the tens part of 91 as 90 = 80 + 10, in order to give 10 to the units part, which will become 1 + 10 = 11.

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Fluency Ex 1

4−6, 8bc, 9b, 10b, 11b

⬚ ⟶ + 13 ⟶ ⬚

If we put 26 into the function machine, will we need to regroup when we add the units?

If we put 18 into the function machine, will we need to regroup when we add the units?

Describe which type of numbers will involve regrouping when put into this function machine.

This function machine is designed to add seventeen. ⬚ ⟶ + 17 ⟶ ⬚

If we put 26 into the function machine, will we need to regroup when we add the units?

If we put 21 into the function machine, will we need to regroup when we add the units?

Describe which type of numbers will involve regroup when put into this function machine.

This function machine is designed to subtract twenty-three. ⬚ ⟶ − 23 ⟶ ⬚

If we put 57 into the function machine, will we need to regroup before subtracting the units?

If we put 45 into the function machine, will we need to regroup before subtracting the units?

Describe which type of numbers will involve regrouping when put into this function machine.

This algorithm is designed to perform addition of two 2-digit numbers together:

a b

Instructions Add the units. If the result in Step 1 is at least 10, regroup into tens and units. Add the tens, including any from Step 2. Write the ﬁnal answer by adding the units to the result of Step 3.

Work through the algorithm to add 41 and 38, identify the result for each step: i

Step 1

Step 2

iii

Step 3

Work through the algorithm to add 38 and 27, identify the result for each step: i

4–6, 8bc, 9b, 10b, 11a

This function machine is designed to add thirteen.

Step 1 2 3 4

Ex 2

4, 6−7, 8a, 9a, 10, 11a

Step 1

Step 2

iii

Step 3

109

Step Instructions 5 Write the ﬁnal answer by adding the hundreds, tens and units. a

What two numbers will be added in Step 1?

What do we need to do at Step 2?

iii

What do we need to do at Step 4?

Write the ﬁnal answer.

If we use the algorithm to calculate 62 + 93: i

What two numbers will be added in Step 1?

What do we need to do at Step 2?

iii

What do we need to do at Step 4?

Write the ﬁnal answer.

If we use the algorithm to calculate 89 + 65:

If we use the algorithm to calculate 43 + 56:

What do we need to do at Step 2?

What do we need to do at Step 4?

iii

Write the ﬁnal answer.

This algorithm is designed to perform addition: Step Instructions 1 Add the units. 2 Is the result at least 10? • If no - GO TO Step 3. • If yes - regroup into tens and units, then GO TO Step 3. Step Instructions 3 Add the tens. 4 Is the result at least 100? • If no - GO TO Step 5. • If yes - regroup into hundreds and tens, then GO TO Step 5. Step Instructions 5 Add the hundreds. 6 Is the result at least 1000? • If no - GO TO Step 7. • If yes - regroup into thousands and hundreds, then GO TO Step 7. Step Instructions 7 Write the ﬁnal answer by adding the thousands, hundreds, tens and units. a

Work through the algorithm to calculate 343 + 415, identify the result for each step: i

Step 4

iii

Step 6

Step 7

Work through the algorithm to calculate 428 + 367, identify the result for each step: i

110

Step 2 Step 2

Step 4

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Step 6

Step 7

Ex 3

This algorithm is designed to perform subtraction: Step Instructions 1 Is the units digit of the first number less than the units digit of the second number? • If no - GO TO Step 2. • If yes - regroup the tens from the first number to give a ten to the units, then GO TO Step 2. Step Instructions 2 Subtract the units of the second number from the units of the first number. 3 Subtract the tens of the second number from the remaining tens of the first number. 4 Write the final answer by adding the results of Steps 2 and 3. a

If we use the algorithm to calculate 97 − 36: i

What do we need to do at Step 1?

What is the result from Step 2?

iii

What is the result from Step 3?

Write the ﬁnal answer.

If we use the algorithm to calculate 71 − 26: i

What do we need to do at Step 1?

What is the result from Step 2?

iii

What is the result from Step 3?

Write the ﬁnal answer.

This algorithm is designed to perform multiplication: Step 1 2 3 4 a

Instructions Split the first number into tens and units. Multiply the second number by the units of the first number. Multiply the second number by the tens of the first number. Add the results of Step 2 and Step 3.

Work through the algorithm to calculate 35 × 6, identify the result for each step: Step 1

i b

Step 2

iii

Step 3

Step 4

Work through the algorithm to calculate 12 × 82, identify the result for each step: Step 1

Reasoning 12

ii 12

Step 2

iii

Step 3

12−13

Step 4

12−14

This algorithm is designed to perform the addition of two 2-digit numbers together: Step 1 2 3 4

Instructions Add the units. Add the tens. Multiply the second number by the tens of the first number. Write the final answer by adding the result of Step 1 and Step 2.

Will this addition algorithm work for all two 2-digit numbers? 3.01 Solve problems using an algorithm mathspace.co

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If you were to adapt the subtraction algorithm for three-digit numbers, what additional steps or modiﬁcations would you include?

You’ve been provided with an algorithm that aims to ﬁnd the sum of two 2-digit numbers. A peer suggests that regrouping is unnecessary when summing two 2-digit numbers. What would be an example where this suggestion does not work?

Problem-solving 15, 18

16, 18

16−19

Alex wrote an algorithm to multiply a number by 7, but he made an error. He added 7 to the number seven times. When he used his algorithm for the number 6, what result did he get?

Using the subtraction algorithm, when subtracting 29 from 65, how many times would you need to regroup?

If a new algorithm multiplies a number by 4 and then subtracts 5, what is the result when the algorithm is applied to the number 10?

Daniel uses the given addition algorithm to sum two two-digit numbers. Step Instructions 1 Add the units. 2 Is the result at least 10? • If no - GO TO Step 3. • If yes - regroup into tens and units, then GO TO Step 3. Step Instructions 3 Add the tens. 4 Is the result at least 100? • If no - GO TO Step 5. • If yes - regroup into hundreds and tens, then GO TO Step 5. Step Instructions 5 Add the hundreds. 6 Is the result at least 1000? • If no - GO TO Step 7. • If yes - regroup into thousands and hundreds, then GO TO Step 7. Step Instructions 7 Write the final answer by adding the thousands, hundreds, tens and units. The result he gets is a three-digit number. Which of the following could be those two numbers? A

55 and 36

42 and 27

48 and 53

34 and 45

Using the subtraction algorithm provided. Step Instructions 1 Is the units digit of the first number less than the units digit of the second number?

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• If no - GO TO Step 2. • If yes - regroup the tens from the first number to give a ten to the units, then GO TO Step 2. Step Instructions 2 Subtract the units of the second number from the units of the first number. 3 Subtract the tens of the second number from the remaining tens of the first number. 4 Write the final answer by adding the results of Steps 2 and 3. Tom wants to calculate the difference between two numbers. He doesn’t need to regroup at Step 1, and the result of Step 2 is 2. What could be the units digit of the second number if the units digit of the ﬁrst number is 4?

3.02 Following algorithms After this lesson, you will be able to… • follow an algorithm. • analyse and correct given algorithms. • apply the concept of algorithms to solve practical problems. • create own examples of algorithms for various tasks.

Follow algorithms When following an algorithm, it is important that the steps are clear and in the correct order. Try to work out what this algorithm is for: Step 1 2 3 4 5 6 7

Instruction Boil the kettle. Put teabag into tea cup. Add milk. Put water into kettle. Take out teabag. Stir with a spoon. Add one teaspoon of sugar.

Could you guess these were instructions for how to make a cup of tea? What was wrong with the algorithm? What do you need to do to improve it? Other examples of algorithms that you could design are: • The route from your school to the nearest park. • How to make a cake. • Create a code for computing. • Instruction on how to grow a sunflower.

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3.02 Practice questions Understanding 1

1−3

Adventurer

Trailblazer

1−3

What could the algorithm be for? Step 1 2 3 4

Explorer

Instruction Weigh out all the ingredients. Mix butter, sugar and eggs together. Stir in the ﬂour. Bake in the oven.

Identify the step in the algorithm that indicates an action being performed. Instruction Measure 250 ml of water. Having a clean bowl. Ensure oven is preheated.

Which step in the algorithm is unnecessary? Step 1 2 3 4 5

Fluency Ex 1

Instruction Choose a pair of socks. Put one sock on your left foot. Put one sock on your right foot. Make sure both socks are from the same pair. Adjust for comfort.

4−7, 9

4−6, 8−9

Which can be accomplished through a deﬁned algorithm? a

Writing a novel

Painting a room

Feeling happy

Changing a car tire

Alana has the algorithm for wrapping a present but it is in the wrong order: • Stick paper down with sticky tape. • Choose wrapping paper. • Decorate with a bow. • Wrap the paper around the present. a

What should be the ﬁrst step?

What would be the last step?

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The given instructions are in the wrong order: • Turn off the tap. • Rinse the soap away with water. • Rub your hands. • Put soap on your hands. • Turn on the tap. a

What should be the second step?

What would be the last step?

The steps for setting up a board game are jumbled. Which step should come ﬁrst? • Roll the dice to start the game. • Place the board on a ﬂat surface. • Distribute cards to each player. • Decide on the number of players.

Ex 2

116

Using a common algorithm for making tea, which steps come after adding the tea bag? A

Pour in cold milk.

Heat water in the kettle.

Add sugar (if desired).

Boil the water.

For each algorithm, work out the output at each step and complete the table: a

Step 1 2 3

Instruction Start with the number 100. Subtract 22 from 100. Round the result to the nearest ten.

Output

Step 1 2 3

Instruction Start with the number 70. Double 70. Add 21.

Output

Step 1 2 3

Instruction Start with the number 16. Add 5 to 16. Double the result.

Output

Step 1 2 3

Instruction Start with the number 2. Add 2 to 2. Round your answer to the nearest ten.

Output

Step 1 2 3

Instruction Start with the number 36. Halve 36. Add 3 to the result.

Output

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Step 1 2 3

Instruction Start with the number 12. Multiply 12 by 10. Add 5 to the result.

Output

Step 1 2 3

Instruction Start with the number 6. Subtract 1 from 6. Double the result.

Output

Reasoning 10

10a-c, 11−13

10−13

This algorithm is designed as a mind reading trick. The magician says: • Think of any number. • Multiply the number by itself. • Subtract the number. • Divide the result by your number. • Now tell me your answer. From this answer, the magician is always able to tell the audience member what their starting number was.

Run through the algorithm for the numbers 5, 6 and 7.

What is the pattern?

Predict the answer for a starting number of 122.

Can you describe how the magic trick works?

Correct the sequence of the algorithm used to create an email account. Justify your answer for each step: Step 1 2 3 4 5

Instruction Start sending emails to friends. Choose an email service provider. Verify your email account through a conﬁrmation email. Fill out the signup form with personal details. Select a unique email username and password.

If an algorithm for making tea instructs you to ﬁrst boil water, then add the tea leaves, but you reversed the steps, how might this affect the outcome?

A student claims that all tasks in life can be broken down into algorithms. Can you think of any tasks or scenarios where this might not be true? Why?

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Problem-solving 14 14

Description For files less than 10 MB, the price is $5. For files between 10 MB and 50 MB, the price increases by $0.20 for every MB over 10 MB. For files between 50 MB, there’s a flat fee of $15 plus $0.10 for every MB over 50.

2 3

A game company uses the given algorithm to score players based on their performance. If a player answered a sequence of questions as (Correct, Correct, Correct, Wrong, Correct, Correct, Correct, Correct), what would be their total score using this algorithm? Step 1 2 3

15−16

A software company uses the given algorithm to calculate the price of a digital product based on its size. What would the price be of a file size of 57 MB? Step 1

14−15

Description For every correct answer, players earn 5 points. For every wrong answer, 2 points are deducted. For 3 consecutive correct answers, they earn an additional bonus of 10 points.

A transportation algorithm determines the fare for a taxi ride based on the given rules: Step 1 2 3 4

Description The base fare is $3. For the ﬁrst 5 kilometres, the cost is $1.50 per kilometre. For kilometres 6 to 10, the cost is $2.00 per kilometre. For kilometres beyond 10, the cost is $2.50 per kilometre.

Using this algorithm, calculate the taxi fare for 12 kilometres.

3.03 Branching After this lesson, you will be able to… • identify and analyse branches in algorithms. • create simple branching algorithms using “if” statements. • visualise branching through flow charts and understand their representation.

Branching Branching is a term which means our algorithm can go in different directions, based on a decision that is made. Branching is used when you need to decide what to do next based on certain conditions. 118

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Conditional is a term which means that a decision needs to be made based off a condition. A conditional shows that there is branching. To determine if a branch exists in an algorithm, the key word to look for is “if”. Look at this algorithm. Choose a number between 1 and 20. If

Then Divide by 2 Add one and then divide by 2

It is even It is odd

The conditional is ‘if the number is even or if something is odd’. We can display a branching algorithm in a flow chart. START

Choose a number between 1 and 20

Divide by 2

YES

Is the number even?

END

Add 1

Divide by 2

Example 1 START

Using the flow chart:

Start with the number

Halve it

a Flow through the chart using the starting number of 8. What is the result?

END

YES

Is the number even?

Subtract 1

Halve it

Create a strategy Flow through the chart. The first decision is, “Is the number even?”.

Apply the idea Since 8 is a multiple of 2, it is an even number. So, we need to follow the condition to halve it. Result = 8 ÷ 2 =4

Halve the given number Evaluate

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3.03 Practice questions Understanding

Adventurer 1−2, 3bd

Trailblazer 1−2, 3cd

What happens when an algorithm has branching?

In the context of coding and algorithms, what does a conditional refer to?

Which algorithms require branching? a

Step 1 2 3

Instruction Start with the number 100. Subtract 22 from 100. Round the result to the nearest ten.

Step 1 2 3

Instruction Begin with a number of 5. If the number is bigger than 40, add 1. If the number is less than 40, double it.

Step 1 2 3

Instruction Begin with a number of 82. If the number is less than 90, multiply it by 10, then add 8. If the number is more than 90, add 8.

Step 1 2 3

Instruction Start with the number 70. Double 70. Add 21.

Fluency Ex 1

Explorer 1, 3ab

4−5, 7−8, 10a-c

5−6, 8−9, 10b-d

6−9, 10d-f

Follow the algorithm for each number: a

Start with the number 10.

If The number is even The number is odd

Start with the number 11.

Then Halve it Subtract one and halve it

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Follow the algorithm for each number: a

Start with the number 14.

Start with the number 35.

b START Start with the given number

Subtract 10

Is the number bigger than 20?

YES

Add 10

END

Follow the algorithm for each number: a

Start with the number 45. Step 1 2 3

Instruction Begin with chosen number. If the number is more than 35, add 4. If the number is less than 35, double it.

Follow the algorithm for each number: a

Start with the number 11.

If The number is even The number is odd 8

Start with the number 27.

Start with the number 16.

Then Double it Double it then add 5

Follow the algorithm for each number: a

Start with the number 8.

Start with the number 35.

START Start with the number

Add 7

YES

Double it

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Is the number even? END

Double it

Add 7

Follow the algorithm for each number: a

Start with the number 240.

If The number is less than 100 The number is more than 100 Ex 2

Then Multiply it by 10 then add 10 Add 10

Given the rounding algorithm: If The digit after the decimal point is 5 or more The digit after the decimal point is less than 5

Then Round up to the next whole number Write the number without the decimal part

Use the algorithm to round 7.2

Use the algorithm to round 3.5

Use the algorithm to round 8.4

Use the algorithm to round 6.7

Use the algorithm to round 24.9

Use the algorithm to round 26.0

Reasoning 11

Start with the number 97.2

11−12

Given the algorithm: If The number is bigger than 30 The number is less than 30

Then Subtract 15 Add 15

Can a number ever remain unchanged after completing the algorithm? 12

Given the algorithm: If

Then

The number is even The number is odd

Add 10 then double it Double it then add 10

How would the results differ if the operations for even and odd numbers were swapped? Use the example of 10 to illustrate your point.

Problem-solving 13−14 13

13, 15

13, 15−16

Given the algorithm: If The number is even The number is greater than 30

Then Add 3 then multiply it by 10 Add 2 then divide it by 2

Identify what is wrong with this algorithm. Justify your answer using the example of 32.

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Imagine a scenario where you’re designing an algorithm for a parking fee system. The system works as follows: If

Then

A car stays for less than 3 hours A car stays for 3 to 5 hours A car stays more than 5 hours

The fee is $5 The fee is $10 The fee is $20

Calculate the fee for a car that stays for 4 hours and 30 minutes. 15

A game has an algorithm to assign scores as follows: If

Then

A player collects a gold coin A player collects a silver coin A player collects a mystery box

They get 10 points They lose 5 points Their score doubles

Given that a player collected 3 gold coins and 2 silver coins and then a mystery box, what is their ﬁnal score? 16

An algorithm adjusts temperatures as follows: If The temperature is below freezing (0 °C) The temperature is above boiling point (0 °C) The temperature is between freezing and boiling point

Then It adds 5 °C. It subtracts 10 °C It does not change

Adjust the temperature that is currently at 105 °C.

3.04 Iterations After this lesson, you will be able to… • identify iterative steps in real-life examples and mathematical contexts. • analyse and follow algorithms containing iterations and perform necessary calculations. • apply knowledge of iterations to solve problems and create algorithms.

Iterations Iterations are steps that repeat within an algorithm.

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An example of an iteration are the steps involved in brushing your teeth:

START Squeeze a small amount of toothpaste onto toothbrush Lift toothbrush to mouth Move to next tooth Have all teeth been brushed?

Brush 1 tooth

Yes Take toothbrush out of your mouth Wash remaining toothpaste from toothbrush END

Another example of an iteration could be steps to write out the 4 times tables: Step 1 2 3

Instruction Start with the number 0. Add 4 to this number. Repeat Step 2 eleven more times.

The results would be 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48. The 12 iterations means that we have added four, 12 times. Iterations are useful when programming computers because computers can perform repetitive tasks multiple times using only one set of instructions.

Example 1 Nadia follows an algorithm to eat her breakfast. Step 1 2 3 4 5 6 7

Instruction Place bowl and spoon on the table. Pour cereal into the bowl. Pour milk into the bowl. Spoon a mouth of cereal into your mouth. Chew the cereal. Repeat Steps 4 and 5 until the cereal is finished. Place bowl and spoon in the sink.

Does this algorithm contain an iteration?

Create a strategy

Apply the idea

Look for a step that tells to repeat a prior step.

Since Step 6 tells us to “repeat” Steps 4 and 5, the algorithm contains an iteration.

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3.04 Practice questions Understanding

Adventurer 1−3, 4bc

Trailblazer 1−3, 4cd

In the context of algorithms, what are the repeated steps called?

In an algorithm, if there have been 6 iterations of a step, how many times has that step occurred?

Why might iterations be especially useful when programming computers to perform tasks?

Identify which algorithm contains iterations. a

Step Instruction 1 Begin with a number of 7. 2 Multiply the number by 3. 3 Add 15 to the result.

Step 1 2 3 4

Step Instruction 1 Begin with a number of 6. If the number is bigger than 2 50, subtract 2. If the number is less than 50, 3 add 5.

Step Instruction 1 Start with the number 50 000. 2 Divide this number by 5. 3 Repeat Step 2 three more times.

Fluency Ex 1

Explorer 1−3, 4ab

5, 6a, 7abd, 9a, 10a

Instruction Start with the number 45. Subtract 3 from this number. Repeat Step 2 two more times. Double the number.

6, 7b-e, 8, 9a, 10b

6, 7c-f, 8−9, 10b

David follows two algorithms to complete his gardening tasks. Algorithm 1: Planting plants Step 1 2 3 4 5 6 7 8 9 10

Instruction Dig a hole for the first plant. Place the first plant into the hole. Cover the first plant with soil. Dig a hole for the second plant. Place the second plant into the hole. Cover the second plant with soil. Dig a hole for the third plant. Place the third plant into the hole. Cover the third plant with soil. Water all planted plants.

Algorithm 2: Watering plants Step 1 2 3 4 5 6

Instruction Fill the watering can from the tap. Walk to the ﬁrst plant. Water the plant evenly. Move to the next plant. Repeat Steps 3 and 4 until all plants are watered. Return the watering can to its place.

Which algorithm shows an iteration?

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Determine and draw the shape that has been mapped out if the given algorithms are followed: a

Ex 2

Step Instruction Stand in the middle of a 1 room. 2 Take ﬁve steps forward. 3 Turn left. Repeat Steps 2 and 3 three 4 more times.

Step 1 2 3 4 5 6

Instruction Stand in the middle of a room. Take three steps forward. Turn right. Take six steps forward. Turn right. Repeat Steps 2, 3, 4, and 5 one more time.

Complete the following tables by writing the results of each the algorithm: a

Step 1 2 3

Instruction Start with the number 7. Add 4 to this number. Repeat Step 2 three more times.

Result

Step 1 2 3

Instruction Start with the number 8. Multiply this number by 6. Repeat Step 2 three more times.

Result

Step 1 2 3 4

Instruction Start with the number 54. Add 6 to this number. Repeat Step 2 three more times. Round this number to the nearest 10.

Result

Step 1 2 3 4

Instruction Start with the number 35. Add 2 to this number. Repeat Step 2 two more times. Half the number.

Result

Step 1 2 3

Instruction Start with the number 40 000. Divide this number by 10. Repeat Step 2 three more times.

Result

Step 1 2

Instruction Start with the number 5. Double this number. Repeat Step 2 until the number is greater than 40.

Result

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For the algorithm. Step Instruction 1 Start with the number 7. 2 Add 7 to this number. 3 Repeat Step 2 three more times.

What is the ﬁnal answer?

Write this algorithm as a sum.

What is the ﬁnal answer?

Write this algorithm as a product.

How many times was Step 2 performed?

Step 1 2 3

Instruction Start with the number 2. Add 5 to the number. Repeat Step 2 until the answer is greater than 30.

Step 1 2 3

Instruction Start with the number 37. Subtract 4 from the number. Repeat Step 2 until the answer is less than 20.

For each algorithm: i

Complete the table by writing the results of the algorithm.

Write the multiplication that represents the problem in the table.

Step 1 2 3

Instruction Start with the number 6. Add 6 to this number. Repeat Step 2 three more times.

Result

Step 1 2 3

Instruction Start with the number 13. Add 13 to this number. Repeat Step 2 three more times.

Result

Reasoning 11

How many 7’s have been added together?

For each algorithm: i

11–12

12−13

11−13

This algorithm shows the steps to write out the 7 times tables. If you wanted to see all the times tables up to 7 × 12, what step would you have to change? Justify your answer. Step 1 2 3

Instruction Begin with the number 0. Add 7 to this number. Repeat Step 2 six more times.

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This algorithm shows the steps to write out the 4 times tables. If we limited our iterations to 3 instead of 6, how would the final listing be different? Step 1 2 3

Instruction Begin with the number 0. Add 4 to this number. Repeat Step 2 five more times.

A farm collects eggs using an algorithm: Step 1 2 3 4 5

Instruction Feed the hens. Collect eggs from 5 hens. Store eggs in a container for 7 minutes. Transfer to storage room. Repeat Steps 2 to 4 until all hens have been attended to.

If Steps 2 to 4 are designed for ﬁve hens and the farm has 25 hens, what does this suggest about the farm’s egg collection capacity using this algorithm?

Problem-solving 14−15 14

15−17

15−18

Jane is writing an algorithm to reach a specific number. She starts with the number 3 and adds 4 repeatedly. Here is her algorithm: Step 1 2 3

Instruction Start with the number 3. Add 4 to the number. Repeat step 2 three times.

What is the final result of this algorithm?

The final result is supposed to be 19. What is the mistake in Step 3?

Rewrite Step 3 so that the final result is 19.

A computer program runs an iteration that doubles a given number until it is bigger than 1000. If the starting number is 5, how many times will the iteration run?

Tom follows an algorithm to water his garden plants: Step 1 2 3 4 5

Instruction Fill the watering can. Water a plant. Move to the next plant. Repeat Steps 2 and 3 until all plants are watered. Reﬁll the watering can if needed and go back to Step 2.

If Tom has 8 plants and his watering can can water only 3 plants at a time, how many times will he need to reﬁll the can?

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A bakery follows an algorithm to bake cookies: Step 1 2 3 4 5

Instruction Set the oven to 180 °F. Place 12 cookies on a baking tray. Bake for 10 minutes. Remove and cool. Repeat Steps 2 to 4 until all cookie dough is used.

If the bakery has dough for 60 cookies, how many times will Steps 2 to 4 be repeated? 18

An algorithm multiplies a given number by 4 and then subtracts 3. This process is repeated until the result is greater than 200. If the starting number is 10, what will be the ﬁnal result after applying the algorithm?

3.05

Flow charts

After this lesson, you will be able to… • identify and use basic components of flow charts including ovals, rectangles, and diamonds. • analyse and interpret linear sequences in flow charts. • use SET, PRINT and loops in flow charts.

Flow charts Flow charts are visual diagrams that use different shapes, lines, arrows and other symbols. They show the steps of a process in the order that they happen. Simple flow charts use 3 different shapes: ovals, rectangles and diamonds. Ovals are used to show the beginning and end. START

END

Rectangles are used to state actions. Add 3

Eat lunch

Halve

Move 3 steps left

Open the door

Answer the phone

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Diamonds are used to indicate decisions that need to be made. You can think of these as yes or no questions.

Is it even?

Are you cold?

Is the number >10?

Is the phone ringing?

START

We connect these shapes with arrows to show the order we move through the flow chart.

Choose a number between 1 and 10

This is an example of a linear sequence, it moves from beginning to end in one straight line. There are no decisions in this flow chart.

Multiply by 10

This particular flow chart is describing one of the multiplication strategies we can use if we want to multiply by 5.

Halve

END

Using the flow chart let’s multiply 7 by 5: START

Choose a number between 1 and 10

Multiply by 10

132

What is the answer to 7 × 5?

7 × 10 = 70

Halve

Half of 70 is 35

END

Our answer is 35

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Branches Whenever we need to make a decision our ﬂow chart branches out. START

The time is 8 am

Eat breakfast

END

This flow chart shows the decision in the diamond. The answer to a decision can be yes/no, or true/false etc. This creates a branch in our flow chart. In this case, our true (T) answer resulted in an action and the false (F) answer resulted in the end.

Example 2 What is the output of this algorithm if the input is A = 2 and B = 5? START

Input A Input B

C=B−A

Is A > B?

Yes

C=A−B

Output C

END

Create a strategy Use the given numbers for A and B and follow the ﬂow chart.

Apply the idea A, B = 2, 5

Input A = 2 and B = 5

Is 2 greater than 5? Then the maths line can be: Is 2 > 5 = No C=5–2 Output = 3

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Decide if A > B Find C = B − A Output C

3.05 Practice questions Understanding

Adventurer 1, 3−5

3, 5

Trailblazer 1−5

How is branching different to a loop?

Imagine a flow chart where you want to determine if a number is even or odd. Which shape would you use to ask the question about the number’s nature?

In the context of a flow chart, what does the action “PRINT” typically mean?

Given the statement SET number = 2 followed by SET number = SET number + 2, what value would the variable “number” have after the second statement is executed?

Look at these shapes: • Shape A: • Shape B: • Shape C:

Which shape represents a decision in a ﬂow chart?

Which shape represents where the ﬂow chart begins?

Which shape represents an action in a ﬂow chart? 6−7, 10

8−11

Input a number

Multiply the number by 5

Add 3 to the number

Output the number

End

Output the number

End

Output the number

End

What is the output of this algorithm if the input is 8? Start

6−7, 10−11

What is the output of this algorithm if the input is 6? Start

• Shape D:

Fluency Ex 1

Explorer

Input a number

Multiply the number by 10

Subtract 3 from the number

What is the output of this algorithm if the input is 10? Start

Input a number

Square the number

Divide the number by 2

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Ex 2

What is the output of this algorithm if the input is A = 4 and B = 8?

Start Input A Input B C=B−A

Is A > B?

Yes

C=A−B

Output C End

Ex 3

Look at these flow charts: • Flow chart A: • Flow chart B: Start

Start

Let i = 1

Choose a number between 1 and 10

Evaluate i + 5

Add 5 to the number

End Halve the number End

• Flow chart C: START

Let A = 5, B = 8

C=B−A

Is A > B?

Yes

C=A−B

PRINT C

END

138

Which ﬂow chart is not linear?

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Which ﬂow chart shows branching?

Look at these flow charts: • Flow chart D: • Flow chart E:

Start

SET number = 1

Choose a number between 1 and 10 Multiply the number by 10

SET number = number +1

Is number >10?

Halve the number

Yes End

End

Which ﬂow chart is linear?

Reasoning 12

Which ﬂow chart contains a loop?

Look at the flow chart:

Start

Mary claims that when f is equal to 7, the flow chart will output “f is less than 7”. Is she correct? Why or why not?

SET f = 7

PRINT f is 7 or greater

Is f < 7?

Yes

PRINT f is less than 7

End

Susan claims that if h starts at 6, the message “Negative value detected” will be printed after 4 iterations. Is she correct? Justify your answer.

Start

SET h = 6

Is h < 0?

h=h−2

Yes PRINT Negative value detected

End

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Problem-solving 15a 14

14−15

15−16

Based on a flow chart for an online quiz game: Start

SET a = answer Provide another question Is a correct?

Deduct 1 point

Yes Add 2 points

End

If you’ve answered three questions wrong consecutively, what would be your score change based on the flow chart before the quiz ends? 15

Imagine a fitness tracker app that sends a celebratory notification once the user has taken 10 000 steps in a day. The app continuously counts each step the user takes. If the user hasn’t reached 10 000 steps, the tracker increments the step count by one and keeps checking until the goal is reached. Once 10 000 steps are hit, the celebratory notification is sent.

Draw a flow chart that describes this process.

Based on your flow chart, if you’ve taken 9999 steps for the day, how many more steps would you need to take before receiving a celebratory notification?

The Euclidean algorithm is a method for finding the highest common factor of two numbers. It is based on the principle that the HCF of two numbers also divides their difference. Here is a brief outline of the algorithm: 1.

Start with two numbers, a and b, ensuring a > b.

2. Divide a by b to get the quotient and remainder r. 3.

If r = 0, then b is the HCF of a and b, and the algorithm ends.

If r ≠ 0, update a to be the old b, and b to be r.

5. Repeat steps 2 through 4 until an exact division is found, meaning r = 0. 6. The value of b when r = 0 is the HCF of the original numbers a and b.

140

Create a flow chart that represents the steps of the Euclidean algorithm.

Find the highest common factor for each pair of numbers: i

70 and 35

iii

33 and 12

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60 and 40

3.06

Algorithms in code

After this lesson, you will be able to… • recognise common notations in pseudocode such as INPUT, OUTPUT, SET, IF-THEN-ELSE. • translate a flow chart into pseudocode. • apply the IF-THEN-ELSE statement to make decisions in code. • interpret pseudocodes and understand what they represent.

Pseudocode and its common notations Before translating an algorithm into a specific programming language, or language that a computer can understand, the steps of the algorithm can be written as pseudocode. Pseudocode is a way of writing code that is human-friendly. With pseudocode, we don’t need to worry about writing an algorithm in the specific way the computer needs. Some common notations for pseudocode are: • INPUT - indicates a user will be inputting something. • OUTPUT or PRINT - indicates that an output will appear on the screen. • SET - assign an initial value to an expression. START

This algorithm (flow chart) can be written in pseudocode as shown.

SET a = 2

SET b = 3 PRINT a + b

END

START

SET a = 2

SET b = 3

PRINT a + b

END

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Example 1 Consider the pseudocode. 1

START

SET a = 73

SET b = 44

SET c = a + b

PRINT c

END

What value is printed at the end of the code?

Create a strategy Add a and b to ﬁnd the value of c.

Apply the idea c=a+b

Write the equation for c

= 73 + 44

Substitute a = 73 and b = 44

= 117

Evaluate

117 is printed at the end.

Example 2 Consider the rectangle shown in the image. The dimensions of the rectangle are W metres and L metres. W L

a What does this pseudocode calculate?

142

START

SET W = 3

SET L = 6

SET A = W * L

PRINT A square metres

END

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When would pseudocode use the ELSE statement?

What does the PRINT command do in pseudocode?

Why do you think that algorithms are commonly represented using pseudocode in the initial stages of development?

Fluency Ex 1

5ab, 6

5, 7−8

5c-f, 7−8

Determine the value that is printed at the end of the pseudocodes: / means division and * means multiplication a

START

SET a = 75

SET a = 26

SET b = 25

SET b = 11

SET c = a + b

SET c = a * b

PRINT c

END

START

SET a = 2

SET a = 7

SET b = a * a * a

SET b = 5

PRINT b

SET m = (a + b) / 2

END

PRINT m END

START

SET a = 4

SET a = 78

SET b = 3

b = a + 73

SET c = 9

PRINT b

SET d = a + b * c

END

PRINT d END Ex 2

Consider the rectangle shown in the image. The dimensions of the rectangle are W metres and L metres.

For the pseudocode: START

A C

SET W = 10 SET L = 12 SET A = W * L

What does the code calculate?

Volume Area

B D

Units Perimeter

What does the code PRINT?

PRINT A m2 END

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Consider the rectangle shown in the image. The dimensions of the rectangle are W metres and L metres.

For the pseudocode:

START

What does the code calculate? A C

SET W = 7 SET L = 9 b

SET P = 2 * W + 2 * L

Volume Area

B D

Units Perimeter

What does the code PRINT?

PRINT P m END Ex 3

START

What statement is printed at the end of the pseudocode?

SET a = 11 IF a / 2 is whole number THEN PRINT

“a is even”

ELSE PRINT

“a is odd”

END IF END

Reasoning 9

9, 11

10−11

The pseudocode to find the sum of two numbers is shown: Step A B C D E

Rimma believes that the pseudocode is in the correct order and Ivana says that the order is incorrect. Who is correct? Justify your answer.

Code SET a = 2 END SET b = 3 PRINT a + b START

If a pseudocode has the line “IF is set to 0?

For the given pseudocode, why is there a multiplication by 2 for both W and L?

146

is integer”, what could be a potential problem if variable b

START

SET W = 8

SET L = 7

SET P = 2 * W + 2 * L

PRINT P m

END

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Problem-solving 13 12

12−13

12−15

Modify the pseudocode such that it calculates the average of x and y and prints the result. 1

START

SET x = 5

SET y = 10

SET z = x + y

PRINT z

END

Design a pseudocode that calculates and prints the square of a number n.

You are provided with two pseudocodes. One calculates the area of a triangle with base b and height h, and the other calculates its perimeter with sides a, b, c. Write the two pseudocodes.

Design a pseudocode that takes in two numbers a and b, swaps their values, and prints the new values of a and b.

3.07

Loops After this lesson, you will be able to… • recognise the structure and function of a WHILE loop. • analyse and interpret pseudocode with iterations. • apply the concept of loops to create and modify algorithms. • connect the understanding of loops to real-world scenarios and problem.

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Loops Loops (or iterations) are steps that are repeated a number of times in an algorithm. For example, the steps involved in eating breakfast could be:

START Fill bowl with cereal. Pour milk onto cereal. Place spoon in bowl and scoop up cereal and milk. Lift spoon to mouth and empty contents of the spoon into mouth. Eat spoonful of cereal and milk. Place spoon in bowl and scoop up cereal and milk. Lift spoon to mouth and empty contents of the spoon into mouth. Eat spoonful of cereal and milk. Place spoon in bowl and scoop up cereal and milk. Lift spoon to mouth and empty contents of the spoon into mouth. Eat spoonful of cereal and milk. Place spoon in bowl and scoop up cereal and milk. Lift spoon to mouth and empty contents of the spoon into mouth. Eat spoonful of cereal and milk. END

This algorithm could be rewritten using a loop:

START Fill bowl with cereal. Pour milk onto cereal. Eat spoonful of cereal and milk.

Is there cereal in the bowl? No END

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Yes

Place spoon in bowl and scoop up cereal and milk.

The loop is Steps 4 to 6 which is represented by the indent in the algorithm. When writing algorithms in pseudocode we can use a WHILE loop. A WHILE loop states a condition at the start of the loop which determines how many times a loop is repeated in an algorithm. Look at this algorithm written in pseudocode, and the explanation of each line of code: 1 START 2 SET a = 2 3 SET sum = 0 4 WHILE a < 5

The loop repeatedly adds the current value of variable a to the sum and then increments a by one until a is no longer less than five.

5 sum = sum + a 6 a = a + 1 7 END WHILE 8 PRINT sum 9 END This loop contains two steps:

sum = sum + a a=a+1

These will be repeated until a is no longer less than 5. The values of the variables a and sum for each loop are: initially after first loop after second loop after third loop

sum 0 0+2=2 2+3=5 5+4=9

a 2 2+1=3 3+1=4 4+1=5

At the end of the third loop the value of a is 5 and so the condition for the loop to end has been reached. The ﬁnal value of sum is 9. This is the value that will be printed (OUTPUT) from the algorithm.

Example 1 Look at this pseudocode: 1

START

SET a = 2

SET sum = 0

WHILE a < 9

5 sum = sum + a 6 a = a + 1 7

END WHILE

PRINT sum

END

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a What is the initial value of a?

Create a strategy Look for the value SET for a.

Apply the idea From line number 2, we can see that the initial value SET for a is 2.

b What is the OUTPUT of this code?

Create a strategy Make a table of values for the loop in Steps 5 and 6.

Apply the idea Making a table of values for each loop until reaching a = 9 we have: initially after first loop after second loop after third loop after fourth loop after fifth loop after sixth loop after seventh loop

sum 0 0+2=2 2+3=5 5+4=9 9 + 5 = 14 14 + 6 = 20 20 + 7 = 27 27 + 8 = 35

a 2 2+1=3 3+1=4 4+1=5 5+1=6 6+1=7 7+1=8 8+1=9

After the seventh loop a = 9 of 9, so the loop stops. At this point the sum is 35. Since the last action is to PRINT sum, the output of this code is 35.

Example 2 Consider the algorithm represented by the ﬂow chart:

START sum = 0 a=1 a=5 Yes PRINT sum END

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sum = sum + a

a=a+1

3.07 Practice questions Understanding

Explorer 1−2

What does a loop do in a pseudocode?

For the pseudocode:

Adventurer

Trailblazer

1−2

START

What type of loop is used?

SET a = 2

What is the condition of the loop?

SET sum = 0

What does the sum = sum + a line in the pseudocode do?

WHILE a < 9 sum = sum + a a= a+ 1 END WHILE PRINT sum END

Fluency 3

3, 5a, 7a, 8

Given the ﬂow chart:

3−4, 5a, 6, 7a-c, 9, 10a

4, 5b, 6, 7c-e, 9, 10b, 11

Start

What will be the initial value of a?

If the initial value of sum is 0, what will be the ﬁnal value of sum?

sum = 0

How many times is the loop executed?

a=5 a < 10

Yes

sum = sum + a

No PRINT sum End

For the pseudocode:

START

Find the initial value of a.

SET a = 9

Find the ﬁnal value of a.

SET sum = 0

Find the final value of sum.

WHILE a > 3 sum = sum + a a= a− 1 END WHILE PRINT sum END

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a=a+1

Ex 1

For each pseudocode: i

Find the initial value of a.

Find the OUTPUT of the code.

START

SET a = 3

SET sum = 0

WHILE a < 11

SET n = 0

sum = sum + a

WHILE a < 12

a= a+ 1

sum = sum + a

END WHILE

a= a+ 1

PRINT sum

n= n+ 1

END

END WHILE PRINT sum / n END

START

Find the ﬁnal value, of n.

SET a = 4 SET n = 0 WHILE a > 1 sum = sum + a a= a/ 2 n= n+ 1 END WHILE END 7

Determine the final output of each pseudocode: a

START

SET a = 19

SET a = 5

SET a = 1

WHILE a > 5

WHILE a < 7

WHILE a < 24

a= a– 1

PRINT a

END WHILE

START

a= a+ 1

a= a* 2

PRINT a

END WHILE

END

START

SET a = 12

SET a = 5

WHILE a > 2

WHILE a < 20

PRINT a

a= a− 2

a = a/5

END WHILE

END

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Ex 2

For the ﬂow chart: Start

a = 10

a>4

Yes

PRINT a

a=a−1

No End

Arrange these lines of pseudocode in the correct order for the algorithm. Step A B C D E

Code WHILE a > 4 END WHILE PRINT a SET a = 10 a= a− 1

For the ﬂow chart: Start sum = 0 a=2 a=6

sum = sum + a

a=a+1

Yes PRINT sum End

Arrange these lines of pseudocode in the correct order for the algorithm. Step A B C D E F G 154

Code a= a+ 1 SET a = 2 WHILE a ≠ 6 PRINT sum sum = sum + a END WHILE SET sum = 0

Mathspace Victorian Curriculum – Year 7 mathspace.co

The flow chart shows the algorithm for finding the sum of the first n natural numbers (positive whole numbers). Start Sum = 0 a=1 sum = a + sum a = n? Yes

a=a+1

PRINT sum End

Find the output of this algorithm if n = 4.

Arrange the lines of pseudocode for adding the ﬁrst 50 counting numbers in the correct order for the algorithm. Step A B C D E F G

Code sum = a + sum PRINT sum SET a = 1 a= a+ 1 GO TO LINE 3 SET sum = 0 IF a = 50 THEN GO TO LINE 7

The factorial of a non-negative integer n is given by n! = n × (n − 1) × (n − 2) × ... ×2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The ﬂow chart represents the algorithm for ﬁnding n!: Start p=1 a=1 p=a∗p a = n? Yes

a=a+1

PRINT p End

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Find the output of this algorithm if n = 6.

Arrange the lines of pseudocode in the correct order for ﬁnding 50!. Step A B C D E F G

Reasoning 12

Code p = p * a a = a + 1 PRINT p SET a = 1 IF a = 50 THEN GO TO LINE 7 SET p = 1 GO TO LINE 3

12−13

What does the condition a > 5 do in this algorithm’s loop?

12−14

START SET a = 11 SET sum = 0 WHILE a > 5 sum = sum + a a= a− 1 END WHILE PRINT sum END

Why is the value of a increased by 1 in each iteration?

START SET a = 3 SET sum = 0 WHILE a < 10 sum = sum + a a= a+ 1 END WHILE PRINT sum END

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How would the output change if the condition in the second pseudocode were a ≤ 14 instead of a < 14?

START SET a = 5 SET sum = 0 SET n = 0 WHILE a < 14 sum = sum + a a= a+ 1 n= n+ 1 END WHILE PRINT sum / n END

Problem-solving 15 15

15−16

Change the pseudocode so that it works out a product (multiplication) instead of a sum. Change the name of the sum variable and change the operation in the fifth line.

16−17

START SET a = 7 SET sum = 0 WHILE a > 3 sum = sum + a a= a− 1 END WHILE PRINT sum END

Modify the pseudocode to calculate the sum of squares of the numbers.

START SET a = 9 SET sum = 0 WHILE a > 3 sum = sum + a a= a− 1 END WHILE PRINT sum END

Design a pseudocode that not only calculates the sum of a series of numbers but also records the smallest and largest number in the sequence.

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3 Chapter review 1

What sort of task could this algorithm be the instructions for? Step 1 2 3 4

Instruction Prepare the soil. Dig a small hole. Place the ﬂower seed or seedling in the hole. Water the plant gently.

In an algorithm, if there have been 7 iterations of a step, how many times has that step been executed?

Given the statement SET number = 5 followed by SET number = SET number + 3, what value would the variable “number” have after the second statement is executed?

For the pseudocode:

START

What is the condition of the loop?

SET a = 3

What does the sum = sum + a line in the pseudocode do?

SET sum = 0 WHILE a < 15 sum = sum + a a= a+ 2 END WHILE PRINT sum END

This function machine is designed to add twelve to a number. ⬚ ⟶ +12 ⟶ ⬚

If we put 29 into the function machine, will we need to regroup when we add the units?

If we put 16 into the function machine, will we need to regroup when we add the units?

escribe which type of numbers will involve regrouping when put into this function D machine.

This algorithm is designed to perform addition: Step 1 2

Instructions Add the units. Is the result at least 10?

• If no - GO TO Step 3. • If yes - regroup into tens and units, then GO TO Step 3. Step 3 4

Instructions Add the tens. Is the result at least 100?

• If no - GO TO Step 5. • If yes - regroup into hundreds and tens, then GO TO Step 5. 158

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Step 5 6

Instructions Add the hundreds. Is the result at least 1000?

• If no - GO TO Step 7. • If yes - regroup into thousands and hundreds, then GO TO Step 7. Step 7 a

Work through the algorithm to calculate 257 + 314, identify the result for each step: i

Step 2

Step 4

Step 2

Step 4

Step 6

Step 7

iii

Step 6

Step 7

Using a common algorithm for assembling a bookshelf, which step comes after arranging the pieces on the ﬂoor? a

Place books on the shelves.

Fix the backboard.

Connect the sides to the bottom shelf.

Hang the bookshelf on the wall.

For each algorithm, work out the output at each step and complete the table: a

Step 1 2 3

Instruction Start with the number 120. Subtract 25 from 120. Round the result to the nearest ten.

Output

Step 1 2 3

Instruction Start with the number 18. Add 6 to 18. Double the result.

Output

Step 1 2 3

Instruction Start with the number 40. Halve 40. Add 4 to the result.

Output

Follow the algorithm starting with the number 48: If The number is more than 45 The number is less than 45

iii

Work through the algorithm to calculate 581 + 192, identify the result for each step: i

Instructions Write the ﬁnal answer by adding the thousands, hundreds, tens and units.

Then Add 5 Double it

Follow the algorithm starting with the number 260: If The number is less than 110 The number is more than 110

Then Multiply it by 12 then add 15 Add 15

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Complete the given tables by writing the results of each algorithm: a

Step 1 2 3

Instruction Start with the number 10. Add 5 to this number. Repeat Step 2 three more times.

Result

Step 1 2 3 4

Instruction Start with the number 50. Add 7 to this number. Repeat Step 2 three more times. Round this number to the nearest 10.

Result

Step 1 2 3

Instruction Start with the number 30 000. Divide this number by 5. Repeat Step 2 three more times.

Result

For the algorithm: Step Instructions 1 Start with the number 9 2 Add 9 to this number. 3 Repeat Step 2 three more times.

How many 9’s have been added together?

Write this algorithm as a sum.

What is the ﬁnal answer?

Calculate 9 × 5.

What is the output of this algorithm if the input is 11? Start

Input a number

Multiple the number by 10

Subtract 3 from the number

Output the number

What is the output of this algorithm if the input is A = 6 and B = 12? START Input A Input B

C=B−A

Is A > B?

Output C

END

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Yes

C=A−B

End

Determine the value that is printed at the end of the pseudocodes: a

START

SET

SET a = 3

SET a = 6

SET a = 90

a = 50

SET b = 2

a = a + 50

SET b = 35

SET b= a∗ a∗ a

SET c = 7

PRINT a

SET c = a + b

PRINT b

END

PRINT c

END

SET d= a+ b∗ c PRINT d

END

END 16

b 17

START

For the pseudocode:

SET W = 10

What does the code calculate? A

Volume

Units

SET L = 10

Area

Perimeter

SET P = 2 ∗ W + 2 ∗ L PRINT P m

What is the output of this code?

END START

What statement is printed at the end of the pseudocode?

SET a = 14 IF a / 2 is whole number THEN PRINT a is even ELSE PRINT a is odd END IF

END

For each pseudocode: i

Find the initial value of a.

Find the OUTPUT of the code.

START

SET a = 4

SET sum = 0

WHILE a < 13

SET n = 0

sum = sum + a

WHILE a < 14

a= a+ 1

sum = sum + a

END WHILE

a= a+ 2

PRINT sum

n= n+ 1

END

END WHILE PRINT sum / n END

Determine the final output that is printed at the end of each pseudocode: a

START

SET a = 15

SET a = 2

SET a = 7

SET a = 4

WHILE a > 8

WHILE a < 30

WHILE a < 10

WHILE a < 50

a = a − 1

PRINT a

END WHILE

a= a∗ 3

a= a+ 1

a= a∗ 3

PRINT a

END WHILE

END

END Chapter 3 review mathspace.co

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Given the algorithm: If The number is even The number is greater than 40

Then Add 4 then multiply it by 12 Add 3 then divide it by 3

Identify what is wrong with this algorithm? Justify your answer using the example of 42. 21

Mia wrote an algorithm to multiply a number by 8, but she made an error. She added 8 to the number eight times. When she used her algorithm for the number 5, what result did she get?

Using the subtraction algorithm, when subtracting 47 from 82, how many times would you need to regroup?

A software company uses the given algorithm to calculate the price of a digital product based on its size. What would the price be of a ﬁle size of 63 MB? Step 1 2 3

Description For files less than 15 MB, the price is $7. For files between 15 MB and 55 MB, the price increases by $0.25 for every MB over 15 MB. For files between 55 MB, there is a flat fee of $17 plus $0.15 for every MB over 55.

Picture designing an algorithm for a library late fee system. The system works in this manner: If A book is returned within 5 days A book is returned between 5 to 10 days A book is returned after 10 days

Then No fee Fee $2 Fee is $5 plus $0.50 for each day beyond 10.

Compute the ﬁne for a book that is returned after 12 days. 25

Lisa follows an algorithm to feed her ﬁsh: Step

Instruction

Prepare the ﬁsh food.

2 3 4 5

Feed a fish. Move to the next fish tank. Repeat Steps 2 and 3 until all fish are fed. Prepare more fish food if needed and go back to Step 2.

I f Lisa has 5 ﬁsh tanks and one preparation of food can feed only 2 tanks at a time, how many times will she need to prepare the ﬁsh food?

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Based on a flow chart for an online quiz game: If you’ve answered five questions wrong consecutively, what would your score change be based on the flow chart before the quiz ends?

Start

SET a = answer Provide another question Is a correct?

Deduct 2 points

Yes Add 3 points

End

You are provided with two pseudocodes. One calculates the area of a rectangle given length, l and width, w, and the other calculates the perimeter. Write the two pseudocodes.

Construct a pseudocode that inputs two characters X and Y, switches their positions, and displays the swapped characters X and Y.

Design a pseudocode that calculates the product of a series of numbers and also determines the smallest and largest number in the sequence.

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Big ideas The number line serves as a model to relate integers to statements of order in real-world contexts.

Curriculum Outcomes VC2M7N08 compare, order and solve problems involving addition and subtraction of integers

4 Integers

Chapter outline 4.01 Integers and the number line 4.02 Compare integers Investigation: Life-size number line 4.03 Add and subtract integers Chapter 4 review

166 173 178 180 188

4.01 Integers and the number line After this lesson, you will be able to… • plot integers on the number line. • determine the location of points on the number line. • identify opposite integers.

Integers on a number line Negative

Positive

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1

2 3 4 5 6 7 8 9 10

The integers are made up of positive and negative whole numbers, as well as the number 0. {…, −3, −2, −1, 0, 1, 2, 3, …} The integers to the right of 0 are positive integers and the integers to the left of 0 are negative integers. 0 is neutral; it is neither negative nor positive. The arrows at each end indicate that the line extends inﬁnitely in the positive and negative directions.

−5 −4 −3 −2 −1 0

2 3 4 5

On this number line, we can see that the point is at the integer −3. This point is 3 units to the left of 0.

−6

−3

On this number line, the location of the point has not been labelled. We can see that there are three gaps between the labels of 0 and 3. This means each gap will be equal to 1 unit, so the location of the point is 4. This point is 4 units to the right of 0.

Interactive exploration

Discover this concept in action online

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mathspace.co

We can also use the number line to ﬁnd the location of a point after it has been moved. If a point located at 1 is then moved 3 units to the right, it will end up at 4. −5 −4 −3 −2 −1 0

If the point at 1 is moved 3 units to the left, it will end up at −2. −5 −4 −3 −2 −1 0

Moving the point to the left will decrease the integer value, and moving the point to the right will increase the integer value. Moving to the left is called moving in the negative direction. Moving to the right is called moving in the positive direction. If we move the point at 0 by 4 places in the positive direction, we get 4. −5 −4 −3 −2 −1 0

If we move the point at 0 by 4 places in the negative direction, we get −4. −5 −4 −3 −2 −1 0

Example 1 Where is the point plotted on the number line? −12

−8

−4

Create a strategy

Apply the idea

Since there are 4 units between every second tick, each tick is 2 units apart.

The point is two units to the left of −4. Number = −6

Example 2 A point is plotted on the number line. Find the number that is 7 units to the left of this plotted point. −10

−5

Create a strategy Find the number represented by the point, then move 7 steps to the left from the point.

4.01 Integers and the number line mathspace.co

167

0 is an integer. Is it negative, positive or neither?

Explain what opposite integers are; give an example. 6a-e, 7a-d, 8, 10, 12, 14ac, 15, 16a-d, 17

Fluency Ex 1

−4 −3 −2 −1

−3

−10

−5

−8

−4

−8

−4

−5

−8

−10

−14

Identify the integer that is 6 units to the right of the plotted point: −5

Identify the integer that is 9 units to the left of the plotted point: −5

Identify the integer that is 6 units to the left of the plotted point: −5

Identify the integer that is 7 units to the right of the plotted point: −5

Identify the integer that is 8 units to the left of the plotted point: −6

−8

Plot these integers on a number line:

−10

170

h 3

−10

Ex 2

−4

−4 −3 −2 −1

−10

−8

−6

−5

6f-j, 7e-h, 11–14, 15cd, 16e-h, 18

Identify the integer where the point is plotted on the following number lines: a

6c-g, 7a-d, 9, 10, 12, 13, 14bce, 16a-d, 17, 18

−4

Mathspace Victorian Curriculum – Year 7 mathspace.co

−2

Identify the integer that is 12 units to the right of the plotted point: −16

Ex 3

−8

−12

−4

Identify the integers that represent these statements: a

Two units to the right of 7

Three units to the left of 12

Two units to the left of −6

Four units to the right of −9

Three units left of 2

Ten units right of −3

For each of the following number lines: i

Identify the integer that has been plotted.

Identify the opposite of the integer that has been plotted.

a −10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

b −14 −12 −10 −8 −6 −4 −2

−20

c −16

−12

−8

−4

d −21 −18 −15 −12 −9 −6 −3

Identify the integer that is the opposite of the following: a

−8

−5

−10

402

−88

123

Consider the integer −4. a

Plot −4 on a number line.

On the same number line, plot its opposite.

The opposite of an integer is −16. Identify the original integer.

Reasoning

20, 22

20, 22–24

19–21, 23, 24

True or False: “The opposite of a number is always greater than the number itself”. Explain your answer.

Explain the importance of 0 on the number line.

Explain why the opposite of 0 is 0.

Explain why the statements “cycling 8 km due south” and “cycling 8 km due north” can be thought of as opposite integer statements.

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The invention of the integer zero, or the concept of nothingness as a number in around 500 AD, was a signiﬁcant development in the history of mathematics. Why do you think zero wasn’t invented earlier?

The number line stretches to positive and negative inﬁnity. Is inﬁnity an integer? Explain your answer.

Problem-solving 25 25

Identify the integer that would represent each situation: i

172

27–29

A gain of $400 can be represented by the integer 400. a

26–28

Gaining another $400

Losing $400

Identify the integer that represents your answer in part (a).

Consider the following statement: “Arriving 14 minutes late.” a

Identify the opposite of “Arriving 14 minutes late”.

Suppose “Arriving 14 minutes late” is represented by the number 14. State the integer that would represent its opposite.

Construct a statement and identify the integer that represents the opposite of each: a

“Descending 2 ﬂoors” is represented by the number −2.

“A weight gain of 11 kg” is represented by the number 11.

“A temperature of 10 °C below freezing point” is represented by the integer −10.

A temperature drop of 21 °C can be represented by the integer −21. a

Identify the integer that represents the opposite of a temperature drop of 21 °C.

Identify the integer that represents the opposite of the opposite of a temperature drop of 21 °C.

A mountain peak is measured to have a height of 5100 m above sea level. An ocean trench is found to be the same distance below sea level. a

If the height above sea level of the mountain peak is represented by the number 5100, identify the integer that represents the distance below sea level of the trench.

In this scenario, what does the integer 0 represent?

Mathspace Victorian Curriculum – Year 7 mathspace.co

b Is this inequality true or false?

Create a strategy Plot the numbers onto the number line to compare their values.

Apply the idea Here are 7 and −5 on the number line. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Since 7 is to the right of −5, it shows that 7 is greater than −5. The inequality is true.

Example 2 Insert an inequality sign, < or > , between the two given numbers to make the statement true.

Create a strategy

−9 ⬚ 9

Plot the numbers onto the number line to compare their values.

Apply the idea Here are −9 and 9 on the number line. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Since −9 is to the left of 9, it shows that −9 is less than 9. The inequality symbol to be used is <. −9 < 9

Example 3 The melting point of krypton is −157 °C. The melting point of radon is −71 °C. Write an inequality comparing the two melting points.

Create a strategy

Apply the idea

A negative integer with a smaller number has a greater value than a negative integer with a larger number.

−71 is a negative integer with the smaller number which means it has a greater value than −157. −157 °C < − 71 °C

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6–10, 11ab, 12a-d, 13a, 14a

Fluency 6

6b, 7b, 8cd, 9cd, 11cd, 12a-d, 13a, 14a

6b, 7b, 8cd, 9cd, 11c-f, 12e-h, 13b, 14b

For each number line: i

Arrange all the plotted integers in ascending order.

Identify the largest number.

a −10

−5

b −20

−16

−12

−8

−4

For each number line: i

Arrange all the plotted integers in descending order.

Identify the smallest number.

a −20

−15

−10

−5

b −12 −10

Ex 1

Ex 2

13, −14, 2

−7, −17, −12

8, −8, 7

35, −36, −40

−14, −4, −21

−7, −24, 10

1, −1, 5

34, −36, −2

8 is less than 10

2 is greater than −4

−5 is less than −1

−3 is greater than −8

For each of the following inequalities i

Write the inequality in words.

Determine whether the inequality is true or false

−7 < −2

8 > −6

7≥4

3≤2

0 > −1

23 < −47

Identify the inequality sign, < or > , that should appear between each pair of numbers to make each statement true: 7 ⬚ −4

39 ⬚ −72

b f

−8 ⬚ 8

−12 ⬚ −14

c g

0 ⬚ −8

−45 ⬚ 22

Write a statement using < or > that compares the two temperatures. a

176

e 13

−2

The statement “7 is greater than 4” can be written as 7 > 4. Write each mathematical statement using an inequality sign:

Ex 3

−4

Identify the smallest number in each set: a

−6

Identify the largest number in each set: a

−8

−19 °C and 4 °C

Mathspace Victorian Curriculum – Year 7 mathspace.co

−3 °C and −17 °C

d h

−2 ⬚ −5

−1 ⬚ −30

Write a statement using < or > that compares the two integers. a

−15 and −2

Reasoning

15, 17

−10 and 0

16, 17

Gian and Kim were asked whether −7 or −5 was the larger integer. Gian said −5 while Kim said −7 because 7 is the bigger number. Explain which person is correct and how you worked out the answer.

Before computers, accountants used red ink to record losses and black ink to record proﬁts. Even today, the saying “$8000 in the red” means an $8000 loss.

Explain what “$8000 in the black” means.

If it were your business, would you rather be $784 in the red or $225 in the black? Explain why.

The boiling point of the element krypton is −153 °C. The boiling point of the element radon is −62 °C. Which of these elements has the lower boiling point? Explain how you worked out your answer.

Problem-solving 18–20, 21abce 18

18–20, 21abce

18–20, 21bdf

Arrange these temperatures in order from coldest to hottest: −15 °C, 30 °C, −11 °C, 2 °C, −9 °C

The numbers −3, 4 and 2 can be written in ascending order using inequality notation like this: −3 < 2 < 4. Arrange this list of numbers in ascending order using inequality notation. 10, −25, 16, −11, 27

The numbers −3, 4 and 2 can be written in descending order using inequality notation like this: 4 > 2 > −3. Arrange this list of numbers in descending order using inequality notation. 3, −13, 11, −9, 15

Complete each number pattern: a c e

2, 4, 6, ⬚, ⬚, ⬚

−16, −12, −8, ⬚, ⬚, ⬚

6, 3, 0, ⬚, ⬚, ⬚

b d f

2, 1, 0, ⬚, ⬚, ⬚

−4, −1, 2, ⬚, ⬚, ⬚

−17, −12, −7, ⬚, ⬚, ⬚

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What number are you standing at now? How can you represent all of the moves you have made so far through addition or subtraction of integers? What movement would you need to make to reach 0? How do you know?

If you had walked a different combination of forwards and backwards on the number line, how might that change the end result?

Investigate Consider the following questions after you have completed the above. 1.

Take turns to move along the number line to model each of these problems. When it is not your turn, write down the answer your partner found. −6 + 6 + 3 −4 + 0 −2 − 1 −2 − 3 − 6 1 − (−6) −14 + (−3) 8 + (−5)

Which movements gave you the same answer? Explain why.

Is there any way you can rewrite the last two problems from the previous question using different signs? Explain your answer.

Work with your partner to create three more of your own problems similar to those you just worked through in the first question. Use the number line to find the answers.

Discussion Define what absolute value means in your opinion. Use the number line that you created as a reference.

Investigation: Life-size number line mathspace.co

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4.03 Add and subtract integers After this lesson, you will be able to… • add and subtract positive and negative integers using the number line. • solve addition and subtraction problems involving positive and negative integers.

Addition and subtraction using a number line The number line can help us to add and subtract integers. Both +3 and −3 have a magnitude of 3. +3 would be 3 jumps to the right on the number line. −3 would be 3 jumps to the left on the number line. The magnitude of an integer is how far that number is away from 0.

Interactive exploration

Discover this concept in action online

mathspace.co

Add a positive number Adding positive numbers will mean from whatever your starting point is, you will move right on the number line.

Example 1 Calculate −1 + 4.

Create a strategy Start on the number −1 and move 4 spaces to the right.

Apply the idea When we add a positive number, we move to the right. −1 + 4 = 3

180

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Mathspace Victorian Curriculum – Year 7 mathspace.co

Subtract a positive number Subtracting a positive number will mean from whatever your starting point is, you will move left on the number line. “Negative is the Opposite to Positive” We can think of the negative sign as meaning “opposite”. The negative sign causes us to move in the opposite direction.

Example 2 Calculate −1 − 4.

Create a strategy Start on the number −1 and move 4 spaces to the left.

Apply the idea When we subtract a positive number, we move to the left. −1 − 4 = −5

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Reflect and check We can think of the negative sign as meaning “opposite”, −5 is the opposite of 5. The negative sign causes us to move in the opposite direction.

Add a negative number Adding a negative number will mean from whatever your starting point is, you will move left on the number line. Adding a negative number is the same as subtracting that number.

Example 3 Calculate 4 + (−3).

Create a strategy Start on the number 4 and move 3 spaces to the left.

Apply the idea If we want to calculate 4 + (−3) we are adding a number, but because it is negative, we move left instead of right. 4 + (−3) = 1

−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

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181

4.03 Practice questions Understanding

Adventurer

1–5

Trailblazer

1–5

In which direction do we move on a number line if we are adding a positive integer?

In which direction do we move on a number line if we are subtracting a positive integer?

In which direction do we move on a number line if we are adding a negative integer?

In which direction do we move on a number line if we are subtracting a negative integer?

The expression 9 + (−4) can be thought of as “start with 9 and move 4 units to the left”. For each expression, name the starting number, the number of units to move, and whether you move left or right on the number line. a

3 – 12

Fluency Ex 1

Explorer

−5 + 9

−2 + (−8)

6, 7a-fk, 8, 9a-h, 10a-h, 11

−4 − (−11)

6bdfh, 7d-j, 8, 9c-j, 10e-l, 11

6bdfh, 7f-l, 8bdeh, 9e-l, 10i-p

Use the plotted point on the number line to add these integers: a

5+9 −5

−15

−10

−5

−15

−10

−5

−15

−10

−5

−3 + 8

−6 + 3

−7 + 7

3+9+7 −5

−5

−7 + 7 + 4 −10

−8 + 6 + 9 −10

−5

−10 + 5 + 12 −15

−10

−5

4.03 Add and subtract integers mathspace.co

183

Evaluate: a

Ex 2

−9 + 0

−45 + 45

−39 + 97

−26 + 15

−13 + 8 + 2

−24 + 10 + 54

−88 + 28 + 92

Use the plotted point on the number line to subtract these integers:

8−5 −10

−5

−10

−5

−10

−5

7−9

3−3

−9 − 9 −20

−15

−10

−5

−15

−10

−5

−10

−5

−2 + 7 − 4

10 − 7 − 2

14 − 18 + 1 −10

−5

−25

−20

−15

−10

−5

−1 − 15 − 3

Evaluate: a

12 – 3

9 – 13

11 – 11

−13 − 13

−32 – 41

36 – 53

−9 − 5 – 8

−12 − 3 + 16

27 − 19 – 10

0 − 11 + 36

−7 − 8 – 15

14 − 49 + 35

Evaluate: a

14 + (−5)

−7 + (−8)

2 − (−3)

5 − (−7)

−12 − (−12)

17 + (−14)

29 − (−52)

−30 − (−32)

12 + 17 + (−15)

−12 + 16 + (−18)

17 − 27 + (−17)

−5 + (−2) + (−7)

m −9 − 2 − (−8)

−5 − (−2) − (−10)

−17 + 23 + (−20)

34 − 63 − (−34)

5 ÷ (20 − 15)

7 × (4 − 2) – 3

27 ÷ (5 − 2) + 4

Evaluate: a

184

−15 + 15

−21 + 0 + 21

−15 + 6

Ex 3, 4

−7 + 15

14 + 7

−3 + (7 × 5)

Mathspace Victorian Curriculum – Year 7 mathspace.co

Three contestants are heading into the ﬁnal round of a game show. Eileen has 39 points and Roxanne has −3 points. If Oprah has 14 more points than Eileen, calculate how many points separate Roxanne and Oprah.

An integer has been assigned to each letter below. A word must be formed using these letters and each letter can only be used once. The table shows the assignment of the integers: Letter Integer

−3

−2

−1

The word score is the sum of the integers assigned to its letters. Determine the score of these words: a 26

NET

EAR

RAIN

CARES

Each bottle of orange juice is supposed to have a capacity of 250 mL. Ten bottles are inspected by the quality assurance department. The results of the inspection are shown in the table: Bottle Amount below or above the required capacity (mL) a b

−7

−11

−1

−5

Bottle 10

Determine the actual capacity of: i

Bottle 1

Bottle 2

iii

Bottle 6

Calculate the total capacity contained in the 10 bottles.

The motion of a robot to the left or right along a straight line is determined by whether the robot is facing left or right, and whether it is stepping forwards or backwards. Use a number line where 1 unit represents 1 step and the positive direction is to the right to answer these questions: a

Facing Right Right Left Left

Stepping Forward Backward Forward Backward

Moving Right Left Left Right

Determine the integer that represents the final position: i

The robot starts at 0, facing right, and takes 5 steps forward.

The robot starts at 0, facing left, and takes 13 steps forward.

iii

The robot starts at 0, facing left, and takes 11 steps backwards.

Suppose the robot performs these instructions in the given order: • Start at −5, facing right. • Take 7 steps forward. • Take 21 steps backward. • Turn to face left. • Take 22 steps backward.

Determine the integer representing the ﬁnal position.

4.03 Add and subtract integers mathspace.co

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4 Chapter review 1

Determine whether each number is an integer or a non-integer: a

8.2

−7.35

−2.850

−4

1250

If you have two integers plotted on a number line, how do you know which integer is the smaller?

Identify the integer plotted on each number line: a

−5

−4 −3 −2 −1

0 0

−6

−3

−8

−4

d 2

Identify the integer that is three units to the right of 4.

Identify the integer that is four units to the left of 15.

Identify the integer that is three units to the left of −3.

Identify the integer that is five units to the right of −11.

Identify the integer that is 5 units to the right of −2.

Identify the integer that is 3 units to the left of 1.

For each number line: i

Arrange all the integers plotted in ascending order.

Identify the largest number.

−12

−8

−4

−25

−20

−15

188

−18, −6, −29

−9, −28, 12

0, −3, 7

38, −40, −5

Identify the inequality sign, < or >, that should appear between each pair of numbers to make each statement true: a

−5

Identify the smallest number in each set: a

−10

−10 ⬚ 10

−2 ⬚ −35

b f

Evaluate:

9 ⬚ −6

45 ⬚ −80

c g

−3 ⬚ −7

−17 ⬚ −15

d h

0 ⬚ −10

−50 ⬚ 25

18 + 9

−20 + 7

−32 + 14

−50 + 50

−35 + 85

−17 + 11 + 3

−30 + 15 + 45

−90 + 35 + 97

Mathspace Victorian Curriculum – Year 7 mathspace.co

Evaluate: a

18 − 5

−17 − 8

15 − 19

−22 − 36

7 − 13 + 29

−16 − 5 + 21

−9 − 7 − 12

20 − 45 + 25

Evaluate: a

8 − (−10)

−9 + (−12)

−14 − (−14)

18 + 20 + (−24)

−7 + (−4) + (−9)

−11 − 3 − (−10)

−18 + 22 + (−24)

40 − 72 − (−36)

8 ÷ (24 − 16)

5 × (7 − 3) − 4

36 ÷ (8 − 5) + 7

Evaluate: a

−5 + (9 × 6)

Consider the following statement: “climbing 18 metres upwards”. a

State the opposite of “climbing 18 metres upwards”.

Suppose “climbing 18 metres upwards” is represented by the number 18. Identify the integer that should represent its opposite.

Arrange the following altitudes in order from lowest to highest: −20 m, 25 m, −8 m, 5 m, −13 m

Arrange the following numbers in ascending order, separated by inequality notation. 8, −18, 22, −5, 13

Complete each number pattern: a c

3, 6, 9, ⬚, ⬚, ⬚

−5, −2, 1, ⬚, ⬚, ⬚

−14, −10, −6, ⬚, ⬚, ⬚

8, 5, 2, ⬚, ⬚, ⬚

One afternoon in Rome, the temperature fell to −5 °C. By late morning the next day, the temperature was forecast to be 12 °C. Calculate the expected rise in temperature.

A caver is 80 m below ground level. They climb 35 m, then descend 45 m before taking a break. Determine the depth of the caver below ground level after they have taken a break.

A factory starts the day with 15 boxes of materials available. At the first delivery, 7 boxes are used and 10 boxes are delivered to the factory. At the second delivery, a further 5 boxes are used, and 12 more boxes are delivered. Calculate the number of boxes available at the factory after the second delivery.

Each container of nuts is supposed to contain a weight of 500 g. Nine containers are inspected and the results of the inspection are shown in the table: Container Amount below or above the required weight (g) a

−5

−12

+10

−2

−8

Container 10

Determine the actual weight of: i

Container 1

Container 2

iii

Container 6

Calculate the total weight contained in the 10 containers.

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189

Big ideas Fractions are essential for developing proportional thinking and reasoning.

Curriculum Outcomes VC2M7N03 find equivalent representations of rational numbers and represent positive and negative rational numbers and mixed numbers on a number line VC2M7N06 use the 4 operations with positive rational numbers, including fractions and decimals, to solve problems using efficient mental and written calculation strategies VC2M7N10 use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts such as ‘best buys’; formulate problems, choosing representations and efficient calculation strategies, designing algorithms and using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation VC2M7N05 multiply and divide fractions and decimals using efficient mental and written strategies, and digital tools

5 Fractions Chapter outline 5.01 Identify, order and compare fractions 5.02 Negative fractions 5.03 Equivalent and simplified fractions Investigation: Fraction flowers 5.04 Add and subtract fractions Investigation: Egyptian fractions 5.05 Fractions of a quantity 5.06 Multiply fractions 5.07 Divide fractions 5.08 Order of operations with fractions Chapter 5 review

192 200 206 216 218 226 228 234 240 246 251

5.01 Identify, order and compare fractions After this lesson, you will be able to… • identify fractions and parts of a fraction. • differentiate between improper fractions and mixed numbers. • compare and order fractions.

Parts of fractions Part of this shape is shaded and part of it is not. Notice that the shape is made of ﬁve equal parts, and only three of them have been shaded. We can call this amount .

The point marked on this number line is halfway between 0 and 1. That is, the point represents the number that is as far from 0 as it is from 1. Notice that the number line is split into two equal parts, and the point lies at the end of the ﬁrst part. We can call this number .

These numbers are examples of fractions. The top part of a fraction is called the numerator. This tells us how many equal parts of the whole are present. The bottom part is called the denominator. This tells us the number of equal parts that are in one whole. The line in the middle is called the vinculum.

4 5

Numerator Vinculum Denominator

For example, consider

In this fraction, the numerator

is 4 and the denominator is 5. We can call this fraction 4 on 5, 4 over 5, 4 out of 5, or 4 ﬁfths.

Rational numbers can be made by dividing an integer by an integer. All fractions are a division of one integer by another integer, so all fractions are rational numbers.

192

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Improper fractions and mixed numbers Fractions like

make up less than one whole. We know this because the numerator is less than

the denominator. We call these proper fractions. What about a fraction like ? Notice that the numerator is greater than the denominator. This means that the fraction is greater than a whole. We call these improper fractions. Each rectangle in this image has been split into ﬁve equal parts, so 5 is the denominator. Eight parts have been shaded, so 8 is the numerator. Since the number of shaded parts is more than the number of parts in one whole, a complete rectangle and three more parts have been shaded. We can write this number as

, which we call one and three ﬁfths.

Numbers like this are called mixed numbers or mixed numerals. Mixed numbers and improper fractions can also be represented on a number line. For

each whole number on the number line is

split into 5 equal parts. 0

For the improper fraction, we would count 8 tick marks from 0, and place the point there. For the mixed number, we would count 3 tick marks from 1.

Example 3 What number is plotted on the number line? Give your answer as a mixed number. 3

Create a strategy For the fraction part, count the number of equal spaces between the two whole numbers, then count how many spaces after the previous whole number the point is located.

Apply the idea The point is between 3 and 4. It is greater than 3 but less than 4. So the whole number part is 3. There 10 equal spaces between 3 and 4, so each space represents right of 3, so the fraction is The number plotted on the number line is

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Mathspace Victorian Curriculum – Year 7 mathspace.co

The point is 9 spaces to the

5.01 Practice questions Understanding 1

Identify the numerator.

Identify the denominator.

Write each word as a fraction: a

One quarter

Three eighths

Sixteen hundredths

Two ninths

One and ﬁve tenths

Six and three ﬁfths

4, 5a-c, 6ace, 7a-c, 8a-c, 9a-d, 10a-d, 11a-c

4, 5c-e, 6ace, 7c-e, 8c-e, 9c-f, 10c-f, 11c-e, 12ab

4c, 5d-f, 6bdf, 7d-f, 8d-f, 9e-h, 10e-h, 11d-f, 12cd

What fraction of each shape is shaded? a

Ex 2

Trailblazer 1d, 2d, 3bdf

Write each fraction in words:

Fluency

Ex 1

a 3

Adventurer 1b, 2b, 3bdf

For each fraction: i

Explorer 1ac, 2ac, 3ace

What fraction is plotted on each number line? a

b 1

d 1

f 1

5.01 Identify, order and compare fractions mathspace.co

197

Ex 3

For each group of shapes: i

Write the fraction that is shaded as a mixed number.

Write the fraction that is shaded as an improper fraction.

What number is plotted on each number line? Give your answer as a mixed number. a

c e

198

d 9

b 4

Ex 4

f 11

Plot these on a number line and then order them in ascending order: a

Express these mixed numbers as improper fractions: a

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 5

For each pair of fractions, determine which fraction is greater: a

Fill in the missing inequality symbol to make the statements true. a e

⬚

b f

Is each statement true or false? a

Reasoning

⬚

13, 15, 16

14–16

⬚

17, 18

Which fraction is greater, one third or one half?

Which fraction is smaller, one fourth or one ﬁfth?

Identify which fraction is greater:

If a number line from 0 to 2 is divided into eight equal parts, what is the fraction represented by the point that is halfway between 1 and 2? Explain your reasoning.

Which is greater,

? Explain your reasoning.

? Explain your reasoning and use a visual representation, such as a

number line or area model, to support your answer. 18

If a number line is divided into 15 equal parts, and point A is located number line, and point B is located

of the way along the

of the way along, what fraction represents the point

that is equally distant from A and B? Explain your reasoning using a number line.

Problem-solving 19ab, 20bd, 21 19

For each group of fractions, arrange them from largest to smallest: a

19cd, 20ac, 22

For each group of fractions, arrange them from smallest to largest: a

19bc, 20bc, 23

Petria has read

of her book and Suma has read

d of theirs. Who has read more of their book?

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Tristian is looking at data from 3 different surveys showing how many people agreed with the focus question. Which survey group shows the most support? a

20 people agreed from a total of 26 surveyed

21 people agreed from a total of 26 surveyed

21 people agreed from a total of 27 surveyed

Alice is preparing a series of 3 fruit smoothies for her friends. The smoothie recipes have

different concentrations of fruits to liquids. The fraction of fruits in the smoothie are apple smoothie,

for the mango smoothie and

for the

for the strawberry smoothie. Order the

smoothies from least to the most fruit content in their recipes.

5.02 Negative fractions After this lesson, you will be able to… • compare and order positive and negative fractions on a number line. • interpret a fraction’s relative values.

Fractions on the number line On the number line shown, each tick is labelled with a multiple of the fraction the point furthest to the left is plotted at the fraction

We can see that

, and the point furthest to the right is plotted

at the fraction

It is common to see number lines where only the integers are labelled, with ticks between each integer that represent a fraction of one whole.

−4

−3

−2

−1

A point has been plotted on this number line. Can you see which fraction the point lies on?

We can ﬁnd the denominator of the fraction using the number of ticks between each integer. To go from 0 to 1, we need to move 3 ticks, so each tick represents To ﬁnd the numerator of the fraction, we can count the ticks from 0 to the point, which gives 8. Since we moved in the negative direction, the point is plotted at the fraction Another way to identify the fraction is to see that the point is two thirds to the left of the integer −2. So, we can write it as the mixed number

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Fractions with different denominators If we are given two fractions that do not have the same denominator, seeing which is greater may not be obvious. For example, how can we compare

and

In the last lesson, we used fraction bars to compare fractions, but we can also use number lines. First, we can plot the point that represents the fraction −1

by dividing each whole into 6 equal

parts. Then, we move 5 ticks to the left of 0. The point on this number line is at Next, we can plot the point that represents the fraction

−1

by dividing each whole into 3 equal

parts. Then, we move 2 ticks to the left of 0. The point on this number line is at

If we compare the number lines, we notice that the integers are aligned and that than

Because

is further left

is closer to 0, we know it is the greater fraction. Another thing to notice is that these denominators are related. Since 6 is a multiple of 3, we can use the same number line to compare these fractions. The longer ticks are marked at thirds, but the shorter ticks are marked at sixths.

Using the number line, we can see that

−2

202

−1

is the same as

Mathspace Victorian Curriculum – Year 7 mathspace.co

and

is the same as

Now that we have common denominators, we can plot the points on the same number line as shown.

5.02 Practice questions Explorer

Understanding

1–3

Trailblazer

2, 3

Identify which of these fractions are negative and explain your reasoning:

What fraction does each tick represent on this number line? −3

−2

−1

Compare and contrast the representation of

Fluency Ex 1

Adventurer

4–9, 10bd, 11bc

and

on a number line.

4–9, 10bc, 11ab

4, 6–9, 10ac, 11ad

Identify the fraction plotted on each number line: a

b −2

−1

−4

−3

−2

−1

−3

−2

−1

For each pair of fractions: i a

204

Plot the fractions on the number line. and

and

−1

−2

Identify which one is closer to zero. 0

−1

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Identify the greatest fraction plotted on each number line: a −2

−1

b −1

Identify the smallest fraction plotted on each number line: a −7

−6

−5

−4

−3

−2

−1

b −4

Ex 2

−3

−2

−1

Arrange these fractions in ascending order:

Arrange these fractions in descending order:

Identify the greatest fraction in each set of fractions. Use number lines to assist you. a

Identify the smallest fraction in each set of fractions. Use number lines to assist you. b

Reasoning

12, 13

12, 14

Explain why the fractions that represent “hiking north” would be on opposite sides of 0.

Which of these negative fractions is closest to zero? Explain your reasoning.

Arrange these negative fractions in ascending order. Explain your reasoning.

Problem-solving 15 15

km due south” and “hiking

A toy car is moving along a straight track. Its initial position is at and it moves to a new position at

km due

15, 16

from the starting position,

from the start. How far did the toy car move?

Express the distance as a positive fraction. 16

A submarine is located 600 metres below sea level. It descends another is the submarine’s new depth in negative metres?

of its depth. What

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5.03 Equivalent and simplified fractions After this lesson, you will be able to… • find equivalent fractions. • simplify fractions. • compare fractions.

Equivalent fractions Sometimes different fractions can represent the same amount. Consider the shaded area of the hexagon below. We can see that 3 of the 6 parts have been shaded, so the area represents

of the whole shape.

We can also see that one half, , of the shape has been shaded. Since the same area is shaded for both fractions must be equal.

, these two

These are called equivalent fractions, since they have different numerators and different denominators but are still equal.

Interactive exploration

Discover this concept in action online

mathspace.co

We can see equivalent fractions in action by representing them visually and seeing how we can create them. Consider this grid: This grid is divided into 6 equal parts as shown by the solid grid lines. 1 out of 6 parts is shaded, so we can represent this area as

of the grid.

If we separate each of the 6 parts into 3 more equal parts, as shown by the dashed lines, we can create an equivalent fraction.

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Simplify fractions We can also create equivalent fractions by decreasing the denominator. To decrease the denominator, we can remove common factors in both the numerator and denominator. Since the numbers become smaller, we call this simplifying the fraction. Previously, we saw that

From here, we can remove the common factor of 3:

This shows:

The fraction

has no common factors between the numerator and denominator (other than 1), so it

is fully simpliﬁed.

Example 2 In the grid shown,

parts have been shaded. Use the grid to simplify

Create a strategy Find the fraction of columns that is shaded.

Apply the idea The grid can be divided into 8 identical columns as pictured shown.

5 out of 8 columns are shaded. So,

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When comparing fractions, we want to make their new denominators equal to the lowest common multiple of their current denominators. Since 24 is the lowest common multiple of 12 and 8, we want to find the equivalent fractions of

and that have a denominator of 24.

We can do this by multiplying the numerator and denominator of denominator of

Since

by 2, and the numerator and

by 3.

is greater than

, we can see that

is greater than .

We could also add some gridlines to the fraction grids so that they both have 24 equal parts in order to compare them.

10 parts out of 24

9 parts out of 24

We can see that after adding the gridlines, the grid for now has

shaded parts. This tells us that

now has

shaded parts and the grid for

is greater than .

Example 4 Arrange these fractions from greatest to smallest:

Create a strategy Find a common denominator of the fractions.

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Complete the solution to simplify the fraction

Which of these statements are correct? A

A fraction is in its simplest form if the numerator and denominator have common factors other than 1.

A fraction is in its simplest form if the numerator and denominator have no common multiple.

A fraction is in its simplest form if the numerator and denominator have no common factors other than 1.

A fraction is in its simplest form if the numerator is 1.

Fluency Ex 1

Ex 2

6a-f, 7a-d, 8ab, 9, 10a, 11aceg, 12ac, 13a, 14a

6c-h, 7c-f, 8bc, 9, 10b, 11cefg, 12ac, 13a, 14a

6e-l, 7e-h, 8cd, 10bc, 11bdeh, 12bd, 13b, 14b

Complete each equivalent fraction statement: a

Use the diagrams provided to simplify these fractions: a

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Ex 3

Determine whether each fraction is equivalent to a

Simplify: a

For each pair of fractions, identify which fraction is greater: b

a Ex 4

Identify all the equivalent fractions from each list: a

For each group of fractions: i

Find the lowest common denominator of the fractions.

Rewrite all the fractions with this lowest common denominator.

iii

List the fractions in descending order.

For each group of fractions: i

Find the lowest common denominator of the fractions.

Rewrite the fractions with this lowest common denominator.

iii

List the fractions in ascending order.

Reasoning

15, 16

15, 17

17, 18

Are

Determine an equivalent fraction to

Which of these pairs of fractions are equivalent? Explain your reasoning.

If the numerator and denominator of a fraction are both prime numbers, is the fraction in fully simplified form? Explain your answer.

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and

equivalent fractions? Explain your reasoning. that has a denominator of 12. Explain your reasoning.

5.04 Add and subtract fractions After this lesson, you will be able to… • add and subtract fractions with the same denominators. • add and subtract fractions with different denominators. • add and subtract mixed numbers.

Add and subtract fractions with the same denominator Suppose we want to ﬁnd

. Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded. Notice that the parts of each circle are the same size. Since the parts are the same size, we can place one circle on top of the other. Now, we can see 5 sevenths of the circle are shaded. So, we can conclude that

When the denominators are the same, we are adding quantities of the same size. This means we can add the number of shaded pieces, without needing to change the number of parts in the whole. Mathematically, we are adding the numerators but keeping the same denominator. Suppose we want to ﬁnd

. Using the same circles as above, we can take 2 sevenths away from 3 sevenths. The part that remains is 1 seventh of the circle. So, we can conclude that

When the denominators are the same, we are subtracting quantities of the same size. Again, the number of shaded pieces is changing, but the number of parts in a whole stays the same. Mathematically, we are subtracting the numerators but keeping the same denominator.

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Now, we can rewrite the fractions with a denominator of 36.

The fractions can now be divided into the same number of parts. After shading the correct amount of pieces, the fractions look like this.

Since the sizes of the parts are the same, we can add the fractions together.

This shows that

When the denominators are different, we create equivalent fractions with the same denominator, then add or subtract the numerators.

Example 2 Evaluate

Create a strategy Find the lowest common multiple of the two denominators.

Apply the idea The lowest common multiple of 4 and 8 is 8. Since the denominator of

is already 8, we only need to ﬁnd the equivalent fraction of . Multiply the numerator and denominator by 2

Evaluate Substitute the new fraction

220

Subtract the numerators

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5.04 Practice questions Understanding 1

Explorer 1–4

Adventurer

Trailblazer

1–4

3, 4

Which is correct?

Describe how to add fractions with common denominators.

What is the first step when adding or subtracting fractions with different denominators?

One method to add mixed numbers such as and then add the fraction parts.

is to first add the whole number parts

Describe another method to add mixed numbers.

Fluency 5

5ac, 6b-df, 7, 8abef, 9–12, 13a-f

5d, 6e-h, 8cdgh, 10, 12, 13e-h, 14bd

Find the lowest common denominator of each set of fractions: a

Ex 1

5bc, 6aceh, 7ac, 8abef, 9–12, 13e-h, 14ac

Evaluate and simplify:

Complete each statement: Rewrite with a common denominator

Evaluate the sum b

Rewrite with a common denominator

Evaluate the sum c

Rewrite as a mixed number Rewrite with a common denominator

Evaluate the difference

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Rewrite with a common denominator

Evaluate the difference Ex 2

Rewrite as a mixed number

Evaluate and simplify: a

Fill in the boxes to complete the working out: Group the whole number parts and fraction parts

Add the whole number parts and fraction parts

Evaluate the sum

For

Find the sum of the whole number part of each mixed number.

Find the sum of the fraction part of each mixed number.

Simplify the expression.

Fill in the boxes to complete the working out: Rewrite as improper fractions

Rewrite with a common denominator

Evaluate the difference

Ex 3, 4

For a

Rewrite the mixed numbers as improper fractions.

Using part (a), rewrite the improper fractions with the same denominator.

Simplify the expression.

Evaluate and simplify: a

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Reasoning 15

For

15, 16

15, 17

15, 17, 18

Calculate the answer by first converting both mixed numbers to improper fractions.

Calculate the answer by first subtracting the whole numbers and then subtracting the fraction parts.

Which method is less likely to cause errors? Explain your answer.

Teddy and Archie are sharing a pizza while watching movies. Teddy takes , and Archie takes . What fraction of the pizza is left? Explain the steps to solve the problem.

A recipe calls for

cup of sugar, but Wei only has

cup of sugar. How much more sugar

does Wei need to complete the recipe? Is your answer reasonable? Explain. 18

A rectangular cake measures 9 cm by 8 cm. You cut a piece that measures 4 cm by 3 cm. What fraction of the cake is left after cutting this piece? Explain your reasoning using a diagram.

Problem-solving 19–22

19, 22, 23

19, 21, 23, 24

Jack felt dehydrated. He drank 1 third of a litre at ﬁrst, then drank another 1 ninth of a litre. How many litres did he drink in total?

Jane eats

of a cake and Joshua eats

of the cake. What fraction of the cake did they eat

altogether? 21

During a blizzard, it snowed

. When the sun came out the next day,

of snow

melted. How much snow was left on the ground? 22

A market stall is open for

hours in the morning and

hours in the evening. How many

hours is the stall open altogether? 23

Georgio buys 3 cups of sugar to use in his baking. He uses cup in his pastries and accidentally spilled

of a cup in his cake,

of a cup. How much sugar does he have left?

Missy mailed 4 letters that have a total weight of 3 grams. One letter weighed another weighed

of a

of a gram and a third weighed

of a gram,

of a gram. How much did the fourth

letter weigh?

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5.05 Fractions of a quantity After this lesson, you will be able to… • use multiplication to find groups of fractions and fractions of a quantity. • describe parts of a quantity, and learn to use multiplication to find fractions of a quantity. • simplify fractions by removing common factors in the numerator and denominator.

Groups of fractions Suppose we want to ﬁnd 6 groups of . We can imagine having 6 whole circles, then shading of each circle. If we count each of the shaded parts, we will ﬁnd there are 12 thirds, which can be combined to make whole circles.

Similar to whole numbers, we can think about 6 groups of

as repeated addition or multiplication.

Recall that when we add fractions of the same size, we only add the numerator because the number of parts in a whole does not change. This means we only multiply the numerator by 6:

Example 1 Find 3 groups of

Create a strategy The phrase “groups of” means multiplication.

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This gave us the number of squares in each group. Since we were interested in ﬁve of the groups, we multiplied the number of squares in each group by 5: 5 × 3 = 15 Just like before, we could have multiplied the whole number by the numerator and simpliﬁed the fraction: Multiply the numerator by the whole number

Evaluate the multiplication

Rewrite using a common factor of 12

Remove the common factor

Whether you remove the common factor first or last, both methods lead to the same answer.

Example 2 Evaluate

Create a strategy Multiply the numerator by the whole number.

Apply the idea Multiply the numerator by the whole number

Evaluate

Simplify

Example 3 Evaluate

of 150 L.

Create a strategy of 150 L is the same as

, when multiplying by a mixed number we first need to

convert to an improper fraction.

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Apply the idea Convert

to an improper fraction.

Reflect and check As a quick check for reasonableness we can check what 2 groups of 150 L and 3 groups of 150 L is.

is between 2 and 3. So our answer should be: • greater than 2 groups of 150: 2 × 150 = 300 L • smaller than 3 groups of 150: 3 × 150 = 450 L

Example 4 Find

of 5 weeks in days.

Create a strategy The word “of” means multiplication. Use the fact that 1 week = 7 days.

Apply the idea Convert 5 weeks to days: 5 weeks = 5 × 7 days = 35 days

Multiply 5 by 7 Evaluate

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231

Ex 3

Ex 4

Evaluate: a

Find these quantities in the given units: a

of $1 in cents

of 6 minutes in seconds

of 4 weeks in days

of 6 L in millilitres

of 3 kg in grams

of 3 cm in millimetres

of 5 m in centimetres

of 36 g in milligrams

of 3 hours in minutes

of 12 km in metres

of $69 in cents

of 1200 mL in litres

of 8 hours in minutes

of 45 days in hours

of 27 weeks in days

of 300 g in kilograms

Reasoning 8

8–10

Justin and Georgia are handing out ﬂiers to advertise their new concert. Justin has handed out 120 of his 400 ﬂiers while Georgia has handed out

9–11

of her 600 ﬂiers.

What fraction of his ﬂiers has Justin handed out?

How many ﬂiers has Georgia handed out?

Who has handed out a higher fraction of their ﬂiers?

How is the process for ﬁnding the fraction of a quantity in parts (a) and (b) different?

A school has 480 students, and

of the students are in the drama club. How many students

are in the drama club? Explain your reasoning. 10

A jar contains 500 marbles, of which are blue and the rest are red. How many red marbles are in the jar? Explain your reasoning.

A fruit bowl contains 12 pieces of fruit. of pears is

are apples, and there are 4 bananas. The number

of the number of bananas. The rest are oranges. How many oranges are in the

bowl? Yulia’s work is shown. Identify the row with the error then correct the step to get the correct answer. 1

Number of apples

Number of pears

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Problem-solving 12, 14, 16

13, 15–17

13, 14, 17–19

Xavier received $10 pocket money, and spent did he spend?

Sinead read

In a class of 27 students, in the class?

Charlie won $40 000 in the lottery. He gave did Charlie keep?

A Year 6 class planned to raise $400 for a local charity by Christmas. If they’ve already raised

of a 450 page book. How many pages has she read? of the students have dark hair. How many students have dark hair

of his winnings to his sister. How much money

of the money, how much more do they still need to raise?

A biased coin lands on tails

of it on mobile games. How much money

of the time.

Why is the coin unfair?

If the coin was ﬂipped 40 times, how many times would it be expected to land on tails?

If a fair coin was ﬂipped 40 times, how many times would you expect it to land on tails? Why?

The Fireﬂies scored 77 points in their basketball game. Fe scored

scored

of the points. Sarah was the only other scorer. How many points did Sarah score?

In the morning, a dam contains 1710 L of water. Throughout the day,

of the points, and Jamie

lost through evaporation,

through cattle drinking from it and

of the total water is

is used for watering crops.

How many litres of water are lost throughout the day?

At the end of the day, how many litres of water remain in the dam?

5.06 Multiply fractions After this lesson, you will be able to… • visualise multiplication of fractions using diagrams and models. • multiply mixed numbers with fractions.

Multiply fractions We’ve seen how to ﬁnd fractions of quantities, but can we ﬁnd fractions of parts?

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In the previous lesson, we found each of 6 circles.

Now suppose we want to ﬁnd

of 6 by shading

of . We can apply the same ideas. We will start with a diagram of . Our goal is to shade

in each of the already shaded ﬁfths.

To do this, we need to split each of these ﬁfths into thirds. Notice that the circle is now divided into ﬁfteenths (that is, 3 × 5) and twelve parts have been shaded (3 × 4). We know from equivalent fractions that this tells us

Then, we can shade 2 thirds of each of the originally shaded ﬁfths. Now, we see that

If you look closely, you can see that the numerators of the two original fractions have been multiplied to get the numerator of the answer, and the denominators of the two original fractions have been multiplied to get the denominator of the answer.

In general, whenever we want to multiply two fractions, we multiply the numerators and the denominators separately. Sometimes we might have to simplify the resulting fraction afterwards.

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Multiply with mixed numbers Recall that a mixed number is a number consisting of an integer and a fraction. We can write mixed numbers as a whole number added to a fraction. For example,

To multiply a mixed number by a fraction, we could use the distributive law of multiplication. For example:

From here, we would need to evaluate the multiplication and add the resulting fractions. We could do this since these are skills we have been practising, but there is a faster and simpler way. Remember that mixed numbers can also be written as improper fractions. If we do this conversion ﬁrst, we can do the multiplication in fewer steps: Convert the mixed number to an improper fraction

Evaluate the multiplication

Notice that we can use the same steps for multiplying fractions that we previously saw, with one added step. That is, we can convert mixed numbers to improper fractions ﬁrst.

Example 3 Evaluate

Create a strategy Rewrite the mixed numbers as improper fractions.

Apply the idea Rewrite mixed numbers as improper fractions

Remove common factor of 3

Remove common factor of 7

Evaluate

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4aceg, 5aceg, 6aceg, 7acdg, 8–13

Fluency 4

Ex 1

Ex 2

Ex 3

4acefg, 5aceg, 6aceg, 7bceg, 10–13

4bdefh, 5bdefh, 6bdfh, 7e-h, 10–14

Evaluate: a

Evaluate:

Simplify:

of a cake was left after a party. Hank ate

of what was left. What fraction of the whole

cake did Hank eat? 9

Georgia was following a recipe which requires

of a cup of ﬂour. However, she only

wants to make half of the quantity that the recipe is for. What fraction of a cup of ﬂour should she use? 10

In a milkshake shop, a medium milkshake needs

of milk. A small milkshake needs

the amount of a medium milkshake. How much milk does the small milkshake need? 11

A farmer has

of an acre of land. If he wants to plant crops on

of the land, how much of

the land will he use for planting? How much land will be left unused? Ex 4

Stephen is years old. He has spent he spent living in Sydney?

In Wings Station, one batch of chicken wings requires is making

of his life living in Sydney. How many years has

cups of ﬂour. If the Wings Station

batches this Sunday, how much ﬂour will they need?

Natalia is starting her jewelry business. She bought

bags of beads. If a single bag weighs

grams, what is the total combined weight of the beads Natalia bought?

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Reasoning

15, 16

A recipe calls for

16, 17

cup of sugar, but you only want to make one-third of the recipe.

Describe the steps you could take to ﬁnd how much sugar you need. Cynthia explains that if each juice box contains

will contain

cup of juice, then a pack of 6 juice boxes

times that amount of juice. She multiplies

cup by

to ﬁnd the total amount

of juice in the pack. Amount of juice in the pack Is Cynthia correct? Why or why not? A rectangular garden measures

metres by

metres. What is the area of the garden in

square metres? Will 10 square metres of turf cover the garden? Explain your reasoning.

Problem-solving 18, 19 18

Lianne had have left?

Opie ordered

18–20

L of orange juice to sell. She sold

18–20

of it. How much orange juice does she

of apples to sell. He found that

of the apples were not fit for sale.

What weight of apples could he sell? 20

A recipe for a cake requires

cup of sugar to make 1 cake.

How much sugar is needed to make 5 cakes?

Josie ate

If 10 cups of sugar costs $2.50, how much will it cost to buy enough sugar to make 5 cakes?

of these cakes, how many cups of sugar is this?

5.07 Divide fractions After this lesson, you will be able to… • divide fractions by whole numbers. • divide fractions by other fractions. • divide fractions by mixed numbers.

Divide fractions by whole numbers When we divide whole numbers, we split the whole number into equally sized parts and ﬁnd the size of one part.

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For example, consider 6 ÷ 3.

This is similar to asking, “If we divided 6 into 3 groups of the same size, what would be the size of each group?” Since there are 3 equal groups of 2 in 6 we can say that 6 ÷ 3 = 2.

Recall that fractions represent division, so 6 ÷ 3 can be written as . In the previous lessons, we saw

This means that dividing by 3 is the same as multiplying by .

Now consider

. This is similar to asking, “If we divided 1 2

into 3

groups of the same size, what would be the size of each group?” Looking at the whole, we see it is divided into 6 pieces. So, the size of each group is .

Just like we saw above, we can say This isn’t a coincidence. Dividing by any number is the same as multiplying by the reciprocal of that number. The reciprocal of a whole number is the fraction of one over that whole number. If we think of the whole number as a fraction over one, that is, numerator and the denominator.

, the reciprocal is found by swapping the

Example 1 Evaluate

. Give your answer as a fraction.

Create a strategy Rewrite the whole number as a fraction. Then multiply the ﬁrst fraction by the reciprocal of the second fraction.

Apply the idea Rewrite the whole number as fraction

Multiply by the reciprocal of

Multiply the numerators and denominators

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What is the reciprocal of: a

Write the problem

as a multiplication.

Can you divide a fraction by itself? If so, explain what the result will be.

Ex 2

Ex 3

5aceg, 6a-f, 7abfh, 8–11, 13

Fluency Ex 1

5aceg, 6a-f, 7c-f, 10–14

5bdfh, 6g-l, 7d-g, 10–14

Evaluate and simplify: a

Evaluate and simplify:

A family drinks family?

A glass holds of a litre of lemonade. How many glasses of lemonade can be ﬁlled with 6 litres of lemonade?

of a carton of milk per day. How many days will 6 cartons of milk last the

of a tart is shared equally between 5 people, what fraction of the tart would each person

receive? 11

Yang eats

of a packet of biscuits each day. If he buys 12 packets, how many days will the

packets last? 12

Christine has

of a metre of ribbon. She uses

s of the ribbon to wrap a present. How

many presents can she wrap with her ribbon? Ex 4

Sheldon is planning to sell banana bread at the upcoming school fair. If a loaf requires cups of sugar, how many banana bread loaves can Sheldon make if he has

cups of sugar

on hand? 5.07 Divide fractions mathspace.co

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Jean is making bows for her gift wrapping business. A bow requires

ribbon. If she has

Reasoning

centimetres of

centimetres of ribbon, how many bows can Jean make? 15, 16

15, 16

16, 17

If you need to divide

You have a pizza that is divided into 8 equal slices, and you want to give

by , which fraction do you take the reciprocal of and why? of a slice to each

of your friends. Describe the steps you could do to ﬁnd how many friends you can share your pizza with. Sam and Ruby are given the following problem:

If it takes book?

of an hour to read

of a book, how many minutes will it take to read the entire

Sam says it will take 67.5 minutes and Ruby says it will take 53.3 minutes. Explain who is correct and why.

Problem-solving 18

18, 19

A water tank has a capacity of 1500 litres. If the tank is currently ﬁlled to

of its capacity,

how much more water needs to be added to the tank to reach

of its total capacity?

Benjamin wants to engrave his name on a metal plate. There is

cm of space on the plate.

There are three sizes of letters to choose from: a

cm wide and

cm wide.

Find the maximum number of letters that Benjamin can engrave on the plate using the following sizes: i

cm wide,

cm wide

iii

cm wide

Which letter sizes could he use for his name?

5.08 Order of operations with fractions After this lesson, you will be able to… • simplify expressions with multiple operations involving fractions. • apply the order of operations to fractions. • solve expressions involving fractions.

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Order of operations with fractions Recall the order of operations is a set of rules that help us know how to correctly simplify an expression with multiple operations. The order of operations with fractions is the same as the order of operations with whole numbers: • Evaluate operations inside grouping symbols such as brackets. • Evaluate multiplication or division going from left to right. • Evaluate addition or subtraction going from left to right. There are a few things to keep in mind when using the order of operations with fractions: First, it is best to write integers as fractions with a denominator of 1. Second, it is best to convert mixed numbers into improper fractions before doing any calculations. Lastly, it is important to remember that when adding and subtracting fractions we need to have common denominators, but this is not the case for multiplying and dividing.

Example 1 Evaluate and simplify

Create a strategy Evaluate the multiplication, then evaluate the addition.

Apply the idea Evaluate the multiplication

Multiply the numerators and the denominators

Evaluate the addition

Example 2 Evaluate and simplify

Create a strategy Follow the order of operations. Evaluate the operation in the brackets, then evaluate the subtraction (going from left to right).

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5.08 Practice questions Understanding 1

Explorer 1–2, 3a-e

Adventurer 2, 3a-e

Trailblazer 3cdf-h

Arrange these steps into the correct order of operations: • Evaluate addition or subtraction from left to right • Evaluate expressions inside the brackets • Evaluate multiplication or division from left to right

How do the brackets in

For each expression, what operation should be performed first? a

Fluency Ex 1

Ex 2

change the order of operations from the expression

4a-e, 5a-ek, 6a-d, 7adeh, 8ab, 9a-c

4c-g, 5d-i, 6c-f, 7adef, 8ac, 9cef

4d-h, 5f-l, 6e-h, 7bcfg, 8bd, 9def

Evaluate and simplify: a

Evaluate and simplify:

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Ex 3

Evaluate and simplify: a

Reasoning

10, 11

11, 12

Simplify

Riley and Reese are given the following problem:

11, 12

. Explain what operation you did ﬁrst and why.

There are 12 students in a scout group.

of them will be going on a trail walk. 20 students

from another scout group are meeting them at the trailhead. In total, how many students are going on the walk? Riley uses the expression

to solve the problem and Reese uses the expression

to solve the problem. Will Riley and Reese get the same result? Explain your reasoning. 12

To ﬁnd the area of the composite shape shown, Ruby uses the expression

1 2

Jade uses the expression 3

1 4 2

Will Ruby and Jade get the same result for the total area of the composite shape? Explain your reasoning.

Problem-solving 13 13

13, 14

A survey shows that a quarter of the people in a town prefer chocolate ice cream, a third prefer vanilla, and the remaining 300 people prefer strawberry. How many people were surveyed?

A school choir needs to divide If they spend

hours of rehearsal time between three different songs.

of their time on the ﬁrst song and

of their remaining time on the second

song, how much time do they have left to work on the third song? 250

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5 Chapter review 1

For each shape identify the fraction that is shaded and the fraction that is unshaded: a

For each group of shapes: i

Write the fraction that is shaded as a mixed number.

Write the fraction that is shaded as an improper fraction.

What fraction is plotted on each number line? a

b 1

d 7

Plot each fraction on the number line: a

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251

Is each statement true or false? b

a 6

Plot the fractions on the number line. and

Determine which one is closest to zero.

and

Complete each equivalent fraction statement: a

Determine if each faction is equivalent to a

What is the smallest fraction in each set of fractions? a

For each pair of fractions: i

Select the equivalent fractions from these lists: a

c 11

For each group of fractions: i

Find the lowest common denominator of the fractions.

Rewrite all the fractions with this lowest common denominator.

iii

List the fractions in descending order.

Evaluate and simplify: b

a 13

For

Find the sum of the whole number part of each mixed number.

Find the sum of the fraction part of each mixed number.

Simplify the expression.

After a heavy rainstorm, out the next day,

252

of water accumulated on the ground. When the sun came of water evaporated. How much water was left on the ground?

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Mishy mailed 4 letters that have a total weight of another weighed

of a gram and a third weighed

grams. One letter weighed

of a gram,

of a gram. How much did the fourth

letter weigh? 16

Evaluate: a

At Sweet Delights, one batch of chocolate chip cookies requires Delights is making

cups of sugar. If Sweet

batches this weekend, how much sugar will they need?

Evaluate and simplify: a

A large container of lemonade can fill

cups. If each serving at a party requires

cups of

lemonade, how many servings can be provided with the large container? 20

A recipe for a chocolate cake requires

cups of cocoa powder. If Sarah has

cups of

cocoa powder available, how many chocolate cakes can she make? 21

A car’s fuel tank can hold

litres of petrol. If the car consumes

litres of petrol for every

100 kilometres, how many kilometres can the car travel with a full tank of petrol? 22

To make a batch of homemade pizza dough, John needs cups of flour. If he has of flour, how many batches of pizza dough can John make?

Evaluate and simplify: a

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cups

253

Big ideas Decimals represent quantities that are a combination of whole numbers and fractional parts. They provide a way to represent and manipulate numbers between whole numbers, offering a more precise and flexible system for dealing with realworld quantities, measurements, and calculations.

Curriculum Outcomes VC2M7N04 round decimals to a given accuracy appropriate to the context and use appropriate rounding and estimation to check the reasonableness of computations VC2M7N06 use the 4 operations with positive rational numbers, including fractions and decimals, to solve problems using efficient mental and written calculation strategies VC2M7N05 multiply and divide fractions and decimals using efficient mental and written strategies, and digital tools

6 Decimals Chapter outline 6.01 Decimals and place value 6.02 Add and subtract decimals 6.03 Convert between fractions and decimals 6.04 Multiply decimals 6.05 Divide decimals 6.06 Round decimals Chapter 6 review

256 262 268 277 283 288 296

6.01 Decimals and place value After this lesson, you will be able to… • recognise the place value of digits in decimal numbers. • compare decimals using place value tables. • multiply and divide decimals by powers of ten.

Place value of decimals Decimals are used to represent values that fall between whole numbers. A decimal point is used to separate the whole number from the decimal or fraction part.

ne O

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When we work with decimals, we need to extend our understanding of place value to include tenths, hundredths, thousandths, and beyond. Each digit after the decimal point represents a fraction of the whole number.

1000

100

1 10

1 100

1 1000

In 312.45, the digit 3 represents 3 hundreds, and the digit 4 represents 4 tenths. As with whole number place values, the farther to the left a column is, the larger its place value. Hundreds 3

Tens 1

Ones 2

. .

Tenths 4

Hundredths 5

Example 1 What is the numeric value represented by the digit 7 for 48.07?

Create a strategy Write the number in the appropriate place value column.

Apply the idea Tens 4

Ones 8

. .

Tenths 0

Hundredths 7

The digit 7 is in the hundredths place. So, it represents 7 hundredths or 0.07.

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Thousandths

This time, let’s look at 73.6 ÷ 100. Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

We have to put in a zero if there is no digit in the ones place.

Dividing by 100 decreases the place of each digit by two places, thus 73.6 ÷ 100 = 0.736. You may have to add or remove zeros, and you may have to add or remove the decimal point after you have ﬁnished moving the values. This depends on which columns the values end up in.

Example 4 What is 0.65 × 10? Write your answer in decimal form.

Create a strategy

Apply the idea

Use a place value table to visualise the value of each digit in the number. When we multiply a number by 10, each digit becomes 10 times larger and will move to the left one column.

Here is 0.65 in a place value table. Ones 0

. .

Tenths 6

Hundredths 5

Multiplying it by 10 means moving each digit one column to the left. Ones 6

. .

Tenths 5

Hundredths

0.65 × 10 = 6.5

Example 5 What is 8.4 ÷ 100? Write your answer in decimal form.

Create a strategy

Apply the idea

Use a place value table to visualise the value of each digit in the number. When we divide a number by 100, each digit becomes 100 times smaller and will move to the right two columns.

Here is 8.4 in a place value table. Ones 8

. .

Tenths Hundredths Thousandths 4

Dividing it by 100 means moving each digit two columns to the right. Use 0 as a placeholder. Ones 0

. .

Tenths Hundredths Thousandths 0 8 4 8.4 ÷ 100 = 0.084

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Write these numbers in descending order. a

Ex 4

Ex 5

0.54, 0.504, 0.5, 0.054, 0.0054

7.28, 7.82, 7.078, 7.168, 7.8

Evaluate, writing your answer in decimal form. a

0.56 × 10

5.2 × 10

0.35 × 100

6.74 × 100

0.6489 × 1000

5.761 × 1000

0.3784 × 10

0.003 × 10

9.3 × 100

59.43 × 100

0.04 × 1000

0.008 × 1000

6.75 ÷ 10

4.8 ÷ 100

Evaluate, writing your answer in decimal form. a

34 ÷ 10

0.832 ÷ 10

953.5 ÷ 100

7583.5 ÷ 100

0.57 ÷ 10

3.9 ÷ 10

79.66 ÷ 100

423 ÷ 1000

423.3 ÷ 1000

2.2 ÷ 1000

Reasoning

12, 13, 15

13–15

Consider the decimal numbers 3.45 and 3.045. Explain the process you would follow to determine which number is smaller. Discuss the role of place value in your comparison process.

Consider the product 0.973 × 100. a

Indicate which direction the digits must be moved to calculate the answer.

Identify the number of places the digits must be moved. Explain your answer.

Evaluate the product.

Explain why the process of multiplying and dividing decimal numbers by powers of 10 is different from multiplying and dividing by other numbers. Use examples to support your explanation.

Consider these two numbers: 0.672 33 and 0.8. Explain why the misconception “longer is larger” does not hold true in this case.

Problem-solving 16 16

16–18

A weather forecaster records the maximum temperature in Melbourne on ﬁve consecutive weekdays starting on Monday. She records the following temperatures: 17.5 °C, 23.8 °C, 19.6 °C, 21.2 °C, 15.9 °C

Which day was the hottest?

On which two days was it colder than the previous day?

Four students in a Year 7 class complete a quiz on decimals. Nicky scored 84.5, Daniel scored 91.3, Bernice scored 78.9, and Dylan scored 86.2. a

Order the students in descending order of score.

Cheska completes the quiz the following day and scores 91.25. What is Cheska’s rank in the class?

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6.03 Convert between fractions and decimals After this lesson, you will be able to… • use place value to convert fractions with denominators that are powers of 10 to decimals. • use either the equivalent fraction method or division to convert fractions to decimals. • convert decimals to fractions using the place value method and simplifying the fraction.

Fractions to decimals using place value

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We already know that in the place value table, we have columns for tenths, hundredths, thousandths, and so on. When we change a fraction to a decimal, it’s easiest if the denominator is a power of 10, so it matches our place value table.

1000

100

1 10

1 100

1 1000

To convert the fraction to a decimal, we need to write the digits in the correct column of the place value table, starting from the left. For example, if the denominator is 100, we write the digits in the hundredths column. If the denominator is 1000, we write the digits in the thousandths column, and so on. Then we read the decimal number from left to right.

Example 1 Write the fraction

as a decimal.

Create a strategy Use a place value table.

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Example 2 Write the fraction

as a decimal.

Create a strategy Multiply the numerator and denominator of the fraction by a number that will make the denominator a power of 10.

Apply the idea Multiply both numerator and denominator by 4

Evaluate

Write as a decimal

Example 3 Convert

to a decimal rounded to the thousandths place.

Create a strategy Use short division, adding extra zeros as required to the numerator.

Apply the idea Set up the division including zero placeholders, up to one more place value than we are rounding to. In this case, 4 zeros. 11 does not go into 7 so we put a 0 above the 7 and copy the decimal point to the top. 11 goes into 70 six times since 11 × 6 = 66, so we put a 6 above the first zero and carry the remaining 4 to the next zero. Now, 11 goes into 40 three times, so we put a 3 above the second zero and carry the remaining 7 to the next zero. Continuing this pattern two more times, 11 goes into 70 six times, so we put a 6 above the third zero and carry the remaining 4 to the next zero. Then, 11 goes into 40 three times, so we put a 3 above the fourth zero. So if we round to the thousandths place, we have

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Convert decimals to fractions We can use place value to help us convert between decimals and fractions. With this method, we can transform any decimal into a fraction with a denominator of 10, 100, 1000, and more. Using place value, we know that 0.1 is equal to

because the 1 is in the tenths place. We can use

the same place value reasoning to see that 0.17 is equal to

We often prefer to write fractions in the simplest form. To simplify the fraction, we need to divide both the numerator and denominator by the highest common factor. For example, if we write 0.36 in a place value table like this: Ones 0

. .

Tenths 3

Hundredths 6

We can identify the place value of the decimal is hundredths and 0.36 can be written as . We can then simplify this by dividing both the top and bottom of the fraction by the highest common factor of 4:

Example 5 Convert these decimals to fractions: a Write the decimal 0.29 as a fraction.

Create a strategy Identify the place value of the decimal. In this case, it is hundredths.

Apply the idea Since there are two decimal places, the denominator of the fraction is 100. Write as a fraction

b Write the decimal 0.535 as a simplified fraction.

Create a strategy Identify the place value of the decimal. In this case, it is thousandths.

Apply the idea Since there are three decimal places, the denominator of the fraction is 1000. Write as a fraction

Divide both numerator and denominator by 5

Evaluate

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Ex 1

Consider this diagram: a

What fraction of the row of blocks is shaded?

Write this fraction as a decimal.

What fraction of the row of blocks is not shaded?

Write this fraction as a decimal.

Consider this diagram: a

What fraction of the grid is shaded?

Write this fraction as a decimal.

What fraction of the grid is not shaded?

Write this fraction as a decimal.

Write these fractions as decimals: a

Write these mixed numbers as decimals: a

Complete these conversion tables: a

Fraction Decimal

Decimal

0.1

Fraction

Decimal Ex 2

274

Fraction 0.5

Fraction Decimal

Write these fractions as decimals: a

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12 Ex 3

Ex 4

Ex 5

Write these fractions as decimals, rounding to the thousandths place, where needed: a

Write these mixed fractions as decimals, rounding to the thousandths place, where needed: a

Fill in the boxes with the missing numbers: a

Write these decimals as a simpliﬁed fraction: a

0.9

0.4

0.39

0.92

0.09

0.907

0.626

1.9

5.8

7.19

5.62

4.09

m 9.007

3.065

76.25

223.205

If 0.2 of a row of blocks is shaded: a

Find the decimal number that represents the unshaded part of the row.

Write the fraction that is unshaded in its simplest form.

If 0.72 of a grid is shaded: a

Find the decimal number that represents the unshaded part of the grid.

Write the fraction that is unshaded in its simplest form.

A recipe calls for make

times the recipe?

A container holds containers?

Reasoning 20

cups of ﬂour. How much ﬂour is needed in decimal form if you want to

of a litre of water. How much water is needed in decimal form to ﬁll 5

20, 21

21, 22

Identify which statements are true of the rectangle shown and explain why: a

More than half the bar is shaded

0.6 of the bar is unshaded

The bar is shaded between 0.6 and

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6.04 Practice questions Understanding

Explorer 1, 2

Adventurer

Trailblazer

1–3

If you multiply two decimals that are less than 1, will the product be less than, equal to, or greater than the decimals being multiplied? Why?

For these products, estimate if the answer will be: • Greater than both numbers. • Less than both numbers. • In between the two numbers.

3×5

3 × 0.5

0.3 × 0.5

7×4

0.7 × 4

0.7 × 0.4

15 × 8

15 × 0.8

1.5 × 0.8

What are the conditions so that the product of two decimal numbers is: a

In between the two numbers

Fluency 4

Ex 1, 2

Ex 3

4a-h, 5a-f, 6–9, 13

Less than both of the two numbers

4e-l, 5g-l, 6, 8, 10, 11, 13

4e-l, 5g-l, 6, 8, 10–13

For these products: i

Identify the number of decimal places in the answer.

Calculate the answer.

0.3 × 0.5

0.2 × 0.54

0.1 × 0.2 × 0.3

0.8 × 0.05

0.03 × 0.05

0.007 × 0.006

0.9 × 0.5 × 0.02

0.015 × 0.2

4 × 0.009

64 × 0.001

18 × 0.07

38.5 × 0.01

Evaluate: a

4×5

4 × 50

4 × 0.5

0.4 × 0.5

0.6 × 0.3

4.4 × 2

1.6 × 2.4

4.23 × 2

4.8 × 8

62 × 9.6

3.89 × 7

2.8 × 6.8

Consider a rectangle with side lengths of 2.34 and 5 units. a

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 2.34 × 5 as a decimal.

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 46.2 × 6 as a decimal.

0.04

Consider a rectangle with side lengths of 46.2 and 6 units. a

0.3

0.2

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Consider a rectangle with side lengths of 4.92 and 0.6 units. a

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 4.92 × 0.6 as a decimal.

Consider a rectangle with side lengths of 2.3 and 3.1 units: a

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 2.3 × 3.1 as a decimal.

Complete the table by ﬁnding the area of each smaller rectangle as a decimal.

Find 0.87 × 7.8 as a decimal.

0.02

0.3

0.8

0.07

3 0.1

Consider a rectangle with side lengths of 0.87 and 7.8 units. a

0.9

0.6

7 0.8

Use the area method to ﬁnd 0.47 × 0.89 as a decimal.

Use the area method to ﬁnd 4.86 × 0.86 as a decimal.

Lisa purchased a car and is paying $148.40 each month to pay it off. If it takes her 60 months to pay it off, how much will she have spent on car repayments in total?

Reasoning

14, 15

15, 16

If 36 × 42 = 1512, what is the answer to 3.6 × 4.2? Why?

James used a calculator to evaluate 3.179 × 3.96 but forgot to enter the decimal points. The answer he got was 1 258 884. What should the answer have been? Why?

A bakery sells a loaf of bread for $3.50. Each loaf contains 20 slices. a

If a customer buys 2 loaves of bread, how much do they have to pay in total?

Describe the steps to ﬁnd how much money the bakery would make if they sold 100 loaves.

If the price of each loaf of bread is increased by $0.25, and the bakery sells the same amount of loaves in a week, how much more money will they make?

Problem-solving 17–20, 23 17

18–21, 23, 24

19–25

Find the area of a rectangle with these dimensions: a

Length of 0.5 cm, width of 0.8 cm

Length of 0.2 m, width of 0.07 m

A recipe calls for 0.75 cups of sugar and 0.5 cups of butter to make 12 cookies. If you want to make 36 cookies, how much sugar and butter will you need?

Neville trains six days a week. Each weekday, he runs 2.4 km and on Saturday, he runs 4.6 km. How far does he run in a whole week?

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How much would you pay for eight t-shirts if they cost $46.88 each, and you receive a discount of $9.47?

A store sells shirts for $23.99 each. If the customer buys a hat for $12.75 and also buys 4 shirts, what is the total cost of their purchase?

A company sells 2500 shares of stock for $19.35 per share. If the company then earns a further $18 765 on top of this, what is the new price per share, rounded to two decimal places, if the company distributes the profit evenly among all shares?

A local bakery sells two types of bread: whole wheat and sourdough. A loaf of whole wheat bread weighs 0.75 kg, while a loaf of sourdough bread weighs 1.25 kg. If the bakery sells 2.5 times more whole wheat bread than sourdough bread and has sold 80 loaves of whole wheat bread today, how many kilograms of bread have been sold in total?

Calculate the total weight of a shipment of boxes. There are both small and large boxes. Each box, regardless of size, weighs 1.25 kg, and there are 3.5 times more small boxes than large boxes in the shipment. If there are 28 large boxes, how many small boxes are in the shipment, and what is the total weight of all the boxes combined?

A factory produces rectangular metal sheets that are 0.5 metres wide and 1.75 metres long. Each sheet is then cut into smaller squares with side lengths of 0.25 metres. If the factory produces 3.5 times more sheets in a week than in a day, and it produces 120 sheets in a day, how many small square pieces are produced in a week?

6.05 Divide decimals After this lesson, you will be able to… • divide decimals by whole numbers. • divide decimals by decimals. • use short division to divide decimals by whole numbers and decimals.

Divide by a whole number Dividing with decimals uses a similar approach to dividing whole numbers. The only difference is that we work with the digits to the right of the decimal point. To evaluate division problems with decimals, we use short division. Sometimes we may need to rename some of our digits, just as we would when dividing whole numbers. The process of partitioning and renaming is what is used behind the scenes when we perform short division.

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6.05 Practice questions Understanding 1

Give a whole number estimate for the answer to the division problem.

Choose the most reasonable estimate for 2.8 ÷ 7: A

Between 0 and 1

Between 1 and 7

Less than 0

Between 10 and 70

Choose the most reasonable estimate for 82.4 ÷ 4: 21

210

2.1

0.21

793.6 ÷ 8

56.231 ÷ 7

Nine ones

Nine tens

Estimate the value for these divisions: a

979.8 ÷ 2

Which statement is the same as 0.9? A

Nine-tenths

7.776 ÷ 2

Nine

Find 0.9 ÷ 3, giving your answer as a decimal. 5, 6aceg, 7, 8, 9a-ek, 10, 11

Fluency 5

Trailblazer 1–3

Write the problem shown in the short division in the form a ÷ b.

Adventurer 1–4

Explorer 1–4

5, 6b-eg, 7, 8, 9a-ek, 10, 11

5, 6e-h, 8, 9f-jl, 10, 11

To ﬁnd 48.8 ÷ 4, we partition the number as follows: 48.8 = 40 + 8 + 0.8 a

Calculate: i

b Ex 1

286

40 ÷ 4

8÷4

iii

0.8 ÷ 4

Add the answers to part (a) to ﬁnd 48.8 ÷ 4.

Evaluate: a

24.24 ÷ 3

11.54 ÷ 2

52.68 ÷ 6

245.37 ÷ 3

393.68 ÷ 7

588.6 ÷ 9

538.32 ÷ 6

1061.2 ÷ 8

Consider the quotient of 3.36 divided by 0.3 a

By what multiple of ten should both numbers be multiplied by, so that the divisor, 0.3, is a whole number?

Rewrite 3.36 ÷ 0.3 so that the divisor is a whole number.

Calculate 3.36 ÷ 0.3

Consider the quotient of 4.808 divided by 0.04 a

By what multiple of ten should both numbers be multiplied by, so that the divisor, 0.04, is a whole number?

Rewrite 4.808 ÷ 0.04 so that the divisor is a whole number.

Calculate 4.808 ÷ 0.04

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 2, 3

Evaluate: a

56 ÷ 0.7

420 ÷ 0.7

1.8 ÷ 0.09

5.205 ÷ 0.05

12 ÷ 0.003

1.6 ÷ 0.4

8.1 ÷ 0.9

2.94 ÷ 0.07

84.06 ÷ 0.06

10.8 ÷ 0.012

33 ÷ 0.011

6.4 ÷ 0.008

On working days, Josie always buys a coffee on her way to work. Her coffee costs her $5.75 per day. One month, she budgets $120.25 for coffee. How many working days were there that month?

A bag of rice weighs 93.25 kg. A box contains 1585.25 kg of rice. How many bags are in the box?

Reasoning 12

12a-d, 13

12, 13

12, 14

Given that 235 × 42 = 9870, evaluate: a

9870 ÷ 42

9870 ÷ 235

98.70 ÷ 4.2

98 700 ÷ 0.42

0.987 ÷ 420

9.87 ÷ 0.235

Jo and Isla are asked to divide 1.6 ÷ 0.4. Jo multiplies both numbers by 10 before dividing. Isla multiplies both numbers by 1000. Will Jo and Isla get the same answer? Explain why or why not.

Casper is trying to determine the width of three rectangles. He has information about the area and the length of each rectangle, which are given to two decimal places. Without a calculator, he decides to round these measurements to the nearest whole number to estimate the width. a

The ﬁrst rectangle has an area of 19.73 square centimetres and a length of 4.23 cm.

Casper estimates the width using an area of 20 square centimetres, and a length of 4 cm. He reasons that this will be an overestimate of the true width.

What answer did Casper get for the width?

Explain why his answer is an overestimate?

The second rectangle has an area of 30.22 square centimetres and a length of 5.83 cm. Casper estimates the width using an area of 30 square centimetres, and a length of 6 cm. He reasons that this will be an underestimate of the true width. i

What answer did Casper get for the width?

Explain why his answer is an underestimate?

The third rectangle has an area of 39.65 square centimetres and a length of 7.91 cm.

Casper estimates the width using an area of 40 square centimetres, and a length of 8 cm. He cannot decide if this will be an underestimate or an overestimate. i

What answer did Casper get for the width?

xplain why Casper has diﬃculty deciding whether this is an underestimate or an E overestimate?

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Problem-solving 15–17

15–17

15, 17, 18

Juliet bought 2.2 kg of bananas for $8.60 and Luna bought 2.5 kg of bananas for $10.10. Who got the better value based on dollars per kilogram?

A water tank holds 20 L of water. Oliver is going to ﬁll the tank by ﬁlling a 600 mL bottle and emptying it into the tank. How many times will he need to do this?

A juice bar sells orange juice in 230 mL cups. One day they make 36.8 L of orange juice. How many cups could they sell?

Juliet makes metal window frames for a company that sells house and land packages. She makes two sizes of frames, one being 2.25 m wide by 1.5 m deep, and the other being 2.25 m wide by 3.2 m deep. a

Calculate the perimeter of: i

the smaller frame.

the larger frame.

Juliet currently has 150 m of metal for making frames. i

hat is the largest number of the smaller frames that she can make and how much W metal would she have left over?

hat is the largest number of the larger frames that she can make and how much W metal would she have left over?

iii

I f she wants to make the same number of each type of frame, what is the largest number of each type she could make, and how much metal would be left?

6.06 Round decimals After this lesson, you will be able to… • round decimals to a place value. • round numbers to a specific number of places. • use estimation to approximate solutions and check the reasonableness of answers.

Round to a specific decimal place Rounding decimals is when we approximate a decimal number by shortening it to a specific number of decimal places. When indicating that a number is an approximation, we either use words such as, “rounded to a certain number of decimal places”, or we use the approximation symbol, ≈. This can be particularly useful when dealing with money, measurements, and calculations where exact values may not be practical. Let’s look at what it means to round a decimal to the nearest whole number.

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When we round a decimal number to the nearest whole number, we look at the digit to the right of the decimal point (tenths place). If this digit is 5, 6, 7, 8, or 9 we round up to the next whole number. If the digit is 0, 1, 2, 3, or 4, we round down to the current whole number. There are situations that require us to round numbers to a specific decimal place. For example, we round money to the nearest cent, length measurements to show accuracy to the nearest thousandths, and our school marks to the nearest whole number or tenth. To round a decimal number to a specific decimal place, we need to follow these steps: 1. Identify the decimal place we want to round to. 2. Look at the digit to the right of the identified decimal place. 3. Apply the rounding rules.

Example 1 Round 2.19 to the nearest whole number.

Create a strategy

Apply the idea

Use a place value table to look at the digit to the right of the decimal point (tenths place). If this digit is 5 or greater, we round up to the next whole number. If the digit is less than 5, we round down to the current whole number.

Ones 2

. .

Tenths 1

Hundredths 9

The number in the tenths place is 1 which is less than 5, so we round down to the current whole number. The rounded answer is 2.

Reflect and check 2.19 is closer to 2 than 3. 2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

Example 2 Round 328.864 to the nearest hundredth.

Create a strategy Use a place value table to look at the digit to the right of the hundredths place. If this digit is 5 or greater, we round up to the next hundredth. If the digit is less than 5, we round down to the current hundredth.

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Ex 3

Ex 4

Round these numbers to one decimal place: a

0.41

8.35

19.38

0.3009

3.55

19.95

13.88

1.0099

Round these numbers to two decimal places: a

2.697

393.586

0.435

7.3582

79.997

2.0029

149.955

0.099

Round these numbers to three decimal places: a

11.8483

2.159 76

4.504 13

1.199 69

90.0335

0.5987

41.5005

9.9996

12.006 65

39.999 95

Round these numbers to four decimal places: a

Ex 5

5.659 64

What is the most reasonable estimate for how much Stephanie drives each day? A

10 km

50 km

100 km

Will this estimate be higher or lower than the actual average?

To the nearest km, what is the average kilometres travelled per day?

Trevor needs to read 564 pages of his book before it is due back at the library in 8 days. a

What is a good estimate for the number of pages he should read each day?

Will that estimate be higher or lower than the actual average? Explain how you know.

To the nearest whole page, what is the average number of pages Trevor should read each day to finish the book in time?

Reasoning 15

5.600 96

Stephanie drives a total of 756.25 kilometres over 7 days. a

15–17

Opie wants to write

15–17

16–18

rounded to three decimal places. Explain how many decimal places he

needs to calculate in order to do this correctly. 16

Josie wants to write 0.321 46 rounded to three decimal places. She says that the 5th decimal place, the 6, would cause the 4 in the 4th decimal place to become a 5, and then that would cause the 1 in the 3rd decimal place to become a 2. a

Explain what Josie has done wrong.

Round 0.321 46 to: i

294

Four decimal places.

Three decimal places.

Consider rounding 4.449. a

Round 4.449 to two decimal places.

If you were asked to round 4.449 to one decimal place, should you use the answer to part (a) or go back to the original number in the question? Explain.

Mathspace Victorian Curriculum – Year 7 mathspace.co

6 Chapter review 1

Which decimal is larger?

Write these list of numbers in descending order.

1.315 or 1.351 0.37, 0.073, 3.7, 0.0073, 0.000 73 3

Evaluate: a

0.5 + 0.2

0.45 + 0.29

3.314 + 1.686

120.317 + 879.683

0.12 + 0.92 + 0.67

25.4 + 42.06 + 12.005

Evaluate: a

7.8 − 6

6.2 − 3.1

12.5 − 5.8

0.65 − 0.42

9.63 − 4.3

8.47 − 1.6

0.672 − 0.245

95.36 − 42.15

Tyrone spends $4.50 at the grocery store. If he pays for it with a ten dollar note, how much change will he receive?

Write these fractions as decimals:

Write these decimals as a simplified fraction. a

0.001

0.37

0.12

0.625

4.6

3.68

8.012

59.75

On a scale drawing, a nail measures the drawing are

cm. In real life, we know that all the measurements on

times larger. What is the length of the nail in real life written as a decimal

to two decimal places? 9

296

Evaluate: a

0.4 × 0.7

0.02 × 0.06

0.3 × 0.6 × 0.03

5 × 0.008

1.8 × 2.2

4.35 × 3

75 × 8.3

3.21 × 6

Evaluate: a

36.36 ÷ 4

15.24 ÷ 3

68.34 ÷ 7

320.49 ÷ 4

481.44 ÷ 8

720.9 ÷ 9

675.84 ÷ 7

1360.8 ÷ 8

Evaluate: a

63 ÷ 0.9

2.4 ÷ 0.6

630 ÷ 1.05

6.15 ÷ 0.06

15 ÷ 0.003 75

12.96 ÷ 0.0144

36.3 ÷ 0.0121

7.2 ÷ 0.009

Mathspace Victorian Curriculum – Year 7 mathspace.co

On working days, Laura always buys a smoothie on her way to work. Her smoothie costs her $4.50 per day. One month she spent a total of $99.00 on smoothies. How many working days were there that month?

Round these numbers to four decimal places: a

3.245 73

7.831 28

10.056 94

22.999 52

Three packages with weights of 2.34 kg, 6.15 kg and 3.52 kg are to be shipped, but they exceed the shipping carrier’s 10.83 kg weight limit. By how much does the total weight exceed the limit?

Place each set of numbers in ascending order: b

a 16

A bookstore sells novels for $16.99 each. If the customer buys a bookmark for $3.50 and also buys 5 novels, what is the total cost of their purchase?

A recipe calls for 0.6 cups of cocoa powder and 0.4 cups of milk to make 10 servings of hot chocolate. If you want to make 30 servings of hot chocolate, how much cocoa powder and milk will you need?

Last year, a local bakery sold 1000 cupcakes at a price of $2.50 each, which only covered their costs, resulting in no profit. This year, the bakery aims to make a total profit of $4500 while maintaining the same production of 1000 cupcakes. What should be the new price per cupcake to achieve this goal?

A fish tank has a volume of 32 L. Sally is going to fill the tank by filling a 750 mL container and emptying it into the tank. How many times will she need to do this?

A candy store sells gummy bears in 120 g packets. One day they make 14.4 kg of gummy bears. How many packets could they sell?

Ally has $420 to buy a desk and a chair. The desk is advertised at $255.25 and the chair at $164.99. To see if she has enough money, Ally rounds both prices to the nearest dollar. a

What is Ally’s estimate of the price of the desk and the chair?

According to her estimate, does she have enough money?

Does she actually have enough money?

Is it appropriate to round down in this situation?

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Big ideas Percentages, like fractions and decimals, are a way of expressing a part of a whole or set. They provide a flexible system for dealing with real-world quantities, measurements, and calculations.

Curriculum Outcomes VC2M7N03 find equivalent representations of rational numbers and represent positive and negative rational numbers and mixed numbers on a number line VC2M7N07 find percentages of quantities and express one quantity as a percentage of another, with and without digital tools VC2M7N10 use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts such as ‘best buys’; formulate problems, choosing representations and efficient calculation strategies, designing algorithms and using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation

7 Percentages Chapter outline 7.01 Fractions, decimals and percentages 7.02 Compare percentages 7.03 Express a quantity as a percentage 7.04 Percentages of a quantity 7.05 Percentages with technology Chapter 7 review

300 308 311 316 323 329

7.01 Fractions, decimals and percentages After this lesson, you will be able to… • define percent. • convert between fractions and percentages. • convert between decimals and percentages.

Meaning of percentage Percentage means ‘per hundred’. If we say 50% of something, it means Fractions

, which is the same as

Decimals

Percentages

1.0

100%

0.5

50%

or 0.5

These common fraction, decimal and percentage conversions are called benchmark values.

0.3 0.25

25%

0.2

20%

0.125 0.1

10%

0.05

0.04

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To convert any fraction to a percentage, we can do the following: 1. Change the denominator of the fraction to 100 if possible and write as a percent. For example: 2. If the denominator isn’t easy to change to 100, such as , then convert the fraction to a decimal ﬁrst by dividing the numerator by the denominator. 3. Multiply the answer by 100 and then attach the % symbol. For example: To convert any percentage into a fraction: 1. Write the percentage as a fraction with denominator of 100 and remove the % symbol. 2. If there is a decimal within the fraction, like in

, then multiply both the numerator and

denominator by a power of 10 so that the numerator becomes an integer, like

3. Simplify the fraction.

Example 4 Write as a percentage.

Create a strategy To write as a percentage, convert to a decimal first then multiply by 100%.

Apply the idea Convert to decimal

Multiply by 100%

Evaluate

Example 5 Write

as a percentage.

Create a strategy To write a mixed number as a percentage, convert to an improper fraction first, then to a decimal and lastly, multiply by 100%.

Apply the idea Convert to an improper fraction

Convert to decimal

Multiply by 100%

Evaluate

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To convert any percentage to a fraction: 1. Write the percentage as a fraction with denominator of 100 and remove the % symbol. 2. If there is a decimal within the fraction, like in

, then multiply both the numerator and

denominator by a power of 10 so that the numerator becomes an integer, like 3. Simplify the fraction.

7.01 Practice questions Understanding 1

Explorer 1–4

Adventurer 1–4

Trailblazer 1–4

Complete these statements using the words ‘divide’ or ‘multiply’. a b

To convert a decimal into a percentage, you ⬚ by 100%.

To convert a percentage into a decimal, you ⬚ by 100%.

Complete this statement: The ﬁrst step in converting a percentage such as 45% into a fraction is to write it as a fraction with a denominator of ⬚.

Out of 100 students, 55 lived locally. What percentage did not live locally?

15 out of the 30 class members were away ill. What percentage were away?

Fluency

5ace, 6a-d, 7a-d, 8a-d, 9ab, 10a-d, 11acde

5a-c, 6ce-g, 7acef, 8e-h, 9a-d, 10b-df, 11bf-h, 12a-c

5b-d, 6cdgh, 7cdgh, 8i-l, 9cd, 10d-h, 12–13

For each of these number patterns: i

Rewrite the pattern as percentages.

Rewrite the pattern as decimals.

e Ex 1

Ex 2, 3

Write each percentage as a decimal: a

18%

57%

23.4%

5.5%

120%

303%

500%

6897%

Write each decimal as a percentage: a

0.07

0.857

5.3

13.04

0.041

0.0035

1.375

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8 Ex 4, 5

Express each fraction as a percentage, rounding your answer to two decimal places where necessary: a

Express each fraction as a percentage: b

a 10

Ex 6

Ex 7

Express each percentage as a decimal, rounding your answer to three decimal places where necessary: a

Write each percentage as fractions in its simplest form: a

75%

40%

10%

13%

68%

36%

Write each percentage as a fraction or mixed number in its simplest form: a

7.5%

2.5%

0.2%

68.5%

109%

450%

360%

285%

Convert between decimals, percentages and fractions to complete the table, expressing the fractions in simplest form:

Decimal

Percentage

Fraction

0.53

0.6%

430% 2.8 11.25

Reasoning 14

306

14, 18

14–15, 18

15–18

Josie heard an athlete say that he was going to give 110% in the ﬁnal. She says this is impossible because you cannot have more than 100%. a

Is Josie correct in the context of her statement?

Is Josie’s statement correct in the context of a test result?

Can you think of a context in which what Josie said was wrong and where you could get more than 100%?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Arrange these numbers in ascending order: 0.24,

In the decimals shown in the table, x, y and z represent different digits. A x. yz

B 0.0zx

C 0. xyz

D 0. zyx

Match each percentage to a decimal equivalent in the table: a 17

xy. z%

xyz%

Suppose you were converting

zy. x%

z. x%

to a percentage. Describe a strategy for doing this that

involves ﬁnding equivalent fractions with denominators that are powers of 10. 18

Jasper won a match in a table tennis competition. The winner of a match was the ﬁrst player to win 4 games and the winner of each game was the ﬁrst player to win 11 points. The scores were 11 − 7, 8 − 11, 11 − 5, 11 − 4, 2 − 11, 11 − 8. a

What fraction of the games did Jasper win?

How many points were played in total?

What fraction of the points did Jasper win?

What percentage of the points did Jasper lose?

Problem-solving 19–21

19–21, 23

19–24

In a survey, 75% of the respondents said they prefer coffee over tea. 0.1 of the respondents expressed no preference. What percentage prefer tea over coffee?

A bag of toffee has 50 pieces in it. If 15 pieces have been eaten, what percentage of the original pieces of toffee remain?

In a game, a player scored 24 out of the 30 points scored by her team. Express the points scored by the rest of her team as: a

A simplified fraction

A percentage

A decimal

A bag has a variety of coloured marbles. It is known that 30% of the marbles are red and that there are 87 red marbles. How many marbles are there in the bag that are not red?

On her ﬁrst Mathematics test, Juliet scored 16 out of 25. On the second Mathematics test, she scored 27 out of 40. On which of the tests did she get the higher percentage?

A 5-a-side soccer team has a 20 match season. After 16 games, they had won 12 matches. a

What is the largest possible win percentage they could achieve for the season?

What is the smallest possible win percentage they could achieve for the season?

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7.02 Compare percentages After this lesson, you will be able to… • convert between fractions, decimals and percentages. • define and use benchmark conversions for fractions, decimals and percentages. • compare and order numbers presented in different formats.

Compare fractions, decimals, percentage When comparing values that have different representations, like fractions, decimals or percentages, one common strategy is to convert them all to the same representation. Arranging numbers in ascending order means smallest to largest; descending order means largest to smallest.

Example 1 In this question, we will be working with the numbers , 60% and 0.3. a Convert

to a percentage.

Create a strategy Write as a fraction with a denominator of 100 first, then as a percent.

Apply the idea

Write as an equivalent fraction with a denominator of 100 Write as a percentage

b Convert 0.3 into a percentage.

Create a strategy To convert a decimal to a percentage, we need to write the decimal as hundredths.

Apply the idea Write as hundredths

Write as fraction

Write as percentage

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James is offered three deals after an Easter egg hunt: • Deal 1: He keeps 12% of the total eggs found. • Deal 2: He keeps 0.65 of the total eggs found. • Deal 3: He keeps

of the total eggs found.

How can James work out which deal to take? 4ab, 5bd, 6a-d, 7, 8a, 9ac

Fluency 4 Ex 1

Rewrite the group as percentages.

Order the original group from smallest to largest. b

69% ⬚ 0.65

⬚ 131%

71% ⬚ 0.59

51% ⬚ 0.87

⬚ 62%

⬚ 0.88

Are these statements true or false?

d h

0.052 ⬚ 4.9% 0.095 ⬚

Are these statements true or false? a

Determine which inequality sign, < or >, would make the statement true: a

Identify whether these values are the same, or determine which number is larger: a

4cd, 5ac, 6e-h, 8b-d, 9d

For these groups of numbers:

a 5

4bc, 5bd, 6e-h, 7, 8b, 9a-c

91% > 0.91

63.2% < 0.72

Arrange each of these sets of numbers in descending order:

a c

92%, 0.1, 0.36, 60.1%

Reasoning

11–13

10–12

10, 12, 13

Write 3 different percentages that lie between

Write 3 different decimals that lie between 69.2% and 69.4%.

Write 3 different fractions that lie between 75% and 80%.

Write 3 different integer values for a and b so that

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and

is equal to 68%.

Problem-solving 14, 15 14

14, 15

15, 16

Josie, Juliet and Jasper had a box of chocolates to share. Josie ate

of the contents of the

box, Juliet ate 30% of the contents of the box, and Jasper ate 0.15 of the contents. There were 12 chocolates left in the box. How many chocolates were in the box originally? 15

Bernie wants to buy a laptop. It is priced at $1500 and is being offered on sale at 3 different shops. Shop A is offering 25% off, Shop B is offering

off, and Shop C says that the

purchase price is 0.7 of the normal price. Which shop is offering the best discount? 16

Romeo, Ollie and Stella won some money in the lottery and decided to share it in proportion to the amounts that they contributed towards the purchase of the tickets. Romeo had paid of the purchase price and Ollie had paid 0.28 of the purchase price. Who should get the largest share of the winnings?

7.03 Express a quantity as a percentage After this lesson, you will be able to… • find parts of a whole. • express a quantity as a percentage.

Quantities as percentages When finding quantities as percentages, we always want to compare our quantity to the whole that it is a part of. When writing this as a fraction, we always put the quantity as the numerator and the whole as the denominator. Once we have our quantity written as a fraction of the whole, all we need to do is convert it into a percentage, by multiplying by 100 and adding the percentage sign.

Example 1 What is 27 as a percentage of 150?

Create a strategy Write the quantity as a fraction of the whole, then multiply by 100%.

7.03 Express a quantity as a percentage mathspace.co

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Which of these shows how to calculate 30 as a percentage of 65? B

3aceg, 4ae, 6a, 7ace, 8-10, 11aceg

Fluency 3

3a-d, 4ac-e, 5, 6ab, 7ace, 8, 10, 11bdfh

3dfh, 4bfgh, 5, 6cd, 7bdf, 8, 10, 11bdfh

Express these percentages as a fraction: i

With a denominator of 100.

In simplest form.

40%

65%

30%

16%

25%

62.5%

75%

35%

36 out of 45

Express these quantities as a: i

Fraction

21 out of 50

Percentage

4 out of 80

28 out of 56

42 out of 24

44 out of 110

64 out of 40

21 out of 175

What percentage of the total triangles are shaded?

iii

What percentage of the total squares are shaded? i

What percentage of one whole circle is shaded in each set? i

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Write these quantities as percentages: a

Ex 1

36 out of 200

10 out of 16

9 out of 27

30 as a percentage of 150

81 as a percentage of 150

34 as a percentage of 56

12 as a percentage of 32

6 as a percentage of 60

8 as a percentage of 12

Calculate 28 cm as a percentage of 3 m.

A class has 32 students, and 12 of them are left-handed.

17 out of 6

Find:

Ex 2

How many of the students in the class were not left-handed?

What percentage of the class is left-handed?

What percentage of the class is not left-handed?

What percentage of 2 hours is 48 minutes?

Ex 3

Find: a

443.22 km as a percentage of 890 km

164 cm as a percentage of 8.2 m

$6 as a percentage of $375

$966 as a percentage of $27 600

45.5 kg as a percentage of 65 kg

8550 mL as a percentage of 28.5 L

54 min as a percentage of 6 hrs

108.9 mL as a percentage of 330 mL

Reasoning

12, 14

13, 14

13, 15

In a pond, there are 35 ducks, 12 geese, and 10 swans. What percentage of the birds in the lake are ducks? Give your answer to two decimal places.

A class has g girls and b boys.

Write an expression for the percentage of girls in the room.

Write an expression for the percentage of boys in the room.

Consider these sales: • Sale 1: A jacket normally priced at $120 was on sale for $102 • Sale 2: A pair of trousers normally priced at $78 was on sale for $65 Which of these represents the largest percentage reduction?

A group of p people attended a conference; q of them were from the United States. a

Write an expression for the percentage of attendees who were from the United States.

Write an expression for the percentage of attendees who were not from the United States.

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Problem-solving 17

16, 18, 19

16, 18, 20

A car weighs 1650 kg and a truck weighs 3 tonnes. Express the mass of the car as a percentage of the mass of the truck.

A box of fruit contains 8 apples, 8 mandarins, and 4 pineapples. a

Find the total number of pieces of fruit inside the box.

What percentage of the box are apples?

What percentage of the box are mandarins?

What percentage of the box are pineapples?

In a factory producing light bulbs, it is found that 27 out of a batch of 300 light bulbs are faulty. What percentage of the light bulbs are faulty?

Deborah scored 40 out of 50 in her history test and 57 out of 75 in her mathematics test.

What percentage score did Deborah get for her history test?

What percentage score did Deborah get for her mathematics test?

On which test did Deborah perform better?

Justin and Georgia handed out flyers to advertise for their new concert. Justin handed out 120 of his 400 flyers while Georgia handed out 360 of her 600 flyers. a

What percentage of his flyers did Justin hand out?

What percentage of her flyers did Georgia hand out?

Who handed out a higher percentage of their flyers?

7.04 Percentages of a quantity After this lesson, you will be able to… • find percentages of quantities using different strategies.

Find percentages by using smaller percentages We know that we can use fractions to express parts of a whole. We can use this to help us ﬁnd percentages of quantities. The benchmark values table gives us fractions that we can use to ﬁnd percentages: Fractions

316

Decimals

Percentages

1.0

100%

0.5

50%

Mathspace Victorian Curriculum – Year 7 mathspace.co

5a-c, 6, 7, 8a-cl, 9a-ce-g, 10ace

Fluency Ex 1

5–7, 8degk, 9ace-h, 10ace

5–7, 8fh-j, 9cd, 10bdf

Follow the steps in each case to determine the given quantity: a

75% of 28 i

How many groups of 25% go into 75%?

What is 25% of 28?

iii Find 75% of 28. b

140% of 35 i

How many groups of 10% go into 140%?

What is 10% of 35?

iii Find 140% of 35. c

28% of 225 i

How many groups of 4% go into 28%?

What is 4% of 225?

iii Find 28% of 225. d

7% of 300 i

How many groups of 1% go into 7%?

What is 1% of 300?

iii Find 7% of 300. 6 Ex 2

7 Ex 3

Ex 4

Determine the value of these quantities by first converting the percentage to a decimal: a

90% of 20

220% of 10

60% of 60

21% of 40

Determine the value of these quantities by first converting the percentage to a fraction: a

15% of 200

250% of 12

16% of 275

175% of 20

Evaluate the percentages without using a calculator: a

10% of 90

25% of 16

1% of 900

2% of 400

500% of 11

350% of 28

60% of 30

45% of 160

200% of 13

150% of 18

80% of 40

50% of 320

Find: a

10% of 80 g

20% of $60

30% of $200

2% of 300 mL

90% of 80 kg

70% of 20 km

25% of 120 L

75% of $24

62% of 46

27% of 230 or 13% of 230

Which is greater? a

62% of 38

70% of 30

250% of 18

45% of 180 or

13% of 17

29% of 13

36% of 48 or 24% of 63

7% of 700

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Reasoning

11, 14

11, 12, 14

13, 14

Show two different ways to correctly calculate 25% of 140.

a b c

hen finding a percentage of 82 that is above 100%, explain whether the result will be W smaller or larger than 82. When finding a percentage of 82 that is below 100%, explain whether the result will be smaller or larger than 82. Show two different ways to find: i

60% of 82

120% of 82

Opie mowed 40% of his lawn in 36 minutes. How long would it take him to mow the whole lawn if he continues to mow at the same constant rate?

a b c d

Calculate 40% of 70. Calculate 70% of 40. What do you notice about 40% of 70 and 70% of 40? Does this result apply in general? Investigate by writing expressions for calculating p% of q and q% of p.

Problem-solving 17–20

15, 17, 18, 20

15–17, 19, 21

Josie calculated 180 ÷ 9 ÷ 4 × 6. What percentage of 180 did she produce? Give your answer to two decimal places.

A water tank is filled to 15% of its capacity in 10 minutes. How long will it take to fill the tank if, after the first 10 minutes, the rate of flow is double than it was in the first 10 minutes? Give your answer in minutes and seconds.

Lisa scored 95% on her geography exam, which was marked out of 160. What was Lisa’s actual mark out of 160?

A theatre has a capacity of 3000 people. On the opening night, the theatre was 97% full. On the second night, it was 78% full.

322

How many people attended the theatre on the ﬁrst night?

How many people attended the theatre on the second night?

How many people attended the theatre in total during the ﬁrst 2 nights?

A stadium can hold 5000 people. The stadium is 70% full and 11% of the audience are children. a

How many people are in the stadium?

How many children are in the stadium?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Describe the process on how to find the values of each statement. You do not need to find the actual answer. a

29 out of 60

40% of 12

130.8% of 9

The number where 17 is its 80%

Fluency 3

3bceh, 4, 5bcfh, 6a-c, 8–10

Ex 1

5 Ex 2

8 Ex 3

44 out of 64

c 899 out of 725

d 5.9 out of 0.32

76 out of 91.2

Use a calculator to convert these to decimals: b

Use a calculator to convert these to decimals: a e

3adeg, 4, 5adeg, 6bdef, 7

Use a calculator to express these as percentages: a

3bcef, 4, 5bdeg, 6a-c, 9, 10

b 135% × 24.6

9.25%

53% × 85%

70% of 617

of 65.86

72% ÷ 3

126% of 87.2

Use a calculator to find: a

23% of

91.6% of 97.3

209% of

47.3% of 1.764

317% of

Use a calculator to determine the value of x: a

70% of x is equal to 42.7

60% of x is equal to 39.54

60.1% of x is equal to 24.04

46% of x is equal to 35.88

Joanne scored 80% on her Maths exam, which was marked out of 160. What was Joanne’s actual mark out of 160?

A city has a yearly water supply of 27 000 000 megalitres. It gets 14% of this water supply from melted snow. Find the amount of water that comes from melted snow each year.

Ryan scored 60% on his Maths exam, which was marked out of 110. What was Ryan’s actual mark out of 110?

Reasoning 11

11, 12

12, 13

Give some examples of when it might be quicker not to use a calculator in percentage calculations.

7.05 Percentages with technology mathspace.co

327

Bernie works at a clothing store. He buys a jacket that is 25% off in the sale. He also receives a staff discount of 12%. a

To calculate the price he needs to pay, he multiplies the original cost of the jacket by 0.75, and then by 0.88. Explain his calculation.

Betty tells Bernie that 0.75 × 0.88 = 0.66; so he could have just multiplied the price of the jacket by 0.66. Verify that Betty is correct.

Would Bernie get the same result if he applied the two discounts in the reverse order, i.e. a discount of 12% followed by a discount of 25%?

Casper got a promotion at work. His new post meant that his salary increased by 8%. He then received his annual cost of living rise of 4%. a

Casper reasons that since 8% + 4% = 12%, he should multiply his original salary by 1.12. Will this give him the correct value for his new salary?

Casper’s manager tells him that he is doing the calculation incorrectly. She suggests that he should multiply the original salary by 1.1232. Explain how she arrived at this ﬁgure.

Who has the correct calculation, Casper or his manager?

Would Casper be on the same new salary if the cost of living rise of 4% was applied before the promotional rise of 8%?

Assuming that his manager’s calculation is correct, what is Casper’s salary after both increases if his original salary was $72 000? Give your answer to the nearest dollar.

Assuming that his manager’s calculation is correct, what was Casper’s original salary if, after both increases, his salary was $89 856? Give your answer to the nearest dollar.

Problem-solving 15 14

328

14, 15

14–17

A bag of potatoes weighs 50 kg. When more potatoes are added, the weight of the bag increases by 35%. a

Find the added weight in the bag of potatoes.

Find the new weight of the bag of potatoes.

Duncan is going to buy a hat that is marked as 25% off. The original price was $30. a

Find the discount amount.

Find the discounted price that Duncan will pay for the hat.

The 64 girls at Victoria’s party represent 80% of all the guests. a

How many guests represent 10% of all the guests?

Find the total number (100%) of guests at the party.

Jane paid $750 to get a dress made. $450 of this was for the materials and the rest was for the labour required. a

Find the percentage of the cost that paid for the materials.

Find the percentage of the cost that paid for the labour.

Mathspace Victorian Curriculum – Year 7 mathspace.co

7 Chapter review 1

Write each percentages as decimals: a

10%

93%

23.4%

4.5%

160%

409%

700%

7894%

Write each decimals as percentages: a

0.03

0.021

0.457

6.2

15.04

0.0035

3.325

Express each fraction as percentages: b

a 4

Convert between decimals, percentages and fractions to complete the table, expressing the fractions in simplest form:

d Fraction

Decimal

Percentage

1.51

Express each percentages as both a fraction and a decimal: a

20%

Rewrite the group as percentages.

Order the original group from smallest to largest. b

For each of these expressions, determine which inequality sign, < or >, would make the statement true: a

225%

For these groups of numbers:

a 7

Are these statements true or false? a

4.6% < 170%

Write these quantities as percentages: a

12 out of 100

15 out of 24

22 out of 33

39 out of 60

Chapter 7 review mathspace.co

329

Find: a

33 as a percentage of 150

27 as a percentage of 75

12 as a percentage of 20

54 as a percentage of 100

18 L as a percentage of 45 L

Find: a

350 g as a percentage of 1 kg

15 cm as a percentage of 60 cm

$12 as a percentage of $80

5.5 hrs as a percentage of 8 hrs

80 pages as a percentage of 320 pages

A car weighs 1050 kg and a truck weight 3 tonnes. Express the weight of the car as a percentage of the weight of truck.

A box of fruit contains 7 apples, 7 mandarins, and 6 pineapples. a

Find the total number of pieces of fruit inside the box.

What percentage of the box are apples?

What percentage of the box are mandarins?

What percentage of the box are pineapples?

In a survey of 500 people, 175 people prefer tea over coffee. What percentage of people prefer tea?

Out of 200 apples, 15 apples are rotten. What percentage of apples are rotten?

Write the fraction or number that completes these statements: a 5% of a quantity is the same as ⬚ of that quantity. b 200% of a quantity is the same as ⬚ of that quantity.

330

Determine the value without using a calculator: a 25% of 16 b 500% of 11 c e 45% of 160 f 200% of 13 g

350% of 28 150% of 18

d h

60% of 30 80% of 40

Which is greater? a

30% of 30 or 250% of 14

70% of 30 or 450% of 16

13% of 17 or 29% of 13

36% of 48 or 24% of 63

Calculate 25% of 80.

Calculate 80% of 25.

What do you notice about 25% of 80 and 80% of 25?

Does this result apply in general? Investigate by writing expressions for calculating r% of s and s% of r.

A theatre has a capacity of 3000 people. On the opening night the theatre was 79% full. On the second night it was 76% full. a

How many people attended the theatre on the first night?

How many people attended the theatre on the second night?

How many people attended the theatre in total during the first 2 nights.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Big ideas The concept of ratios and their applications allows for effective analysis and comparison of relationships between quantities, helping to solve real-world problems and making informed decisions.

Curriculum Outcomes VC2M7N09 recognise, represent and solve problems involving ratios VC2M7M06 use mathematical modelling to solve practical problems involving ratios of lengths, areas and volumes; formulate problems, interpret and communicate solutions in terms of the situation, justifying choices made about the representation

8 Ratios

Chapter outline 8.01 Identify ratios 8.02 Equivalent and simplified ratios 8.03 Find part or whole measures Chapter 8 review

334 339 346 354

8.01 Identify ratios After this lesson, you will be able to… • use ratios to express relationships between two quantities. • describe part-to-part and part-to-whole relationships.

Ratios A ratio compares the relationship between two (or more) values. It tells us how much there is of one thing compared to another. If we want to describe the relationship between the number of blue dots and the number of green dots, we could say that there is 1 blue dot for every 3 green dots. We could also express this as a ratio, which we would write as 1 : 3. It is important that both quantities are in the same units; otherwise, we cannot compare them with a ratio. In the example above, we are comparing one blue dot to three green dots, so a single dot can be thought of as our unit. A ratio can express a “part to part” relationship, like we saw in the dot example above. We can also describe a “part to whole relationship”. For example, if we wanted to describe the ratio of green dots to all the dots, we could write it as 3 : 4, because there are 3 green dots and 4 dots in total. The order that the words are written corresponds to the order of the values in the ratio, so it is important that we don’t jumble them around. We can also express fractions, decimals, and percentages as ratios.

Example 1 Write a numerical ratio for the number of circles to squares. Give your answer in the form a : b.

Create a strategy As the unit here is shapes, we can compare circles to squares directly. Count the number of circles and squares. Write the ratio as the number of circles to squares.

Apply the idea There are 5 circles and 14 squares. We can write the ratio as 5 : 14.

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8.01 Practice questions Understanding 1

4:3

3:4

5:7

What does the ﬁrst number in the ratio represent?

What does the second number in the ratio represent?

Express the ratio of cats to dogs in the form a : b. 3, 4a, 5, 6aceg, 7a-d, 8a, 9a-d, 10

3, 4a, 5, 6aceg, 7c-f, 8a, 9b-e

The number of circles to squares:

6:9

3, 4b, 5, 6bdfh, 7e-h, 8b, 9c-f

The number of squares to circles:

336

Write a numerical ratio for: a

Trailblazer 1cd, 2

Given a ratio of 3 cats to 5 dogs:

Fluency Ex 1

Adventurer 1bc, 2

A ratio can be described as a word phrase. E.g. the ratio 1 : 2 could represent 1 pencil and 2 pens. Write your own word phrase to represent each given ratio: a

Explorer 1ab, 2

For each diagram: i

Write the ratio of shaded squares to non-shaded squares.

Write the ratio of shaded squares to total squares.

Mathspace Victorian Curriculum – Year 7 mathspace.co

For each diagram: i

Write the ratio of shaded sectors to non-shaded sectors.

Write the ratio of shaded sectors to total sectors.

Ex 2

Express each pair of quantities as a ratio in the form a : b a

$5 to $7

47 volts to 43 volts

13 minutes to 17 minutes

37 lollies to 17 lollies

5 seconds to 26 seconds

13 newtons to 29 newtons

15 oranges to 76 oranges

130 days to 171 days

Express each pair of quantities as a ratio using the same units: a

9000 metres to 8 kilometres

$1 to 31c

2 centuries to 409 years

21 minutes to 2 hours

6 minutes to 659 seconds

43 weeks to 1 year

37 months to 2 years

13 hours to 5 days

Write the ratio of triangles to squares.

For each diagram: i

Write the ratio of circles to squares.

Write each statement as a ratio: a

There are 11 cats to 6 dogs.

In a classroom, there are 12 girls to 5 boys.

When making pasta, you need 5 cups of plain ﬂour to 8 eggs.

There are 3 red marbles to 2 blue marbles.

The Great Dane weighed twice as much as the Labrador.

In a working day, a bricklayer lays 50 bricks every hour.

In a bag there are 20 marbles. If there are 5 green marbles, 6 blue marbles, 2 red marbles and 7 yellow marbles, find the ratios of the coloured marbles: a

Green to red

Blue to red

Green to blue

Yellow to the total number of marbles

8.01 Identify ratios mathspace.co

337

8.02 Equivalent and simplified ratios After this lesson, you will be able to… • identify equivalent ratios. • simplify ratios to their most basic form. • understand the relationship between equivalent and simplified ratios.

Equivalent ratios Consider a cake recipe that uses 1 cup of milk and 4 cups of flour. The ratio of milk to flour is 1:4 What if we want to make two cakes? We would need to double the amount of milk and flour we use. Now the ratio of milk to flour is 2:8 We say that 1 : 4 and 2 : 8 are equivalent ratios. 2:8

1×2:4×2

2 sets of 1 : 4

Two ratios are equivalent ratios if both numbers in one of the ratios can be multiplied or divided by the same number to make the second ratio. 1

When the ratio 1 : 2.2 is multiplied by 10, we get the equivalent ratio 10 : 22.

2.2

× 10

10 :

8.02 Equivalent and simplified ratios mathspace.co

339

Example 1 Complete the table of equivalent ratios and use it to answer the following questions: a

Dogs 9 18 27 45

to : : : : :

Cats 5 10

Create a strategy To complete the table, we need to find equivalent ratios to 9 : 5. Equivalent ratios will be multiples of each other so we need to find what multiple to use to turn 9 : 5 into 27 : ⬚.

Apply the idea ×?

×3

×?

×5

×?

× 10

9: 5 27 : 9: 5 27 : 15 9: 5 45 : 9: 5 45 : 25 9: 5 : 50 9: 5 90 : 50

×?

9 × 3 = 27 so multiply all parts of the ratio by 3

×3

So 9 : 5 is equivalent to 27 : 15

×?

9 × 5 = 45 so multiply all parts of the ratio by 5

×5

So 9 : 5 is equivalent to 45 : 25

×?

5 × 10 = 50 so multiply all parts of the ratio by 10

× 10

So 9 : 5 is equivalent to 90 : 50

The complete table of equivalent ratios: Dogs 9 18 27 45 90

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to : : : : :

Cats 5 10 15 25 50

Example 3 Simplify the ratio 35 : 5

Create a strategy

Apply the idea

Find the highest common factor of the two numbers and then divide both numbers by it.

The HCF of 35 and 5 is 5, as 5 is the highest number that can divide both 35 and 5 with no remainders. Divide both numbers by the HCF. 35 ÷ 5 = 7 5÷5=1 So the simpliﬁed ratio is 7 : 1.

Example 4 a Simplify the ratio

Create a strategy

Apply the idea

Find a number that we can multiply to both

Multiply both parts of the ratio by the denominator of the fraction, which is 4.

and 2 to get a ratio involving only whole numbers.

So the simpliﬁed ratio is 3 : 8. b Simplify the ratio

Create a strategy Find a number that we can multiply to both

and

to get a ratio involving only whole numbers.

Apply the idea Get a common denominator with the fractions, which is 15.

So the simpliﬁed ratio is 9 : 2.

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Fluency

5a-e, 6–8, 9a

3–4, 5e-il, 6, 7, 9

3, 4, 5h-l, 6, 7, 9

Ex 1b

Tricia and Luigi contribute money for a birthday present in the ratio 4 : 9. If Tricia contributed $24, how much has Luigi contributed?

Ex 2

The ratio of students to teachers competing in a charity race is 9 : 4. If 54 students take part in the race, how many teachers are there?

Ex 3

Write as simplified ratios: a

Ex 5

10 : 20

20 : 40

35 : 7

100 : 2000

8:4

15 : 40

36 : 48

16 : 24

45 : 35

18 : 22

450 : 350

10 : 100

Write down three ratios that are equivalent to these ratios: a

Ex 4

4 : 10

3:1

6 : 10

2:5

Write as simplified ratios: a

7 : 6.3

1.5 : 2.4

0.5 : 2.25

2.4 : 3.9

2.8 : 1.6

0.4 : 2.08

$0.60 : $2.00

$5.20 : $9.50

$8.20 to $11.60

5.4 km to 420 m

120 cm to 5.1 m

11 years to 28 months

20 days to 5 weeks

Write as simplified ratios: a

Reasoning

10, 11, 13ab, 14a

12, 13a-c, 14a

12, 13d, 14b

At a sporting event, the ratio of security guards to spectators is 1 : 120. If 8400 spectators attend the event, explain how to ﬁnd how many security guards will be required.

Suppose you are explaining to a classmate who missed the lesson how to simplify the ratio 620c to $4.00. Describe how you would simplify the ratio.

Given the ratio of apples to oranges is 3 : 5, explain whether these statements are true or false.

344

If the number of apples are increased, the number of oranges will also increase.

If the number of apples are doubled, the number of oranges are halved.

If the number of apples are doubled, the number of oranges are doubled.

If there are no apples, there are no oranges.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Hermione is making a patterned lid for a wooden box. She knows that for every mahogany, she will need 70 centimetres of oak. a

She states the ratio of mahogany to oak is why not.

Write the ratio of mahogany to oak using fractions.

Simplify this ratio.

How many metres of mahogany would Hermione need if she wants to use 14 metres of oak?

Is Hermione correct? Explain why or

In these sets of equivalent ratios, one ratio is incorrect. Find the incorrect ratio in each set. Explain why you think it doesn’t ﬁt in the set and replace one of its numbers to make it equivalent to the others. a

18 : 24,

3 : 4,

6 : 9,

15 : 20,

30 : 40

5 : 10,

20 : 40,

14 : 30, 2 : 4,

8 : 16

Problem-solving 15ab, 16ab 15

metres of

15–17

A painter wants to create a certain colour by mixing two different colours of paint, Vespa and Nitro, in the ratio 4 : 1. He uses 4 litres of the Nitro colour. a

How many litres of the Vespa colour must he use?

How many litres of paint will he have altogether once the two colours are combined?

A litre of paint can cover an area of 8 m2. Suppose that the rooms in a house each have 65 m2 of wall space. How many rooms can the painter paint with the new colour?

The ratio of students who play badminton to football to softball is 2 : 5 : 9. a

If there are 72 students who play softball, how many students play badminton?

Using your answer from part (a), determine the total number of students that play these sports.

A local business wants to sponsor the badminton and football teams by buying their team shirts. If shirts are $15 each, can the business provide enough shirts for the teams with a budget of $800?

Two kinds of pine trees, Bristlecone and Aleppo, are planted in rows. In each row, the ratio of Bristlecone to Aleppo is 10 : 9. There are a total of 1140 trees planted. Determine the number of each Bristlecone and Aleppo that were planted.

8.02 Equivalent and simplified ratios mathspace.co

345

8.03 Find part or whole measures After this lesson, you will be able to… • apply your knowledge of equivalent and simplified ratios to calculate shares of a quantity and determine a quantity from a given share.

Find a share of a quantity Suppose that Opie and Josie contribute some funds to a business in the ratio of 7 : 4. The total amount invested is $88 000. We consider the $88 000 as the total 11 parts. This means that Opie contributed 7 parts and Josie contributed 4 parts. There is $88 000 in total, which needs to be equally shared into 11 parts. $88 000

7 parts

4 parts

Find the size of each part: $88 000 ÷ 11 = $8000. $88 000

$8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000 $8000

7 parts

• Opie invested the value of 7 parts: $8000 × 7 = $56 000 • Josie invested the value of 4 parts: $8000 × 4 = $32 000

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4 parts

Example 1 Ben always buys melons and bananas in the ratio 7 : 3. If he buys 30 pieces of fruit in total, how many melons did he buy?

Create a strategy Divide the amount of fruit into equal parts. The ﬁrst part of the ratio refers to melons, so 7 parts will be melons.

Apply the idea The ratio 7 : 3 means we have 7 + 3 = 10 parts. 30 pieces of fruit

7 parts

3 parts

7 of those parts are melons. 30 pieces of fruit M

7 parts

30 ÷ 10 = 3 7 × 3 = 21

3 parts

Divide 30 into 10 parts 7 parts are melons

Ben buys 21 melons (and 9 bananas).

8.03 Find part or whole measures mathspace.co

347

Find a quantity from a share A recipe uses oats and flour in the ratio of 3 : 4. These are known as the dry ingredients. If we know we are to use 180 g of oats, how much flour should we use? And what is the total amount of the dry ingredients? In this case, we know the amount of oats and we want to work out the amount of flour. The 3 in the ratio 3 : 4 refers to the oats, and this is 180 g. 180 g

3 parts

4 parts

Oats

Flour 3

One part would therefore be 180 ÷ 3 = 60 g. Flour is then 4 parts, so 4 × 60 = 240 g of ﬂour should be used. 180 g

60 g

3 parts

60 g

Oats

4 parts

60 g

3 : 4 180g :

Flour

8.03 Find part or whole measures mathspace.co

349

The total of the dry ingredients (oats and ﬂour) is then 3 + 4 = 7 parts, so the total of the dry ingredients is 7 × 60 = 420 g. 420 g

180 g

60 g

240 g

60 g

3 parts 60 g

60 g

4 parts 60 g

Oats

60 g

3 : 4 180 g : 240 g

60 g

Flour

Example 3 Juliet wants to make some cakes. She has lots of ﬂour, but only 2 kg of butter. The ratio of butter to flour in the recipe is 4 : 5. How much ﬂour should she use?

Create a strategy Create a ratio diagram that shows the ratio of butter to flour (4 : 5).

Apply the idea We need to divide the 2 kg of butter into 4 parts. 2 kg

4 parts

2 ÷ 4 = 0.5

5 parts

2 kg divided into 4 parts 2 kg

0.5 kg

4 parts

5 × 0.5 = 2.5

350

5 parts

Each part is worth 0.5kg. Multiply the 5 parts of the flour by 0.5.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 2

The ratio of adults to children on a train is 2 : 7. The train is carrying 360 passengers. a

Ex 3

How much money does James get?

Simplify the ratio.

How much did Xavier pay for the groceries?

How much money does Sophia get?

A car and a truck are driving down the highway. Their speeds are in the ratio 3.1 : 2.9. If the car is travelling at 93 km/hr, how fast is the truck travelling? 11, 12

11, 12

9–12

Mara is mixing paint for a mural. She uses two colours in the ratio of 3 parts of blue paint for every 6 parts of yellow paint. a

Write the simplified ratio of blue paint to yellow paint.

Describe how to determine the quantity of yellow paint needed if Mara uses 9 parts of blue paint.

Avril is making a scarf that uses two materials. She wants to use the materials in the ratio of metres of red for every

How many children are on the train?

Xavier and Ursula split the cost of the groceries in the ratio 0.5 : 1.5. Altogether, the groceries cost $160.

Reasoning 9

James and Sophia share $880 in the ratio 2 : 9. a

How many adults are on the train?

metres of green.

Write the simpliﬁed ratio of red material to green material.

If she needs 33 metres of red material, describe how to ﬁnd how many metres of green material she will need.

Kathleen and Luke invest money in a business in the ratio 3 : 10. A student reading the statement above concludes that if Kathleen invests $3000 in the business, that means Luke invested $7000 for a total of $10 000 invested.

What is the mistake the student may be making with their statement.

Rewrite the statement so that the student’s conclusion is correct.

Two kinds of pine trees, Paperbark and Gumtree, are planted in rows. In each row, the ratio of Paperbark to Gumtree is 9 : 8. Altogether, 340 trees are planted. Explain two different ways to determine how many Gumtree trees are planted.

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8 Chapter review 1

Express each pair of quantities as a ratio: a

60 : 40

90 hours to 150 hours

45 : 15

525 : 375

Write as simplified ratios: a

18 apples to 54 apples

Write as simplified ratios: a

24 pounds to 42 pounds

0.75 : 1.5

3.6 : 2.7

1.2 : 3.3

16 years to 48 months

30 days to 6 weeks

Write as simplified ratios: a

7.2 km to 600 m

Emma and Marco contribute money to an investment in the ratio 3 : 7. If Emma has invested $3000, how much has Marco invested?

The ratio of players to coaches participating in a community sports event is 7 : 3. If 49 players take part in the event, how many coaches are there?

A train travels 120 kilometres in 4 hours and continues for another 180 kilometres in 6 hours. Write the distance travelled in the first 4 hours to the total distance travelled as a ratio.

An artist wants to create a unique shade by blending two different colours of paint, azure and tangerine, in the ratio 3 : 2. She uses 6 litres of the tangerine colour.

How many litres of the azure colour must she use?

How many litres of paint will she have altogether once the two colours are combined?

A litre of paint can cover about 10 m2. Assume that the walls in a studio each have 50 m2 of surface area. How many studios can the artist paint with the new shade?

The ratio of students who play basketball to soccer to volleyball is 3 : 4 : 7. a

If there are 84 students who play volleyball, how many students play basketball?

Using your answer from part (a), find the total number of students that play soccer and volleyball.

A local company wants to sponsor the basketball and soccer teams by buying their team jerseys. If jerseys are $20 each, can the company provide enough jerseys for the teams with a budget of $1000?

Two varieties of apple trees, Gala and Honeycrisp, are planted in an orchard. In each row, the ratio of Gala to Honeycrisp trees is 5 : 6. There are a total of 1210 trees planted. Find the number of each type of tree that was planted.

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Big ideas Algebraic expressions serve as the fundamental units of algebra, enabling the representation and analysis of real-world scenarios.

Curriculum Outcomes VC2M7A01 recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown VC2M7A02 apply the associative, commutative and distributive laws to aid mental and written computation, and formulate algebraic expressions using constants, variables, operations and brackets VC2M7A05 generate tables of values from visually changing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane

9 Algebra Chapter outline 9.01 Introduction to variables 9.02 Build algebraic expressions 9.03 Substitution 9.04 Collect like terms Chapter 9 review

358 363 369 374 380

9.01 Introduction to variables After this lesson, you will be able to… • write algebraic expressions using variables. • combine like variables in algebraic expressions.

Write expressions with variables The basic concept of algebra is using symbols, called variables or pronumerals, in place of numerical values that we don’t know. We normally use letters of the alphabet as our variables. We can use algebra tiles to help us visualise algebraic expressions.

Let this equal x

How many oranges in the box?

The tile x represents an unknown number. The tile + 1 represents adding one unit and −1 represents subtracting one unit. Positive

Negative

Variable tiles

−x

Unit tiles

−1

This table demonstrates how expressions can be built using the tiles: Word expression

Algebraic expression

three more than x

x+3

three less than x

x–3

Representative with algebra tiles +x +1

+1 +x

−1

−1 +x

the product of x and three

x×3

+x +x

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−1

Fluency 6

6, 7, 8a-c, 9a-c

a b c

Ex 1, 2

7cd, 8cd, 9cd

Consider this diagram:

6–9

Fill in the boxes to write an algebraic expression that represents the diagram: ⬚ + ⬚ + ⬚ + ⬚ + ⬚.

Fill in the boxes to write an algebraic expression that represents the diagram: 5 × ⬚. Write the answer to part (b) in another way.

Write a simpliﬁed algebraic expression for these diagrams: a

Each single box represents p.

Each single box represents b.

Each single box represents y.

ach purple box represents p and each E green box represents g.

Write a simpliﬁed algebraic expression for these diagrams: a

+1 +1 +1 +x

+x −1

+x +x

−x

−1

9.01 Introduction to variables mathspace.co

361

Use these algebra tiles to draw a diagram that represents each expression:

Positive

Negative

Variable tiles

−x

Unit tiles

−1

x+2

Reasoning

2x + 1

10, 11

3x − 4

−3x + 2

12, 13

Show that a + a + a + a = 4a for three different values of a.

Show that x + x + x + y = 3x + y for three different values of x and y.

Using the given key, explain why it is not possible to simplify a + b any further using algebra.

Key

contains a apples

contains b apples

Is 2x + 3 the same as 3 + 2x? Explain your answer.

Problem-solving 14, 15, 16a 14

14, 15, 16ab

15, 16bc, 17

Consider this list of colours of pens: blue, blue, red, red, red If b is the number of blue pens and r is the number of red pens:

What does the expression b + r represent?

What are the values of b and r?

Blue marbles are represented by b, green marbles by g, red marbles by r and yellow marbles by y. b + g + r + y represents the total number of marbles in the marble jar. What are the values of b, g, r and y?

Complete the pattern in these tables: a

362

1 2 3 4 x 2x 1 z+2

2 2z + 4

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1 2y + 1

2 4y + 2

Consider the key: Key contains a oranges contains b oranges

If James has 5 of the boxes containing b oranges and this totals 100 oranges, what is the value of the variable b?

If Denise has 3 of the boxes containing a oranges and this totals 33 oranges, what is the value of the variable a?

Sasha has 6a + 2b oranges. How many oranges is this?

9.02 Build algebraic expressions After this lesson, you will be able to… • recognise and identify coefficients, variables, and constant terms in algebraic expressions. • translate word expressions into algebraic expressions and simplify them when necessary.

Terms, variables and coefficients We use algebraic expressions when we want to write a number sentence but don’t know all the numbers involved. For example: What is the total weight of a cat and a dog? In this case, let’s use c for the weight of the cat and d for the weight of the dog.

c+d

c + d is called an algebraic expression. c and d are called variables or pronumerals. They are used in place of a numeric value. Coeﬃcients are used to show how many lots of a variable we Total weight = cat plus dog = c + d have. The variable u with a coeﬃcient of 3 is written as 3u, which means 3 × u.

3u = 3 × u The variable is u with a coefficient of 3

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Basic operations in algebra The basic operations work almost the same for algebraic terms as they do for numbers. Between variables and numbers, we have: Word expression Three more than x Three less than x

Algebraic expression x+3 x−3

The quotient of x and three

x÷3

The product of x and three

x×3

Simpliﬁed algebraic expression x+3 x−3

Between variables and other variables, we have: Word expression y more than x y less than x

Algebraic expression x+y x−y

The quotient of x and y

x÷y

The product of x and y

x×y

Simpliﬁed algebraic expression x+y x−y

Notice that division is written as a fraction, for example:

Example 2 What does the expression 8x mean?

Create a strategy

Apply the idea

A coeﬃcient is multiplied by the variable next to it.

The expression 8x means 8 is multiplied by x.

Example 3 Write an algebraic expression for the following phrase, “eight more than the quotient of 9 and x”.

Create a strategy

Apply the idea

Translate the terms into mathematical symbols and operations.

The phrase “eight more than” indicates that we need to add 8. The “quotient of 9 and x” indicates division, which we can write as a fraction, with 9 as the numerator and x as the denominator. We can combine the whole description into a single expression:

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A single component of a product.

The result of adding two or more values or algebraic expressions.

A single component of a sum or subtraction.

The result of subtracting a value or algebraic expression from another. 7ab, 8, 9ab, 10a-c, 11a, 13–16

Fluency Ex 1

the coefficient of x

the constant term

8y + 2x − 3y + 8

5x − 2y + 4

13x − 13y + 13z − 13

x−6

e 8

Ex 3

p+7

Write an algebraic expression for these statements without using a multiplication or division sign: a

Five times the sum of x and ten

Three lots of the sum of a and b

Seven lots of the difference between n and 8, then increased by 4

Eleven more than the product of 4 and the difference between m and 9

Write an algebraic expression for these statements without using a multiplication or division sign: a

Seven multiplied by x

Eight more than 5x

Three less than 8y

The product of 9a and b

The quotient of 8u and 9

Four times y take away 9

7 less than the quotient of 5 and x

The sum of a and b, divided by 6

4(n + 1) − 11

Write a word statement for these expressions: a

−x + y − z

Write a word statement for these expressions: a

7b-e, 8, 9cd, 10c-h, 11b-d, 12bdfh, 15, 16

For each expression, identify:

Ex 2

7b-d, 8, 9a-c, 10ac, 11a-c, 12aceg, 13, 15, 16

2(b + 9)

x (3 − y)

For these expressions: i

Identify the order in which the operations should be performed.

Simplify the expression.

3×u÷v+8

5+y×x−4

(4 − k) ÷ 5

7 − 12 ÷ (3 × a)

8+3×3÷m

(n + 3 − 2) ÷ 7

(15 − 3) × (m + 1)

(3 × p + 5 − 4) ÷ 6

The cost of a movie ticket is $12. Write an algebraic expression for the cost of n movie tickets.

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The total cost of buying x apples is $4. Write an algebraic expression for the cost of one apple.

A shipping company charges a ﬂat fee of $25 plus $0.50 per kilogram for shipping packages. Write an algebraic expression for the total cost of shipping a package that weighs w kilograms.

The length of a rectangle is 4 more than its width. Write an algebraic expression for the length of the rectangle in terms of its width, w.

Reasoning

17, 18

17–19

17–20

Why does 5 × 7 + 2 need to have a multiplication sign, however 3b + 2 doesn’t?

Write “the product of 3 and k, then divided by b” in three different ways.

If m is a positive odd number:

Explain why m + 2 and m + 4 are also odd.

Write the third odd number after m.

Write the sixth odd number after m.

Write the pth odd number after m.

Explain how “2 groups of 8 more than x” is different to “8 more than 2x.”

Problem-solving 21

21, 22

22–25

A recipe for a cake calls for 2 cups of ﬂour for every 3 eggs. Write an algebraic expression for the number of cups of ﬂour needed for n eggs.

The sum of the ages of a mother and her daughter is 42 years. If the mother’s age is q years, write an algebraic expression for the daughter’s age.

Write down an algebraic expression for:

The total value of m coins, where each coin is worth p cents.

The length of string remaining when two pieces of length K cm are cut from a piece of string originally L cm long.

The term “consecutive integers” means whole numbers that follow each other on the number line. For example: 5, 6, 7 are three consecutive integers. a

Write algebraic expressions for three consecutive integers if the smallest integer is x.

Write algebraic expressions for three consecutive integers if the largest integer is p.

Write algebraic expressions for three consecutive integers if the middle integer is y.

A square has an area of h2 square units. Write an algebraic expression for the perimeter of the square.

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9.03 Substitution After this lesson, you will be able to… • use substitution to evaluate algebraic expressions. • complete tables of values and find unknown values in a table.

Evaluate expressions To evaluate an algebraic expression, we ﬁrst replace the variable with a number. We call this substitution. If p = 2 then: • p + 4 is equal to 2 + 4 which is 6. • 10 − p is equal to 10 − 2 which is 8. • 3p is equal to 3 × 2 which is 6. Each time we replace the p with the number 2 and then evaluate.

Example 1 Find the value of 9 + m when m = 3.

Create a strategy We can substitute m = 3 into the equation by replacing m with 3.

Apply the idea 9+m=9+3 = 12

Substitute m = 3 into the equation Evaluate the addition

Example 2 Find the value of

when u = 54.

Create a strategy We can substitute u = 54 into the equation by replacing u with 54.

Apply the idea Substitute u = 54 into the equation

Evaluate the division

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and the 105th number (x = 105): y = 3x +1

Write the equation

= 3 × 105 + 1

Substitute x = 105 into the equation

= 315 + 1

Evaluate the multiplication

= 316

Evaluate

Example 5 Consider the table: m n

10 3

11 4

12 5

a Use the formula n = m − 7 to complete the table.

Create a strategy We can ﬁnd the n-values by substituting the m-values into the formula.

Apply the idea n=m–7

Write the formula

n = 13 – 7

Substitute m = 13

n=6

Evaluate the subtraction

n = 14 – 7

Substitute m = 14

n=7

Evaluate the subtraction

n = 15 – 7

Substitute m = 15

n=8

Evaluate the subtraction m n

10 3

11 4

12 5

13 6

14 7

15 8

Reflect and check Notice the pattern that is created in the table. Each time m changes, the n value changes by the same amount.

b What is the value of n when m = 20?

Create a strategy We can substitute m = 20 into the equation by replacing m with 20.

Apply the idea n=m−7

Write the formula

n = 20 − 7

Substitute m = 20

n = 13

Evaluate the subtraction

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8 Ex 1, 2, 3

Find the value of: a

9 + m when m = 8

14 + v when v = 15

n − 2 when n = 5

n − 8 when n = 44

u × 5 when u = −6

when n = 14

when u = 54

2x + 9 when x = 4

when x = 8

3x − 4 when x = 7

n × 9 when n = 0.8

Ex 4

Ex 5

6x − 9 when x = 10

when x = 2

2(x + 7) when x = 4

+ 6 when x = 63

14x +

2a2 + 6 when a = 4

a2 + 6b2 when a = 2 and b = 3

Find the value of: a

m + n when m = 7 and n = 12

2x + 3y + 1 when x = 6 and y = 5

8a + 4b − 9 when a = 2 and b = 5

when x = −3 and y = 6

a3 + bc when a = 3, b = 8 and c = 7

x3 + y2 + z when x = 4, y = 8 and z = 10

Complete these tables: a

p 1 2 3 p+7 8 9 10

p 1 2 3 6p + 4 10 16 22

p 1 2 3 5p − 3 2 7 12

Reasoning 11

when

p 3p

x 2x + 4

11, 12

p 11p + 5

11–13

If p = 2, calculate the value of:

2 × 12p

Explain your results from part (a).

Evaluate 24 × (p + p) when p = 1.

Is this result the same as previous results? Explain.

3 × 8p

iii

2 6

3 9

1 6

2 8

3 10

1 16

2 27

3 38

11–13

1 3

6 × 4p

1 × 24p

If x = 5 and y = 3, write three different expressions including both variables that each evaluate to 23.

If a and b are integers between 0 and 10, show that 10a + b is always a two-digit number.

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Problem-solving 14, 16, 19

14, 16, 17, 20, 21

14–18, 21

When a whole number is substituted for x in the expression 5x + 1, the result is a number greater than 56. What is the smallest value that the whole number could be?

If t is an integer greater than 0 and less than 7, ﬁnd the minimum possible value of 9 − 4t.

Describe what happens to the value of the expression when: i

the value of x increases

the value of x decreases

x + 6

15.6 − x

Describe what happens to the value of the expression when:

the value of x doubles

the value of x halves

10x

Describe what happens to the value of the expression when:

the value of both x and y is doubled

both x and y are multiplied by 3

2x + 3y

5xy

Valentina’s hens produce 5 eggs each day.

If Valentina collects the eggs from her hens for y days, write an algebraic expression for the total number of eggs she collects.

Find the number of eggs Valentina will have after 30 days.

John breeds dogs for sale. Each of his d dogs gives birth to p puppies. a

Write an algebraic expression for the total number of puppies.

If he has 3 dogs, ﬁnd the total number of puppies.

In a Year 7 class, students are either twelve years old or thirteen years old.

If there are k twelve year-old and m thirteen year-old students, write an algebraic expression for the total number of students in the class.

If there are 4 twelve year-old and 16 thirteen year-old students, ﬁnd the number of students that are in the class.

9.04 Collect like terms After this lesson, you will be able to… • identify like terms. • combine like terms using addition and subtraction in order to simplify expressions.

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Are these expressions equal to 11y? a

Ex 2

9 + 8r

5r + 4t + 8

17r 8 + 4rt + 5rt

10y + 1

9r + 8 − r + 1

10r + 2 − 2r + 7

9r + 8t

17rt

Simplify the expression.

5u + 7 + 8u + 8

4x − 6 + 5x

12 + 6x + 7y − 8

2b + 3b

7 + 6a

9 + 7m − 2p + 3q

7u + 2v − 7u

Simplify these expressions where possible: 2a + 5a

12y − 3y + 1

4x + 7x − 5

3c + 4c + 7c

15x − 6x − 2x

19b − 12b − 6b

−3n + 6n + 3n

8x − 3x + 7x

4xy + 10xy − 9xy

15b − 7b + 10 + 1

9 + 7m − 2 + 3m

10x + 12 + 6x − 7

Simplify: a

11m + 8n + 14m

6p + 8q − 6p

7m + 8n + 3m + 2n

8x + 9y − 5x + 10y

7a + 11a + 9b − b

13m + 2n − 8m + 10n

2ab − 7c + ab + 9c

−3s + 6t + 9s − 4t

9xy + 20 + 12yx − 3

19a + 8b − 4c − 8a + b

8x + 6y − 2y − 4x + 2 + 3

o 12

Name any pairs of like terms.

m 18x − 9y + 7x + 11y + 11

Ex 4

9 + y + 10y − 9

m 10ab + 3ab − 8ba n 11

For these expressions:

Ex 3

6y − 5y

Are these expressions equal to 9rt + 8? a

Are these expressions equal to 8r + 9? a

5y + 6y

8x + 10 − 7y − 6z + 10z + 3

10m + 7 + 9n + 5m + 10n

11y − 6z + y + 6z + 12 − 10

12j + 7 + 9k − 7k + 11j − 2

Simplify: a

x+x+x+x

2n + n + n

3m + 2m + m

4k + k + 3k

y×2

9×l

15 × h

21 × x

18z ÷ 3

36a ÷ 6

Simplify: a

5x − 2y + 8x + 4 + 6z + 3y + 4z + 2

2xy + 5yz + 4xz + 8xy + 2xz − 3yz + 3

20 + 9x − 4y + 6y + 7x + 21

9mn − 8m − 7n + 7n + 10m + 2nm

11s + 6t + 3st − 2ts + t + 10 − 4s – 5

13ab − 8bc + 6ba − 9b + 11cb

Reasoning

14, 17

14, 16, 17

14–17

Why do you think we write y and not 1y when using algebra?

Are these terms like terms: 5abc and − 9acb? Explain your answer.

Substitute the values x = 5, x = 100 and x = 0.5 to show that 6x + 2x − x is equal to 7x.

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Simplify: i

5x + 5x + 5x

4p + 4p + 4p + 4p + 4p

iii

−3c − 3c − 3c − 3c

0.5k + 0.5k + 0.5k

Review your results for part (a) and explain how the answer to 7 × 5m can be calculated.

Problem-solving 18, 20 18

19, 21, 22, 24

Complete these statements by filling in the blanks: a c e

19–21, 23

2x + ⬚ = 8x

7x − 5x + 10 − ⬚ = 2x + 1

7x + ⬚ + 10y − ⬚ = 11x + 6y

Consider the triangle shown.

b d f

3y + 3 + ⬚ + 5 = 8y + 8

9x + 4y + ⬚ = 9x + 6y

8ab + ⬚ + 9b + 6ab = ⬚ + 11b

Write an expression for the perimeter of the triangle in terms of x.

Simplify the expression.

10x − 3

3x + 9

x+6

Each side of a square is 3a metres in length. Calculate the distance around the outside of the square.

A library has a courtyard in the shape of an equilateral triangle with sides of length 5y metres. Calculate the distance around the outside of the courtyard.

Sam and Max each exercise for x minutes per day. Sam burns 8 calories per minute and Max burns 10 calories per minute.

Write an expression for the number of calories Sam burns in one day.

Write an expression for the number of calories Max burns in one day.

Write a simpliﬁed expression for the total number of calories Sam and Max burn in one day.

Ali, Ben and Mohammed each work for h hours per day. Ali earns $15 per hour, Ben earns $20 per hour and Mohammed earns $35 per hour. a

Write an expression for the amount Ali earns in one day.

Write an expression for the amount Ben earns in one day.

Write an expression for the amount Mohammed earns in one day.

Write a simpliﬁed expression for the total amount the three employees earn in one day.

If they work 8 hours in one day, how much do they earn altogether?

Sarah and Emily each read n pages per day. Sarah reads 15 pages per hour and Emily reads 20 pages per hour. a

Write an expression for the number of hours Sarah spends reading in one day.

Write an expression for the number of hours Emily spends reading in one day.

Write a simpliﬁed expression for the total number of hours Sarah and Emily spend reading in one day.

If they read a total of 120 pages, how many hours did they spend reading altogether? 9.04 Collect like terms mathspace.co

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9 Chapter review 1

Write a word statement for these expressions: a

z+4

Write an algebraic expression for these statements without using a multiplication (×) or division (÷) sign: a

Three times p plus 8

The difference of m and n, divided by 5

6 more than the quotient of 9 and r

(9 − l) ÷ 4

Simplify: a

2×w÷x−7

6+z×y−3

A courier company charges a flat fee of $15 plus $0.75 per kilogram for delivering parcels. Write an algebraic expression for the total cost of delivering a parcel that weighs p kilograms.

The length of a rectangle is 6 less than twice its width. Write an algebraic expression for the length of the rectangle in terms of its width, x.

The sum of the ages of a father and his son is 60 years. If the father’s age is r years, write an algebraic expression for the son’s age.

Find the value of: a

4y − 5 when y = 6

3b2 + 8 when b = 3

Find the value of: when x = 2 and z = 5

a c 9

p2 + qr when p = 4, q = 3 and r = 6

u2 + v3 + w when u = 5, v = 2 and w = 9

Complete these tables: a

1 6

r+5 b

1 4

4r c

2 7

r 5r + 3

2 8 1 8

3 8

3 12

3 18

2 13

When a whole number is substituted for y in the expression 7y + 2, the result is a number greater than 65. What is the smallest value that the whole number could be?

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If s is an integer greater than 0 and less than 8, find the minimum possible value of 12 − 3s.

Lisa breeds cats for sale. Each of her c cats gives birth to k kittens. If she has 4 cats, write an expression for the total number of kittens in terms of k.

Simplify these expressions where possible: a

5xz + 12xz − 7xz

15cd + 6cd − 11dc

15x − 4y + x + 4y + 8 − 6

3 + 9p + 3q − 5p + 8q − 9

Simplify: a

3 + 8b

26 × 2x

81a ÷ 6

Simplify: a

15r + 8u + 4ru − 3ur + u + 12 − 6r − 7

11cd − 7de + 8dc − 10d + 13ed

A park has a square-shaped lawn with sides of length 4x metres. Write an expression for the distance around the entire lawn in terms of x.

Jenny and Tony each exercise for y minutes per day. Jenny burns 6 calories per minute, while Tony burns 9 calories per minute. Write a simplified expression for the total number of calories Jenny and Tony burn in one day in terms of y.

Sara, Alex, and Jane each work for k hours per day. Sara earns $25 per hour, Alex earns $30 per hour, and Jane earns $40 per hour. Write a simplified expression for the total amount Sara, Alex and Jane will earn per day in terms of k.

John and Karen each read p pages per day. John reads 10 pages per hour, and Karen reads 15 pages per hour. a

Write an expression for the total number of hours John and Karen read in one day in terms of p.

Using the expression from part (a), calculate the total number of hours they spend reading if they each read 180 pages.

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Big ideas Creating, solving and manipulating equations helps us to model the relationships in the world around us.

Curriculum Outcomes VC2M7A03 solve one-variable linear equations of increasing complexity with natural number solutions; verify equation solutions by substitution VC2M7A02 apply the associative, commutative and distributive laws to aid mental and written computation, and formulate algebraic expressions using constants, variables, operations and brackets VC2M7A05 generate tables of values from visually changing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane

10 Equations Chapter outline 10.01 Introduction to equations 10.02 Write rules for relationships 10.03 One-step equations 10.04 Two-step equations Chapter 10 review

384 391 400 405 415

Ex 4

Ex 5

Write a corresponding word sentence that describes the mathematical relationship for each equation: a

8x = 32

5y = 60

6x + 4 = 60 − 20

8x + 7 = 72 − 25

Write an equation that represents each word sentence: a

The quotient of g and seven is equal to eleven.

Sixteen less than m is equal to twenty six.

Twelve less than q is equal to twenty ﬁve.

The product of seven and y is equal to x less than twenty ﬁve.

The quotient of p and ten is equal to seven.

The product of ﬁve and q is equal to p less than twenty nine.

The product of seven and p is taken away from sixty ﬁve. The result is equal to twelve.

The product of twelve and r is taken away from seventy two. The result is thirty six.

Identify if p = 8 is a solution to each equation: a

23 + p = 31

29 + p = 35

35 − p = 27

p − 14 = 6

p − 11 = −3

6p = 48

11p = 96

13p = 104

Identify whether the given value of x is a solution to each equation: a

17 + x = 40, x = 20

c e

x − 30 = 21, x = 48

2x = 10, x = 5

8 − x = 10, x = 2

7x = 33, x = 5

g 13

Identify if the given solution for each equation is correct: a

u + 3.3 = 4.8, u = 2.2

v + 20.7 = 51.41, v = 20.71

Identify if y = 2.2 is a solution to each equation: a

2.3 + y = 4.5

3.2 − y = 1.2

5y = 1.2

4y = 8.08

4.5 − y = 3.2

y + 3.5 = 4.5

Consider the expression m − 17. a

Complete the table: m

m − 17 b

What value of m will make the equation m − 17 = 12 true?

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Consider the expression 4 × k. a

Complete the table: k

4×k b

What value of k will make the equation 4 × k = 96 true?

Reasoning 17

17, 18

Consider the equation 56 − t = 39. a

Which side of the equation is bigger when substituting t = 10?

Should we try a smaller or larger number than 10 to solve the equation?

Which side of the equation is bigger when substituting t = 20?

What can be said about the solution of the equation?

Consider the equation 2u = 70. a

Which side of the equation is bigger when substituting u = 30?

Should we try a smaller or larger number than 30 to solve the equation?

Which side of the equation is bigger when substituting u = 40?

What can be said about the solution of the equation?

Problem-solving 19 19

17, 18

19, 20

19–22

A company charges $95 per hour for their services. If a customer uses their services for x hours and pays $855, write an equation to ﬁnd the hourly rate. Which equation would match the scenario? A

855x = 95

x + 95 = 855

95x = 855

There are 20 cupcakes on the table. Soleil can bake 12 cupcakes per hour. Write an equation to ﬁnd the number of cupcakes (C) per hour (h).

The price of a kilogram of apples is p dollars. If John bought 10 kilograms of apples for $30, write an equation that matches the scenario.

Bob is currently b years old. In 16 years’ time his age will be 42. Write an equation to represent Bob’s current age.

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Fluency 3

3–6, 8ace, 9ab

Figure 2

Figure 3

Complete the table: Figure number (F ) Number of matchsticks (M )

By how many matchsticks is the number increasing each time?

Write the rule for the number of matchsticks, M, in terms of the ﬁgure number, F.

Hiro is making a sequence of shapes out of matchsticks: Figure 2

Hiro makes a table comparing the ﬁgure number to the number of matchsticks needed to construct it as shown:

Figure 3

Figure 4

Figure number (F ) Matchsticks (M )

1 5

2 6

Complete the table for the above pattern.

By how many matchsticks is the number increasing each time?

Write the algebraic rule for the number of matchsticks, M, in terms of the ﬁgure number, F, for this pattern.

Dave is constructing a continuing pattern of squares using matchsticks: Figure 1

Figure 2

Figure 3

Dave wants to continue the pattern. Determine the number of matchsticks he needs to add for each extra square.

Dave made a table comparing the ﬁgure number to the number of matchsticks required. Describe any pattern that you notice.

Complete the statement: The number of matchsticks is equal to ⬚ times the ﬁgure number plus ⬚.

d 396

Figure 1

4, 6, 7, 8bdf, 9bdf

Yvonne has constructed the ﬁrst three triangles of a pattern using matchsticks: Figure 1

Ex 1

4–7, 8ace, 9ace

Figure no. (F ) Matchsticks (M )

1 4

2 7

3 10

4 13

Write the algebraic rule for the number of matchsticks, M, in terms of the ﬁgure number, F.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Vanessa is making a sequence out of tiles. She creates a table to record the ﬁgure number of a shape and the number of tiles needed to make it:

Figure 1

Figure 2

Figure number (F )

No. of tiles (N )

Figure 3

Write an equation for the number of tiles, N, in terms of the ﬁgure number, F.

Calculate the number of tiles required for F = 12.

James is making snowflakes out of hexagonal tiles:

James creates a table comparing the width of a snowﬂake to the number of tiles needed to make it: Width (W ) No. of tiles (T )

1 1

3 7

5 13

7 19

Write an equation for the number of tiles, T, in terms of the snowﬂake’s width, W.

Calculate the number of tiles, T, when, W = 100.

For these tables: i

Find the change in y for each increase in x by 1.

Write an equation for y in terms of x.

x 1 2 3 4 y 9 10 11 12

5 13

x y

6 1

7 2

8 3

9 4

10 5

x 1 2 3 4 y 7 14 21 28

5 35

x y

1 7

2 12

3 17

4 22

5 27

x 1 2 3 4 y 14 11 8 5

5 2

x y

3 1

5 5

7 9

9 13

11 17

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For each table of values, write an equation for y in terms of x: a

x 6 7 8 9 y 3 4 5 6

10 7

x y

10 50

11 55

12 60

13 65

14 70

x 6 7 8 9 y 13 14 15 16

10 17

x y

5 12

6 14

7 16

8 18

9 20

x 5 7 9 11 y 2 8 14 20

13 26

x y

72 8

81 9

90 10

99 11

108 12

Reasoning 10

10, 11

Explain how to ﬁnd the missing value in this table using two different methods: a b

4 6

5 10

6 14

8 22

Is the rule y = 5x + 2 the same as the rule y = 2 + 5x? Explain your thinking.

Consider the pattern of triangles shown with the rule M = 3T, which translates to “the number of matchsticks is equal to three times the number of triangles”.

Explain the signiﬁcance of the number 3 in the rule.

Write a rule for a similar pattern made with squares.

Write a rule for a similar pattern made with hexagons.

Problem-solving 13, 14, 16 Ex 2

10–12

15–17

Gwen sells bananas in bunches of 4. a

Complete the table: No. of bunches (x) No. of bananas (y)

b c

398

13, 15, 16

Complete the statement: The number of bananas is equal to ⬚ times the number of bunches.

Write an equation for the number of bananas, y, in terms of the number of bunches, x.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Quentin buys some decks of playing cards that contain 52 cards each. a

Complete the table: No. of decks (d) No. of cards (c)

b c 15

Complete the statement: The number of cards is equal to ⬚ times the number of decks. Write an equation for the number of cards, c, in terms of the number of decks, d.

Complete the statement: The number of marbles will be equal to ⬚ times the number of bags plus his original ⬚. Write the algebraic rule for the number of marbles Liam will own, m, in terms of the number of bags he buys, b.

Kendra opens a bank account and deposits $200. At the end of each week, she adds $15 to her account. a

Complete this table which shows the balance of Kendra’s account over the ﬁrst four weeks: Week (W ) Account total (A)

Liam already owns 5 marbles. He then buys some bags of marbles containing 4 marbles each. a

0 $200

1 $215

Write the algebraic rule for Kendra’s account total, A, in terms of the number of weeks W, for which she has been adding to her account.

Vanessa makes the ﬁrst four entries of a sequence out of coloured tiles as shown:

Figure 1

Figure 2

Figure 3

She then creates a table to show the relationship between the ﬁgure number and the number of tiles used for that ﬁgure:

Figure 4

Figure number No. of tiles used

1 1

Complete the table of values.

What type of numbers are in the No. of tiles used row?

Write the algebraic rule for the number of tiles, N, in terms of the ﬁgure number, F, for this pattern.

How many tiles would be needed for ﬁgure number 7 for this pattern?

10.02 Write rules for relationships mathspace.co

399

Apply the idea The inverse operation for ×9, is ÷9. So the correct answer is Option A.

b Apply the operation from part (a) to the equation to ﬁnd the solution.

Create a strategy Divide both sides of the equation by 9.

Apply the idea Write the equation

Divide both sides by 9

Evaluate

Example 2 Solve the equation v − 4 = 44.

Create a strategy The reverse of subtracting 4 is adding 4. So we need to add 4 to both sides of the equation.

Apply the idea v − 4 = 44

Write the equation

v − 4 + 4 = 44 + 4 Add 4 to both sides v = 48

Evaluate

Reflect and check Substitute the value of v back into the equation to see if it is correct. v − 4 = 48 − 4 = 44

Example 3 Solve the equation 24 + p = 59.

Create a strategy The reverse of adding 24 is subtracting 24. So we need to subtract 24 from both sides of the equation.

10.03 One-step equations mathspace.co

401

Match the equation with the correct inverse operation. i

Add 8

Subtract 8

iii

8x = 72

x + 8 = 20

3a-c,g, 4ab, 5ab, 6ab, 7ab, 8a-i, 10a-h

Fluency Ex 1

t + 3 = −45

−4.5 + x = 6.8

4.1x = 94.3

x − 1.39 = 8.67

4n = 27.2

36.5 = 7.3n

Use inverse operations to solve: x − 15 = 35

y−7=2

b − 8 = 27

k − 14 = −7

b + 10 = 72

k + 8 = −10

5x = 45

9a = 63

Use inverse operations to solve: 14 + x = 64

12 + y = 21

Use inverse operations to solve: 3x = 90

2d = 24

Use inverse operations to solve: b

Solve: a

q + 8 = 10

s + 8 = 29

r + 22 = 69

u − 3 = −4

v − 4 = 44

u − 22 = 4

4k = 20

8k = 56

y + 5 = −6

24 + p = 59

−48 + x = 30

−27 + m = −32

−10.22 + x = −3.62

m 22 − x = 22

12n = 60

Solve: a

3, 4cd, 5cd, 6cd, 7cd, 8h-p, 9, 10el-n

s − 2 = 35

a 8

x−8=5

a Ex 4

7p = 63

a 6

Multiply by 8

a Ex 3

3a-h, 4bc, 5bc, 6bc, 7bc, 8c-i, 10i-k

Identify the inverse operation that should be applied to both sides in order to solve each equation:

i Ex 2

Divide by 8

x + 5.6 = 7.4

x + 16 = 16

For each statement: i

Write an equation, where p is the unknown number.

Solve the equation.

When 10 is added to a number, the result is 12.

When −9 is added to a number, the result is 21.

When 20 is subtracted from a number, the result is 5.

When 13 is subtracted from a number, the result is −15.

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403

The sum of an unknown number and 39 is 83.

75 is equal to the sum of an unknown number and 42.

17 less than an unknown number is 25.

The quotient of an unknown number and 8 is 12.

When a number is multiplied by 10, the result is 60.

When a number is divided by 8, the result is 9.

The product of 3 and an unknown number is 18.

When a number is multiplied by 12, the result is 84.

m When a number is divided by 11, the result is 11. n

7 multiplied by an unknown number is 84.

Reasoning

11, 12

12, 13

If x + 5 = 21, and you want to ﬁnd x, would you add or subtract 5 from both sides of the equation? Why?

Consider the equation x − 8 = −5. Is it possible to solve this equation by adding 8 to both sides? Why or why not?

Samantha is planning to buy a new laptop that costs $900. She has saved $600 so far and plans to earn the remaining amount by doing freelance work. a

Samantha charges $30 per hour for her freelance work. Write an algebraic equation to represent the number of hours (h) she needs to work to earn the remaining amount needed to buy the laptop (r).

How many hours does Samantha need to work to earn the remaining amount needed to buy the laptop?

Samantha’s friend, Lisa, also wants to buy the same laptop but has already saved $780. How many hours of freelance work does Lisa need to do to earn the remaining amount needed to buy the laptop? Assume Lisa gets paid at the same rate as Samantha.

What are some reasons why Samantha might charge more or less per hour for her freelance work compared to Lisa?

Problem-solving 14 14

14, 15

Write an equation, and then solve: a

A bag of lollies contains 80 lollies. The lollies are shared evenly between 10 children. How many lollies (n) will each child receive?

404

Justin divides a deck of cards into 7 piles. There are 12 cards in each pile. How many cards (c) are in the deck?

Tricia gives Tom 25 marbles from her collection. Tricia now has only 44 marbles in her collection. How many marbles (m) did Tricia have initially?

Lisa bought a hat and a dress for a total of $80. Find the cost of the hat (h) if the cost of the dress is $56.

Mathspace Victorian Curriculum – Year 7 mathspace.co

A company has a total of 500 employees, of which 60% are over 35 years old and the rest are 35 years and under. a

If the company plans to increase its under 35 workforce by 25%, how many new under 35 year old employees should they hire to achieve this goal?

What percentage of the total workforce will be 35 and under after the new hires?

10.04 Two-step equations After this lesson, you will be able to… • use flowcharts to analyse the structure of a two-step equation, identifying the operations used to construct the equation. • determine the inverse operations required to solve a two-step equation based on the flowchart analysis. • apply the inverse operations in the reverse order to both sides of the equation in order to solve for the variable. • solve a variety of two-step equations, demonstrating the ability to apply the learned techniques effectively and accurately.

Flow charts A ﬂow chart can help us understand how an equation has been built and what inverse operations, in what order are required to solve the equation. Consider the equation 6y + 9 = 33 Looking at the left-hand side, we can determine what the order of operations are, that is, what is happening to the y in order. y is ﬁrst multiplied by 6 and then we add 9: 6 y

9 6y 9

To undo these operations we must backtrack. We must apply the inverse operations in the reverse order to return to the variable y: 6 y

9 6y 9

6y 6

This tells us that to solve the equation we must ﬁrst subtract 9 from both sides and then divide both sides by 6.

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405

If we were to solve the equation using the inverse operations we just identified, it would look like this: Subtract 9 from both sides

Divide both sides by 6

Evaluate

Example 1 By following the operations given, complete the entries. 4

Create a strategy Multiply 8 by 4 for the ﬁrst step, then add 1 for the second.

Apply the idea 4 8

1 32

Example 2 Identify the inverse operations and then backtrack to complete the entries. 5

16 41

Create a strategy Identify the inverse operations for each step, and write them underneath the flow chart. After that, use the inverse operations to fill in the blanks.

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Apply the idea 5

41 5

Example 3 For 7x − 24 = 18: a Create a flow chart.

Create a strategy Create a flow chart of the left-hand side to identify the order of operations.

Apply the idea 7 x

24 7x

7x 24

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The inverse operations in reverse order to solve the equation will be: 1. First divide both sides by 7. 2. Then subtract 6 from both sides. Divide by 7 Subtract by 6

7 ( y + 6) = 49 y+6=7 y=1

Divide by 7 Subtract by 6

Example 4 Solve the equation

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression on the left-hand side was built by applying the operations ‘add 7’ and ‘divide by 2’ to the variable u. So the order of reverse operations will be: 1. Multiply by 2 2. Subtract 7 Write the equation

Multiply both sides by 2

Evaluate

Subtract 7 from both sides

Evaluate

Example 5 Solve the equation 3(s − 2) = 21.

Create a strategy Apply the inverse operations in the reverse order.

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Apply the idea We can see from the equation that the expression on the left-hand side was built by applying the operations ‘subtract 2’ and ‘multiply by 3’ to the variable s. So the order of reverse operations will be: 1. Divide by 3 2. Add 2 Write the equation

Divide both sides by 3

Evaluate

Add 2 on both sides of equation

Evaluate

Example 6 Solve the equation

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression on the right-hand side was built by applying the operations ‘divide by 5’ and ‘subtract 3’ to the variable t. The order of reverse operations will be: 1. Add 3 2. Multiply by 5 Write the equation

Add 3 to both sides

Evaluate

Multiply both sides by 5

Evaluate

Swap sides

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Ex 3

For each equation: i

Create a flow chart of the left side of the equation to identify the order of operations.

Solve by identifying the inverse operations.

5y + 10 = 20

e 7

Ex 6

8y + 9 = 1

The answer is 12. Write this as an equation.

Write the two inverse operations required to solve this equation in the correct order.

I have a number, k. I add 8 and then divide the result by 5. Write this as an expression.

The answer is 6. Write this as an equation.

Write the two inverse operations required to solve this equation in the correct order.

I have a number, x. I divide it by 2 and then add 5. Write this as an expression.

The answer is 19. Write this as an equation.

Write the two inverse operations required to solve this equation in the correct order.

Solve: a

2w + 9 = 37

4x + 9 = 25

7m + 18 = 39

6y + 4 = 34

13x − 2 = 24

5t − 15 = 15

29m − 8 = 21

2p − 17 = 3

Solve:

Solve: a

5(t + 9) = 60

2(p + 14) = 58

9(x + 2) = 27

6(x + 5) = 54

5(n − 15) = 35

3(s − 16) = 21

9(p − 3) = 0

4(s − 8) = 16

−13 = 7z − 48

48 = 6(p + 8)

−17 + 8s = 7

Solve:

3(x − 12) = 90

32 = 8(2 + s)

−15 + 5x = 10

Soledad has x marbles. After receiving 18 marbles from her brother Kiko, her classmate Rhian gave her additional marbles so that her number of marbles was doubled. Soledad now has 72 marbles. a

Complete the ﬂow chart to represent the situation. 18

b 412

6(s − 9) = 24

I have a number, n. I multiply it by 2 and then add 6. Write this as an expression.

a Ex 5

5(q + 3) = −10

i Ex 4

How many marbles did Soledad have initially?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Reasoning

16–18

Explain why subtracting 3 is the incorrect first step to solving 21 = 7(m + 3).

Write an equation involving p, division and subtraction, that has a solution of p = 8.

Consider this equation where A and B are positive integers, less than 12. Ax + B = 12 Write the reverse operations required to solve the equation for x.

Consider this equation where C, D, and E are positive integers. C (x − D) = E a

Write the reverse operations required to solve the equation for x.

Write the equation with x as the subject.

Problem-solving 20

19, 20, 24

19–26

A taxi charges a ﬂat rate of $10 plus $2 for each kilometre driven. If the total cost of the ride was $98, how many kilometres did the taxi travel?

A gardening service charges a base fee of $50 and an additional $15 for each hour of work. If the total cost for a gardening job was $230, how many hours did the gardener work on the job?

Jenna has a balance of $1500 in her account. Each month, she makes a deposit of $40. Additionally, she has made a one-time withdrawal of $20. How many months did it take for Jenna to accumulate her current balance, assuming she started with a balance of $0?

A social media account gains 400 followers every week but loses 1000 followers due to inactivity. If the account currently has 17 000 followers, how many months did it take to reach this number?

A charity organisation has a current fund of $1120. They receive monthly donations of $150 and spend $80 on one of their charitable activities. How many months did it take for the charity organisation to reach their current fund balance?

The area of a rectangle is given by the formula A = length × width. a

If the length of a rectangle is 9 m and its width is 3 m, ﬁnd its area.

The area of a particular rectangle is 45 m2. If its length is 9 m, determine the width of the rectangle.

Width

Length

10.04 Two-step equations mathspace.co

413

10 Chapter review 1

Determine whether these equations have x = 12 as a solution: a

25 − x = 13

17 + x = 39

Determine whether these equations have y = 3.5 as a solution: a

x − 24 = 12

6y = 21

A company charges $115 per hour for their services. If a customer uses their services for x hours and pays $1035, which equation would match this scenario? A

1035x = 115

x + 115 = 1035

115x = 1035

Billy is b years old. In 12 years time his age will be 39. Write an equation using b that could be used to solve for Billy’s current age.

For each table of values, write an equation for y in terms of x: a

x 1 3 5 7 y 5 11 17 23

x y

2 13

3 21

4 29

5 37

6 45

Identify the inverse operation that should be applied to both sides in order to solve each equation: a

3x = 342

x + 8.5 = 25.6

x − 5 = 29

x + 7 = 48

4x = 164

x + 14 = 31

8x = 112

Use inverse operations to solve: a

x + 5 = 35

11x = 176

x − 9 = 47

Solve: a

−15 + x = 28

−19 + x = −43

9x = 117

x + 13 = 31

−12.2 + x = −2.8

Solve: a

9 29

For each statement: i

Write an equation, where n is the unknown number.

Solve the equation.

15 multiplied by an unknown number is 225.

19 less than an unknown number is 76.

The quotient of an unknown number and 12 is 24. Chapter 10 review mathspace.co

415

Write an equation, and then solve: a

A box of chocolates contains 90 chocolates. The chocolates are distributed evenly amongst 6 friends. How many chocolates (c) will each friend receive?

Samantha splits a bundle of pencils into 5 equal groups. There are 7 pencils in each group. How many pencils (p) are in the bundle?

Alex gives Lisa 20 books from his library. Alex now has only 30 books in his library. How many books (b) did Alex have initially?

Solve: a

4x − 2 = 14

5m + 15 = 30

7n − 21 = 0

5 (x − 4) = 0

3 (y − 6) = 9

7 (z − 5) = 56

Consider the pattern. a

Describe how the pattern changes over each step.

Determine the number of circles in the next step of the pattern.

2c + 10 = 16

Solve: a

4b + 3 = 19

Solve: a

5a + 7 = 22

Step 1

Step 2

Step 3

Step 4

Write the algebraic rule for the number of circles, C, in terms of the step number, S, for this pattern.

James has y books. After receiving 12 books from his sister, his friend Nathan gave him additional books so that his number of books was tripled. James now has 54 books. How many books did James have initially?

A shuttle bus charges a flat rate of $15 plus $3 for each kilometre driven. If the total cost of the ride was $120, how many kilometres did the shuttle bus travel?

The area of a kite is given by the formula

, where p and q are the lengths of the

diagonals. A particular kite has a short diagonal length p = 7 m and area A = 42 m2. Find the value of q.

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Big ideas The coordinate plane provides a visual representation of the relationship between two quantities by plotting them as ordered pairs on a set of perpendicular axes.

Curriculum Outcomes VC2M7A05 generate tables of values from visually changing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane VC2M7A04 investigate, interpret and describe relationships between variables represented in graphs of functions developed from authentic data

11 Number plane

Chapter outline 11.01 The number plane 11.02 Directed numbers on the number plane 11.03 Graphs of linear relationships Chapter 11 review

420 429 438 448

Example 1 What are the coordinates of the points A, B and C?

4 A

3 2 1

−5 −4 −3 −2 −1 −1

x 1

2 3 4 5 C

−2 −3 B −4 −5

Create a strategy The coordinates are in the form (x, y), where x is the value on the x-axis (horizontal) and the y is the value on the y-axis (vertical).

Apply the idea For point A: • The x-coordinate is -5, and y-coordinate is 3. So the coordinates of point A is (-5, 3). For point B: • The x-coordinate is -2, and y-coordinate is -4. So the coordinates of point B is (-2, -4). For point C : • The x-coordinate is 3, and y-coordinate is -1. So the coordinates of point C is (3, -1).

Example 2 In which quadrant does the point (3, −2) lie?

Create a strategy

Apply the idea

Recall the characteristics of each quadrant:

Since the coordinates have positive x and negative y, the point (3, −2) lies in 4th quadrant.

• 1st quadrant: positive x and positive y. • 2nd quadrant: negative x and positive y. • 3rd quadrant: negative x and negative y. • 4th quadrant: positive x and negative y.

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Reﬂect and check We can plot the point to check which quadrant it is in. Quadrant 2

Quadrant 1

3 2 1 −4 −3 −2 −1

−1

x 1

4 (3, −2)

−2 −3 Quadrant 3

−4

Quadrant 4

This is Quadrant 4.

Example 3 What is the distance between A (6, 8) and B (−3, 8)?

Create a strategy Since the y-coordinates are the same, find the difference of the x-coordinates.

Apply the idea Distance = 6 - (-3)

Subtract -3 from 6

=6+3

Combine the adjacent signs

= 9 units

Evaluate

Reﬂect and check We can plot both points on the plane and then see if we can determine the distance. If the points are in a horizontal or vertical line the distance is the difference between the x or y-values. We can see that the points are in a horizontal line. The total distance is 3 units on the left of the y-axis and 6 units on the right of the y-axis, which is a total of 9 units.

y (−3, 8)

(6, 8)

8 6 4 2

−8 −6 −4 −2 −2

x 2

−4 −6 −8

11.02 Directed numbers on the number plane mathspace.co

431

Ex 1

For each equation: i

Complete the table of values provided.

Plot the points from the table on a number plane.

y = 4x x y

y=x−5 x y

−1

y = 2x + 7 x y

Consider the given table of values: a

Plot the points in the table on a number plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 4, find the corresponding value of y.

Consider the given table of values: a

Plot the points in the table on a number plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 8, find the corresponding value of y.

Consider the given table of values: a

Plot the points in the table on a number plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 3, find the corresponding value of y.

−2

x y

0 0

1 5

2 10

3 15

x y

−8 −2

−4 −1

0 0

4 1

x y

−1 −3

0 1

1 5

2 9

A farmer sells apples by weight. The farmer charges $5 per kilogram of apples. a

Fill in the table of values to show the total cost for different weights of apples. Weight (Kg) Total Cost ($)

y = x + 3 x y

−1

Plot the points from the table on a number plane.

Is the relationship linear or non-linear?

Let the height of a pot plant be y cm. As the pot plant grows, the height increases according to the equation y = 2t + 9, where t is the elapsed time in weeks. a

Complete the table of values: Time (t weeks) Height of pot plant (y cm)

Plot the points from the table on a number plane.

Is the relationship linear or non-linear?

11.03 Graphs of linear relationships mathspace.co

443

Ex 2

Diane is making a sequence of shapes out of matchsticks. Figure 1

Figure 2

Figure 4

Complete the table of values by indentifying the number of matchsticks for each figure. Figure number (x) Number of matchsticks ( y)

Figure 3

Plot the points in the table on a number plane.

Reasoning

12–13

12–14

12–15

What is the minimum number of points you need to be able to graph a linear relationship?

Consider the three points A (3, 3), B (5, 5), and C (7, 7). Without plotting the points, name three other points that lie on the same line formed by these points.

Consider the three points X (−3, −7), Y (−3, −9), and Z (1, −2). Without plotting the points, explain whether the points form a linear relationship.

Consider the given table of values: Does the ordered pair (−3, −5) satisfy the linear relationship between x and y?

Problem-solving 16, 18–19 16

−2 −2

−1 1

0 4

1 7

17–18, 20–23

After each week, the number of apples on an apple tree increases according to the diagram: a

Complete the table of values: Week No. of apples

444

16, 18–22

x y

Week 1

Week 2

Week 3

Week 4

Plot the number of apples against each week on a number plane.

Is the relationship between the number of weeks and the number of apples linear?

If the relationship between the number of weeks and the number of apples is linear, write the equation for the relationship where x is the number of weeks and y the number of apples.

Mathspace Victorian Curriculum – Year 7 mathspace.co

A staircase has 3 steps where each step has a height of 15 cm and a length of 10 cm. a

Complete the table of values: No. of steps Height of staircase (cm)

10 cm

15 cm

Plot the height of the staircase against the number of steps on a number plane.

Is the relationship between the height of the staircase and the number of steps linear?

If the relationship between the height of the staircase and the number of steps is linear, write the equation for the relationship, where x is the number of steps and y is the height of the staircase.

Is it possible to extend the staircase to exactly 105 cm high, with steps that have the same height and length as in the image?

Complete the table of values: 1 10

Plot the length of the staircase against the number of steps on a number plane.

Is the relationship between the length of the staircase and the number of steps in a staircase linear?

Is it possible to have a staircase 70 cm long, with steps that have the same height and length as in the image?

A bakery records the number of loaves sold each day over a week and the total revenue generated from the sales. For a sale of 2 loaves, the bakery earns $4. For a sale of 3 loaves, the bakery earns $6 and for 5 loaves, the bakery earns $10. a

Complete the table of values with the number of loaves sold and the total revenue for each day. Number of loaves, l Revenue, r

Which equation represents the relationship between the number of loaves and the revenue? A

446

No. of steps Length of staircase (cm)

1 15

l=2

l = 2r

l = 2r + 2

Plot the points from the table on a number plane.

Is the relationship linear?

Determine the bakery’s revenue if they sell 4 loaves of bread.

Mathspace Victorian Curriculum – Year 7 mathspace.co

l=r+2

Suppose a factory produces n widgets in h hours. The number of widgets produced is always equal to the square of the number of hours that have passed. a

Complete the table of values: Hours (h) Widgets produced (n)

Which of these equations represents the relationship between the number of hours passed, h, and the number of widgets produced, n? A

n=h

n = h2

n = 2h

n=h+2

Plot the points from the table on a number plane.

Is the relationship between hours passed and widgets produced linear or non-linear?

Olivia is 18 years old and Emily is 12 years old. Let O represent Olivia’s age and E represent Emily’s age. a

Write an equation for O in terms of E.

Identify whether these ordered pairs, in the form (E, O), satisfy the relationship between Olivia and Emily’s ages:

(12, 18)

(34, 28)

iii

(15, 21)

(22, 28)

(13, 19)

vii (35, 42)

(20, 20)

Plot the points from part (b) that satisfy the relationship between their ages on a number plane.

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Big ideas Geometry uses standard vocabulary and symbols to communicate facts and relationships about geometric figures.

Curriculum Outcomes VC2M7M04 identify corresponding, alternate and co-interior relationships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons VC2M7M05 demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles VC2M7SP02 classify triangles, quadrilaterals and other polygons according to their side and angle properties; identify and reason about relationships VC2M7SP04 design algorithms involving a sequence of steps and decisions that will sort and classify sets of shapes according to their attributes, and describe how the algorithms work

12 Angles, lines, and shapes Chapter outline 12.01 Measure angles 12.02 Parallel lines and transversals 12.03 Triangles 12.04 Quadrilaterals Investigation: Sort shapes with algorithms Chapter 12 review

454 463 475 485 494 501

12.01

Measure angles

After this lesson, you will be able to… • identify different types of angles. • understand the relationships between angles, including adjacent angles, vertically opposite angles, complementary angles, and supplementary angles.

Angle types An angle is formed between two lines, rays, or segments whenever they meet. These lines are called the arms of the angle. The point at the centre of the angle is called the vertex. An important angle in geometry is the angle 90°. We call this special angle a right angle and we draw it with a small square rather than a circular arc. A right angle represents a quarter of a turn around a circle. When lines form a right angle, we say they are perpendicular. A right angle is 90°. The symbol for a perpendicular line is ⊥.

We draw all other angles with a circular arc. An angle that is smaller than a right angle is called an acute angle. An acute angle is an angle that is between 0° and 90°.

An angle that is greater than a right angle but smaller than a straight angle is called an obtuse angle. An obtuse angle is an angle that is between 90° and 180°.

Two right angles together form a straight angle. This is the angle on a straight line. A straight angle is 180°.

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Mathspace Victorian Curriculum – Year 7 mathspace.co

Two obtuse angles that are adjacent form another obtuse angle.

If two angles form a full revolution, then one of them must be greater than or equal to 180°.

Complementary angles that are adjacent form a straight angle.

Fluency Ex 1

3–6

3–6, 8

Classify each as either a: • Reﬂex angle • Straight angle • Revolution

• Right angle • Acute angle • Obtuse angle

3, 4, 6–8

Classify each pair of angles using any of the terms in the list that apply: • Adjacent • Complementary • Supplementary • Vertically opposite angles a

Ex 2

Approximate these angles in degrees, and then label each one as acute, right, obtuse, straight, reflex or revolution. a

12.01 Measure angles mathspace.co

459

For each pair of angles, decide if they are: • Complementary angles • Supplementary angles • Neither a

160°

19°

25° 65°

140° 40°

34° 55°

Given that lines BE and CF are straight lines, use the diagram to identify: a

A pair of adjacent angles.

A pair of vertically opposite angles.

A pair of complementary angles.

D F

In the diagram, ﬁnd: a

An angle that is vertically opposite to ∠CXD.

An angle that is supplementary with ∠AXB.

A P

R C

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Reasoning 9

9, 10

9, 11

Decide whether each diagram shows the given angle pair. Explain your reasoning. a

Complementary angles

Vertically opposite angles 28° 28°

68° 12°

Supplementary angles

71° 109°

If ∠ABC and ∠DBF have the same measure, can we say that ∠ABC and ∠DBF are vertically opposite angles? Explain your answer.

Consider the question which was written poorly: Find the measure of ∠Q. S

37° 22° Q

Two students are debating on the answer to the question. Presley thinks ∠Q should be 59°. Qing thinks ∠Q should be 22°. Determine the issue with this question and explain why the two students chose different answers.

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Problem-solving 12a-e, 13ab Ex 3

12, 13a-d

12–13

Find the value of x in each diagram: a

x°

x° 40°

50°

x°

83°

130°

94°

f 150° x°

x°

h 10° x°

40°

x°

Find the value of the variable in each diagram: a

x° x°

x°

y°

y° y°

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d 50°

z° z° z° z°

z°

50° 50°

55°

x°

z° z°

x°

x° x°

x° 160°

x° x°

x°

12.02 Parallel lines and transversals After this lesson, you will be able to… • identify and explain the concept of parallel lines and transversals in geometry. • define and recognise co-interior, alternate, and corresponding angles formed by a transversal crossing parallel lines. • apply the knowledge of angle relationships to solve problems in geometry.

Parallel lines and transversals Parallel lines are like train tracks that never meet. They are straight lines that are always the same distance apart and go in the same direction.

12.02 Parallel lines and transversals mathspace.co

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The lines A and B are parallel and lines C and D are parallel. a Name the angle alternate to angle 1. b Name the angle corresponding to angle 4. c

If angle 2 is 56°, what is the size of angle 3?

If angle 4 is 113°, what is the size of angle 1?

2 1

3 4

The lines A, B and C are parallel. a Name all the angles that are equal in size to angle 1. b

Name all the angles that are equal in size to angle 4.

Name four pairs of vertically opposite angles.

If angle 4 is 112°, what is the size of angle 8?

If angle 5 is 105°, what is the size of angle 9?

Determine the size of angle 6.

1 5 6 9 13

∠NBP and ∠DCR

∠BCQ and ∠TDA

∠NBA and ∠LAD

∠MAL and ∠TDS

3 4 7

A 8

10 14

Which of these pairs of angles are equal? If they are, justify by naming the angle relationships. A

N M

B L

C D

T S

Which of these pairs of angles are equal? If they are, justify by naming the angle relationships. A

∠NBA and ∠QCB

∠BAD and ∠ADC

∠ADC and ∠DCR

∠ABC and ∠BAD

N M

B L

C D

T S

12.02 Parallel lines and transversals mathspace.co

471

For this diagram: a

Find the value of x.

Find the value of y.

Find the sum of the 3 interior angles of the triangle. 33°

y°

For this parallelogram: a

Find the value of x.

Find the value of y.

x°

(5x + 3)°

148°

(6y)°

Find the values of x, y and z.

x° 133° y°

53°

47°

z°

12.03

Triangles

After this lesson, you will be able to… • classify triangles based on their properties. • apply angle sum and exterior angle properties of triangles to solve problems.

Classifications of triangles A triangle is a 2D shape with three straight sides and three angles. Triangles can be classiﬁed according to their interior angles or sides. Classiﬁcation by interior angles: • If all the angles are acute, the triangle is an acute triangle. • If one of the angles is a right angle, the triangle is a right-angled triangle. • If one of the angles is obtuse, the triangle is an obtuse triangle.

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Find the values of x and y. a

x°

84° y°

65°

72°

44°

65° y° x°

56°

y° 77°

33°

120° x° y° 55°

While camping, Jena was told to have her supporting ropes at an angle of 25°. A picture of her tent is shown.

She knows the tent is on level ground so ∠BCA = 90° and ∠ABC = 65°. Determine whether Jena has pitched her tent correctly, so the supporting ropes are at an angle of 25°. A

For this diagram: a

Identify the relationship between ∠JQR and ∠JNP.

Identify the relationship between ∠MNK and ∠JNP.

Use your answers from parts (a) and (b) to ﬁnd the value of x.

Q J

K L 156°

x° N

115° R

For this diagram: a

Identify the relationship between ∠JHM and ∠KMH.

Use your answer from part (a) to ﬁnd the value of x.

J 68°

92° P x° K

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12.04 Quadrilaterals After this lesson, you will be able to… • classify quadrilaterals as convex or concave based on their internal angles. • identify special quadrilaterals. • describe the properties of different types of quadrilaterals.

Quadrilaterals A quadrilateral is a 2D shape that has four vertices (corners), four sides and four internal angles.

Quadrilaterals can always be split into 2 triangles by connecting a pair of opposite vertices. Since the angle sum of a triangle is 180°, the angle sum of a quadrilateral is the angle sum of two triangles, or 360°. Quadrilateral

Split into two triangles

Quadrilaterals can be classiﬁed as convex or concave: Convex

All angles less than 180°. All four vertices point outwards.

Convex

Concave

One internal angle greater than 180°. One vertex points inwards.

Concave

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A kite has two pairs of equal adjacent sides, and one pair of equal opposite angles. The properties are summarised in this diagram: Trapezium One pair of parallel sides

Parallelogram Two pairs of parallel sides

Rhombus All sides equal

Kite Two pairs of adjacent equal sides

(Opposite sides are equal) Rectangle All angles right

Square

12.04 Practice questions Understanding 1

Explorer 2–4

Adventurer 1, 3, 4

Trailblazer 1, 3, 4

Are these statements true or false? a

All quadrilaterals are parallelograms.

All rectangles are parallelograms.

All parallelograms are rhombuses.

A parallelogram with four equal sides is a square.

A rectangle with two equal connected sides is a square.

A rhombus is a parallelogram.

All quadrilaterals have an angle sum of 360°.

A quadrilateral can have exactly 3 right angles.

Can a quadrilateral have exactly 2 right angles? If so, name a quadrilateral that could have two right-angles.

In the quadrilateral ABCD, side AB has a length of 6 units. State the other side of ABCD which must have a length of 6 units.

D C

12.04 Quadrilaterals mathspace.co

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State whether each quadrilateral is convex or concave: a

5–8, 10, 11

5–11

Fluency Ex 1

5a-d, 6a-d, 7, 11a-c

Find the value of x. a

120°

60°

80°

60°

95° x°

155°

x° 35°

70°

x°

46°

123°

74°

112° 131°

63°

x°

68° 136°

35°

x°

108° 107°

Ex 2

57°

Classify each quadrilateral using all the terms that apply out of: rectangle, rhombus, kite, parallelogram, square, trapezium. a

490

83°

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 3

Decide whether each quadrilateral is a trapezium, parallelogram, or neither: a

59° 97° 73°

121°

d 66°

66°

114°

116°

114° 64°

66°

Sketch each parallelogram, labelling all sides and angles accordingly: a

A parallelogram that is neither a rectangle nor a rhombus.

A parallelogram with a pair of connected sides that are equal, and a pair of acute angles.

A parallelogram with a right angle.

Which two sides are parallel in this trapezium?

B 99°

81°

80°

100°

12.04 Quadrilaterals mathspace.co

491

Find the values of angles a, b and c. a

R Q c

a a

77°

111°

S J

c b M

B C a

b Y

60°

b a

D 105° X

c W

For each quadrilateral: a

Find the values of x and y. x°

Find the values of a, b and c. a°

y°

117°

73°

81°

c°

b°

Find the values of a and b. a°

Find the values of p and q. 6

b° p

101° q−3

Reasoning

13, 14, 16, 17

13–16

12–15

What is the greatest number of internal reﬂex angles that a quadrilateral can have? Explain your answer.

What is the greatest number of internal obtuse angles that a quadrilateral can have? Explain your answer.

What is the greatest number of internal right angles that a quadrilateral can have? Explain your answer.

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Sheldon draws a quadrilateral and covers it up. He tells Wanda that the quadrilateral has opposite sides that are equal in length. From this information, determine the most specific classification that Wanda can determine for the figure. Explain your answer.

Classify each quadrilateral. Explain your reasoning. a

91°

15 8

97°

83°

97°

8 15

Explain why some quadrilaterals may be classiﬁed as more than one type of quadrilateral.

Problem-solving 18bc 18

18a-c, 19

18–20

Find the value of x. a

(3x + 2)° 114°

b 55°

128°

x°

97°

64° x°

P 51° (28x − 11)° R

26°

∠CDE is 24°. Populate the rectangle ABCD with all the remaining angle values stating reasoning.

A E

Find the value of x if the perimeter of this parallelogram is 48 m. 8 8x

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d 45°

x°

83°

65° x° 45°

e 42°

x°

68°

Find the value of the missing angle marked with x. a

80°

60°

x° 155°

x° 45°

45°

30°

x°

86°

129° 64°

109° x°

Chapter 12 review mathspace.co

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Big ideas • The formula for the area of a triangle is derived from the area of a rectangle, then applied to find the area of composite shapes and surface area. • The circumference of a circle is the distance around its boundary or the perimeter of the circle.

Curriculum Outcomes VC2M7M01 establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem-solving VC2M7M03 describe the relationship between π and the circumference, radius and diameter of a circle

13 Length and area Chapter outline 13.01 Units of length 13.02 Perimeter 13.03 Perimeter of composite shapes 13.04 Area of rectangles and squares 13.05 Area of triangles 13.06 Area of parallelograms 13.07 Area of composite shapes Investigation: Explore circles 13.08 Circumference of a circle Chapter 13 review

508 514 524 530 539 546 552 562 565 573

13.01

Units of length

After this lesson, you will be able to… • recognise the common units of length, including millimetres, centimetres, metres, and kilometres. • convert lengths between units of length. • compare different lengths by converting them to the same units. • perform two-step conversions.

Conversion factors The most common units of length we will encounter are: • millimetres (mm): a grain of sand has a length of around 1 mm. • centimetres (cm): the length of an ant is around 1 cm. • metres (m): the length of a desk is around 1 to 2 m. • kilometres (km): the Sydney Harbour Bridge is just over 1 km long. To convert from one unit to another, we will make use of these relationships: 1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm The conversion factors are summarised in the conversion chart below: ÷ 10

÷ 100

× 10

÷ 1000

× 100

Example 1 Convert 6.52 centimetres to millimetres.

Create a strategy There are 10 mm in 1 cm so to convert from cm to mm use ×10.

Apply the idea 6.52 cm = 6.52 × 10 = 65.2 mm

508

Multiply 6.52 by 10 Evaluate

Mathspace Victorian Curriculum – Year 7 mathspace.co

× 1000

13.01 Practice questions Understanding 1

a e

The length of a bus = 12 ⬚

The width of a shoe box = 10 ⬚

⬚ m = ⬚ km

⬚ cm = ⬚ m

⬚ cm = ⬚ mm

⬚ mm = ⬚ m

When converting from a small unit of measurement (such as m) to a large unit of measurement (such as km), we ⬚ by the conversion factor.

Divide

When converting from a large unit of measurement (such as km) to a small unit of measurement (such as mm), we ⬚ by the conversion factor.

Complete:

d e f

To convert from m to cm, you multiply by ⬚.

To convert from cm to mm, you multiply by ⬚. To convert from km to m, you multiply by ⬚. To convert from mm to m, you divide by ⬚. To convert from cm to m, you divide by ⬚.

To convert from mm to cm, you divide by ⬚.

Fluency

5a-f, 7–9, 11, 14

5, 6a-f, 10, 13–14

5g-l, 6g-l, 10c, 12

Convert: a c e g i k

The height of a chair = 1.2 ⬚

Ex 3, 4

The distance between two cities = 13 ⬚

The length of an ant = 3.7 ⬚

Multiply

Ex 1

The thickness of a book = 2.5 ⬚

Fill in the correct answer to make these statements true:

Trailblazer 3–4

Fill in the boxes to show the relationships between these pairs of units. a

Adventurer 1, 3–4

Identify the appropriate unit of measurement to use for each measurement: c

Explorer 1–4

3 cm to millimetres 6 m to centimetres 6 km to metres 88.2 mm to centimetres 14.9 cm to metres 4.6 cm to millimetres

b d f h j l

60 mm to centimetres 800 cm to metres 9000 m to kilometres 7.33 m to centimetres 12 226 m to kilometres 5.69 km to metres

b d f h j l

3484 mm to metres 866 800 cm to kilometres 1 528 400 mm to kilometres 150 mm to metres 45 300 cm to kilometres 320 000 mm to kilometres

Convert: a c e g i k

24 m to millimetres 6 km to centimetres 9 km to millimetres 9.03 m to millimetres 7.43 km to centimetres 6.71 km to millimetres

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The Qinghai-Tibet railway in China includes the Tanggula railway station. At 5068 metres high, it is the world’s highest railway station. How many kilometres high is the station?

A city is 125 metres above sea level. Shifts in the Earth’s plates make it useful to know the height of the city above sea level in centimetres. Find the height of the city in centimetres.

Ex 2

Purnjay is 176 cm tall. Nadia is 1.51 m tall. Find the shorter person by converting both lengths to the same unit of measurement.

Find the taller person in these pairs: a

Haonan is 165 cm tall and Tina is 1.53 m tall.

Hermione is 1.23 m tall and Sandy is 161 cm tall.

Joanne is 1270 mm tall and Ellie is 1.41 m tall.

A coffee table is 1.17 m long. A whiteboard is 147 cm long. Find the longer item by converting both measurements to the same unit.

Plant 1 grows 3.6 cm in a week. Plant 2 grows 29 mm in the same week. Which plant grew more during the week?

The table shows the different skyscrapers around the world and their heights: Name One World Trade Center Shanghai Tower Petronas Twin Towers Zifeng Tower Burj Khalifa

Height (metres) 541.3

Height (kilometres) 0.632

451.9 0.45 828

Complete the conversions for the given skyscrapers in the table.

Which skyscraper is the tallest?

A desk is 1.39 m long. A rug is 120 cm long. a

Determine whether the desk or the rug is longer.

Calculate the difference between the lengths in centimetres.

Reasoning

15–16

15–17

When you are building a house, all house plan dimensions and building materials are measured in mm. Give a reason why this is the case.

Australia oﬃcially changed all of its units of measurement to the Decimal system in 1974. Prior to this, the Imperial system was used, which included conversions such as 1 foot = 12 inches and 1 mile = 5280 feet. Why do you think the Australian Government made the change to the Decimal system?

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hy can’t metric units such as km be used as a reliable measure of distance in space W exploration and astronomy?

What unit of measurement is used to measure the distance between galaxies, and how is it calculated?

Problem-solving 20–22, 26

18, 23–25, 27, 29

19, 23–25, 27–30

Arrange these in descending order of length: 5.23 m, 499 cm, 495 000 mm.

Arrange these in ascending order of length: 0.19 km, 19 150 cm, 192 m.

A rectangular table is measured to be 586 centimetres long and 3.64 metres wide. Calculate the difference between the length and the width in metres.

A car is 4 m long. A truck is 934 cm long. How many metres longer is the truck than the car?

A sail boat is 6 m long. A yacht is 956 cm long. How many centimetres longer is the yacht than the sail boat?

A rectangular table is measured to be 5.23 metres long and 499 centimetres wide. Calculate the difference in metres between the length and the width of the table.

If a ﬂy is 1.16 cm long and a bee is 19 mm long, how much longer is the bee than the ﬂy in centimetres?

If a mosquito is 1.23 cm long and a wasp is 13 mm long, how much longer is the wasp than the mosquito in millimetres?

Michael wants to lay a row of square pavers along the edge of his 3 m garden bed. Find the side length of each paver in centimetres.

Paver Paver Paver

Garden

Paver Paver

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13.02 Practice questions Understanding 1

Trailblazer 2bd

Find the perimeter of these rectangles, where each square is 1 unit. a

Complete the steps to ﬁnd the perimeter of this triangle: P = 55 + ⬚ + ⬚ = ⬚ cm

518

Adventurer 1, 2bc, 3

In these diagrams, each grid square represents 1 square unit. Find the perimeter of each shape: a

Explorer 1, 2ac, 3

Mathspace Victorian Curriculum – Year 7 mathspace.co

55 cm 48 cm 73 cm

Fluency

Ex 1

4–5, 6a-c, 7, 8ab, 9, 10, 11ab, 14ab, 15, 16a

4ace, 5, 6cd, 7c-e, 8–10, 11bc, 12, 13, 14bc, 15b, 16ab

4bde, 5b, 6bd, 7c-e, 8bc, 10, 11c, 12, 14cd, 16c

Find the perimeter of each rectangle: a

21 m 6m

21 m

3 cm

10 cm

31 mm

6 mm

Find the perimeter of a rectangle with these dimensions. Draw a diagram if needed: a

31 m

13 m

3 cm

Length: 22 m, Width: 8 m

Length: 2 m, Width: 13 m

Find the perimeter of each square: a

7 cm

18 m

7 cm

5 cm

4 cm

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Find the perimeter of each triangle: a

9 mm

15 cm 8 cm

9 mm

16 cm

14 cm

21 cm

15 cm

9 mm

10 mm

10 cm

8 cm

Find the perimeter of each triangle. Draw a diagram if needed: a

A scalene triangle with side lengths 13 mm, 17 mm and 18 mm.

An equilateral triangle with a side length of 4 m.

An isosceles triangle where the 2 equal side lengths are 13 mm each, and the third side measures 4 mm.

Ex 2

A square has a perimeter of 12 mm. Find the side length of the square.

Ex 3

Find the perimeter of this shape:

8 mm

Find the perimeter of these polygons: a

A regular quadrilateral where the length of each side is 7 cm.

A regular pentagon where the length of each side is 6 mm.

A regular octagon where the length of each side is 6 m.

Given the perimeter, P, of these rectangles, ﬁnd the value of the variable: a

b P = 24 m

c P = 20 cm 8 cm

n P = 66 cm

12 cm 520

Mathspace Victorian Curriculum – Year 7 mathspace.co

A rectangle has a perimeter of 26 cm. If it has a length of 9 cm, ﬁnd its width.

Given the perimeter, P, of these triangles, ﬁnd the value of the variable: a

P = 61 cm

P = 29 m 10 m

21 cm

17 cm

d 5 cm P = 16 cm

P = 35 cm 7 cm

Ex 4

Find the side length of these shapes, where P is the perimeter: a

P = 32 cm

P = 20 m

Find the side length of these polygons: a

A regular hexagon with a perimeter of 66 cm.

An equilateral triangle with a perimeter of 21 m.

A regular decagon with a perimeter of 150 mm.

Reasoning

17, 20–22

17–18, 20, 22

18–19, 22–23

Consider a rectangle with a length = 8 cm and a width = 5 cm. Use two different methods to ﬁnd the perimeter of the rectangle.

The perimeter of a square is 24 cm. Will the perimeter increase by 4 times if the length of each side is increased by 4 cm? Explain your reasoning.

The perimeter of a square is 40 cm. Will the perimeter increase by 4 times if the length of each side is increased by 4 times? Explain your reasoning.

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If each side of a rectangle is doubled in length, will this mean the perimeter is doubled? Support this with an example.

A square has a side length of a cm. Write an algebraic expression for the perimeter of the square.

A regular hexagon has a side length of p mm. Write an algebraic expression for the perimeter of the hexagon.

A rectangle has a length of 3x mm and a width of 2x mm. Write an algebraic expression for the perimeter of the rectangle.

Problem-solving 24, 26, 28, 29, 32

25–27, 29, 30, 33, 35

29–34, 36

Find the other side length of a triangle that has a perimeter of 18 m, and two side lengths that both measure to be 5 m.

Find the side length of one of the equal sides of an isosceles triangle that has a perimeter of 22 cm, and a third side length that measures to be 4 cm.

A rectangular race track has a length of 25 m and a width of 10 m. What distance is covered in one lap of the track?

A 44 cm long piece of wire is bent into the shape of a square. What is the side length of the square?

Hebah is tiling her bathroom with equilateral triangle tiles. She knows the perimeter of each tile, but she wants to ﬁnd the length of each side. If each tile has a perimeter of 36 cm, how long is each side?

A square has the same perimeter as an equilateral triangle. If the triangle has sides of length 8 mm, ﬁnd the side length of the square.

The length of a rectangle is three times its width. Find the dimensions of the rectangle if the total perimeter is 40 m.

Nadia wants to build a thin wire frame for a photo that is 13 cm long and 6 cm wide. How much wire will she need to go around the entire photo?

A company is building a rectangular soccer ﬁeld with a length of 100 m and a width of 60 m. If the cost of fencing per metre is $15, how much will it cost to fence the ﬁeld?

A construction company wants to build a square parking lot with a side length of 120 m. If the cost of fencing per metre is $21, what is the cost of fencing the parking lot?

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A present is contained in a cube shaped box. Ribbon is wrapped around the present as shown in the ﬁrst ﬁgure: a

If the side length of the box is 13 cm, ﬁnd the shortest length of the ribbon needed to neatly go around the box without overlap.

The bow requires 17 cm of ribbon. How much ribbon is needed altogether to wrap the box in this way?

Find the total length of ribbon needed if we wrap the present with two lengths of ribbons, as shown in the second ﬁgure. Assume a single bow is tied.

In a small town, the local park has a rectangular playground area. The length of the playground is represented by the algebraic expression y + 3, while the width is represented by the expression 2x.

y+3

To install a new fence around the playground, the park’s management needs to calculate the perimeter of the area.

Find an algebraic expression for the perimeter of the playground.

Simplify the expression.

A landscape architect is designing a garden that consists of a rectangular ﬂower bed and an adjoining triangular lawn area. The length of the rectangular ﬂower bed is represented by the algebraic expression 4y − 1 3x − 2, and the width is represented by the expression 4y − 1. The triangular lawn has two equal sides with lengths represented by the expression 3y − 2 each.

3x − 2 3y − 2

3y − 2 3x − 2

The landscape architect wants to install a border around the entire garden. a

Find an algebraic expression for the perimeter of the combined garden area.

Simplify the expression.

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13.03 Perimeter of composite shapes After this lesson, you will be able to… • understand the concept of perimeter and how it applies to composite shapes. • determine missing side lengths of composite shapes. • find the perimeter of composite shapes.

Perimeter of composite shapes A composite shape is a figure formed by combining two or more basic shapes, such as rectangles, triangles, circles, etc. To ﬁnd the perimeter of a composite shape, we add the sides around the outside of the shape. Sometimes we need to determine missing side lengths before ﬁnding the perimeter.

Example 1 Find the perimeter of this shape.

12 cm

16 cm

4 cm 8 cm

8 cm

Create a strategy

Apply the idea

Add the lengths of all the given sides.

Perimeter = 8 + 8 + 16 + 12 + 4 = 48 cm

Add the sides Evaluate

Example 2 7m

Consider the figure: 5m

y 2m

17 m

a Find the length x.

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Create a strategy The side length x will be equal to the difference between the other two vertical side lengths. Use the fact that the vertical side on the left has the same length as the two vertical sides on the right added together.

7m x

y 2m

17 m

Apply the idea 5=x+2 5−2=x+2−2 x=3m

Write the relationship in an equation Subtract 2 from both sides Evaluate

b Find the length y.

Create a strategy The side length y will be equal to the difference between the other two horizontal side lengths. Use the fact that the horizontal side on the bottom has the same length as the two horizontal sides on the top added together.

7m 5m

y 2m

17 m

Apply the idea 17 = y + 7 Write the relationship in an equation 17 − 7 = y + 7 − 7 y = 10 m

Subtract 7 from both sides Evaluate

c Calculate the perimeter of the ﬁgure.

Create a strategy The perimeter of a ﬁgure is the sum of all of its sides. Use the results found in parts (a) to (b).

Apply the idea Perimeter = 7 + x + y + 2 + 17 + 5

Add the sides

= 7 + 3 + 10 + 2 + 17 + 5

Substitute x and y

= 44 m

Evaluate

Reflect and check The perimeter can also be worked out by 2 × 17 cm (for the width) +2 × 5 cm (for the height). Can you explain why?

13.03 Perimeter of composite shapes mathspace.co

525

Fluency Ex 1

6–9, 10a, 11ace

6, 8, 10, 11acd

8, 10–11

Find the perimeter of each composite shape: a

9 cm

24 m

8 cm

18 m

18 m 5 cm

8 cm 12 cm

16 cm

6 cm 30 m

4 cm 20 cm

30 m

10 cm

Calculate the perimeter of this ﬁgure:

3 mm x 7 mm 14 mm

Ex 2

Calculate the perimeter of these ﬁgures: a

18 mm 4 mm

13 m

12 mm

4m 16 m

3 mm

y 3 mm

Calculate the perimeter of this ﬁgure:

16 m x

3m y

12 m 14 m

z 3m

Find the perimeter of each figure: a

6 cm

17 cm

12 mm 2 mm

4 mm

3 cm 4 cm

6 cm

5 mm

3 mm

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Find the perimeter of each composite shape: a

2 cm

13 m 4m

8 cm 14 cm 12 m 5 cm 5m

17 m

13 m

6 cm

11 m 9m

35 m

7 cm

10 cm 5 cm 7 cm

14 cm

8 cm

1 cm

6 cm

Reasoning 12

12–13

12–14

Calculate the perimeter of the compound shape by adding the side lengths together.

Find 2 × (17 + 13)

Are your answers to the ﬁrst two parts the same? Explain why or why not.

12–14 9 cm 5 cm 13 cm

17 cm

Use two different methods to calculate the perimeter of this figure:

9m 5m 13 m 528

Mathspace Victorian Curriculum – Year 7 mathspace.co

Show that the perimeter of the compound shape is the same as the perimeter of a rectangle with sides 6 mm and 5 mm.

2 mm 1 mm 6 mm

Problem-solving 15–17 15

16–19

2 mm

16–17, 19–20

alculate the outside perimeter (shown in blue) of the plot of land on this site plan. C All measurements are given in metres. eet

Cedar Str 127 11.5

Main Stree

Line of wall

Building

17.5 17.5

Property Line 127

b 16

If fencing costs $35.50 per metre, calculate the cost of fencing the plot of land.

A swimming pool has a 2 m wide path around its edge. The outer width of the path is 20 m and the length is 30 m. a

Find the dimensions of the pool.

Calculate the perimeter of the pool.

20 m

30 m

Alexander draws the rectangular plot of land which contains his house and garden. a

Find the perimeter of the plot of land.

Find the perimeter of the house.

Find the perimeter of the garden.

18 m 11 m

Garden 23 m House

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A rectangular swimming pool measures 10 m by 5 m. If a 3 m wide walkway is constructed around the outside of the entire pool, calculate the perimeter of the pool and walkway combined.

A farmer wants to fence a rectangular ﬁeld that measures 50 m by 30 m. The fence is to be placed 2 metres inside from the edge of the ﬁeld.

Calculate the total length of the fence required.

If fencing costs $18.75 per metre, calculate the cost of fencing the ﬁeld.

A square tablecloth measures 2 m by 2 m. A decorative trim of width 10 cm is added around the edges of the tablecloth. a

Calculate the perimeter of the tablecloth and trim combined.

If trim costs $5.40 per metre, calculate the cost of trimming the tablecloth.

13.04 Area of rectangles and squares After this lesson, you will be able to… • understand the concept of area and square units. • find the area of rectangles and squares using the formula A = l × w or A = s2. • use the formula to find a missing length or width given the area of a rectangle or square. • solve problems involving area of rectangles and squares.

Area of rectangles and squares The area of a rectangle is the amount of space inside the rectangle. We measure area in square units: 1 square millimetre = 1 mm2 1 square centimetre = 1 cm2 1 square metre = 1 m2 1 square kilometre = 1 km2

2 cm 4 cm

530

This rectangle contains 8 squares, each with a side length of 1 cm. So this rectangle has an area of 8 cm2. Another way to ﬁnd the area is to multiply the length and width, 4 × 2 = 8 cm2.

Mathspace Victorian Curriculum – Year 7 mathspace.co

The area of a rectangle is given by: Area = length × width A=l×w

The area of a square is given by: Area = side × side A = s × s = s2

Interactive exploration

Discover this concept in action online

mathspace.co

Example 1 Find the area of this rectangle: 9 cm 12 cm

Create a strategy Use the formula for the area of a rectangle.

Apply the idea A=l×w

Write the formula

= 12 × 9 = 108 cm

Substitute the values 2

Evaluate

Example 2 Find the area of this square: 8 cm

Create a strategy Use the formula for the area of a square.

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531

For each pair of shapes: i

Find the area of Shape A.

iii

Which shape is smaller?

Shape A

Find the area of Shape B.

Shape A

Shape B

For each pair of shapes: i

Find the area of Shape A.

iii

Which shape is larger?

Shape A

Shape B

Mathspace Victorian Curriculum – Year 7 mathspace.co

Find the area of Shape B.

Shape A

Shape B

Shape A

Shape B

Shape A

Shape B

Shape A

Shape B

Shape A

534

e 23 mm 6 mm

10 m

9 mm

10 m

5m Ex 2

3 mm

Find the area of each square: a

9 mm

18 m

9 mm

4 cm

9 mm

6 mm 23 mm

For each rectangle: i

Find the perimeter.

3 cm

Find the area.

4 cm 8 cm

2 cm

6 cm

d 3 cm

5 cm

7 cm

Find the area. Draw a diagram if needed. a

A square whose side is 7 cm.

A rectangle whose length is 10 cm and width is 3 cm.

A rectangle whose length is 12 cm and width is 5 cm.

Find the side length of this square:

A = 25 mm²

A square has a perimeter of 32 cm. a

536

Find the side length of the square.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Find the area of the square.

Ex 3

Find the length of the missing side of this rectangle:

Area = 30 mm²

5 mm

Find the length of the missing side in these rectangles: a

A rectangle that has an area of 36 m2 and a width of 4 m.

A rectangle that has an area of 18 m2 and a length of 9 m.

Reasoning 16

16–17, 19

16–19

16–21

Find the area of the square.

Double the side lengths of the square and ﬁnd the area.

When we double the side lengths of a square, do we also double the area of the square?

7 cm

Investigate whether doubling each side of the rectangle also doubles the area of the rectangle.

From the results of the last two questions: a

When the sides of the square or rectangle were doubled, by what factor was the area increased?

Explain why the area is increased by this number.

If the sides of a square or rectangle were multiplied by 5, by what factor would the area increase?

If the sides of a square or rectangle were multiplied by x, by what factor would the area increase?

Determine a quadrilateral that has its area equal to its perimeter.

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Explain how the area of a rectangle could be used to ﬁnd the area of this triangle. 55 cm 48 cm 73 cm

For a rectangle with an area of A = 50 m2: a

Complete the table, showing the possible dimensions of rectangles with an area of A = 50 m2. Length ⬚ ⬚

Width ⬚ ⬚

Area 50 m2 50 m2

Length ⬚ ⬚

Width ⬚ ⬚

Area 50 m2 50 m2

Complete the table, showing the perimeter of each rectangle. Perimeter ⬚ ⬚

Using the table from part (b), explain why some rectangles with the same area cannot have the same perimeter.

Problem-solving 22, 24–25

23–24, 26

23, 25–28

A rectangular swimming pool has a length of 6 metres and a width of 4 metres. What is the area of the pool in square metres?

A square piece of cardboard is cut into 4 smaller squares, each with an area of 4 square centimetres. What is the area of the original square cardboard in square centimetres?

A company has a rectangular warehouse that measures 50 metres by 30 metres. They want to install a new ﬂoor in the warehouse that costs $15 per square metre. What will be the total cost of the new ﬂoor?

A wall in a room has a rectangular shape with a length of 3 m and a height of 2.5 m. If each square metre of wall needs one litre of paint and the paint costs $20 per litre, what will be the cost of painting the wall?

A bathroom tile has a square shape with an area of 0.16 square metres. If a bathroom ﬂoor measures 6 metres by 4 metres, how many tiles are needed to cover the entire ﬂoor?

A square painting is mounted in a frame that is 3 centimetres wide on all sides. If the painting has an area of 36 square centimetres, what is the total area of the painting and frame?

Emma wants to lay new tiles in her kitchen which measures 4.5 m by 3.8 m. The tiles she plans to use come in boxes of 20 tiles, each measuring 30 cm by 30 cm. How many boxes of tiles will Emma need to cover her kitchen ﬂoor?

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Example 1 Find the area of this triangle:

7 cm

10 cm

Create a strategy Use the formula for the area of a triangle.

Apply the idea Write the formula

Substitute b = 10 and h = 7

Evaluate

Example 2 Find the area of the given triangle with a base length of 10 metres and a perpendicular height of 8 metres.

10 m

Create a strategy Use the formula for the area of a triangle.

Apply the idea For this triangle, the base is b = 10 m and the height is h = 8 m. Write the formula

Substitute b and h

Evaluate

540

Mathspace Victorian Curriculum – Year 7 mathspace.co

3a-c, 4, 5a-c, 6ac, 7, 8ab, 9a, 10

Fluency 3

3b-d, 4, 5a-d, 6ac, 7, 8ac, 9ac, 10

3d-f, 4, 5c-f, 6bd, 7, 8cd, 9bd, 10

For each figure: i

Find the area of the entire rectangle.

Find the area of the triangle.

5 cm

3 cm

10 cm

d 8m

2 cm 11 cm

10 m

8 cm

16 m

3 cm 4m Ex 1, 2

Find the area of each triangle: a

5m 6 cm 12 m 7 cm

55 cm 48 cm

8 cm

73 cm

6 cm

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Find the area of a triangle which has these dimensions: a

Ex 3

Height = 8 cm and base = 9 cm

Height = 4 m and base = 6 m

Height = 7 mm and base = 8 mm

Height = 3 cm and base = 9 cm

Base = 12 cm and height = 100 mm

Base = 5.2 m and height = 200 cm

Determine whether these dimensions could form a triangle with an area of 20 mm2: a

Base = 10 mm, height = 4 mm

Base = 1 mm, height = 20 mm

Base = 5 mm, height = 8 mm

Base = 2 mm, height = 2 cm

For this triangle with an area of 20 mm2, find the value of b.

5 mm

b mm

Find the base measurement for these triangles given the area and height: a

Area = 18 cm2 and height = 9 cm

Area = 40 cm2 and height = 10 cm

Area = 400 mm2 and height = 4 cm

Area = 12.5 mm and height = 10 mm

Find the height measurement for these triangles given the area and base: a

Area = 90 cm2 and base = 20 cm

Area = 150 cm2 and base = 50 cm

Area = 2500 mm2 and base = 50 cm

Area = 33.75 mm and base = 4.5 mm

Complete this table of base and height measurements for four different triangles, which all have an area of 30 m2:

Reasoning 11

11, 14

Base (m) Height (m) Area (m2)

10 12 30

11, 12, 14, 16

Explain how the area of a triangle is connected to the area of a rectangle using this diagram:

2 30

12, 13, 15, 16

A right-angled triangle has a base of 5 cm and a height of 10 cm. a

If the height is halved, what effect does it have on the area of the triangle?

If the base is doubled, what effect does it have on the area of the triangle?

Two right triangles have the same area. If one triangle has a base of 8 cm and a height of 6 cm, what are the possible dimensions of the other triangle, given the base and height are whole numbers?

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If two right-angled triangles have the same area, does this mean they have exactly the same shape? Explain your reasoning.

A triangle has base k units and perpendicular height p units.

Write an algebraic rule to ﬁnd A, the area of the triangle.

If p = 5 mm and k = 20 mm, calculate the area of the triangle.

The formula to ﬁnd the area of a triangle a

Rearrange the formula to make b the subject (make it b = ).

Rearrange the formula to make h the subject.

Problem-solving 17, 18, 20, 21

18, 19, 21

19, 21–24

A farmer wants to install a triangular shade sail in her paddock to protect her sheep from the sun. The base of the shade sail is 12 m long, and the height of the triangle is 8 m. What is the area of the triangular shade sail in square metres?

A civil engineer needs to design a triangular traﬃc island for a new road intersection. The base of the triangle is 20 m long, and the height of the triangle is 15 m. Calculate the area of the triangular traﬃc island in square metres.

An artist is painting a triangular canvas with a base of 80 cm and a height of 50 cm. Calculate the area of the canvas that needs to be painted: a

in square centimetres

in square metres

An architect is designing a triangular courtyard for a new apartment building. The base of the triangle is 30 m long, and the height of the triangle is 20 m. a

Calculate the area of the courtyard.

If paving tiles cost $25 per square metre, calculate the cost of paving the courtyard.

A painter charges $30 per square metre to paint walls. If a room has a triangular section with a base of 8 m and a height of 5 m, how much will it cost to paint that section of the wall?

Joy wants to paint her living room walls. One of the walls measures 3 m high and 5 m wide, while the other wall measures 3 m high and 4 m wide. There is a triangular window on one of the walls that has base of 2 m and height of 1 m. If each litre of paint can cover an area of 6 m2, how many exact litres of paint will Joy need to buy?

Cody is re-tiling his rectangular bathroom ﬂoor. The bathroom measures 2 m by 3 m, but there is a shower cubicle in the corner in the shape of a right triangle with a base of 1 m and a height of 1.5 m. Each tile measures 20 cm by 20 cm. a

Calculate the number of tiles that Cody needs to buy.

If Cody needs to account for 10% wastage, how many tiles does he need to buy?

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A gutter running along the roof of a house has a cross-section in the shape of a triangle: If the area of the cross-section is 40 cm2, and the length of the base of the gutter is 10 cm, ﬁnd the perpendicular height, h, of the gutter.

h cm

10 cm

13.06 Area of parallelograms After this lesson, you will be able to… • define a parallelogram and its relationship with a rectangle. • apply the formula for finding the area of a parallelogram to solve problems.

Parallelograms and rectangles A parallelogram is a quadrilateral with two pairs of opposite sides parallel. It looks like a rectangle that has been pushed over.

We will make use of the base and perpendicular height of the parallelogram to ﬁnd its area.

height

base

Notice that a rectangle is a special type of parallelogram, with all internal angles measuring 90°.

Interactive exploration

Discover this concept in action online

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Mathspace Victorian Curriculum – Year 7 mathspace.co

mathspace.co

Area of a parallelogram The area of a parallelogram is given by Area = base × perpendicular height A=b×h Perpendicular height means that the height and base are at right angles to each other.

Example 1 Find the area of this parallelogram:

13 m

Create a strategy

Apply the idea

Multiply the base by the perpendicular height.

We are given b = 13 and h = 8. A=b×h

Write the formula

= 13 × 8 = 104 m

Substitute b = 13 and h = 8 2

Evaluate

Example 2 Find the area of this parallelogram:

11 mm

3 mm

Create a strategy

Apply the idea

Multiply the base by the perpendicular height.

We are given b = 3 and h = 11. A=b×h

Write the formula

= 3 × 11 = 33 mm

Substitute b = 3 and h = 11 2

Evaluate

13.06 Area of parallelograms mathspace.co

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Identify the base and the perpendicular height needed to ﬁnd the area of these parallelograms: a

4.2 cm

10 cm 3 cm

3.6 cm

4 cm 2 cm

2.6 cm

3.6 cm 10 mm

8 mm 1.8 cm

Fluency 3

5.5 mm

3–5, 6a-d, 7, 8, 9a, 11

3ac, 5, 6c-f, 7a, 8ac, 9a, 10–12

Redraw this parallelogram as a rectangle.

Find the area of the rectangle.

Find the area of the parallelogram.

3b, 5, 6e-h, 8bd, 9–12

6 cm

12 cm

If this parallelogram is formed into a rectangle, determine: i

the length of the rectangle

the width of the rectangle

iii

the area of the rectangle

6 cm

Find the area of the parallelogram. 10 cm

Determine the length of the base.

Determine the length of the perpendicular height.

What is the size of the angle formed by the base and the perpendicular height?

Find the area of the parallelogram.

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Ex 1, 2

Find the area of each parallelogram: a

11 mm

5 mm

13 cm

8 mm

10 mm

9 cm

f 15 cm

18 mm

50 mm 24 cm 15 mm

8 cm

30 mm

19 cm 14 cm

12 cm

Ex 3

Find the area of these parallelograms with these dimensions: a

The base is 5 mm and the perpendicular height is 2 mm.

The base is 15 cm and the perpendicular height is 7 cm.

Determine whether these could be the dimensions of a parallelogram with an area of 28 mm2: a

Base = 1 mm, perpendicular height = 28 mm

Base = 7 mm, perpendicular height = 4 mm

Base = 4 mm, perpendicular height = 7 mm

Base = 2 mm, perpendicular height = 28 mm

Find the value of the base or perpendicular height in each parallelogram: a

Area = 36 cm2

Area = 35 m2

b cm 5m 3 cm

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Mathspace Victorian Curriculum – Year 7 mathspace.co

Find the base length of a parallelogram with an area of 108 cm2 and a perpendicular height of 9 cm.

Find the perpendicular height of a parallelogram with an area of 99 m2 and a base length of 9 m.

Complete this table of base and height measurements for three different parallelograms with an area of 60 cm2:

Reasoning

13, 15, 18

Base (cm) Perpendicular height (cm) Area (cm2)

13–15, 17

12 60

14, 16, 17

If you are given two parallelograms with the same base but different perpendicular heights, explain how you would determine which parallelogram has the greater area.

A parallelogram has a base of 10 cm and a height of 20 cm. a

If 2 cm is added to the base and the perpendicular height, is the area increased by 2 cm2 also?

If the base and perpendicular height are doubled, is the area also doubled?

If the base and perpendicular height are doubled, by what factor has the area increased?

Explain why this number is the increase factor.

Two parallelograms have the same area. If one parallelogram has a base of 8 cm and a perpendicular height of 4 cm, what are the possible dimensions of the other parallelogram, given the base and perpendicular height are whole numbers?

A parallelogram has a base of 2k units and a perpendicular height of 3p units.

Write the area of the parallelogram as an algebraic expression.

If p = 2 and k = 14, ﬁnd the area of the parallelogram.

The formula to ﬁnd the area of a parallelogram is A = b × h. a

Rearrange the formula to make b the subject (make it b = ).

Rearrange the formula to make h the subject.

A parallelogram has parallel sides of length 7 cm and 5 cm. A rectangle also has parallel sides of length 7 cm and 5 cm. Draw the two quadrilaterals and explain which one will have the largest area.

Problem-solving 19, 20 19

20–22

20–23

A parallelogram-shaped sign is being designed for a park with a base of 4.5 metres and a perpendicular height of 2 metres. Find the area of the sign in square metres.

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A roof in the shape of a parallelogram is being designed for a new house. The roof will be covered with tiles, which cost $10 per square metre. The roof has a base length of 20 metres and a perpendicular height of 8 metres. a

Find how many square metres of tiles are needed to cover the roof.

Find the cost of tiling the roof.

A factory produces parallelogram-shaped cardboard boxes. Each box has a base length of 40 centimetres and a perpendicular height of 20 centimetres. The boxes are wrapped in decorative paper, which costs $0.05 per square centimetre. The factory produces 1000 boxes.

Determine the number of square centimetres of paper needed to wrap all the boxes.

Determine the cost of the decorative paper required.

A rectangular parcel of land has dimensions 200 m by 300 m. A road cuts through the parcel in the shape of a parallelogram with a base of 150 m and a perpendicular height of 50 m. a

Find the area of the parallelogram in square metres.

Find the area of land that is left on either side of the road.

A jewellery maker is creating parallelogram-shaped pendants for a new collection. Each pendant has a base length of 3 cm and a perpendicular height of 40 mm. The pendants will be covered with a thin layer of gold that costs $0.50 per square centimetre. If the jewellery maker needs to make 500 pendants, ﬁnd the total cost of covering them with gold.

13.07 Area of composite shapes After this lesson, you will be able to… • identify composite shapes and visualise them as a combination of simple shapes. • add the areas of simple shapes to find the total area of a composite shape. • apply subtraction to find the shaded or remaining area of composite shapes with holes and cutaways. • solve real-world problems involving composite shapes by applying your knowledge of finding area.

Area of composite shapes using addition We can think of a composite shape as one that is made from a number of smaller shapes. Dashed lines can be used to visualise which simple shapes make up a composite shape. We ﬁnd the total area by adding the areas of each of the simple shapes.

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Example 1 Find the area of this figure: 4 mm

9 mm

15 mm

Create a strategy Calculate the area of the triangle and the rectangle, then add them.

Apply the idea Area of the triangle: Write the formula

Substitute b = 15 and h = 4

Evaluate

Area of the rectangle: A=l×w

Write the formula

= 15 × 9

Substitute l = 15 and w = 9

= 135

Evaluate

Total area: A = 30 + 135 = 165 mm

Add the areas Evaluate

Example 2 Find the total area of this figure: 6 cm

15 cm

19 cm

6 cm

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Fluency Ex 1

4, 6, 7, 8a-d, 9, 10ab

5, 7–9, 10b-d

5, 7, 8c-g, 10cd

For this composite shape: 16 cm

14 cm

16 cm

Ex 2

8 cm

Find the area of the triangle.

Find the area of the rectangle.

Find the total area of the composite shape.

The quadrilateral in the diagram has a diagonal measuring 14 m.

Calculate the total area of this quadrilateral.

14 m

For this composite shape:

4 cm

Determine the dimensions of rectangle A.

Find the area of rectangle A.

Find the area of rectangle B.

Calculate the total area of the given shape.

7 cm 6 cm

A 3 cm

B 14 cm

For each figure: i

Find the area of shape A.

Determine the dimensions of shape B.

iii

Find the area of shape B.

Calculate the total area of the composite shape.

6 cm

17 cm

12 mm 2 mm

B 4 mm

3 cm 4 cm

6 cm

5 mm

3 mm

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Find the area of each composite shape: a

2 cm

13 m

15 mm 2 mm

8 cm

14 mm 14 cm

12 m 4 mm

5 cm

10 cm

3 cm

8 cm

5 cm 7 cm

7 cm

10 cm

14 cm 2 cm

8 cm 8 cm

6 cm

16 cm

8 cm 11 cm 4 cm 13 cm

Ex 3

For these figures: • Figure 1:

•

Figure 2:

• Figure 3: 13 cm

13 cm 4 cm 9 cm

4 cm

9 cm

9 cm 6 cm 6 cm

558

Find the area of the shaded rectangle in Figure 1.

Find the area of the shaded triangle in Figure 2.

Find the area of the shaded region in Figure 3.

Mathspace Victorian Curriculum – Year 7 mathspace.co

The aim is to calculate the shaded area in Figure 1. Use the images in Figures 2, 3 and 4 to write a set of steps explaining how to do this. Note that you do not need to calculate the actual areas. • Figure 1: • Figure 2: 15

3 7

11 4 5

• Figure 3: •

Figure 4: 15

7 4 5 5

Problem-solving 14, 16, 17, 19 14

15, 17, 18, 20

15–18, 20, 21

The dining room ﬂoor is to be tiled with slate tiles. The cost of tiling the room is $10 per square metre. a

Calculate the area to be tiled.

How much will it cost to tile the room?

9m 6m 8m

A children’s book is in the shape of a house. Calculate how much wrapping paper is required to wrap the book as a present, assuming the width of the book is negligible.

4 cm

9 cm 12 cm

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Mathspace Victorian Curriculum – Year 7 mathspace.co

Scott draws the plot of land which contains his house and garden.

18 m

Find the total area of the plot of land.

Find the area covered by the house.

Find the area covered by the garden.

If the cost of turﬁng the garden is $26 per square metre, how much will it cost Scott to turf the garden?

23 m House

11 m

Garden

A swimming pool has a 2 m wide path around its edge. The outer width of the path is 20 m and the length is 30 m.

20 m

Calculate the area of the surface of the pool.

Calculate the area of the path around the pool.

Calculate the cost of a pool blanket if blankets cost $5.55 per square metre.

30 m

A farmer needs to fertilise a rectangular field that measures 500 m by 300 m. The field contains a square shed with dimensions 25 m by 25 m. Fertiliser costs $0.03 per square metre. Calculate the cost of fertilising the field.

A graphic designer designs a logo for a company. If A4 paper is approximately 21 cm × 30 cm, calculate the amount of white space that will be left after the logo is printed.

3 cm

8 cm

6 cm 11 cm

A triangular backyard is to have two smaller garden beds planted in it. The rest of the backyard will be covered with crushed rock.

20 cm

17 m

Find the area of the crushed rock.

If crushed rock costs $12 per square metre, calculate how much it will cost to cover the backyard.

A farmer is going to fertilise his paddock which has this shape: a

Find the area of the paddock.

If it takes 150 kg of fertiliser to fertilise 100 square metres, how many kilograms of fertiliser should the farmer purchase?

5m 10 m 20 m

52 m 47 m

201 m

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13.08 Circumference of a circle After this lesson, you will be able to… • define a circle and its features, including the radius and diameter. • explain the meaning of pi and its relationship to the circumference of a circle. • find the circumference of a circle using the formula C = π d or C = 2π r. • solve problems related to finding the circumference of a circle given the diameter or radius.

Features of a circle A circle is the set of all points that are the same distance from a central point. We call this distance the radius. A line segment joining the central point to some point on the circle is called a radius.

A line segment joining two points on the circle, passing through the central point, is called a diameter.

The diameter is equal to twice the radius, since any diameter can be split into two radii. d = 2r

The circumference of a circle is the distance around its boundary. We can think of the circumference as the perimeter of the circle.

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Circumference π pronounced “pi” is a special number which is the ratio between the circumference and the diameter of any circle. π = Circumference ÷ Diameter If we type π into our calculator and try to evaluate it, the result will look something like this:

π = 3.141 592 653 59 Some calculators will show more or less decimal places than what is shown here, but none of them will show the exact value of π. This is because π is an irrational number, meaning that its decimal expansion continues forever. The formula for π can be rearranged to form the formula for the circumference of a circle: C = πd where C is the circumference and d is the diameter of a circle. As the diameter is equal to two times the radius, the formula can also be written as: C = 2π r where C is the circumference and r is the radius of a circle.

Example 2 A circle has a radius of 46 cm. What is its circumference, rounded to two decimal places?

Create a strategy

Apply the idea

Use the formula C = 2π r.

C = 2π r

Write the formula

= 2π × 46

Substitute r = 46

= 289.026 52…

Evaluate using a calculator

= 289.03 cm Round to two decimal places

Example 3 A circle has a diameter of 7.7 cm. What is the circumference, rounded to two decimal places?

Create a strategy

Apply the idea

Use the formula C = π d.

C = πd

Write the formula

= π × 7.7

Substitute d = 7.7

= 24.190 26…

Evaluate using a calculator

= 24.19 cm Round to two decimal places

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Write down the value of π, rounded to: a

2 decimal places

4 decimal places

1 decimal place

3 decimal places

Write down the formula for the circumference if you know the radius of a circle.

Write down the formula for the circumference if you know the diameter of a circle.

Fluency Ex 1

6, 8a, 9, 10ab, 11ac, 13, 15

6–8, 10ab, 11ac, 12–13, 14ac, 15, 16a-c, 17

Determine the radius of each circle: a

14 cm

7, 8b, 9, 10cd, 11bd, 13, 14b-d, 15, 16b-d, 17

18 cm

Write an algebraic expression for the diameter of this circle. x

Complete these tables: a

Diameter (cm) Radius (cm) Circle 1 22 Circle 2 17 Circle 3 38 Circle 4 110 Circle 5 66

b Circle 1 Circle 2 Circle 3 Circle 4 Circle 5

Diameter (cm) 82

Radius (cm) 25

27 3.4 15.6

The radius of the smaller circle in the ﬁgure is 16 cm. If CD = 4 cm, ﬁnd AD, the diameter of the larger circle. A B

13.08 Circumference of a circle mathspace.co

C D

569

10 Ex 2, 3

Calculate the circumference for each circle. Round your answer to two decimal places. a

13 cm 18 cm

c 11

A circle with a radius of 4.1 cm

A circle with a diameter of 9.5 cm

Calculate the circumference of these circles, rounded to two decimal places: a

A circle with a diameter of 38 cm

A circle with a diameter of 45 cm

A circle with a radius of 27 cm

A circle with a radius of 1.8 cm

For this circle: a

Use the rule places.

to ﬁnd the diameter, rounded to two decimal

Use the rule

to ﬁnd the radius, rounded to two decimal

42 m

places. 13

14 Ex 4

570

A circle has a circumference of 52 cm. a

Calculate the radius of the circle, rounded to two decimal places.

Calculate the diameter, rounded to two decimal places.

Calculate the radius of these circles, rounded to two decimal places: a

A circle with a circumference of 14 cm

A circle with a circumference of 18 cm

A circle with a circumference of 14.9 cm

A circle with a circumference of 98 cm

A circle has a circumference of 36 cm. a

Calculate the diameter of the circle, rounded to two decimal places.

Calculate the radius, rounded to two decimal places.

Calculate the diameter of each circle, rounded to two decimal places: a

A circle with a circumference of 37 cm

A circle with a circumference of 44 cm

A circle with a circumference of 70 cm

A circle with a circumference of 24.5 cm

Mathspace Victorian Curriculum – Year 7 mathspace.co

Complete this table that lists the radii, diameters, and circumferences of various circles. Round all measurements to two decimal places. Circle 1 Circle 2 Circle 3 Circle 4

Reasoning

Radius (cm) 2 8

Diameter (cm) 4

Circumference (cm)

82 340

19, 20

18–20

18, 19, 21

For each circle in the previous question, what do you notice when the circumference is divided by the diameter?

Circumference = 15π cm is called an exact measure of circumference because the value of π has not been rounded. Give a real life example where it might be more beneﬁcial to use the exact value for the circumference of a circle instead of an approximation.

Caitlin and David calculate the circumference of this circle using different formulas. David uses the formula C = 2π r to calculate the circumference, where r is the radius.

10 cm

Caitlin uses the formula C = π d to calculate the circumference. Explain why they will get the same result.

Investigate what happens to the circumference when the radius of a circle is doubled.

Problem-solving 22–27

24–27, 29–30

25–32

The bottom of a ﬂower pot has a radius of 16 cm. Find the circumference of the bottom of the ﬂower pot, rounded to one decimal place.

A scooter tyre has a diameter of 34 cm. Find the circumference of the tyre, rounded to one decimal place.

Find the circumference of the Ferris wheel, rounded to one decimal place. 8m

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13 Chapter review 1

Find the area of each shape: a

7 mm

8 cm 7 mm

7 mm

11 cm 7 mm

Find the length of the missing side in these rectangles: a

A rectangle that has an area of 24 m2 and a width of 3 m.

A rectangle that has an area of 44 m2 and a length of 4 m.

Find the area of these triangles: a

12 cm 5 cm

7 cm 8 cm

Find the base measurement for these triangles, given the area and height: a c

Area = 24 cm2 and height = 6 cm 2

Area = 15 mm and height = 5 mm

Area = 35 cm2 and height = 7 cm

Area = 360 mm2 and height = 6 cm

Find the area of these parallelograms: a

9 mm

11 cm

7 mm 12 cm

Complete this table of base and perpendicular height measurements for three different parallelograms with an area of 80 cm2: Base (cm) Perpendicular height (cm) Area (cm2)

8 80

4 5 80

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Find the area of these composite shapes: a

3 cm

7 cm

9 cm 9 cm

15 cm

6 cm

15 cm

Find the area of the shaded region in these figures: a

11 cm

6m 16 m

3 cm

13 m

Calculate the circumference for each circle. Round your answer to two decimal places. a

21 cm 17 cm

Calculate the radius of these circles, rounded to two decimal places: a

A circle with a circumference of 28 cm

A circle with a circumference of 140 cm

Two parallelograms have the same area. If one parallelogram has a base of 10 cm and a perpendicular height of 5 cm, what are the possible dimensions of the other parallelogram, given the base and perpendicular height are whole numbers?

A rectangular piece of fabric is divided into 6 smaller rectangles, each with an area of 9 square centimetres. What is the area of the original rectangular fabric in square centimetres, assuming no fabric is wasted?

A landscape architect needs to design a triangular garden for a new public park. The base of the triangle is 30 m long, and the height of the triangle is 12 m. Calculate the area of the triangular garden in square metres.

A wallpaper installer charges $25 per square metre to apply wallpaper on walls. If a room has a triangular section with a base of 10 m and a height of 6 m, how much will it cost to apply wallpaper on that section of the wall?

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A company manufactures parallelogram-shaped wooden panels. Each panel has a base length of 60 centimetres and a perpendicular height of 30 centimetres. The panels are coated with a protective varnish, which costs $0.10 per square centimetre. The company produces 500 panels. Determine the cost of the protective varnish required to coat all the panels.

A homeowner plans to install carpet in their bedroom floor. The cost of the carpet installation is $11 per square metre. 11 m 6m 10 m

How much will it cost to install the carpet in the bedroom? 17

Ryujin needs to apply fertiliser to his rectangular lawn that measures 250 m by 150 m. The lawn contains a square gazebo with dimensions 20 m by 20 m. Fertiliser costs $0.02 per square metre. Calculate the cost of applying fertiliser to the lawn.

A triangular garden is to have a pond and a fountain installed in it. The rest of the garden will be covered with artificial grass. a

Find the area of the artificial grass.

If artificial grass costs $12 per square metre, calculate how much it will cost to cover the garden. Pond

Fountain 19 m

5m 10 m

4m 8m 22 m

A toy car tyre has a diameter of 4.4 cm. If the tyre rotates through 90 complete revolutions, find the distance it will travel. Round your answer to one decimal place.

A circular swimming pool has a diameter of 31 m. How many laps around the pool must be completed to swim 2000 m? Round your answer to one decimal place.

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Big ideas • Solids can be visualised in a number of ways for different purposes, nets allow for the surface of the solid to be explored and isometric and perspective views provide the viewer with detail about the image. • The volume of a three-dimensional object is the amount of space that it occupies, measured in cubic units.

Curriculum Outcomes VC2M7SP01 represent three-dimensional objects in 2 dimensions; discuss and reason about the advantages and disadvantages of different representations VC2M7M02 solve problems involving the volume of right prisms including rectangular and triangular prisms, using established formulas and appropriate units VC2M7A06 manipulate formulas involving several variables using digital tools, and describe the effect of systematic variation in the values of the variables

14 Solids and volume Chapter outline 14.01 Nets of solids 14.02 Visualise solids 14.03 Volume using unit cubes 14.04 Units of volume 14.05 Volume of right prisms 14.06 Attributes in measurement formulas Chapter 14 review

578 583 601 605 610 619 633

14.01

Nets of solids

After this lesson, you will be able to… • identify nets for different solids, including cubes, rectangular prisms, triangular prisms, and pyramids.

Nets We can never see every part of a three-dimensional object at once - there is always part of it that is behind the view we are looking at.

Interactive exploration

Discover this concept in action online

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To better think about a solid object, we sometimes represent it with its net. Each face of the solid is laid flat on the same surface, breaking it along the edges and folding it out. This way, we can think about folding it back up along its edges to recover the original shape. There are many ways to unfold a net from a solid. We will investigate nets of prisms and pyramids. Here are some prisms: Prisms

Triangular

Square

Rectangular

Pentagonal

Hexagonal

Octagonal

Prisms have rectangular sides, and the shape on the top and the base is the same. The name of this shape gives the prism its name. Here are some pyramids: Pyramids

Triangular

578

Square

Rectangular

Mathspace Victorian Curriculum – Year 7 mathspace.co

Pentagonal

Hexagonal

Octagonal

Fluency 3

Ex 2

Name the 3D shapes formed by these nets: b

A rectangular prism has been sliced through a plane parallel to its base. What two-dimensional shape is formed by the slice?

582

Name the solid that can be formed from the given sets of 2D shapes: a

Ex 3

3–8

Draw a net for these solids:

3–8

Name the solid formed from these nets: a

Ex 1

3–8

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 4

A solid is cut parallel to one of its faces to give this cross-section: What could the solid possibly be?

Reasoning

9, 10

9–11

Describe how to use the net of a solid to find the solid’s surface area.

Explain the similarities and differences between a prism and a pyramid.

Jack wants to make a cardboard box in the shape of a rectangular prism with dimensions 10 cm × 8 cm × 6 cm. Sketch a net that Jack could use to make the box without using a measuring tool. Explain your reasoning.

Problem-solving 12 12

12, 13

Draw a net for each of the following items: a

Cereal box

Tissue box

Camping tent

6-sided die

Identify the relationship between the number of faces in a net and the number of edges and vertices in its corresponding 3-dimensional solid.

14.02 Visualise solids After this lesson, you will be able to… • identify and draw different views of solids and understand how they can be useful in various fields. • visualise 3D shapes from aerial views and footprints of structures.

Views of isometric drawings and interpreting perspective An isometric drawing refers to a 2D representation of a 3D object. This type of drawing has various views. Three-dimensional objects can be represented with the side view (the view on the right side of the object specifically), front view, and top view clearly indicated on a two-dimensional surface. We can then ask about the view from each of these elevations.

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Top view

Side view Front view 3D view

Front view

Side view

Top view

As we can see from the images, the front, side and top views are all 2D shapes that match the faces of the prism visible from each view when looked at directly. The plan view is another name for the top view and is used in architecture when referring to the plan of a building, which is how the building looks when viewed from directly above. Since the 2D views of a solid only show what the solid looks like directly from one angle, these views cannot show depth like a 3D view can. For example, if we look at this hexagonal prism from the front, we will be able to see these three faces.

However, two of these faces are sloped and will appear thinner in the front view than they actually are. As a result, the front view will look like this.

A 2D view might also show a side that we can’t see from the 3D view. Top view

Front view

When looking at a triangular prism from the 3D view, we notice that we can only see two of the five faces. The bottom and back faces of the prism won’t show up on any of our 2D views, but the last hidden face will be visible from the top view.

Side view

We find that the top view of this triangular prism looks like this. The fact that we can see two faces from the top view is shown by the line dividing the view into two rectangular faces.

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14.02 Practice questions Understanding

Adventurer

1, 2

Trailblazer 1, 2

What is an isometric drawing?

Which type of drawing is more realistic, an isometric drawing or a perspective drawing?

Fluency Ex 1

Explorer

3a-c, 4a-c, 5, 6a-c, 7–11, 12a-d

3c-e, 6c-e, 9, 12c-f

3de, 4b-d, 6d-f, 9, 12g-j

For each solid formed from cubes: i

Draw the diagram for the front view.

iii

Draw the diagram for the top view.

Top view

Front view

Draw the diagram for the side view.

Top view

Side view

Side view Front view

Top view

Front view

Side view

Top view

Front view

Side view

Top view

Front view

Side view

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Select the solid that matches the given views. The side view is on the right of the shape. a

Front view

Side view

Top view

Top

Side view

Front view

Top

594

Front view

Mathspace Victorian Curriculum – Year 7 mathspace.co

Side view

Front view

Which of these diagrams shows the top view of this collection of cylinders?

Name the solid that would have these views: a

Front view

Side view

Top view

Front view

Side view

Top view

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Front view

Top view

Side view

Front view

Top view

Side view

Front view

Top view

Name the solid that would have this shape as one of its views: a

Ex 2

Side view

Choose the solid that has this rectangle as its front view:

Top view

Side view

Side view Front view

Front view 596

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Top view

Side view Front view

Choose the solid that has this rectangle as its side view: A

Top view

Side view Side view Front view

Front view

Top view

Side view

Front view

Side view

Choose the solid that has this as its top view:

Top view

Front view

Top view

Side view

Side view Front view

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Top view

Side view

Front view Front view

Choose the solid that has this as its top view:

Top view

Side view Side view

Front view Front view

Top view

Side view Side view Front view

Front view

598

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Ex 3

For each solid: i

Sketch the front view.

Top view

Sketch the side view.

iii

Sketch the top view.

Top view

Front view

Side view

Front view

Top view

Side view

Front view

Side view Front view

Top view

Side view Side view

Front view Front view

Top view

Side view

Side view Front view

Front view

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Yashi is studying visualising solids. Yashi states that these solids have the same front view, but different side and top views. Explain Yashi’s thinking.

For this solid, explain why a side view alone is an insufficient visualisation.

Problem-solving 16

16, 17

Choose a real-world object. Draw the diagrams for its side view, front view, and top view.

Draw any solid, oriented in two different ways.

14.03 Volume using unit cubes After this lesson, you will be able to… • calculate the volume of rectangular prisms by using the concept of counting the number of rows of similar layers.

Volume using unit cubes The unit cube

1 unit3 of volume

1 unit 1 unit

The space inside a 3D shape is called its volume. We measure volume by imagining how many unit cubes, each with sides that are 1 unit long, can fit inside the shape. One of these small cubes has a volume of 1 cubic unit, written as 1 unit3.

1 unit

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14.03 Practice questions Understanding 1

Volume

20 unit cubes

4 unit cubes

25 unit cubes

There are 2 layers and in each layer, there are 9 cubes.

There is 1 layer and in each layer, there are 16 cubes.

There are 6 layers and in each layer, there are 6 cubes.

There are 11 layers and in each layer, there are 9 cubes.

There are 14 layers and in each layer, there are 10 cubes.

16 unit cubes

Identify how many different rectangular prisms can be made using the numbers of cubes given, and provide their dimensions. 12 cubes

Fluency 5

Unit cube

Identify the total number of unit cubes for these statements:

Ex 1

Trailblazer 1, 2, 3c-e, 4cd

Sketch a rectangular prism that is made up of: a

Adventurer 1, 2, 3c-e, 4ab

Explain what these terms mean: a

Explorer 1, 2, 3a-c

b 5

48 cubes

72 cubes

120 cubes

In these figures, 1 block = 1 cm3. Find the volume of each solid figure. a

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Reasoning

7, 8

Will the volume of a rectangular prism change if it is turned on its side, changing its length, width, and height? How do you know?

Explain how to use 1 centimetre cubes to fill a 1 metre cube-shaped cardboard box.

How can we use unit cubes and a rectangular prism to explain how the formula for the volume of a rectangular prism, V = l × w × h, works?

Problem-solving 9 9

604

Imagine you have a large cube that is 5 units long on each side and made up of smaller cubes, each with a side length of 1 unit. If you remove all the inner cubes to only leave the shell, what is the volume of the remaining structure?

Mathspace Victorian Curriculum – Year 7 mathspace.co

14.04 Units of volume After this lesson, you will be able to… • estimate volume and capacity. • identify appropriate units for measuring volume and capacity.

Units of volume Volume is how much space something takes up, like a 3D shape or even a solid, liquid, or gas. Capacity is different - it’s about how much a container can hold, like how much juice can fit in your cup. We usually talk about capacity when we’re dealing with liquids in containers. Common units for volume from smallest to largest are: • cubic millimetres (mm3) • cubic centimetres (cm3) • cubic metres (m3) Common units for capacity from smallest to largest are: • millilitres (mL) • litres (L) • kilolitres (kL) • megalitres (ML) Just as with lengths and areas, to determine the correct unit to use and to be able to estimate the volume or capacity of an object, it is useful to identify some common items that are about the size of the units above. Such as: • 1 cubic millimetre, 1 mm3, is very small. This is a cube with length, width and height 1 mm — about the size of 10 grains of salt. • 1 cubic centimetre, 1 cm3, is the size of a cube with length, width and height 1 cm — this is about the size of a die. • 1 cubic metre, 1 m3, is the size of a cube with length, width and height 1 m — about the size of a fridge. • 1 millilitre, 1 mL, is the same as 1 cm3, and a teaspoon is 5 mL. • 1 litre, 1 L. It is very common for milk to come in 1 and 2 litre cartons. • 1 kilolitre, 1 kL, is one thousand litres. This is equal to 1 m3. • 1 megalitre, 1 ML, is one million litres — this is a lot of liquid. An Olympic swimming pool holds on average 2.5 ML.

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14.04 Practice questions Understanding

Trailblazer 1, 2

Explain the difference between volume and capacity.

Give examples of real-world objects that could have the given volume or capacity: a

1 mm3

1 cm3

1 m3

1 kL

1 ML

3–8

7–12

6–12

Identify the most appropriate unit for measuring the volume of: a

Sand pit

Wardrobe

Glass of water

Which units can we measure with a jug like the one in the picture?

Identify the most appropriate unit for measuring the capacity of a coffee cup.

Identify the most appropriate unit for measuring the capacity of a fish bowl.

A landscaper needs to move soil across the garden. Choose the best estimate for the volume of soil that can be carried by a wheelbarrow in one trip. A

608

1 mL

Determine whether these units are appropriate to use to measure the water needed to sustain the population of a small city. a

Ex 2

Adventurer 1, 2

Fluency Ex 1

Explorer 1, 2a-e

771 mL

21 ML

71 L

Consider this matchbox: a

Identify the most appropriate unit for measuring the volume of a matchbox.

Estimate the volume of the matchbox.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Air in a hall

Ex 3

This suitcase has a total capacity of 88 500 cm3. Estimate the volume of the shirts contained in the suitcase.

Estimate the volume of liquid in this container if it is able to hold 3.6 L when full.

Estimate the volume of liquid in this container if it is able to hold 3.8 litres when full.

Reasoning

13, 14

Estimate the volume of liquid in this container if it is able to hold 2400 mL when full. Explain the steps you took to decide.

The volume of the bookshelf shown is 0.64 m3. Estimate the volume of books on the bookshelf and explain how you calculated your estimation: a

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Problem-solving 16

15–17

A town’s reservoir contains 10 million megalitres of water, and the daily demand for water is 50 000 megalitres. If the demand remains constant, how many days will the reservoir supply the town’s water needs without further water supply?

A full jug of milk has a total capacity of 1.5 litres. If 300 mL of milk is used to make a cake, how much milk is left in the jug?

A 500 mL bottle of soda is sold for $2.50, while a 2 L bottle of the same soda is sold for $7.80. a

Which is a better value buy?

What is the cheapest way to buy 3 L of this soda?

14.05 Volume of right prisms After this lesson, you will be able to… • find the volume of a variety of right prisms and composite shapes using appropriate formulas and methods.

Volume of a rectangular prism Interactive exploration

Discover this concept in action online

mathspace.co

Since the area of the cross-section parallel to its base is the same size throughout a rectangular prism, we find the area of the base and multiply by the number of rows, or the height of the prism. The volume of a rectangular prism is given by: Volume = area of base × height or

Heigth

Width Length

Volume = length × width × height or V = l × w × h

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A cube is a special type of rectangular prism, one that has all sides equal in length. The volume of a cube is given by: Volume = area of base × height or

Side

Volume = side × side × side or

Side

V = s × s × s = s3

Side

In general, the formula for the volume of a right rectangular prism or a cube is given by V = Abase × h where Abase is the area of the prism’s base and h is the height of the prism.

Example 1 Find the volume of the rectangular prism: 12 cm

5 cm 7 cm

Create a strategy Calculate the area of the base of the rectangular prism, then multiply that by the height.

Apply the idea Based on the diagram above, we are given l = 12, w = 7 and h = 5. A=l×w

Write the formula for the area of the base

= 12 × 7 = 84 cm

Substitute the values 2

Evaluate

Now, when we multiply the area of the base by the height of the rectangular prism, we have the volume of the rectangular prism: V = Abase × h

Write the formula

= 84 × h

The Abase = 84

= 84 × 5 = 420 cm

The height (h) is 5 3

Evaluate

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14.05 Practice questions Explorer

Understanding

Adventurer

Trailblazer

Describe how the volume of a composite solid could be calculated.

Explain why every cube a rectangular prism but not every rectangular prism is a cube.

Fluency 3 Ex 1

3–7, 9–11

3cd, 5, 6, 8, 10–12

3cd, 5, 6, 8, 11–13

Find the volume of these prisms: a

9 mm

10 mm

8 mm

50 cm

9 cm

5 cm

9.9 m 12 m 4m

8.7 m 7.5 m

4 cm 2 cm

11 cm 8 cm

4 cm

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Find the volume of these cubes: a

6 cm 12 cm

Find the volume of a cube that has a side length of 14 m.

Find the side length of a cube that has a volume of 27 cm3.

A container is a rectangular prism with dimensions 50 cm, 30 cm, and 60 cm. Find the volume of the container.

A regular box of cereal is a rectangular prism and measures 25 cm by 7 cm by 28 cm. a

Find the volume of the box of cereal.

The company that makes these cereal boxes also makes a jumbo size box, which is one and a half times as long, wide, and tall as the regular size boxes. i

Find the volume of the jumbo box of cereal.

Is the volume of the jumbo box one and a half times the volume of the regular box?

A coffee table has the shape of a triangular prism with dimensions of a base area of 980 cm2 and a height of 70 cm. Find the volume of the table.

A box is shaped like a triangular prism, with a base length of 50 cm and a height of 30 cm for its triangular face. The height of the prism is 60 cm. Find the volume of the box.

Ex 2

The composite solid can be broken down into two smaller rectangular prisms:

Ex 3

6 cm 6 cm

6 cm 7 cm

6 cm

14 cm

Find the volume of the top prism.

Find the volume of the bottom prism.

Then, find the total volume of the composite solid.

6 cm 7 cm 6 cm 7 cm 14 cm

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The composite solid can be broken down into two smaller rectangular prisms: 2 cm

14 cm 2 cm 7 cm 8 cm

Find the volume of the top prism.

Find the volume of the bottom prism.

Then, find the total volume of the composite solid.

Find the volume of these composite solids: a

1 cm

13 cm

7 16

4 cm 10 cm

10 cm 2 cm

d 2 cm

7 cm 2 cm 24 cm

2 cm

Reasoning 14

15, 16

14, 16

Find two different ways to calculate the total volume of the composite solid and explain your thinking. Determine which of the two methods could be considered more efficient and why. 7 cm

16 cm

A box is 2 m long, 30 cm high and 40 cm wide. Write the steps to calculate the volume of the box in cubic centimetres.

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This solid consists of a rectangular prism with a smaller rectangular prism cut out of it. a

Explain how to find the volume of the solid, then calculate the volume of the solid.

Is there another way the volume could be found? If so, explain.

Problem-solving 17, 18

12 cm 4 cm

17–20

7 cm

6 cm

18 cm

18–21

A tank has a length of 9 m, width of 4 m and depth of 8 m. Find the volume in cubic metres and then use that to find the capacity of the tank in litres.

Consider the aquarium: a

Calculate the maximum volume of water that the aquarium can hold, and use that to find the total capacity in litres.

Explain what reasonable volume of water would be needed to fill the aquarium, and how you determined your answer.

50 cm 30 cm

25 cm

A rectangular prism has a volume of 120 cubic units. It is known that the length, width and height are all whole numbers. If the length and width of the prism are in a ratio of 3: 4, what is the height of the rectangular prism?

A juice box is in the shape of a rectangular prism with a length of 8 cm, a width of 4 cm, and a height of 12 cm. If the juice box is filled with 250 mL of juice, what is the height of the juice in the box?

Hanni is constructing a tent in the shape of a triangular prism in the woods. The base of the tent measures 2.4 m and its height is 3.4 m, while the total length of the tent is 4 m. If Hanni estimates that her friends will occupy space in the tent?

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of the tent, what is the volume of the remaining

14.06 Attributes in measurement formulas After this lesson, you will be able to… • manipulate the area formula, formula for volume of a rectangular prism, and formula for speed, distance, and time to investigate the effect of changing variables. • produce a table of values and draw a graph of the values to understand the linear and non-linear relationships between the variables.

Manipulate the area formula Interactive exploration

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Consider the formula for the area of a rectangle: Area = length × width

Length

or A=l×w Width

where l and w are variables.

When the length or width of a rectangle is a constant, we called that value fixed. The relationship between the area of a rectangle and the product of the length and width as one of the dimensions changes then becomes a linear relationship.

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Example 1 Consider these rectangles and their dimensions: 1 cm

2 cm

3 cm

A=1×3 A = 3 cm2

3 cm

A=2×3 A = 6 cm2

A=3×3 A = 9 cm2

a For the rectangle above, with a fixed width of 3 cm, complete the table for various lengths. Width Length Area

3 1

3 2

3 3

3 4

3 5

3 6

3 7

3 8

3 9

3 10

Create a strategy We can use the formula A = l × w to find the area of the rectangle.

Apply the idea Width Length Area

3 1 3

3 2 6

3 3 9

3 4 12

3 5 15

3 6 18

3 7 21

3 8 24

3 9 27

3 10 30

b Now consider the length fixed as 5 cm. Complete the table for various widths. Width Length Area

Create a strategy Multiply the length and width, given the constant length of 5 cm.

Apply the idea Width Length Area

620

1 5 5

2 5 10

3 5 15

4 5 20

5 5 25

Mathspace Victorian Curriculum – Year 7 mathspace.co

6 5 30

7 5 35

8 5 40

9 5 45

10 5 50

c Graph the functions for area varying the length and varying the width on the axes shown, labelling each carefully.

Area

45 40 35 30 25 20 15 10 5 1

Create a strategy Plot the points on a number plane by using the values from the table. Use the value of the length as the x-coordinate and the corresponding area as the y-coordinate to plot each point. Connect the points to form a line or a curve.

Length / width 7 8 9

Apply the idea 50

Area

45 varying width length = 5 cm

40 35 30 25 20

varying length width = 3 cm

15 10 5

Length / width 1

2 3 4 5 6 7 8 9 10

d Determine the formula that can be used to calculate the area of a rectangle with a fixed width of 3 cm.

Create a strategy

Apply the idea

Use the formula A = l × w. Substitute the value of the constant dimension.

For w = 3, A = l × 3, or A = 3l

e Determine the formula that can be used to calculate the area of a rectangle with a fixed length of 5 cm.

Create a strategy

Apply the idea

Use the formula A = l × w. Substitute the value of the constant dimension.

For l = 5, A = 5 × w, or A = 5w

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Apply the idea V=1×3×2

Substitute the values for length, width and height

Evaluate

The volume is 6 cm3.

b What is the volume of the block with length 2 cm?

Create a strategy Use the formula V = l × w × h to find the volume of the block.

Apply the idea V=2×3×2

Substitute the values for length, width and height

= 12

Evaluate 3

The volume is 12 cm .

c What is the volume of the block with length 3 cm?

Create a strategy Use the formula V = l × w × h to find the volume of the block.

Apply the idea V=3×3×2

Substitute the values for length, width and height

= 18

Evaluate 3

The volume is 18 cm . d Complete the table: Length Volume

Create a strategy To find the volume of a rectangular solid, use the formula V = l × w × h. Keep in mind that the width (w) and height (h) values are constant.

Apply the idea Since the width and height are both 2 and 3, respectively, we can calculate the volume by multiplying the length (l) by 6 to complete the table. Length Volume

1 6

2 12

3 18

4 24

5 30

6 36

7 42

8 48

9 54

10 60

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e Graph the relationship between the length of these blocks and their volume on the axes.

Volume (cm)3

50 45 40 35 30 25 20 15 10 5

Length (cm) 1

Create a strategy Plot the points on a number plane by using the values from the table. Use the value of the length as the x-coordinate and the corresponding volume as the y-coordinate to plot each point. Connect the points to form a line or a curve.

Apply the idea 50

Volume (cm)3

45 40 35 30 25 20 15 10 5

Length (cm) 1

Find a formula relating the length and the volume in the form V = kl

Create a strategy Since we know that the height (h) is 2 and the width (w) is 3, we can substitute these values into the formula.

Apply the idea

624

V=l×3×2

Substitute the values of width and height

V = 6l

Evaluate

Mathspace Victorian Curriculum – Year 7 mathspace.co

Example 3 Consider the rectangular prism. l = 1 cm w = 3 cm

h = 1 cm

a What is the volume of this rectangular prism?

Create a strategy Use the formula V = l × w × h.

Apply the idea Let l = 1, w = 3, and h = 1 V=1×3×1

Substitute the values of length, width and height

Evaluate

The volume of the rectangular prism is 3 cm3.

b Suppose that we are going to increase the volume by adding extra blocks to increase the height while keeping the width and length the same. Select the diagram that shows the next rectangular prism in the sequence. A

Create a strategy To increase its height, we would add more identical prisms of dimensions 1 unit long, 3 units wide, and stack them on top of the existing prism.

Apply the idea Add more identical rectangular prisms (or blocks) of the same dimensions to the existing prism to increase its height. The correct diagram is C.

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Manipulate the formula for speed, distance and time Speed, distance and time are important concepts in understanding how things move. They are connected by the formula: Speed = Distance ÷ Time or • Speed (S) tells us how fast something is moving. It can be measured in different units, like kilometres per hour (km/h) or metres per second (m/s). • Distance (D) is the space between two points and is measured in units like kilometres or metres. • Time (T) is how long it takes to travel that distance and is usually measured in hours, minutes, or seconds. Using this formula, we can find the speed of any object if we know the distance it travelled and the time it took to travel that distance.

Example 4 Consider a journey of 720 km. a Calculate the average speed of the journey for each time given in the table: Time (hours) Speed (km/hr)

Create a strategy Use the formula

Apply the idea Divide the distance D = 720 by the time T to complete the table. Time (hours) Speed (km/hr)

1 720

2 360

3 240

4 180

6 120

b Graph the relationship between the time taken and the average speed on the axes shown.

8 90

700 650 600 550 500 450 400 350 300 250 200 150 100 50

9 80

10 72

Speed (km / hr)

Time (hrs) 1

9 10

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Consider a rectangle which has a fixed width of 6 cm and a variable length. a

Complete the table for various lengths: Length Area

From the table, write a rule that can be used to find the area of a rectangle with a fixed width of 6 cm.

Plot the points from the table and join them, to graph the function for area with fixed width and varying length on the axes shown:

d Ex 2

60 55 50 45 40 35 30 25 20 15 10 5

Area

Is this function linear or non-linear?

Consider the volume of the rectangular prisms. They all have a length of 4 m and a width of 2 m. l=4m

w=2m

l=4m

w=2m

Find the volume of the block with a height of 1 m.

Find the volume of the block with a height of 2 m.

Find the volume of the block with a height of 3 m.

Complete the table: Height Volume

w=2m

Graph the relationship between the height of these blocks and their volume on the axes shown.

Find a formula relating the height and the volume in the form V = kh.

80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5

Volume (m)3

Height (m) 1

630

2 3 4 5 6 7 8 9 10

Consider a rectangular prism which has a length of 5 cm, a width of 2 cm and a height of 3 cm. a

Find the volume of the rectangular prism.

Suppose we are going to increase the volume by adding extra blocks to increase the height while keeping the width and length constant.

Mathspace Victorian Curriculum – Year 7 mathspace.co

l=4m

h=3m

h=2m

h=1m

Ex 3

Length 2 3 4 5 6 7 8 9 10

Complete the table for various heights: Width Length Height Volume

2 5 4

2 5 5

2 5 6

2 5 7

2 5 8

2 5 9

2 5 10

Find a formula for this rectangular prism, relating the height and the volume in the form V = kh, where k is a constant.

Find the volume of this rectangular prism if the height is 35 cm.

A car travels a fixed distance of 360 km The average speed of the car is given by the rule:

Ex 4

Find the average speed of the car for each time given in the table: Distance (km) Time (hours) Speed (km/h)

360 1

360 2

360 3

360 4

Use the second and third row of the table of values to graph the relationship between time and speed for the fixed distance on the axes shown:

Is this relationship linear?

Find the speed of the car when the time taken is 15 hours.

360 5

60 55 50 45 40 35 30 25 20 15 10 5

360 6

10, 11

360 9

360 10

Speed (km / h)

Time (hours) 1

Reasoning

360 8

2 3 4 5 6 7 8 9 10

10, 11

The number 5 is an example of a constant, and w is an example of a variable. Why do you think the words constant and variable were chosen to represent these mathematical terms?

A square has a fixed area of 144 cm2. Is the side length of the square a constant or a variable? Explain your answer.

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Problem-solving 12, 13 12

12–14

12–16

Consider the formula to find the area of a rectangle: A = lw where A is the area, l is the length and w is the width. Determine whether these statements are true or false:

If the length is 4 cm and the width is 3 cm, then the area is 12 cm2.

If w = 3 and l increases, A also increases.

If l = 2 and w increases, A decreases.

If A = 5w, the width is a constant value of 5.

Consider the formula of the volume of a rectangular prism: V = lwh where V is the volume, l is the length, w is the width and h is the height. Determine whether these statements are true or false:

If the width and length are constant and h decreases, then V also decreases.

If h is constant and w and l increase, V decreases.

If V = 10lw, the height is a constant of 10.

If the volume is 24 cm3, width is 4 cm, and the length is 3 cm, then the height is equal to 2 cm.

The formula to find the speed, S, over distance, D, travelled in the time taken, T, is

Determine whether these statements are true or false: a

If the speed is 9 m/s and the distance travelled is 81 m, then the time is equal to 9s.

If the distance is constant and T increases, S also increases.

If the time is constant and D decreases, S also decreases.

, the time is a constant with value of 15.

A triangle has a fixed height and varying base. If we want the area to double, by how much should we increase the base?

Peter is driving a fixed distance. If he wishes his time to be reduced by half, by how much should he increase his speed?

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14 Chapter review 1

Name these solids: a

Draw a net for these solids: a

Name the solid formed by each net: a

For the solid formed from cubes: i

Draw the diagram for the side view.

iii

Draw the diagram for the top view.

Top

Front view

Draw the diagram for the front view.

Top view

Side view Front view

How many different rectangular prisms can be made using: a

Side view

20 cubes

24 cubes

Identify the most appropriate unit for measuring the volume of: a

Swimming pool

Refrigerator

Coffee cup

Car’s trunk

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Find the volume of these prisms: a

4 mm 7 mm

10 mm

14 cm 3 cm 5 cm

13 mm

d 8 cm

15 mm 7 cm 5 cm

11 mm

A rectangular prism container has dimensions 25 cm, 35 cm, and 50 cm. Find the volume of the container.

A regular box of cereal is in the shape of a rectangular prism and measures 23 cm by 11 cm by 19 cm.

Find the volume of the box of cereal.

The company that makes these cereal boxes also makes a jumbo size box, which is 1.5 times larger in all dimensions. i

Find the volume of the jumbo box of cereal.

Is the volume of the jumbo box one and a half times the volume of the regular box?

Find the volume of these prisms: a

5.3 cm 10 cm

11.5 cm 2 cm 8 cm 4 cm

10 cm

5 mm 10 mm

18 cm

6 mm 6 cm

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8 cm

The volume of the triangular prism shown is 287.5 m3: 5m

10 m

Find the value of k, to one decimal place. 12

The volume of the triangular prism shown is 247.45 cm3:

7 cm y cm 9.9 cm

Find the value of y to one decimal place. 13

Find the volume of these composite solids: a

11 cm 2 cm

2 cm

2 cm 8 cm 16 cm 2 cm 4 cm 9 cm

d 5 cm 11 cm

4 cm

6 cm

16 cm

4 cm 13 cm

A swimming pool has a length of 12 m, width of 11 m and depth of 6 m. Find the volume of the swimming pool.

A rectangular prism has a volume of 168 cubic units. It is known that the length, width and height are all whole numbers. If the length and width of the prism are in a ratio of 2:3, what is the height of the rectangular prism?

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A rectangular container has a length of 10 cm, a width of 6 cm, and a height of 15 cm. If 450 mL of liquid is poured in, what is the height of the liquid in the container?

Identify these variables as constant or varying: a

Length is 9 cm

Width is k cm

Height is H m

Distance is 25 km

Consider a rectangle which has a fixed length of 6 cm and a variable width. a

Complete the table below for various widths: Length Width Area

6 1

6 2

6 3

6 4

6 5

6 6

6 7

6 8

6 9

6 10

From the table, write a rule that can be used to find the area of a rectangle with a fixed length of 6 cm.

Plot the points from the second and third row of the table and join them to graph the function for area of this rectangle on the axes below: 60 55 50 45 40 35 30 25 20 15 10 5

Area

Width 1

d 19

2 3 4 5 6 7 8 9 10

Is this function linear or non-linear?

Consider a rectangular prism which has a length of 6 cm, a width of 3 cm and a height of 5 cm. a

Find the volume of the rectangular prism.

Suppose that we are going to increase the volume by adding extra blocks to increase the height while keeping the width and length constant. Complete the table below for various heights: Width Length Height Volume

636

3 6 6

3 6 7

3 6 8

3 6 9

3 6 10

3 6 11

3 6 12

Find a formula for this rectangular prism, relating the height and the volume in the form V = kh, where k is a constant.

Find the volume of this rectangular prism if the height is 25 cm.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Big ideas • Rigid transformations involve translating (sliding), rotating (turning), or reflecting (flipping) an object, that can be combined in a sequence without changing the object’s shape or size. • Line symmetry and rotational symmetry are important concepts in geometry that help us identify properties of shapes and develop spatial reasoning and critical thinking skills.

Curriculum Outcomes VC2M7SP03 describe the effect of transformations of a set of points using coordinates in the Cartesian plane, including translations, reflections in an axis, and rotations about the origin

15 Transformations Chapter outline 15.01 Transformation of shapes 15.02 Translations 15.03 Reflections 15.04 Rotations 15.05 Multiple transformations Investigation: Tessellations 15.06 Line and rotational symmetry Chapter 15 review

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15.06 Line and rotational symmetry After this lesson, you will be able to… • identify the two main types of symmetry: line symmetry and rotational symmetry. • identify lines of symmetry in shapes. • determine the order of rotational symmetry of shapes.

Line symmetry A shape has symmetry if it looks the same before and after a transformation. Symmetries are very common in nature as well as many areas of mathematics. The two main types of symmetry we will explore are line symmetry (or reflection symmetry) and rotational symmetry. A line that reflects a shape onto itself is called a line of symmetry (or axis of symmetry).

Interactive exploration

Discover this concept in action online

mathspace.co

A square has four lines of symmetry, all passing through its centre.

1 2

One way to picture lines of symmetry is with reflections. Another way is to picture folding the shape along a line. Whether reflecting or folding, the resulting shape should perfectly overlap the original.

A shape that has no lines of symmetry is called asymmetric. Neither of these shapes have a line of symmetry.

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Example 3 For each shape, determine if it has line symmetry, rotational symmetry, both types or no symmetry at all. a

Apply the idea The shape can be divided into two equal parts by a vertical line, but it does not have rotational symmetry. This shape has line symmetry.

Apply the idea This is an irregular shape. It cannot be divided into two equal parts by a line and does not have rotational symmetry. This shape has no symmetry.

Apply the idea The shape can be divided into two equal halves with a vertical line, a horizontal line and diagonal lines. It has rotational symmetry of order 4. This shape has both line and rotational symmetry.

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Determine whether these letters have rotational symmetry: a

Ex 2

Determine whether these diagrams have rotational symmetry. a

Do these road signs have rotational symmetry? a

Ex 3

Identify whether these diagrams have line symmetry, rotational symmetry, both, or neither: a

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Big ideas The probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.

Curriculum Outcomes VC2M7P01 identify the sample space for single-stage experiments; assign probabilities to the possible outcomes and predict relative frequencies for related experiments VC2M7P02 conduct repeated chance experiments and run simulations with a large number of trials using digital tools; compare predicted with observed results, explaining the differences and the effect of sample size on the outcomes

16 Probability

Chapter outline 16.01 Language of probability 16.02 Probability as a number 16.03 Theoretical probability 16.04 Experimental probability Investigation: Simulations Chapter 16 review

704 712 718 724 732 737

16.01 Language of probability After this lesson, you will be able to… • group outcomes into events and sort them into categories of likelihood. • apply concepts to real-world scenarios, such as spinning a spinner or rolling a die.

The language of probability Probability is the study of chance and prediction. To make sure our predictions are valid, we need to learn and use the right mathematical language. An experiment is any activity that has a possible result. Each result of an experiment is called an outcome. Events or trials can either be a single outcome, or a group of outcomes from an experiment. The list of all possible outcomes of a trial is called the sample space. Experiment Flipping a coin

Sample Space heads, tails

Outcome Getting tails

Rolling a die

1,2,3,4,5,6

Rolling a 6

Event Getting tails (single outcome) Rolling an even number (group of outcomes)

Example 1 If I spin this spinner, what is the sample space?

Create a strategy

Apply the idea

The sample space is the list of all possible outcomes.

When we look at the spinner, we can see that the sample space includes Pig, Ball, and Apple.

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When you flip a coin the possible outcomes are Head or Tail. What probability terminology is used for the set of all possible outcomes?

Which of these is true? a

When rolling a die, rolling an even number is an example of an event.

If the event happens more than half the time, we say it is likely.

If no outcomes correspond to the event, we say it is unlikely.

If an event happens exactly half the time, we say it has an even chance.

Fluency 4

5, 6, 8, 9, 11–14, 16, 17

5–7, 9, 11, 12, 16, 17

List the sample space: a

One spin of the following spinner

Rolling a standard six-sided die

Vincent rolls a twelve-sided die and writes down whether he rolls 6 or more. If he does, he writes “yes”, otherwise he writes “no”. List the sample space that corresponds to the event where he writes “yes”.

Ex 1

4, 5, 7–17

8 6

Determine whether these events are impossible, unlikely, even chance, likely, or certain: a

Winning the lottery

Scoring 140% on an exam

Flipping tails with a coin

Next week will have 8 days

A number greater than 4 is rolled with a standard die

When a standard die is rolled, we get a number from 1 to 6

In a prize draw, 8 red tickets and 8 blue tickets are handed out. Choose the best description of the chance of each event: • Impossible • Unlikely

• Even Chance

The winning ticket being black

The winning ticket being red

The winning ticket being red or blue

• Likely • Certain

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Consider the spinner.

Determine whether each event has an even chance:

Spinning an even number

Spinning 7 or more

Spinning 12 or less

Spinning 7 or less

7 11

There are 20 marbles in a jar: 7 red, 10 green and 3 blue. If a marble is chosen at random: a

What colour marble is most likely to be chosen?

What colour marble is least likely to be chosen?

One marble is drawn out of this jar containing 3 black, 3 red, and 5 blue marbles. a

Identify the colour of the marble that you are most likely to draw.

Describe the likelihood that you draw a marble that is not Blue.

These three cards are shuffled. One card is then drawn at random:

Describe the likelihood of drawing a Heart.

Describe the likelihood of drawing a number card.

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These four cards are shuffled. One card is then drawn at random:

Describe the likelihood of drawing a Jack, Queen or King.

Describe the likelihood of drawing a card with a number less than 4.

Yuri plays a game using this spinner: a

Describe the likelihood that he spins a number less than 15.

Describe the likelihood that he spins an even number.

Describe the likelihood that he spins the number 4.

Describe the likelihood that he spins the same number three times in a row.

Describe the likelihood that he spins a number between 5 and 14.

Reasoning

19, 22

18–20, 22

12 7

19–22

A twelve-sided die is rolled. Determine the likelihood that the outcome is 8 or higher. Explain your reasoning.

A standard six-sided die is rolled. Which outcome is least likely? Justify your answer.

Rolling a prime number

Rolling a number less than 3

Rolling an even number

Consider drawing a card from a standard deck of cards. Order the events from least likely to most likely. Justify your ordering. 1.

Drawing a black card

3. Drawing a diamond

2. Drawing a seven 4. Drawing a card numbered 2–10

Think of an example where multiple events are equally likely to happen, but the events do not have an even chance of occurring. Use your example to explain the difference between even chance and equally likely.

Marie is watching a car race where there are 48 Holdens and 16 Fords. Each car is equally likely to win. Marie says that it’s impossible for a Ford to win. Is Marie correct? Explain your reasoning.

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Problem-solving 23, 24 23

24, 25ab

23–25

James has a bag of eleven counters numbered 1 to 11. Mae has a bag of twenty counters numbered 1 to 20. Each chooses a counter from their own bag without looking. Determine whether these statements are true or false: a

James is more likely than Mae to choose a counter numbered 11.

They are equally likely to choose a number less than 3.

James is more likely than Mae to choose an odd number.

Mae is less likely than James to choose a counter numbered 16.

Kenneth has a six-sided die. Lisa has a twelve-sided die. They decide to roll both their dice and the winner is the person who rolls the highest. Who is more likely to win? 3 11

Rochelle draws a card from a standard 52 card deck, and writes down the suit of the card. a

How many different outcomes could there be?

Are the events equally likely? Explain your reasoning.

Identify different outcomes that are also equally likely.

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16.02 Probability as a number After this lesson, you will be able to… • calculate the probability of an event using the formula. • apply probability calculations to real-world situations. • understand the significance of probability on a number line and how to interpret its value.

Probability as a number The likelihood of an event can be placed on a spectrum from 0 to 1 using fractions or decimals, or from 0% to 100% using percentages: Impossible

Even chance

Certain

Unlikely

0 0%

Likely

0.4 8%

0.5 50%

0.63 72%

100%

A probability can never be less than 0 or more than 1. The larger the number, the more likely it is, and the smaller the number, the less likely it is. To find the probability of an event, we can use the formula:

A favourable outcome is the outcomes we want, or are looking for. If every outcome is favourable, then we have a probability of 1. If there are no favourable outcomes, then probability is 0.

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Charlene spins this spinner: a

List the sample space.

Find the probability of spinning a number greater than 29.

Find the probability of spinning a number in the twenties.

A number is randomly selected from this list: {1, 3, 3, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10}

Ex 3

What is the probability of selecting a 1?

What is the probability of selecting a 3?

What is the probability of selecting an 8?

Which number is most likely to be selected?

A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red. a

If the probability of picking a red marble is , how many red marbles are there in the jar?

Find the probability of picking a blue marble as a percentage.

Reasoning

15–17

Paul has 23 marbles in a bag and 4 of them are black. Paul picks a marble from the bag without looking. Find the probability that Paul picks a marble that is not black. Justify your answer.

A bag of marbles contains different coloured marbles. The probabilities of drawing a particular colour are listed: • Red: 10% • Blue: 20% • Green: 30% • Orange: 40% Determine whether the bag could contain marbles of a colour that is not listed. Explain your reasoning.

Consider this spinner. Identify the two equally likely events. Justify your answer with probability.

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Problem-solving 18–19 18

18–19

18–21

A book has pages numbered from 1 to 100. If the book is opened to a random page, what is the probability that the page number: a

is a multiple of 9?

has the digit 6 in the page number?

Draw a spinner that has four outcomes with 2 of them being likely to occur and 2 unlikely to occur.

A card is drawn from a standard deck of cards:

Find the probability of drawing the Jack of Hearts or the Jack of Clubs.

Rosey draws a 6 of Spades and puts it back in the deck and asks you to draw a card. Find the probability that you will draw a card with a number larger than 6, and the same colour as Rosey’s card.

The 26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is to be picked out of the bag at random, find the probability of picking: J

K, Y or R.

M, K, D, O, H or B.

A letter in the word PROBABILITY.

A letter in the word WORKBOOK.

16.02 Probability as a number mathspace.co

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Consider a full set of scrabble tiles where the last two letters are the two “blank tiles”: A A A B C C E E E F G G I I I L M M O O O R R R T T T V V W

A D E G I N O R T W

A D E H I N O R T X

A D E H J N O R T Y

A D E I K N O S U Y

A E E I L N P S U Z

A B E E E F I I L L N O P Q S S U U

One letter tile is drawn at random:

Find the probability that it is a “G” or an “R”.

Find the probability of drawing a vowel.

Luigi and Danielle enter a raffle where 80 tickets are on sale. Luigi buys 1 ticket and Danielle buys 5 tickets. a

Find the probability that Luigi wins if all the tickets were sold.

Find the probability that Danielle wins if all the tickets were sold.

Find the probability that Danielle wins if only 70 tickets were sold.

A twenty-sided die is rolled 100 times. How many times should we expect to roll a 14 or more?

Reasoning

12–13, 15

12–15

A six-sided die is rolled 24 times. How many times should we expect to roll a 1? Explain your process.

If you flip a fair coin 10 times, do you think it will land on heads exactly 5 times? Explain your reasoning.

Consider a game where a standard die is tossed. The rules of the game are: • If a 1 is rolled, you win $2. • If a 6 is rolled, you win $1. • If anything else is rolled, you lose.

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Would you expect to win money or lose the game?

Rowan plays the game once, and the die lands on 1. He plays again, and the die again lands on 1. Do you think it is possible for Rowan to roll a 1 on his next roll? Explain.

Suppose it costs $1 to play the game. Elsie claims that if she plays the game 3 times, then she is sure to win money. Do you think Elsie should play the game 3 times? Explain your reasoning.

Mathspace Victorian Curriculum – Year 7 mathspace.co

A die is rolled 358 times. a

If it lands on a six 12 times, what is the probability that the next roll will land on a six?

Does the outcome of the next roll depend on the outcome of previous rolls?

Problem-solving 16 16

16, 18–19

16–20

A bag contains 29 yellow marbles, 21 blue marbles and 10 pink marbles. If a marble is randomly selected from the bag 300 times with replacement, find the number of times that you would expect to pick a marble that is: a

Yellow

Pink

Blue or pink

Yellow, blue or pink

Neil takes these cards and shuffles them up. He draws one card, shows it to you, and keeps it. You then draw one card from the remaining seven cards, and you win if the number on your card is higher than Neil’s. Find your chance of winning if Neil draws these cards: a

Consider these dice:

Four-sided die

Ten-sided die

Six-sided die

Twelve-sided die

1 3

Twenty-sided die

Eight-sided die

3 18 12

Identify the die that would give the highest probability of rolling: a

16.03 Theoretical probability mathspace.co

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Amelia selects a card 260 times from a standard deck of 52 cards, she puts the card back into the deck each time before drawing a new one.

How many diamonds can she expect to draw?

How many black cards can she expect to draw?

How many royal cards (Kings, Queens and Jacks) can she expect to draw?

How many times can she expect to draw the King of diamonds?

Sally enters a raffle every week where 130 tickets are sold each week. If there are 52 weeks in a year, find the number of times she can expect to win in a half-year period if she purchases: a

1 ticket every week

10 tickets every week

16.04 Experimental probability After this lesson, you will be able to… • define experimental probability. • calculate experimental probabilities. • predict the number of occurrences of an event in a given number of trials by multiplying the experimental probability by the number of trials. • analyse and evaluate experimental data to make informed predictions about future events.

Experimental probability Theoretical probability describes how likely an event is to occur. Experimental probability describes how frequently an event did occur in an experiment.

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Example 1 A die is rolled 121 times, and the results are displayed in the table: Number Frequency

1 18

2 21

3 20

4 21

5 22

6 19

a What is the experimental probability of rolling a 5 with this die?

Create a strategy Use the formula for experimental probability.

Apply the idea We know that the die was rolled 121 times, as it is given above. When we look at number 5 on the table, we can see that it came up 22 times, so we have the:

Reflect and check Notice that if we add up all the numbers in the second column, we get 121, which is the total number of times we rolled the dice.

b What is the experimental probability of rolling a 4 or higher with this die?

Create a strategy Determine how many times the die showed a number 4 or higher then use the formula for experimental probability

Apply the idea We need to add up the frequency of numbers 4 to 6. This tells us how many times the die rolled 4 or higher. So,

Add the frequencies Evaluate

We already know that the die was rolled 121 times, so we have the:

Example 2 An insurance company found that in the past year, of the 2558 claims made, 1493 of them were from drivers under the age of 25. Give your answers to these questions as percentages, rounded to the nearest whole percent. 726

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a What is the experimental probability that a claim is filed by someone under the age of 25?

Create a strategy Use the formula for experimental probability and convert to a percentage by multiplying by 100%.

Apply the idea Substitute the values

Multiply by 100%

Evaluate

Round to the nearest whole percent

b What is the experimental probability that a claim is filed by someone 25 or older?

Create a strategy Find the numbers of the claims that were made by people who are 25 or older and use the formula for experimental probability.

Apply the idea The number of claims made by drivers 25 or older is 2558 - 1493 = 1065. Substitute the values

Multiply by 100%

Evaluate

Round to the nearest whole percent

Example 3 The experimental probability that a commuter uses public transport is 50%. Out of 500 commuters, how many would you expect to use public transport?

Create a strategy To find the expected number of commuters, multiply the total number of commuters by the probability.

Apply the idea Multiply by a probability

Convert to a fraction

Evaluate

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At a set of traffic lights, the green light is on for 24 seconds, then the orange light lasts for 3 seconds, and then the red light is on for 13 seconds. This cycle then repeats continuously. If a car approaches the traffic light at a random moment, calculate the probability that the light will be: a

Green

Orange

Red

To prepare for the week ahead, a restaurant keeps a record of the number of each main meal ordered throughout the previous week. a

How many meals were ordered altogether?

Calculate the experimental probability that a customer will order a beef meal.

Meal Chicken Beef Lamb Vegetarian

Frequency 54 32 26 45

Give your answer as a percentage, rounded to the nearest whole percent. 9

11 Ex 3

High school students attending an international conference were asked to register what language they speak other than English. The results are recorded in the table: a

How many students attended the conference?

Find the probability that a student chosen at random speaks:

Language French Arabic Spanish Mandarin Other

French

Mandarin

iii

Arabic or Spanish

Spanish or Other

Frequency 20 13 21 19 37

A factory produces tablet computers. In March it produced 8000 tablets, and 240 were found to be faulty. a

Calculate the experimental probability that a tablet produced by the factory is faulty.

The factory plans to produce 9000 tablets in April. How many should they expect to be faulty?

A car manufacturer found that 1 in every 4 cars they were producing had faulty brake systems. If they had already sold 5060 cars, how many of those would they expect to need to be repaired? Uther flipped a coin 14 times. a

Theoretically, how many times would he expect a tail to appear?

After flipping the coin 14 times, Uther noticed that a tail had appeared 4 times. Find the experimental probability of getting a tail.

Derek spun this spinner 20 times. a

How many times would he expect the arrow to land on X?

After he finished spinning, he noticed that the arrow fell on X 8 times. Find the experimental probability of getting an X.

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Five schools compete in a basketball competition. The results from the last season are recorded in the table: Schools competing each game St Trinian’s vs Ackley Bridge Ackley Bridge vs Summer Heights Lakehurst vs Marquess Marquess vs St Trinian’s St Trinian’s vs Summer Heights St Trinian’s vs Lakehurst Ackley Bridge vs Marquess Marquess vs Summer Heights Lakehurst vs Summer Heights Ackley Bridge vs Lakehurst

Winner St Trinian’s Summer Heights Marquess St Trinian’s Summer Heights Lakehurst Ackley Bridge Summer Heights Lakehurst Lakehurst

Calculate the experimental probability that Ackley Bridge wins one of their matches. 15

Georgia is drawing a card out of a deck of 10 cards, labelled from 1 to 10. She drew a card and returned it to the bag 40 times. a

How many times would she expect to get the card with 6 on it?

After she finished, she noticed that the card with 6 on it was drawn out of the bag 21 times. Find the experimental probability of getting a card with 6 on it.

Reasoning 16

16, 18

16–18

Hermione rolled a standard six-sided die 60 times. a

State the theoretical probability of rolling a six.

How many times would she expect a six to appear?

After she finished rolling the die, she noticed that she had rolled a six 24 times. Find the experimental probability of getting a six.

Was the theoretical and experimental probability of getting a six the same? Explain why or why not.

A card is selected at random from a standard deck of cards. The result is recorded and the card is placed back in the deck. This is repeated multiple times. Consider the two tables: Table 2

Table 1 Colour Black Red

Frequency 54 55

Suit Spade Heart Diamond Club

Frequency 21 34 21 33

In which table are the results closest to the expected outcome? Explain your thinking.

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Federica claims that she has created a random number generator which generates a number from 1 to 10. a

She ran the program 10 times and generated this frequency table. Explain whether you think the program is fair or not. Number Frequency

1 1

2 2

3 0

4 0

1 9

2 11

3 12

Problem-solving 19–20

7 1

4 10

5 8

6 10

8 2

9 2

Calculate the experimental probability that a train will be on time on Monday, as a whole percentage.

Which day had the highest experimental probability of a train being on time?

Calculate the experimental probability of a train arriving on time across the entire week, as a whole percentage.

Boxes of toothpicks are examined and the number of toothpicks in each box is recorded in the table.

Contain 89 toothpicks

Contain more than 90 toothpicks

Contain less than 90 toothpicks

7 11

19–20

The table shows the number of trains that arrived on time at the local station during the week.

If the number of toothpicks of another box were counted, find the experimental probability it will:

6 1

10 1

She ran the program 100 times and generated this frequency table. Explain whether you think the program is fair or not. Number Frequency

5 0

8 9

10 10

19–21

Day Monday Tuesday Wednesday Thursday Friday

Number of toothpicks 87 88 89 90 91 92 93

The table shows the fighting style of each competitor in a mixed martial arts tournament fought last year.

9 10

Event Karate Wrestling Judo Taekwondo

Number of trains 29 24 21 27 23

On time 22 23 19 22 20

Number of Boxes 0 6 4 1 1 2 1

Frequency 40 54 47 59

Find the total number of competitors.

Find the experimental probability of a wrestler winning.

Find the experimental probability of a Taekwondo fighter not winning.

If 100 more competitors joined the competition, how many of them would you expect to use Karate as their fighting style?

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The probability actually reduces as we go down the table, so 1 roll is the most likely, followed by 2 and 3 and so on. Yet we can see that the frequency for 8 rolls is greater than that for 7 rolls and that the frequency for 16 rolls is greater than that for 15.

Investigate This sort of discrepancy is to be expected for this number of simulations. 1.

Why would we expect the discrepancy?

Had we done many millions of simulations, so that the frequencies were higher for these numbers of rolls, what might be expect to happen?

What might be the same or different to the above 3000 rolls?

The UK national lottery The jackpot for the national lottery in the UK can be won by selecting 6 winning numbers from 1 to 59. How it works The machine contains 59 balls, numbered from 1 to 59. The first 6 balls to be drawn from the machine constitute the jackpot numbers. So if the balls drawn are 11, 15, 22, 31, 40 and 48 and your ticket has those numbers, then you win the jackpot, possibly making you an instant millionaire. If, as has happened on numerous occasions, more than one ticket has those numbers, the prize is shared. Popular numbers It is estimated that 10 000 people play the numbers 1, 2, 3, 4, 5, 6 each draw. People who do not understand probability say “that will never happen,” but that combination of numbers stands exactly the same chance as any other combination, such as 2, 5, 23, 27, 31 and 58, which was the result on 4th February 2023. The reason many people would not play 1, 2, 3, 4, 5, 6 is not because it has less chance than any other combination, but simply because 10 000 people play that combination. If it did come up, the money would be split 10 000 ways. What are the odds? So, what is the chance of getting the winning numbers? It can be calculated that there are 45 057 474 combinations of 6 numbers chosen from 59, so the probability of any one set of 6 numbers being the winning ones is a bit under 1 in 45 million. With those odds, you might say there is no chance of winning. However, between 15 and 45 million people play each draw. With such large numbers playing, the jackpot is won quite frequently. The longest string of weeks with no winner is 15.

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Law of large numbers This is the law of large numbers. If the chance of winning is 1 in 45 million and only one combination is played, then, on average, it should be won every 45 million weeks. However, it could be won in the first week. The more different combinations of numbers that are played, the greater the chance on average of having a winner each week. We know the chance of winning the lottery is approximately 1 in 45 million. We know that between 15 and 45 million people play the lottery each week. We know that some combinations are popular, 1, 2, 3, 4, 5, 6 being an example, but for simplicity let us assume that 15 million different combinations are played each draw. That gives us a 15 million in 45 million chance of the jackpot being won. This simplifies to a 1 in 3 chance. We need to devise a scheme for using the numbers from the roll of a die. A suitable scheme would be: • 1, 2, jackpot is won (J) • 3, 4, 5, 6, jackpot is not won (known as a rollover) (R)

Investigate Here are 10 rolls of the dice: 1, 3, 1, 5, 3, 5, 1, 1, 5, 5 Using these results and the scheme above, we have obtained the following results: J=4 R=6 1.

Are these results in line with what you would expect?

What percentage of the draws resulted in the jackpot being won?

What is the expected percentage?

What was the longest streak of jackpot wins?

What was the longest streak of rollovers?

Using a die to simulate the UK national lottery Let’s simulate more outcomes. Complete 5 more sets of 10 rolls of the die, and record them in the provided chart. You might use a random number on your calculator or a spreadsheet to simulate rolling a die. Look at each simulation individually, and complete the table in the provided chart. Do the same thing with combined results, including the given rolls above, giving a total of 60 rolls. Remember, you may have to re-count streaks as one simulation might finish JRR and the next one might start RRRJ, giving a streak of 5. When you have 60 rolls, discuss the following questions with a classmate: Is the percentage of jackpots closer to the expected percentage? Investigation: Simulations mathspace.co

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If possible, pool your results with other students. What happens as you get more data? Suppose we used a program to run 6 000 000 simulations like the one in the previous activity. The jackpot percentage from 10 runs of this program were: Run 1 2 3 4 5 6 7 8 9 10

Jackpot percentage 33.354 33.341 33.357 33.350 33.364 33.335 33.347 33.362 33.329 33.330

Law of large numbers • It is interesting to note that the first 8 of the above simulations are higher than the expected percentage and the last two are lower. They are all close to the expected percentage, which we would expect with a large number of simulations. An aspect of the law of large numbers is that we would expect the outcome percentages to be close to the expected percentage with a large number of simulations. However, it tells us nothing about the pattern for the next few weeks.

Investigate Think about another experiment or situation that could be tested or explored using a large number of simulations. Let’s think about the pieces required to run this experiment. 1.

What is your situation? Describe it and how it involves repeated trials.

What do you expect the result to be?

How can you set this up using technology? Describe how you set up a way to run the number of trials needed.

Run the trials, 50 then 100, then 1000, then 5000. Comment on how the results of the simulation changed in each round.

Discussion

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Describe the law of large numbers in your own words.

Think of 3 real world situations where simulations could determine whether or not the situation is fair.

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16 Chapter review

A standard six-sided die is rolled. Which outcome is least likely? A

Rolling a prime number

Rolling a number less than 3

Describe the likelihood that corresponds to each probability as impossible, unlikely, even chance, likely, or certain: a

0.6

80%

A twelve-sided die is rolled.

Find the probability of rolling 6 or more.

Find the probability of rolling less than 6.

Find the probability of rolling a multiple of 4.

12 7

Rolling an even number

8 6

For each spinner, find the probability of getting a star. Write your answer as a percentage. a

A circular spinner is divided into unequal parts. The green sector takes up an angle of 250 degrees at the centre. The red sector takes up an angle of 60 degrees at the centre and the blue sector takes up the remainder of the spinner. What is the probability that the spinner will land on blue?

A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red.

If the probability of picking a red marble is

Find the probability of picking a blue marble.

, how many red marbles are there in the jar?

Determine whether these events have equally likely outcomes: a

Getting a head or a tail from tossing a fair coin

Drawing a random card from a standard deck of cards

Choosing a random letter from the word MATHEMATICS

Rolling a 1 or 2 on a six-sided die

Getting a random marble from a bag containing red and blue marbles of equal number but of different sizes

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A letter is chosen at random from the word COCOONS. a

Which letter is most likely to be chosen?

Find the probability that the chosen letter is a “C”.

A three-digit number is formed only using the digits 6, 7, and 8. It can be three of the same, two of one and one of another, or one of each in any order. a

Find the probability that the number formed is odd.

Find the probability that the number formed is palindromic (the same forwards as backwards).

Consider a full set of scrabble tiles where the last two letters are the two “blank tiles”: A B E F I L O R T V

A C E G I M O R T V

A C E G I M O R T W

A D E G I N O R T W

A D E H I N O R T X

A D E H J N O R T Y

A D E I K N O S U Y

A E E I L N P S U Z

A E E I L N P S U

B E F I L O Q S U

One letter tile is drawn at random: a

Find the probability that it is a “G” or an “O”.

Find the probability of drawing a consonant.

A class of 30 students conducted an experiment where they flipped a coin 10 times each. The total number of heads flipped was 160. Calculate the experimental probability of flipping a head in this experiment.

If the probability of an event is , how many times would you expect the event to occur in 85 trials?

An eight-sided die is rolled 15 times. How many times should we expect to roll a 4?

A die is rolled 670 times. a

If it lands on a six 348 times, what is the probability that the next roll will land on a six?

Does the outcome of the next roll depend on the outcome of previous rolls?

A twenty-sided die is rolled 70 times. How many times should we expect to roll a 12 or more?

Beth enters a raffle every week where 390 tickets are sold each week. If there are 26 weeks in a half-year, find the number of times she can expect to win in a half-year period if she purchases:

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9 tickets every week

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An insurance company found that in the past year, of the 2558 claims made, and 1493 of them were from drivers under the age of 25. Calculate the experimental probability, as a whole percentage, that a claim is filed by someone: a

Under the age of 25

25 years or older

A retail store served 951 customers in December, and there were 26 complaints during that month. Calculate the experimental probability that a customer complains. Give your answer as a percentage, rounded to the nearest whole percent.

The experimental probability that a person uses public transport is 33.33%. Out of 3 people, how many would you expect to use public transport? Round your answer to the nearest whole number.

Hermione rolled a standard six-sided die 60 times.

Determine the theoretical probability of rolling a six.

How many times would she expect a six to appear?

After she finished rolling the die, she noticed that she had rolled a six 24 times. Find the experimental probability of getting a six.

Was the theoretical and experimental probability of getting a six the same? Explain your answer.

To prepare for the week ahead, a restaurant keeps a record of the number of each main meal ordered throughout the previous week. a

How many meals were ordered altogether?

Calculate the experimental probability that a customer will order a chicken meal.

Meal Chicken Beef Lamb Vegetarian

Frequency 52 24 45 36

Give your answer as a percentage, rounded to the nearest whole percent. c 22

What is the probability of ordering a non-vegetarian meal?

The table shows the number of trains that arrived on time at the local station during the week:

Day Monday Tuesday Wednesday Thursday Friday

Number of trains 23 28 24 27 26

On time 16 26 21 26 21

Calculate the experimental probability that a train will be on time on Monday, as a whole percentage.

Calculate the highest experimental probability of a train being on time out of each of the five days, as a whole percentage.

Calculate the experimental probability of a train arriving on time across the entire week, as a whole percentage.

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Big ideas Different representations of data highlight different characteristics of the data.

Curriculum Outcomes VC2M7ST01 acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data VC2M7ST02 create different types of displays of numerical data, including dot plots and stemand-leaf plots, using software where appropriate; describe and compare the distribution of data, commenting on the shape, centre and spread including outliers and determining the range, median, mean and mode VC2M7A04 investigate, interpret and describe relationships between variables represented in graphs of functions developed from authentic data VC2M7ST03 plan and conduct statistical investigations for issues involving discrete and continuous numerical data, and data collected from primary and secondary sources; analyse and interpret distributions of data and report findings in terms of shape and summary statistics

17 Statistics Chapter outline 17.01 Types of data 17.02 Frequency tables 17.03 Summarise data 17.04 Outliers 17.05 Dot plots and column graphs 17.06 Stem-and-leaf plots 17.07 Pie charts 17.08 Line graphs Investigation: Statistical investigations Chapter 17 review

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17.01 Practice questions Understanding

Explorer 1, 2

Adventurer 1, 2

What are the two types of statistical data?

Match the terms to their corresponding definitions: i

Categorical data

Discrete numerical data

iii

Continuous numerical data

These are data that are counted.

These are data that are collected as a set of words.

These are data that are measured and can take any value within a range.

Fluency 3

3, 4a-h, 5a-f

3, 4i-p, 5g-l

Determine whether these data are numerical or categorical: a

Amount owing on lay-by

Types of vegetables

Brands of tablets

Types of folders

Months of the year

Brands of phones

Maximum snowfall

Daily UV index

Types of dogs

Favourite colours

Cost of a PS4 game

Types of fruits

Determine whether these examples of numerical data are discrete or continuous: a

The number of classrooms in your school

The ages of a group of people

Amount of money paid for walking the dog

Lengths of cats’ whiskers

How long it takes you to fall asleep at night

The rainfall in your city

Amount of money owing on a loan

The top speed of different models of motorcycles

Daily humidity

The time taken to run 200 metres

The number of people taller than you in your class

The value of people’s rare coin collections

m The distance from Earth to nearby galaxies

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Trailblazer

The hourly pay rate for a particular job

Number of children in your sports team

The strength of an earthquake

Mathspace Victorian Curriculum – Year 7 mathspace.co

Ex 1, 2

For each survey question, classify the answer as categorical, discrete numerical or continuous numerical data: a

How far do you travel to school each day?

How many siblings do you have?

How many people live in your house?

What languages do you speak at home?

How long did you spend on homework last night?

How many apps do you have on your phone?

Where was your most recent holiday?

What modes of transport do you use to get to school?

How long does it take for you to get to work each day?

How many times have you ridden your bike?

How tall are you?

Do you think recess is long enough?

Reasoning 6

6, 7

7, 8

8, 9

Determine whether these examples of numerical data are discrete or continuous. Explain your reasoning. a

The amount of soft drink you drink in a day

The number of cans of soft drink you drink in a day

The number of views that a YouTube channel has

The world record time for solving ten Rubik’s cubes

For each survey question, classify the answer as categorical, discrete numerical or continuous numerical data. Explain your reasoning. a

How long did you sleep for last night?

What kinds of pets do you own?

How many pets do you own?

How much does your school bag weigh?

The answer to this question is categorical: “Which students scored higher than 70 in your most recent Science test?” How could you rewrite the question so the answer is numerical? Justify your answer.

The answer to this question is numerical: “How many pairs of shoes do you own?” How could you rewrite the question so the answer is categorical? Justify your answer.

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Problem-solving 10, 12 10

11, 12

11–13

Consider the data set that shows the distance to the nearest hospital (in km): 26.4, 28.7, 15.1, 37.5, 11.8, 7.9, 16.7, 6.9, 5, 24.9, 35.3, 45.5

Is this data set continuous or discrete?

What is the farthest distance to the nearest hospital?

Glen surveys his class and asks, “What is your favourite movie genre?” The results are recorded in the table: Genre Students

Drama 5

Comedy 7

Action 6

Horror 2

What type of data is displayed in the table?

Which genre is the most popular among his class?

How many people did Glen survey?

Ten people were asked about the number of hours they spent outside last weekend. The results of the survey are listed: 5, 9, 9, 4, 6, 9, 4, 9, 6, 4

What type of data is this? If it is numerical, specify if discrete or continuous?

How many people spent exactly 6 hours outside?

What was the longest that anyone spent outside?

What was the most common amount of time spent outside?

Ten people were asked about the number and type of pets they own. The results of the survey are recorded in the table: Twelve fish One dog

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Three turtles One dog, one cat

Two cats One cat

What type of data is the type of pet?

How many cats do these 10 people own altogether?

How many pets do these 10 people own altogether?

What is the most common survey response?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Two cats Five ferrets

One dog One dog

17.02 Frequency tables After this lesson, you will be able to… • create frequency tables. • interpret data using frequency tables.

Tables, frequencies, and modes One common way of collecting data is through a survey. Conducting a survey involves choosing a question to ask and then recording the answer. This is great for collecting information, but at the end we are left with a long list of answers that can be difficult to interpret. A table is a convenient way to organise data, making it easier to analyse and interpret. Frequency refers to the number of times an item or value appears in a dataset. The mode is the value that occurs most frequently in the data set. When analysing data, we can use tables to count the frequency of each value, helping us identify the mode and better understand the distribution of the data.

Example 1 Yvonne asks 15 of her friends what their favourite colour is. She writes down their answer. Here is what she wrote down: Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow, Green, Blue, Yellow, Pink, Yellow, Pink, Pink a Count the number of each colour and complete the table. Colour

Pink

Green

Blue

Yellow

Number of friends

Create a strategy Count the number of times Yvonne wrote each colour.

Apply the idea Colour Number of friends

Pink

Green

Blue

Yellow

b Which colour is the mode?

Create a strategy

Apply the idea

Choose the colour that occurs the most frequently.

The mode of the data is pink because it occurred the most often.

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The table shows the students’ responses to their favourite sport. Are these statements true or false? a

Ten students chose cricket as their favourite sport.

Football is the favourite sport of 15 students.

More students chose tennis as their favourite sport than rugby.

Eight students chose swimming as their favourite sport.

Fluency 3

3ac, 4–6, 8, 11

12, 12, 13, 11, 13, 9, 13, 14, 9

14, 2, 5, 2, 2, 5, 5, 7, 12, 13, 13, 14 Score Frequency

21 31

22 37

23 39

24 33

25 38

A gymnast received these scores over 12 rounds of a competition:

3bc, 4, 7, 9, 11

26 34

Score Frequency

Calculate the missing frequency to find how many times the gymnast scored 8. Ex 1

Frequency 10 15 8 12 5

Find the mode(s) of these sets of scores:

3ac, 5, 6, 8, 10

Sport Rugby Football Swimming Cricket Tennis

5 2

6 1

7 1

9 1

Zam asks a number of her friends what their favourite sport is, and writes down their responses: Football, Rugby, Football, Swimming, Netball, Rugby, Rugby, Swimming, Netball, Football, Swimming, Rugby, Cricket, Rugby, Cricket

Ex 2

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Construct a frequency table.

Which sport is the mode?

How many friends did Zam survey?

Zack conducted a survey on the average number of hours his classmates played video games per day and displayed his data in the table:

No. of hours 0 1 2 3 4

Frequency 10 12 8 4 2

Price (in cents per litre) 130.0–134.9 135.0–139.9 140.0–144.9 145.0–149.9

Frequency 4 9 12 5

How many classmates did Zack survey?

Find the mode of the data.

How many classmates played video games for 2 hours or less?

How many classmates played video games for at least three hours?

As part of a fuel watch initiative, the price of petrol at a service station was recorded each day for 30 days, and displayed in the frequency table:

Mathspace Victorian Curriculum – Year 7 mathspace.co

What was the maximum price that could have occurred in the 30 day period?

Find the number of days where the price was above 139.9 cents.

The frequency table shows the resting heart rates of a number of people taking part in a study: a

Find the number of people taking part in the study.

How many people had a resting heart rate between 40 and 59 beats/minute?

How many people had a resting heart rate of below 50 beats per minute?

A group of 50 people were surveyed about how many movies they had watched in the past month. The results are recorded in the table: a

Find the number of people who watched 10 or more movies.

Find the number of people who watched less than 5 movies.

Find the number of people who watched more than 4, but less than 15 movies.

Mrs. Tran asks her students how many bedrooms they have in their house and writes their responses:

Number of movies watched 0–4 5–9 10–14 15–19

Number of bedrooms 1 2 3 4 5

3, 3, 4, 4, 3, 3, 4, 5, 2, 2, 4, 4

Heart rate 30–39 40–49 50–59 60–69

Complete the frequency table.

How many students did Mrs. Tran survey?

What is the modal number of bedrooms?

Frequency 20 22 28 37

Frequency 11 14 18 7

Number of students

Bill asks nine of his friends which pet they would like and writes down their responses: Dog, Fish, Cat, Cat, Fish, Dog, Fish, Fish, Dog Construct a frequency table for these responses.

Reasoning 12

12, 13

13, 14

The frequency table shows the number of apples sold by a fruit vendor in a week. Explain how you would find the total number of apples sold. Number of Apples 10 15 20 25

Frequency 5 8 12 6

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A school nurse is investigating the number of sick days taken by students in a class, and constructs the table: a

Construct a frequency table for the number of sick days.

What is the number of students who had no sick days?

What is the most common number of sick days for the class?

Student Sally Lachlan Tara Xavier Luigi Nadia Lisa Dave Bill Danielle

Sick days 0 2 1 4 2 4 0 1 3 4

Fifteen airline passengers weigh their luggage. The weights in kilograms are listed: 17, 20, 19, 19, 20, 21, 20, 22, 20, 19, 17, 16, 20, 20, 16 a

Construct a frequency table for this data.

How many passengers had luggage that weighed more than 20 kg?

The airline charges each passenger $3 for every kilogram above 20 kg. Calculate the total charge for overweight luggage for this group of passengers.

17.03 Summarise data After this lesson, you will be able to… • calculate the mean of a data set. • identify the median of a data set. • calculate the range of a data set. • use mean, median and range to solve problems.

Mean The mean is the average of the values in the data set. It is a measure of centre. Three friends are planning a trip to Alice Springs. They plan to fly there, and learn that the airline has a rule: each person can only bring 20 kg of stuff in their bags. On the night before the flight they weigh their luggage and find the weights form this data set: 17, 18, 22 One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now?

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Instead we can select the median, which is the middle score. We remove the biggest and the smallest scores to get: $400, $400, $430, $470, $490 Then the next biggest and the next smallest to get: $400, $430, $470 Then the next biggest and the next smallest to get: $430 There is only one number left, and this is the median - so for this data set the median is $430. This weekly income is much closer to the other scores in the data set, and summarises the set better. The median is the number in the middle of a numerical data set. • If the list has an odd amount of numbers, the median is the one right in the centre. • If the list has an even amount of numbers, the median is the number halfway between the two middle ones. Half the numbers in the list will be bigger than the median, and half will be smaller. Sets ordered smallest to largest Median Odd number of scores:

Even number of scores:

Median:

+ 2

Example 2 Find the median of these scores: 3, 18, 10, 19, 12, 5, 6, 20, 7

Create a strategy

Apply the idea

We need to put the scores in order and find the middle score.

The scores in order are: 3, 5, 6, 7, 10, 12, 18, 19, 20 The middle score is 10 because it has 4 scores above it and 4 scores below it. The median of the scores is 10.

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Example 3 Two soccer teams, the Lightning Bolts and the Thunder Strikes, have played 5 games each. The scores for each team are listed below: The Lightning Bolts’ scores: 6, 10, 8, 12, 9 The Thunder Strikes’ scores: 5, 8, 11, 9, 12 a Calculate the mean, median, and mode for both teams.

Create a strategy We need to put the scores for both teams in order. Use the formula for the mean, find the middle scores, and find the scores which occur most often in the data set if applicable.

Apply the idea The scores in order for the Lightning Bolts are: 6, 8, 9, 10, 12 Calculating the mean we have Use the formula

Add the numbers in the numerator

Perform the division

The median is 9 since this is the middle score. There is no mode, as no score occurs more than once. The scores in order for the Thunder Strikes are: 5, 8, 9, 11, 12 Calculating the mean we have Use the formula

Add the numbers in the numerator

Perform the division

The median is 9 since this is the middle score. There is no mode, as no score occurs more than once. b If the Lightning Bolts scored 14 goals in one game instead of 6, how would their mean, median, and mode change?

Create a strategy Change one of the scores of the Lightning Bolts from 6 to 14 to find the new mean and new median.

Apply the idea The new scores in order for the Lightning Bolts are: 8, 9, 10, 12, 14

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17.03 Practice questions Understanding

Explorer 1, 2

Adventurer 1, 2

Trailblazer 2, 3

Describe what the mean measures for a set of values.

Describe what the median measures for a set of values.

A diver measures how long she can hold her breath underwater over several dives. The median time is 3.9 minutes. Are these statements true or false? a

The longest she held her breath was 7.8 minutes.

Most of the time, she held her breath for less than 3.9 minutes.

For half the dives, she was able to hold her breath longer than 3.9 minutes.

The shortest time she held her breath was 1.95 minutes.

Most of the time, she held her breath for longer than 3.9 minutes.

Fluency 4 Ex 1

4a-d, 5, 6a-d, 7, 8, 10–13, 15

4c-f, 5, 6c-f, 7, 8, 11, 12 14, 16

4e-h, 5, 6e-h, 7, 9, 11, 12, 14, 16

Calculate the mean of these data sets: a

8, 5

7, 18, 9, 3

4, 10, 2, 9, 5

16, 4, 7, 2, 5, 19, 6, 11

11, 5, 14, 18, 15

8, 14, 24, 7, 12

13, 27, 9, 6, 8

13, 15, 7, 17, 9

Martha’s grades in her 5 subjects are: 91, 93, 94, 90 and 92 Determine the mean of Martha’s grades.

Ex 2

Ex 3

Determine the median of these data sets: a

1, 4, 6, 8

22, 12, 19, 16

7, 19, 6, 14, 18, 11

6, 9, 3, 5, 4

5, 5, 5, 5

13, 1, 10, 29, 12, 5, 16, 20, 6

5.3, 8.4, 2.6, 4.7, 3.5

49.4, 25.4, 47.2, 34.2, 28.4, 32.2

Two basketball teams, the Red Rockets and the Blue Blazers, have played 5 games each. The scores for each team are: • The Red Rockets’ scores: 75, 82, 88, 90, 85 • The Blue Blazers’ scores: 70, 78, 85, 87, 92 a

Calculate the mean, median, and mode for both teams.

If the score for the Blue Blazers first game of 70, was incorrect and they had actually scored 90 how would their mean, median, and mode change?

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Two sales representatives, Jacob and Sarah, have recorded their weekly sales (in units) for 5 weeks. The sales for each representative are listed below: • Jacob’s sales: 50, 55, 60, 65, 70 • Sarah’s sales: 45, 60, 55, 60, 65 a b

Calculate the mean, median, and mode for both representatives. The next 3 weeks of sales figures come in • Jacob: 40, 35, 38 • Sarah: 45, 48, 50

Does the mean, median and mode go down for both sales agents? 9

Two restaurants, Café Delight and Bistro Bliss, have received customer ratings (out of 10) for 5 weeks. The ratings for each restaurant are: • Café Delight’s ratings: 8, 9, 7, 8, 10 • Bistro Bliss’s ratings: 7, 9, 8, 9, 10

The star ratings look very similar to each other, predict if you think one might have a slightly higher mean than the other?

Calculate the mean, median, and mode for both restaurants.

Consider these scores: 3, 6, 8, 10, 2, 11

Ex 4

What is the largest number?

What is the smallest number?

Determine the range.

Calculate the range of these scores: 10, 16, 6, 18, 17, 11, 9, 15, 14

For each data set, determine: i

The mode

Score 11 14 Frequency 10 13

The range 19 4

22 7

29 12

Score Frequency

14 9

19 14

22 12

25 1

How many calls did the employee make during their shift?

For the length of calls made, determine: The mean

The median

iii

The mode

The range

A student scored these grades on his end of year tests: 87, 90, 85, 84, 89, 80, 85, 87, 87 a

How many tests did the student take? i

760

32 10

A phone support centre tracks the length of calls made each day in minutes. During a shift, one employee made calls of these lengths: 2, 3, 3, 2, 2, 4, 4, 2, 5, 13

i 14

The mean

The median

Mathspace Victorian Curriculum – Year 7 mathspace.co

Determine:

iii

The mode

Which set of data has the smallest range? • Set A: 13, 16, 19, 21, 24 • Set B: 15, 24, 50, 58 • Set C: 6, 8, 16, 35, 40 • Set D: 8, 14, 19, 24, 28

Which set of data has the largest range? • Set A: 101, 105, 118, 129, 136 • Set B: 19, 23, 25, 28, 29 • Set C: 22, 25, 43, 45, 57 • Set D: 104, 107, 113, 128, 141

Reasoning 17

17–19

18, 19

18, 20

Determine whether each statement is always, sometimes, or never true. Explain your reasoning. a

The median of a data set is a value in the set.

The mean of a data set is larger than all values in the set.

The range describes how far the lowest data values lies from the highest data value.

Consider this data set that represents the number of apps on six people’s phones: 110, 113, 117, 121, 127, 132 Explain what would happen to the range if each person downloads another 9 apps.

These are the number of points scored by a basketball team in each game of their previous season: 59, 67, 73, 82, 91, 58, 79, 88, 69, 84, 55, 80, 98, 64, 82 a

Determine the range and explain what it represents in context.

Explain why the range might be important for the team’s coach to know.

The television show “Dragon Duels” ran for three seasons, each with 6 episodes. The average fan ratings (out of 10) for each episode are recorded in the table:

Season 1 Season 2 Season 3

Episode 1 8.2 8.2 8.9

Episode 3 7.9 9.1 9.2

Episode 4 7.9 7.8 9.7

Episode 5 8 8.8 7.4

Episode 6 8.2 8.2 4.3

a b

Which season has the greatest mean rating? Which season has the greatest median rating?

If the rating for Episode 6 of Season 3 was 6.3 instead, would your answers to parts (a) and (b) change? Justify your answer.

Problem-solving 21–25 21

Episode 2 8.4 8.6 9.4

21–23, 25–27

22, 23, 25, 26, 28, 29

The range of a data set is 8, and the greatest value is 19. Determine the least value in the set. 17.03 Summarise data mathspace.co

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A set of 23 scores is arranged in increasing order. In what position does the median score lie?

A set of 34 scores is arranged in increasing order. Between which two scores does the median score lie?

James has played 10 games this season, with a mean of 1.5 goals per game. How many goals did he score in total?

The mean of four values is 21. If three of the values are 17, 3 and 8, what is the fourth value?

In a competition, a contestant must complete 12 challenges earning as many points as possible. Her scores for the first 11 challenges are: 66, 105, 38, 108, 67, 82, 92, 43, 119, 45, 102 Determine her score in the 12th round if the median of all of her 12 scores is 83.5. Counting numbers in increasing order such as 7, 8, 9 are called consecutive numbers.

Identify five consecutive numbers whose median is 6. 28

Identify 3 different sets of four numbers whose median is 12.

Identify four whole numbers whose median is 14 and whose range is 3.

17.04

Outliers

After this lesson, you will be able to… • identify outliers in a given data set. • describe the effect of outliers in the mean, mode, median and range of a data set.

Outliers In statistics, we tend to assume that our data will fit some kind of trend and that most things will fit into a “normal” range. This is why we look at measures of central tendency, such as the mean, median and mode . An outlier is a data point that varies significantly from the body of the data. An outlier will be a value that is either significantly larger or smaller than other observations. There can be more than one outlier.

Interactive exploration

Discover this concept in action online

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Example 2 Consider the following set of data: 53, 46, 25, 50, 30, 30, 40, 30, 47, 109 a Find the mean, median, mode, and range.

Create a strategy To find the mean, use the formula: To find the median find the middle score, to find the mode find the most frequent score. To find the range, use the formula: Range = Highest score − Lowest score

Apply the idea Use the formula

Evaluate the addition

Evaluate the division

To find the median, order the scores: 25, 30, 30, 30, 40, 46, 47, 50, 53, 109 The middle scores are: 40, 46 Find the average of the middle values

Evaluate

To find the mode, choose the score which occurs most often. Mode = 30 To find the range: Range = 109 − 25 = 84

Substitute the values Evaluate the subtraction

b Which data value is an outlier?

Create a strategy Choose the value that is much greater or much smaller than the rest of the data set.

Apply the idea Outlier=109

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17.04 Practice questions Understanding

1, 2

Adventurer 1, 2

Trailblazer 2, 3

What is an outlier?

Describe the effect of removing an outlier from a data set on these measures:

Mode

Range

For each scenario, an outlier was removed. Was the outlier smaller or larger than the values that remain? a

The mean decreased after the outlier was removed.

The mean increased after the outlier was removed.

The median decreased after the outlier was removed.

The median increased after the outlier was removed.

Fluency Ex 1

Explorer

766

4, 5, 7, 8, 9a, 10

4, 5, 7, 8, 9c, 10

4, 6–8, 9b

Identify the outlier for each data set: a

9, 10, 12, 9, 1

54, 52, 99, 50, 57

104, 115, 275, 109, 118, 121

4.0, 3.5, 5.1, 2.4, 1.6, 3.9, 3.5, 3.1

Consider the frequency table: a

What is the mode?

Which weight is the outlier?

If the outlier is removed, what is the new mode?

Compare the mode before and after removing the outlier.

Consider the frequency table: a

Which weight is an outlier?

If the outlier is removed, what is the new mean? Round your answer to one decimal place if needed.

If the outlier is removed, is the new mean higher or lower than the current mean?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Weight in kilograms 14 15 16 17 18 19 20

Frequency 1 0 0 3 6 4 2

Weight in kilograms 12 13 14 15 16 17 18

Frequency 2 5 1 2 0 0 1

For each scenario, determine whether the range would increase, decrease, or stay the same: a

If 25 is added to the data set: 1, 1, 2, 3, 3, 3, 4

If 10 is removed from the data set: 10, 42, 55, 60, 65, 70

If 5 and 27 are removed from the data set: 5, 12, 12, 13, 15, 15, 27

If 1 is added to the data set: 24, 25, 25, 26

Consider the data set in the frequency table: Suppose one score of 8 is changed to a 9. Determine whether each measure would change or not:

9 Ex 2

Mean

Median

Mode

Range

Score 1 2 4 5 6 8 9

Frequency 1 2 2 1 2 3 2

For each set of data: i

Find the mean, median, mode, and range. Round your answers to two decimal places where necessary.

Which data value is an outlier?

iii

Remove the outlier from the set and recalculate the values found in part (i).

Describe how each of the four statistics changed after removing the outlier.

27, 50, 24, 37, 47, 41, 27, 126, 44, 27

4.7, 2.8, 1.9, 0.9, 0.9, 2.2, 2.2, 1.2, 1.5, 0.9

4700, 4700, 4700, 4500, 5300, 4900, 5200, 4800, 1500, 5100

The weight of fish caught in a “weigh and release” fishing competition, in kilograms are recorded: 12.5, 15.1, 13, 14.2, 14.5, 14.9, 12.5, 14.3, 1.5 a

Find the mean weight.

Find the median weight.

Recalculate the mean weight excluding the outlier.

Recalculate the median weight excluding the outlier.

Reasoning

11, 12, 14

12, 13, 15

The five numbers 16, 16, 17, 24, 17 have a mean of 18 and a median of 17. Describe the effect on the mean and median if a new number is added that is larger than 24.

Carl has been recording his spelling test scores for the past semester. His scores were: 14, 16, 2, 15, 15, 16, 15 a

Calculate the median of Carl’s scores.

Calculate the mean of Carl’s scores, to two decimal places.

Which measure of centre most accurately describes the centre of this data set? Explain your answer. 17.04 Outliers mathspace.co

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A journalist wanted to report on road speed cameras being used as revenue raisers. She obtained data that showed the number of times 20 speed cameras issued a fine to motorists in one month. The results were: 101, 102, 115, 115, 121, 124, 127, 128, 130, 130, 143, 143, 146, 162, 162, 163, 178, 183, 194, 977 The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which measure of centre should she use in her article? Explain your answer.

The selling prices of artworks sold at an auction are: $18 000, $11 000, $17 000, $20 000, $18 000, $16 000, $15 000, $218 000 Which measure of centre best identifies the typical selling price of recently sold artwork? Explain your answer.

Ted is an employee who works for a company and currently gets paid $26 000 annually. Here is the breakdown of all the employees’ salaries at the company he works for in a frequency table:

Number of employees 4 10 5 3 1

Would the owner of the company use the mean, median, or mode to describe the company’s annual pay? Why?

Would Ted use the mean, median, or mode to describe the company’s annual pay? Why?

Would a mathematician who is not involved with the company use the mean, median, or mode to describe the company’s annual pay? Why?

Problem-solving 16, 17 16

Annual income $26 000 $58 000 $62 000 $150 000 $400 000

16–18

17–19

A group in a study volunteer to take a test that assesses their reaction time. The participants clicked a button as soon as they heard a sound which was played at random intervals. The reaction time, in milliseconds, of each participant is recorded: 220, 280, 210, 220, 215, 180, 185, 190, 190, 195, 150, 190, 195, 195

Identify any outliers.

Which is the most likely explanation for the outlier? A

The participant was not focused.

The sound took longer to play.

The participant missed the button.

The timer malfunctioned.

The selling prices of six houses are: $467 000, $413 000, $410 000 $456 000, $487 000, $929 000

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What is the median selling price? Round your answer to the nearest thousand dollars.

What is the mean selling price? Round your answer to the nearest thousand dollars.

Which measure of centre most accurately represents the centre of the data?

Which of the selling prices appears to be an outlier?

Recalculate the mean selling price, without the outlier.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Given the data: 4, 5, 6, 7, 8, x What does x need to be for the mean and median to be equal?

The number of three-pointers scored in different basketball games by a single team are recorded in the table. Describe the effect on each measure if the outlier is removed: a

Mean

Median

Mode

Range

Three-pointers 0 1 2 3 4 5 6 10

Frequency 2 4 5 3 2 2 2 1

17.05 Dot plots and column graphs After this lesson, you will be able to… • create and interpret dot plots, including identifying modes, range, and distribution patterns such as positive skew, negative skew, symmetric, and uniform. • create and interpret column graphs, focusing on presenting categorical data, and understanding the significance of the scale in data interpretation. • identify clusters, gaps and outliers in datasets.

Dot plots The dot plot is a useful way to express discrete data in a visually simple manner. • The main advantages of the dot plot are that we can find the mode and range very easily, as well as quickly see how the data is distributed. • The main disadvantages are that we need to count each dot when finding the median and it is often easier to convert to a table to find the mean. • The dot plot is particularly suited to discrete data where the frequency of results are often greater than one. In a dot plot, each dot represents one data point belonging to the result that it is placed above. The modes of a dot plot will be the results with the most dots. Since a dot plot stacks vertically, the highest columns will belong to the modes.

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Sometimes we can describe the data in a dot plot using the shape of the plot. When the dot plot has a negative skew, the tail of the data is to the left on the plot. When the dot plot has a positive skew, the tail of the data is to the right on the plot. Other distribution patterns include symmetric and uniform: Direction of tail

Direction of tail

0 1

2 3 4 5 6 7 8

9 10

Negative Skew

The dot plot shows a negatively skewed distribution.

80 85 Symmetric

The dot plot shows a positively skewed distribution.

10 70

3 4 5 6 7 Positive Skew

Uniform

The dot plot shows a symmetrical distribution.

The dot plot shows a uniform distribution.

Clusters on dot plots are groups of dots that are close together, showing that many data points have similar values. Gaps are spaces with no dots, which means there aren’t any data points with values in that range.

Example 1 A group of adults is asked: “How old were you when you passed your driving test?”. The responses were: 22, 17, 17, 17, 19, 21, 17, 22, 21, 18, 18, 17, 18, 22, 18 This dot plot represents the responses. Responses

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a What is the range of this data set?

Create a strategy To find the range, use the formula: Range = Highest score − Lowest score

Apply the idea Range = 22 − 17 =5

Find the difference Evaluate the subtraction

b What is the mode of this data set?

Create a strategy

Apply the idea

Choose the score which has the most dots above it.

Mode = 17

c What is the median of this data set?

Create a strategy

Apply the idea

Order the scores and find the middle score.

Arrange the scores in order: 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 21, 21, 22, 22, 22 The middle score is: 18 Median = 18

d How many people passed their driving test on or after their 19th birthday?

Create a strategy Count the number of dots that are greater or equal to 19.

Apply the idea No. People = 6

Count the scores

Example 2 Sarah celebrated her 13th birthday at a bowling alley. She invited 20 friends, and they played a game of bowling. The scores for the game were: 12, 17, 23, 31, 35, 42, 45, 49, 49, 49, 49, 53, 56, 65, 69, 75, 75, 83, 83, 300 This dot plot represents the scores.

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

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a Describe the shape of the data.

Create a strategy

Apply the idea

Check where the bulk of the data sits and look at the general shape of the distribution.

The scores have higher-frequency columns on the left (the lower scores), so the distribution is positively skewed. There appears to be an outlier as there is one value far to the right of the others. The data set also shows a large gap between 83 and 300, and clustering around 50.

b Determine the mean and median of the data.

Create a strategy

Apply the idea Add all the 20 scores

To find the mean, use the formula: To find the median, find the middle score.

Evaluate the division

Since the number of scores is even, then the two middle scores are 49 and 49, then this means that: Median = 49

c If we remove the outlier, how does the shape of the data change?

Create a strategy

Apply the idea

Check where the bulk of the data sits after removing the outlier 300 and look at the new general shape of the distribution.

The distribution will become less skewed and more symmetric, as the outlier was pulling the mean towards the higher end of the scale. The median will remain unchanged, as it is not affected by the outlier.

d State the mean and median of the data without the outlier.

Create a strategy

Apply the idea

To find the mean, use the formula:

Add all the 19 scores

To find the median, find the middle score.

Evaluate the division

Since the number of scores is odd, then the middle score is 49. Median = 49

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b How many more mandarins were brought to school than pineapples?

Create a strategy

Apply the idea

Check the graph to see how many pineapples and mandarins were brought to school, then find the difference.

The number of pineapples is 2, and number of mandarin is 8. Difference = 8 − 2 =6

Subtract the 2 from 8 Evaluate

c Complete the table using the information from the graph. Fruit Banana Apple Mandarin Pineapple Orange

Number 2 ⬚ ⬚ ⬚ ⬚

Create a strategy

Apply the idea

Remember, the height of each column represents the number of each fruit that was brought to school.

By checking the height of each column, we can fill in the numbers of each fruit. Fruit Banana Apple Mandarin Pineapple Orange

Number 2 3 8 2 4

Example 5 The table shows the number of people who visited Disneyland between 2011 and 2015. Year 2011 2012 2013 2014 2015

Number of people in hundreds of thousands 165 164 152 159 168

a Represent the data in a column graph.

Create a strategy We assign the values from the table for each year to make the column graph. The Year should go on the horizontal axis and the Number of people should go on the vertical axis.

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If the frequency of each category in a set of data is high, is it better to use a dot plot or a column graph to represent the data?

Fluency 4

4, 5, 7–10, 13, 14, 16–19

9, 17–20

A group of adults are asked: “How old are your kids?” The responses were: 15, 4, 18, 15, 9, 10, 5, 15, 16, 5, 16, 17, 10, 12, 19 a

Plot the responses on a dot plot.

Determine: i

c 5

the range

the mode

iii

Sophia is a casual nurse. She used a dot plot to keep track of the number of shifts she did each week for a number of weeks: a

In this dot plot, what does each dot represent?

How many weeks did she work 5 shifts?

What was the most frequently occurring number of shifts per week?

Describe the shape of the distribution of the number of shifts per week.

Consider the column graph showing the pets that the children in a class have: a

Snakes are the least popular pet.

Two children have dogs as their pets.

Turtles are the second most popular pet.

No child has a cat as a pet.

Number

Are these statements true or false?

5 6 7 Shifts per week

Pets

10 9 8 7 6 5 4 3 2 1 0 Dog

Snake

Crab

Cat

Turtle

Animal

These graphs show the annual net profit (in millions) of a company over a several year period. For each graph, determine the year in which the annual net profit is an outlier: a

Annual Profit 80 70 60 50 40 30 20 10 0

120 Net Profit (millions)

the median

How many people have kids that are 15 or older?

Net Profit (millions)

Ex 1

5, 6, 8–12, 14–17

2005 2006 2007 2008 2009 2010 Year

Annual Profit

100 80 60 40 20 0

2004 2005 2006 2007 2008 2009 Year

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Glen wants to put the information from this column graph into a table:

Favourite Sports Number of People

5 4 3 2 1 0

Cricket Basketball Soccer

Tennis

Netball

Sport

Complete this frequency table to match the information in the column graph.

Ex 2

Sport Cricket Basketball Soccer Tennis Netball

Number of People

During a school field trip, a group of students participated in a laser tag game. The scores for the game were: 5, 10, 15, 20, 25, 30, 35, 40, 40, 40, 40, 45, 50, 55, 60, 60, 65, 70, 75, 200 This dot plot represents the scores.

Ex 3

100

120

140

160

Describe the shape of the data.

Determine the mean and the median of the data.

If we remove the outlier, how does the shape of the data change?

Determine the mean and median of the data without the outlier.

200

A group of people were surveyed on the number of websites they visit on a daily basis. The results are displayed in the dot plot: Number of websites visited daily

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Complete the frequency table based on the dot plot. Number of websites visited daily 0 1 2 3 4 5 6

Frequency

Identify any clusters in the number of websites visited daily.

Are there any gaps in the distribution of websites visited daily? If so, between which values?

This dot plot shows the number of students in each section of the school band: a

Which section has the least members?

Determine the number of people in the band.

How many more people are there in the Strings section than in the Brass section?

Describe the shape of the distribution of students in the band sections.

Band members

Strings

Woodwind

Each judge on a cooking show gives contestants a rating between 1 and 5. The dot plot shows the scores given to the winning contestant: a

Describe the shape of the distribution of ratings given by the judges to the winning contestant.

How many judges gave the contestant a rating of 4?

How many judges gave the contestant a rating of 4 or more?

How many judges gave the contestant a rating of 2 or less?

Contestant rating

The goals scored by a football team in a single season are represented in the dot plot: a

What is the modal number of goals scored?

Determine the median number of goals scored.

Brass Percussion

Goal scores

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In a study, a group of people were shown 30 names, and after one minute they were asked to list the names they could remember. The results are presented in the dot plot: a

How many people took part in the study?

What is the highest number of names someone remembered?

What was the lowest number of names someone remembered?

Determine the range of names remembered.

A student conducts a survey asking their class members what their favourite video game is. The results are displayed in this column graph: a

Which is the most popular game?

Which two games were equally popular?

How many students took part in the survey?

Names Remembered

Favourite Video Game 10 9 8 7

Frequency

6 5 4 3 2 1 0

Blobrox

Underlook

Bash Sisters

Spawn Campers

Elf Fortress

Game

This column graph shows the star ratings by different patrons at a local hotel: a

What is the mode?

How many ratings of 5 were given?

How many more people gave a rating of 3 than a rating of 2?

Frequency

Ex 4

Construct a frequency table using the information from the graph. 1

2 3 4 Hotel Ratings

Yvette has 7 black marbles, 3 blue marbles, 9 yellow marbles, 12 white marbles, and 5 red marbles. Construct a column graph to represent this information.

Ex 5

780

The table shows the number of people (in millions) who visited the Sydney Opera House between 2015 and 2019: a

Use the information from the table to construct a column graph.

A marketing executive examines the graph and says, “There has been a significant increase in the number of visitors from 2015 to 2019”. Is this statement correct?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Year 2015 2016 2017 2018 2019

Number of people in millions 8.2 8.8 9.6 10.0 10.8

The average monthly attendance of a fitness class is recorded in the table: Attendance

August 42

September 41

October 43

November 39

Construct a column graph to represent the given data.

In which month is the average attendance at a minimum?

In which month is the average attendance at a maximum?

Determine the range.

December 40

Aviation glass is manufactured to be 12 mm thick. If the glass is too thin (less than 11.5 mm) or too thick (more than 12.5 mm) the aircraft will fail its glass safety check. A regional airport conducts a glass safety check of all aircraft landing on their runways over a one month period and displays the results in the dot plot: a

How many aircraft were checked for glass safety?

What is the lowest level of glass thickness measured?

What is the modal glass thickness measured?

What percentage of aircraft failed the glass safety check?

Reasoning 21

21, 23

Glass thickness (in mm)

10.5

11.0

11.5

21, 22a, 23

12.0

12.5

13.0

21–23

The dot plot shows the temperature (°C) in a town over several weeks.

Temperature in a town

Which measure of centre would represent this data best? Explain your reasoning.

Consider the column graph. a

Darius says the median score is 48.5 since 48 and 49 are the two scores in middle. Is he correct? Explain your reasoning.

Explain a method for calculating the mean score.

Frequency

10 9 8 7 6 5 4 3 2 1 46 47 48 49 50 51 Scores

17.05 Dot plots and column graphs mathspace.co

781

The table shows the house points earned by four houses at their swimming carnival: a

Construct a column graph using the information from the table.

Would it be reasonable to say that Dundalli had twice as many points as Nemarluk? Explain your reasoning.

Problem-solving 25 24

782

House Truganini Nemarluk Dundalli Umbarra

24, 25

Points 40 30 70 50

24–26

A supermarket manager takes a note every time an employee is late to work, and how late they were (rounded to the nearest half hour). The dot plot shows their results for the last month:

Lateness (in hours)

Determine the median amount of hours that employees were late by.

What fraction of late employees were later than the median amount?

If an employee is more than 1 hour late, the manager fines them $10. How much money did the manager collect in fines over the last month?

These dot plots display how many seconds it took for students to race from their classroom to their locker: Mr. Navra’s Class

Mr. Lee’s Class

50 51 52 53 54 55 56 57 58 59 60

What is the slowest time recorded?

Which teacher’s class had the largest range of times?

Which teacher’s class had the fastest student?

Which teacher’s class had the slowest median time?

In each game of the season, a basketball team recorded the number of ‘three-point shots’ they scored. The results for the season are represented in the dot plot:

How many games did they play during the season?

If the team scored 4 three-point shots in a game, how many points are scored from three-point shots?

What was the total number of points they scored from three-point shots during the season?

Determine the mean number of points they scored per game from three-point shots, rounded to two decimal places.

Mathspace Victorian Curriculum – Year 7 mathspace.co

Number of Three Points Shots

The owner of a business wants to have a company-wide retreat outside of Australia. He narrowed the list down to five countries, but he cannot decide which country to choose. He sends a survey to all 180 of his employees, asking them to choose the country in which the retreat should be held. Would these data displays be appropriate for the owner to use to analyse the results of the survey? a

Dot plot

Fluency 3

Column graph

3–8

Stem-and-leaf plot

4, 6–9

The pie chart shows the results of a class survey where students were asked to nominate their favourite food: a

Which was the most popular food?

Which two foods were equally popular?

Pie chart

7–9 Favourite Food

Nuggets Pizza Noodles Burger

Ex 1

794

The pie chart represents the popularity of each chip brand amongst high schoolers: a

Which brand is the most popular?

Which brand is the least popular?

Which brand is popular with one quarter of the high schoolers?

If 56 students selected Hot Cheezos, how many high schoolers were surveyed?

Edward surveyed a large group of people and asked for their favourite milkshake flavour. The results of the survey are presented in a pie chart. a

What fraction of people chose chocolate?

What percentage of people chose chocolate?

What fraction of people chose strawberry?

What percentage of people chose strawberry?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Tortillos

Hot Cheezos

Snakis

Wrinkles

Milkshake Flavours

Vanilla

Strawberry

Chocolate

Caramel

A florist recorded the roses of different colours sold in his store over time. This is displayed in the pie chart. If 800 roses were sold in a day, determine the number you would expect to be: a

Red

Orange

White

Pink

Pink 10% Orange 25%

Red 50%

White 15%

The results of a senatorial election in a European country, where 4000 people voted, are displayed in a pie chart. a

Who won the election?

After the election, the winner decides to send promotional packages to everyone they suspect voted for Zorg. How many packages should they prepare?

Voting Results Zorg 25% 10% Umbridge Palpatine 65%

Roald has a weekly income of $1200 and this pie chart shows a break-down of his spending. a

Determine how much money Roald spends weekly on: i

Leisure

Clothes

iii

Food

Housing

If he sticks to his savings plan, how much will he have saved after 5 weeks?

25%

30%

5% 10%

30%

Food

Leisure

Savings

Clothes

Housing

A technology company wishes to see what headphones people are using. They took a small sample and found these: • Apple: 156 • Sony: 19 • Bose: 32 • Other: 74 a

Which sector represents Bose?

What percentage of people use a brand other than Apple, Sony, or Bose? Round your answer to one decimal place.

A B

17.07 Pie charts mathspace.co

795

Reasoning 10

10, 11

Consider the pie chart:

10–12 Fruits purchased from the grocery

If we knew the number of grapes that were purchased, how could we use that information to find the number of apples that were purchased?

Apples

12% 8%

Blueberries Oranges

13%

40%

Grapes

27%

Limes

The pie charts show the distribution of nuclear weapons owned by countries around the world: Nuclear Weapons (Not in USA or Russia) Israel North Korea 20 80

Number of Nuclear Weapons

r Othe 5 116

USA 6450

Pa ki 14 stan 0

Russia 6490

France 300

India 130 China 280

UK 215

Which country has the most nuclear weapons according to the pie charts?

Which two countries have around half of the nuclear weapons that are not owned by Russia or the USA?

How many countries have more than 100 nuclear weapons?

Elise thinks these pie charts are confusing and claims a column graph would be better. Do you agree or disagree with Elise? Explain your reasoning.

This pie chart was displayed on a news program: a Describe two mathematical errors with this pie chart.

Would vote for Grylls

Would vote for McGill

70%

63%

60%

Would vote for Flanders 796

Mathspace Victorian Curriculum – Year 7 mathspace.co

Which of the images better represents the same information? A

Would vote for Flanders

Would vote for Grylls Would vote for McGill

Would vote for McGill

Would vote for Flanders

After looking at the data, a student concludes that the majority of people would vote for each of the three candidates. Which is not an explanation for how this is possible? A

The three candidates are running for different positions in government.

Three different populations were surveyed.

he survey questions said, “Would you vote for Grylls if they ran for prime minister? T Would you vote for McGill if they ran for prime minister? Would you vote for Flanders if they ran for prime minister?

he survey question said, “If Grylls, McGill, and Flanders run for prime minister, who T would you vote for?”

Problem-solving 13, 14 13

Would vote for Grylls

13–15

13–16

A teacher surveys 220 students to find their favourite subject and wishes to construct a pie chart to display the data. The survey results are recorded in the table: Subject

Number of students

Maths

English

History

Science

Other

Percentage

Angle size 20% of 360 = 72°

Use your calculator to complete the table and find the percentage of students and angle size for each subject.

Construct a pie chart to represent this data, labelling each sector piece with the name and percentage.

17.07 Pie charts mathspace.co

797

A pharmaceutical company runs trials of a potential virus vaccine. There are 1080 volunteers for the trial, and the results are recorded in the table: Result

Improvement

705

No change

240

Deterioration

135

Use your calculator to complete the table and find the angle size for each result.

Construct a pie chart to represent this data, labelling each sector piece with the name and angle size.

A business wishes to construct a pie chart to display which month holidays are taken over the summer months. They surveyed 180 employees, and the results are recorded in the table: Number of employees

November

December

January

February

March

Angle size

Use your calculator to complete the table and find the angle size for each month.

Construct a pie chart to represent this data, labelling each sector piece with the name and angle size.

A marketing company wishes to construct a pie chart to display the market share of smartphone manufacturers. They surveyed 4000 people, and the results are recorded in the table: Manufacturer

798

Angle size

Month

Number of volunteers

Responses

Apple

2051

Samsung

967

Huawei

531

Other

451

Angle size

Use your calculator to complete the table and find the angle size for each manufacturer, rounded to one decimal place.

Construct a pie chart to represent this data.

Mathspace Victorian Curriculum – Year 7 mathspace.co

What year had the lowest seal population?

One year, due to too much fishing in the local area, a shortage of fish caused the seal population to drop. In which year did the seal population drop?

Number of seals

Seal population at Rocky point 55 50 45 40 35 30 25 20

2013

2014

2015

What is the highest seal population over the five years?

By how much did the seal population increase over the five years?

The highest daily temperature in Sommersville over a week is presented in the line graph: a

Which day had the warmest temperature?

Which two days had the same temperature?

How much warmer was Friday than Monday?

2016

2017

Year

Daily Temperature 26º 24º 22º 20º 18º 16º 14º

Mon

Tue

Wed

Thu

Fri

Sat

Sun

Day

A paper corporation records the amount of paper they produce each month over a year in a line graph. a

802

In which month did the paper corporation produce the most paper?

Paper Produced 4.0 3.8

Mass (tonnes)

Ex 1

The line graph shows the seal population at Rocky Point over a 5-year period:

Temperature ( Degrees Celsius)

3.6 3.4

In which month did the paper corporation produce the least paper?

In which period did paper production increase the most?

In which period did paper production decrease the most?

In which period did paper production not change?

Mathspace Victorian Curriculum – Year 7 mathspace.co

3.2 3.0

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

Month

The line graph shows monthly rainfall data for Berlin and London. a b

804

65 60

Calculate the range in monthly rainfall for Berlin between June and October. In the months between March and July (inclusive), determine the range in London’s monthly rainfall.

55 50 45 40 35 30

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

London

Berlin

Which city has the lowest range over the year?

In which month did the cities have the smallest difference in rainfall?

The price of cans of soft drink over a year is represented on a line graph. a

What was the lowest price of the cans of soft drink?

In which months was the price the lowest?

What was the highest price of the cans of soft drink?

Soft Drink Prices $3.00 $2.75 $2.50 $2.25 $2.00 $1.75 $1.50 $1.25 $1.00

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

Month

In which months was the price the highest?

If 250 cans of soft drink are sold each month, calculate the difference in the value of the sales between the month with the highest price and the month with the lowest price.

Luigi brings a 1.8 L bottle of water to work each day. He records the amount of water in the bottle at the start of each hour on a line graph. a

How much water was in Luigi’s bottle at the start of the day?

What time was Luigi’s bottle first refilled?

How much water did Luigi drink in the hour after first refilling his bottle?

Water Bottle

Water (litres)

Monthly Rainfall (mm) in Major Cities 70

Which city recorded the highest monthly rainfall over the year?

Price

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

9 am 10 am 11 am 12 pm 1 pm 2 pm 3 pm 4 pm 5 pm Time

How many hours were between the first and second time Luigi filled his bottle?

How much water did Luigi drink between the two refills?

How much did the amount of water in Luigi’s bottle change between the start and end of the day?

How much water did Luigi drink in total during the day?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Danielle’s distance from home throughout the day is recorded on a line graph. a b c d

How far was Danielle from her house at 2:00 p.m.? What time did Danielle get back home?

3.5 3.0

What was the furthest distance Danielle was from her house? Determine the total time that Danielle was away from her house.

Reasoning 18

4.0

Distance (km)

2.5 2.0 1.5 1.0 0.5 0.0

11:00 a.m.

11:30 a.m.

12:00 12:30 p.m. p.m.

1:00 p.m.

1:30 p.m.

2:00 p.m.

2:30 p.m.

3:00 p.m.

Time

18, 19, 22

18, 19, 21

19–21

The manufacturer of a dishwasher wants to create a graph showing how one of their dishwasher models work. These are the stages of its operation: 1.

The dishwasher quickly fills to half-way and performs a short wash.

2. The dishwasher empties completely. 3. The dishwasher completely fills up quickly and remains full over the course of the wash. 4. The dishwasher empties completely. Which graph matches the water level inside the dishwasher throughout a cycle? A

Full

Empty 10 20

Time (mins) 30

Empty 10 20

Time (mins) 30

17.08 Line graphs mathspace.co

805

Full

Empty 10 20

Full

Time (mins) 30

Empty 10 20

Time (mins) 30

A kitchen sink starts with 20 L of water in it. It empties at a rate of 4 L per minute and then after 4 minutes the drain gets blocked. No water empties for 2 more minutes while it gets unblocked, then the remaining water is drained out in 1 minute. Construct a line graph that represents this scenario.

A tank is being filled at a constant rate of 4 L per second for 4 minutes. The hose is then turned off for 2 minutes. Finally it is topped up again with water at a rate of 3 L per second for 1 minute. Construct a line graph that represents this scenario.

The travel graph shows the first ten minutes of a student’s journey home from school.

10 9

Describe their travel between these times: 0 to 3 minutes

3 to 6 minutes

c d

6 to 9 minutes 9 to 10 minutes

8 Distance (km)

7 6 5 4 3 2 1 0

Consider the line graph of the sales of a car company: Explain why the statements are misinterpretations of the line graph.

806

The company achieved the target of increasing trend in sales for the year.

The changes in the sales each month are almost consistent.

2 3 4 5 6 7 8 9 10 Time (mins)

Tiger car sales

Cars sold

Mathspace Victorian Curriculum – Year 7 mathspace.co

100 90 80 70 60 50 40 30 20 10 0

3 Month

23–25

Two siblings, Apollo and Artemis, leave home at different times. They are travelling to see their favourite sports team compete in the championship. The stadium is 30 km away and the match starts at 9:00 p.m. a

At what times did they both leave home?

Between which two times were Apollo and Artemis the same distance from home?

Who was travelling faster?

Did Apollo make it to the stadium on time?

24–27

30 Distance (km)

Problem-solving 24, 25

25 20 Artemis

15 Apollo

10 5

2 3 4

5 6 7

8 9 10

Time (p.m.)

This travel graph represents a train’s journey between the airport and a nearby station. a

How far did the train travel over the 10 minute period?

When was the train travelling the fastest?

Distance (km)

10 9 8 7 6 5 4 3 2 1 0

Kenneth works for a delivery company. The graph shows his distance from the warehouse throughout his shift. a

How far was Kenneth from the warehouse at the start?

How far was he from the warehouse after 8 hours?

How many times did he visit the warehouse during this time period?

Calculate the total distance that he travelled between 1 and 4 hours.

When was Kenneth travelling the fastest?

9 8 Distance (km)

1 2 3 4 5 6 7 8 9 10 Time (mins)

7 6 5 4 3 2 1 0

4 5 6 7 Time (hours)

8 9

17.08 Line graphs mathspace.co

807

The Weber family travel 600 km every year for their annual holidays. Their distance from home on the trip this year is recorded in a travel graph. a

When did they stop for a break?

How far from their destination were they after 2 hours?

600 Distance (km)

Identify two time periods when they were travelling at the same speed.

500 400 300 200 100

Castor and Pollux are working on a bushfire prevention team. Over the course of their 10-hour shift they move along a boundary and remove any flammable material they find. The travel graph shows their distance from the base throughout their shift:

808

How far from the base was Pollux when the shift began?

How far apart were Castor and Pollux after 2 hours?

When did Castor and Pollux meet up?

When was Pollux travelling the same speed as Castor’s speed in his first three hours?

Mathspace Victorian Curriculum – Year 7 mathspace.co

Distance (km)

10 9 8 7 6 5 4 3 2 1 0

3 4 5 6 7 Time (hours)

Pollux Castor 1

2 3 4 5 6 7 8 9 10 Time (hours)

Investigate 1.

What graph do you think will best represent your data and why? (dot plot, column graph, bar graph, or histogram)? Graph your data using your choice.

What are the mean, median, mode and range for your data?

Based on your calculations and your graph what would be the best measure of centre for the amount of time you bounced the ball? Why?

What is the average time it took for you to bounce the ball 1 time? If you worked in a group only look at your personal bounce data here.

If you wanted to bounce the ball 1000 times how long would it take you? If you worked in a group only look at your personal bounce data here.

Compare with friends. Have a friend (or another small group) complete the investigation as well, and hang up the final graphs next to one another. Compare and contrast the graphs.

Discussion

810

Which person (or group) had the highest average number of bounces? How do you know?

What can you say about the range of each of the graphs? How do you know?

What centre of spread would be appropriate for each graph? Does it vary from person to person? Why?

How do the different types of graphs used display the data in different ways? Does it affect the way you read the data?

Mathspace Victorian Curriculum – Year 7 mathspace.co

17 Chapter review 1

Identify the type of data described if discrete, continuous, nominal, or ordinal: a

Number of students in each classroom in a school

Names of different colors in a box of crayons

Temperatures recorded each day for a month

Type of data for the ratings (1 – 5 stars) given to a restaurant

A student scored these grades on his end of year tests: 92, 91, 88, 89, 93, 86, 90, 92, 96 a

How many tests did the student take?

Calculate: i

Mean

Median

iii

Mode

Examine the data set presented in the frequency table. Suppose one score of 7 is changed to a 10. Determine if each measure would change or remain the same:

Mean

Median

Mode

Range

Score 2 3 5 6 7 9 10

Range Frequency 1 2 2 1 3 2 1

Olivia has a fitness goal and records her exercise routine. She used a dot plot to track the number of days she exercised each week for a certain period. a

In this dot plot, what does each dot represent?

In how many weeks did she exercise for 5 days?

What was the most frequent number of days she exercised per week?

1 2 3 4 5 6 7 Days per week spent exercising

Chapter 17 review mathspace.co

811

Fish colours in a tank

This pie chart represents the distribution of fish colours in a tank. a

Which colour is the most common among the fish?

Which fish colour makes up one quarter of the fish in the tank?

If there are 28 blue fish, how many fish are there in total in the tank?

This pie chart represents the distribution of Year 10 students participating in various sports. a

Which sport has the most participants according to the pie chart?

Which sport has the least participants according to the pie chart?

How many students participate in football and cricket combined?

The number of books sold by a bookstore over a 12-month period is represented in this line graph. a

How many books were sold in Month 1?

How many books were sold in Month 10?

In which month(s) did the bookstore sell 2100 books?

Were more books sold in Month 7 than in Month 9?

Orange

Blue

Red

Yellow

Year 10 sports by student Basketball

Cricket

30 86 53

Volleyball 27

2350 2300 2250 2200 2150 2100 2050 2000 1950 1900 1850 1800

Water Polo

Football

Number of books

Month 1 2 3 4 5 6 7 8 9 10 11 12

A financial analyst is studying the growth of monthly sales in a retail store. They collected data on the percentage increase in sales for 20 consecutive months. The results were: 2, 2, 4, 4, 5, 5, 7, 7, 8, 8, 9, 9, 10, 13, 13, 14, 16, 17, 20, 46 The analyst wants to highlight the significant growth potential of the retail store. Which measure of central tendency should they use in their report? Explain your reasoning.

Chapter 17 review mathspace.co

813

Two friends, John and Jane, leave home at different times. They are going for a bike ride to a nearby park that is 20 km away, and they plan to meet there at 5:00 p.m.

25 km

At what times did they both leave home?

20 km

Between which two times were John and Jane the same distance from home?

15 km

Who was travelling faster?

Distance 30 km

Did John make it to the park on time?

10 km

John

5 km Time (p.m.) 2 3 4 5 6 7 8 9 10

Mia and Ava are participating in a beach cleanup event. Distance (km) Over the course of their 8-hour shift they move along 10 9 the beach and collect any rubbish they find. This travel graph shows their distance from the starting point throughout their shift:

Jane

8 7

How far from the starting point was Ava when the shift began?

How far apart were Mia and Ava after 2 hours?

When did Mia and Ava meet up?

At what point during her shift was Ava traveling at the same speed as Mia?

5 4 3

Ava Mia Time (hours) 1

Trent was working with a set of 6 data points. He knows 5 of them are: 11, 12, 17, 25 and 28. a

If he knows the mean was 19, what is the missing score?

If it was actually the median that was 19, what is the missing score?

Add two more scores to the data set that would also keep the mean and the median value the same.

Chapter 17 review mathspace.co

815

Answers

2 B y adding the products which are inside each portion.

1.01 Addition and subtraction

12 × 8 = 8 × 10 + 8 × 2 = 80 + 16

Understanding

= 96

1 a Addition

b Addition

c Subtraction

d Addition

e Subtraction f Subtraction

g Addition

h Subtraction i Addition

j Subtraction

k Addition

Fluency 3 a

l Subtraction

2 a 8858

b 2838

c 86

d 528

e 1714

f 750

g 482

h 1485

b 990

c 791

d 9923

e 6679

f 64 695

g 59 240

h 1500

i 957

j 1726

k 13 643

l 11 842

4 a 99 699

b 64 770

c 41 415

d 3607

e 2373

f 91 729

g 59 662

h 8468

4000

150

b 4150

Fluency 3 a 99

4 a 6

3600

48000

4800

1800

180

6 342 + 35 = 300 + 40 + 2 + 30 + 5 = 300 + 77 b True

e True

f True

c True

d False

8 A = 9, B = 0 and C = 7 Problem-solving 9 80 pieces of fruit

b 54 780

600

5 18 + 12 = 10 + 8 + 10 + 2 = 20 + 10

7 a False

5 a

i 832 Reasoning

b 3648

600 8

6 a 900

72000

5600 320

6300

490

b 84 738

7 a 6

b 7

c 36

d 8

10 341 species

e 25

f 12

g 21

h 30

11 130 seconds

i 30

j 77

k 36

l 60

8 a 180

12 30 bricks

1.02 Multiplication Understanding 1

3 5 15 tiles 3 × 5 = 15 15 tiles 5 × 3 = 15

816

Mathspace Victorian Curriculum – Year 7 mathspace.co

b 270

c 261

d 351

e 3888

f 2367

g 2268

h 4380

i 16 665

j 11 106

k 2596

l 10 707

9 a 280 000

b 30 000

c 1 200 000

d 21 000

e 3978

f 15 594

g 12 630

h 10 374

i 23 446

j 46 397

k 246 666

l 475 626

10 a 90 e 120

b 48

c 630

d 48

f 1200

g 240

h 1125

Reasoning

11 Sample answer: 5 × 6 is the same as 5 × 2 × 3, which is the same as 10 × 3. 12 a 8 × 70 + 8 × 5 c 80 × 64 + 2 × 64

b 11 × 90 + 11 × 2 d 70 × 475 + 3 × 475

Problem-solving 13 a 2 and 3

b 2 and 12

c 9 and 4

d 36 and 36

14 a 45 hours

b Tristan

16 $29 190 17 $542 500 18 904 320

9 3×9+1

Problem-solving

19 $56 128 960

10 To solve this problem, we need to find a number that leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 7. One way to do this is to start with the number 30 and add 1 until we find a number that satisfies both conditions.

20 23 760

1.03 Division Understanding

2 a False

8 Dividing by zero is an invalid operation in mathematics that leads to undeﬁned behaviour, causing problems in mathematical models. Therefore, it is crucial to avoid undeﬁned values to maintain the consistency and validity of our mathematical systems. In this expression, 3 is the divisor, 9 is the quotient, and 1 is the remainder. When you multiply 3 by 9, you get 27. Then, adding the remainder gives you 28, which is the original dividend, confirming the correctness of the rewrite.

15 530 m

1 a 25 ÷ 5

7 x ÷ y = y ÷ x when x = y or x = −y and both x and y are not equal to 0.

b 24 ÷ 8

c 49 ÷ 7

When we divide 30 by 3, we get a remainder of 0. Therefore, the next number that leaves a remainder of 2 when divided by 3 is 30 + 2 = 32.

b False

c False

d True

b 18

c 30

d 90

e 91

f 276

g 260

h 257

i 749

j 2393

k 907

l 1023

4 a 40

b 62

c 74

d 120

e 45

f 44

g 314

h 299

11 16 biscuits

i 666

j 606

k 605

l 600

12 50 glasses

Fluency 3 a 9

When we divide 32 by 7, we get a remainder of 4. Therefore, 32 is the smallest number greater than 30 that leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 7.

5 a 177 r. 3

b 781 r. 2

c 821 r. 3

13 17 boxes

d 925 r. 5

e 680 r. 5

f 1019 r. 1

14 44 canned items

g 914 r. 6

h 1862 r. 2

i 3078 r. 2

j 4793 r. 3

k 3734 r. 4

l 10 532 r. 1

b 7 r. 28

c 13 r. 13

d 22 r. 5

e 107 r. 23

f 106 r. 4

g 288 r. 10

h 540 r. 1

i 499 r. 20

j 490 r. 16

k 614 r. 1

l 1286 r. 17

6 a 24 r. 3

15 78 rows 16 140 plants 17 a 476 ÷ 17 = 28

b 434 ÷ 31 = 14

c 560 ÷ 14 = 40

d 651 ÷ 21 = 31

e 816 ÷ 34 = 24

f 837 ÷ 31 = 27

g 924 ÷ 21 = 44

h 999 ÷ 37 = 27

Answers mathspace.co

817

1.04 Order of operations with whole numbers

10 a i No

Understanding

ii Yes

iii No

iv Yes

b The addition is now done before the division.

1 The order of operations when performing arithmetic with whole numbers is as follows: 1. First, perform any calculations inside parentheses or brackets, working from the innermost set of parentheses to the outermost. 2. Next, perform any multiplication or division operations from left to right. 3. Finally, perform any addition or subtraction operations from left to right. 2 a 3×5

b They cause the addition to be evaluated before the multiplication

b 4×2

c 9÷3

d 7−2

11 a i Yes

ii No

iii No

b The subtraction is now done before the multiplication. 12 a i No

ii Yes

iii No

b The subtraction is now done before the division. Problem-solving 13 a ×, −

b ×, ×

c ÷, ×

d +, ×

e ×, +

f ÷, −

14 50 apples

Fluency

15 $22

3 a 14

b 16

c 14

d 16

e 19

f 40

g 78

h 16

16 2 lollies

i 34

j 13

k 27

l 5

n 10

17 22 marbles

4 a 160

b 13

c 1

d 48

e 20

f 60

g 58

h 4

1.05 Check reasonableness of calculations

i 12

j 11

k 42

l 1

n 4

5 a 13 b i 51

ii 13

iii 63

ii 63

iii 88

6 a 88 b i 82 7 a 19 b i 19

Understanding 1 a 20

b 80

c 60

d 350

e 130

f 100

g 80

h 910

2 a 800

b 300

c 900

d 100

e 200

f 700

g 500

h 1000

3 a 1000

b 7000

c 9000

d 5000

f 45 000

g 19 000

h 67 000

e 10 000 ii 15

iii 89

iv 5

8 a 12 − (6 + 6) ÷ 6 = 10

4 • “The product is less than 4000.” is correct. • “The product is more than 2700.” is correct.

b (5 + 10) × 4 – 5 × 4 = 40

• “The product is more than 4000.” is incorrect. The product is less than 4000.

c 4 + 5 × (13 – 5) – 4 = 40 d 5 × 15 − (4 ÷ 2 + 8) = 65

• “The product is less than 2700.” is incorrect. The product is more than 2700.

e 18 × (4 − 1) + 4 − 18 = 40 f (23 − 10) × 3 + 8 × 9 – 2 = 109 g 15 ÷ 3 + 7 × (4 + 2) × 6 = 257

Fluency

h 30 − 4 × 5 ÷ (5 – 3) = 20

5 a Yes, approximately 60 − 30 = 30 b No, approximately 70 − 30 = 40

Reasoning 9 a i No

818

ii Yes

iii No

iv Yes

Mathspace Victorian Curriculum – Year 7 mathspace.co

c Yes, approximately 2880 − 2830 = 80 − 30 if we subtract 2800 from both numbers.

d No, approximately 6870 − 5830 and if we subtract 5800 from both values this equals 1070 − 30 not 70 − 30. 6 a No, 4 × 80 = 320, since 89 > 80 it will be more than 320. b Yes, 240 = 3 × 80 and since 85 is great than 80, so this answer will be larger than 240.

b Yes, because then they will have $22.40 which will cover the cost of two pizzas. 25 Example answer: To find the approximate area of the sticker, you can round the length by the width first to the nearest tens and then multiply them. The dimensions are approximately 60 mm and 110 mm. The approximate area is 60 mm × 110 mm = 6600 mm2

7 a 10

b 100

c 1000

d 1000

8 a Yes

b Yes

c No

d Yes

9 a Yes

b Yes

c No

d No

10 a Yes

b No

c No

d Yes

11 a Yes

b No

c Yes

d No

b 900

c 400

d 500

1.06 Number calculations with technology

d Larger

Understanding

12 No 13 No 14 a 4200 Reasoning 15 a Larger

b Smaller c Larger

26 Example answer: To find the approximate number of students in the school, you can round the number of students to the nearest tens and then add them. The approximate number of students are 120 for Year 7, 190 for Year 8, 210 for Year 9 and 50 for Year 10. The approximate number of students in the school is 120 + 190 + 210 + 50 = 570.

16 a Yes, the estimate will be larger.

1 B

b Yes, the estimate will be smaller.

2 B

c No

3 A

17 a Smaller b Larger

c Smaller d Larger

18 a No

b No

4 A

c Yes, the estimate will be larger.

Fluency

d Yes, the estimate will be smaller.

5 a 514

19 a Smaller b Larger

c Larger

d Larger

d 80

f 509

g 79

h 26 096

c 10 441

d 80

e 490 260

b Yes, the estimate will be smaller. c No

22 a No

c 1172

6 a 720 000 b 9

20 a Yes, the estimate will be larger.

21 a Smaller b Smaller c Larger

e 135 194

b 61 809

d Larger

b No

c Yes, the estimate will be larger. d Yes, the estimate will be smaller.

7 a False

b False

c True

d False

e False

f True

g False

h True

i False

j False

Reasoning 8 No, he swapped the digits in a number.

Problem-solving

9 Yes

23 7 boxes that hold 30 padlocks each.

10 No, he forgot a zero.

24 a No, because he is rounding too much. Two pizzas cost $19.50, so Michael cannot afford two pizzas.

11 No, he added an extra zero. 12 No, he swapped some of the numbers. 13 Yes

Answers mathspace.co

819

14 No, he didn’t use the correct order of operations.

Reasoning 13 a Example answer: 15 apples were initially in a box, and then 2 groups of 4 apples were removed.

Problem-solving 15 28

b Example answer:

16 a 6

b $384

10 cookies were divided into 2 groups, and 3 more cookies were added.

17 9 lollies 18 Alﬁe gets 20 marbles, Barry gets 40 marbles and Corey gets 60 marbles 19 3

1.07 Write mathematical statements Understanding 1 a More than, add, plus, sum, increase, total b Less than, subtract, minus, difference, decrease, take away, deduct c Multiply, times, product, groups of

c Example answer: There were 10 cupcakes, but 4 of them fell onto the floor and had to be discarded. The amount left over were then shared between two people. d Example answer: 8 pencils were initially in the box, and 3 groups of 2 pencils were added to the box. Problem-solving 14 Maximum Kilometres = (425 − (5 × 30)) ÷ 0.25 15 (21 × 76) − (20 × 75)

d Divide, split, quotient, share, ratio 2 a x

b vi

c v

d ix

16 (12 − 4) ÷ 2 + 4

e ii

f i

g viii

h vii

17 (20 ÷ 4) × (20 ÷ 4)

i iv

j iii

Chapter 1 review Fluency

1 a 7 × 3 + (9 − 12) + 2 = 20

3 a 2+7

b 7−6

c 3 × 10

d 12 ÷ 4

b (5 + 10) × 4 − 5 × 4 = 40

e 4×7

f 13 ≠ 11

g 4 < 10

h 8>5

c (9 + 6) ÷ 3 × 4 − 8 = 12

4 a 15 = 7 + 8 c 80 = 8 × 10

b 6 = 16 − 10 d 5 = 50 ÷ 10

e 19 > 7 + 5

2 a 9 b i 20

5 a

b 12 × (10 − 6)

d 11 × 2 − 9 ≠ 10

f 20 ≥ 2(3 + 6)

3 a +, ×

ii 9

iii 24

b ×, +

c ×, ÷

d +, ×

b 9000

c 7000

d 2000

f 44 000

g 18 000

h 70 000

4 $75 5 a 1000 e 10 000

6 a Yes, approximately 80 + 30 = 110

6 C

b No, approximately 80 + 20 = 100. The actual addition is 76 + 15 = 91. The statement is not reasonable because the estimated result 100 is significantly different from the actual result 91.

7 D 8 B 9 A

c Yes, approximately 9000 + 3000 = 12 000. The actual addition is 9134 + 2678 = 11 812 which is close to the actual result.

10 A 11 D 12 a 2 + 80 ÷ 10

820

d (14 ÷ 2) × 4 + 9 − 3 × 2 = 31

b 10

Mathspace Victorian Curriculum – Year 7 mathspace.co

d No, we can round both numbers to the nearest thousand and perform the subtraction: 7500 − 1000 = 6500. The actual subtraction is 7652 − 1389 = 6263. The statement is unreasonable because the estimated result (6500) is not close enough to the actual result (6263) to be a reasonable approximation. A better estimation would be to round the numbers to the nearest hundred: 7700 − 1400 = 6300, which is much closer to the actual result.

2.01 Number operation properties Understanding 1 a Associative law

b Reordering

c Commutative law 2 Answer may look like:

7 a Yes

b Yes

c No

d No

The distributive law allows us to break apart larger numbers to help us calculate larger multiplications or divisions. For example: 1326 ÷ 13 = (1300 ÷ 13) + (26 ÷ 13).

8 a 10

b 100

c 1000

d 1000

3 a Commutative law, reordering

9 a Yes

b No

c Yes

d Yes

10 a 4500

b 1000

c 400

d 500

b Commutative law, reordering c Reordering d Associative law e Distributive law

11 A 12 a False

b True

c True

d False

e True

f True

g False

h False

13 29

f Distributive law g Reordering h Reordering 4 a True

14 a 3

b False

c False

d True

b $768 Fluency

15 12 lollies 16 a 2 + 7

b 7−6

c 3 × 10

d 12 ÷ 4

e 4×7

f 13 ≠ 11

g 4 < 10

h 8>5

17 a The sum of 12 and 8 is equal to 20. b The difference between 8 and 1 is equal to 7.

5 B 6 a 75

b 58

7 a 2×5 b 2 × 16 × 5 = 2 × 5 × 16 or 2 × 16 × 5 = 5 × 2 × 16

c The product of 2 and 7 is equal to 14.

2 × 16 × 5 = 160

d The quotient of 36 and 4 is equal to 9.

8 260

e The quotient of 10 and 5 is not equal to the quotient of 5 and 10.

9 78

18 a

b 12 × (10 − 6)

10 79 11 2

d 11 × 2 − 9 ≠ 10

f 20 ≥ 2(3 + 6)

13 11

b 10

14 a 54 + 26

19 a 2 + 80 ÷ 10

12 a 26

b 12

b 118

20 (12 − 4) ÷ 2 + 4

15 340

21 (20 ÷ 4) × (20 ÷ 4)

16 a 3

b 20

c 805

22 Maximum Kilometres = (325 − (8 × 25)) ÷ 0.25

17 a 30

b 1

c 754

18 a 8982

b 7014

c 1364

d 2472

19 132 ÷ 4 ÷ 11 = 3

Answers mathspace.co

821

20 a 12

b 12

c 32

21 a 21

b 7

c 17

22 a 202

b 99

c 87

factors that are the same, then it is a square number. d 45

3 B 4 a 9

Reasoning

b 2

5 Answer may look like:

23 An example answer: Consider the example 15 ÷ 5. This division problem is asking “how many groups of 5 are there in 15?” The answer to this question is 3. If we applied the commutative law, we would get the expression 5 ÷ 15 which is the same as asking “how many groups of 15 are there in 5?” This would not be a whole number, which shows 15 ÷ 5 ≠ 5 ÷ 15, so the commutative law does not work for division. 24 In step 2, Julietta applied the associative law, but the associative law does not work for subtraction.

If we know the number of tiles that cover the length or width of the floor, then we can use the square of that number to find the total number of tiles needed. This is because the total number of tiles is the same as the area, and the area of a square floor will be the side squared. Fluency 6 The first five square numbers are 1, 4, 9, 16 and 25. 7 a No

b Yes

c Yes

d No

e No

f Yes

g Yes

h No

8 a 16

b 49

c 64

d 144

e 225

f 529

g 2500

h 10 000

9 a 85

b 58

c 130

d 10 100

e 40

f 17

g 108

h 9900

b 16

i 4

j 144

k 400

d 8 × 20 + 8 × 2

l 1 000 000

b 32

o 25

p 64

10 a 14

28 Cost of the shirts: 4 × $15 = 60

e 26

Cost of the pants: 3 × $25 = 75

11 576 m2

25 Regroup the problem so that it is 5 × 200 × 34 which is 1000 × 34 = 34 000 Problem-solving 26 a 160 c 176 27 a 320

d 16 × 18 + 16 × 2

c 288

Calculating the discount: 0.10 × (60 + 75) = 0.10 × 60 + 0.10 × 75 Distributive law

13 324 chairs

Evaluate the multiplication

Reasoning

= 13.50

Evaluate the addition

2.02 Square numbers Understanding 1 Answer may look like: A square number is a number that is the result of multiplying an integer by itself. 2 Answer may look like: We can determine if a number is a square number by looking at its factors. If it has two

Mathspace Victorian Curriculum – Year 7 mathspace.co

b 52

c 25

d 40

f 200

g 5

h 5

14 Answer may look like: It is not possible to do this with whole number side lengths. A non-square whole number is the product of two different numbers. If a square is formed, the side lengths would be decimals. 15 a 4 × 25 = 100 b 100 c Yes since 22 × 52 = (2 × 5)2. d 64 16 a 36 ÷ 4 = 9 b 9

822

n 25

12 81

= 6.00 + 7.50

m 16

This pattern increases by two at each stage starting with one. This makes a sequence of odd numbers.

c Yes, since 62 ÷ 22 = (6 ÷ 2)2. d 16 17 a 16 + 25 = 41

25 a

b 81

Number

Square Number

100

c No, since 42 + 52 ≠ (4 + 5)2. First Difference

18 a 144 − 81 = 63 Second Difference

b 9 c No, since 122 − 92 ≠ (12 − 9)2.

26 a Number

Problem-solving 20 4 21 25 22 15 23 81 cm2 ii 3

iii 5

iv 7

b We obtain the sequence of odd numbers: 1, 3, 5, 7, 9, … c Example answer: To grow the square at each stage we need to add on a new L-shaped layer. 9 7 5 3 1 4 +3

9 +5

16 +7

25 +9

This layer can be viewed as adding two lots of the previous side length plus one corner square. Previous side length

c Consistently, the difference is 2.

Odd numbers will always have 1, 3, 5, 7 or 9 in the units place value. When these numbers are squared, the units place value will be 1, 9, 5, 9, 1 respectively, which are all odd digits.

b The set consists of odd numbers from 3 to 19.

19 Answer may look like:

24 a i 1

Previous side length

Square of number

Final digit of square number

100

Number

Square of number

Final digit of square number

121

144

169

196

225

256

289

324

361

400

c The pattern is a repeating pattern 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. d The ﬁnal digits of the square numbers from 21 to 30 are 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 respectively. e Yes, because when you square a number, the digit in the units place is the only one that affects the digit in the square number’s units place. The values in any location other than

Answers mathspace.co

823

the units place (including the tens place) have no effect on the ﬁnal digit. So, when we square the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, the units of the square number will always be 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 respectively.

15 8 m Reasoning to

16 Any square root from Example justiﬁcation for 62 = 36.

and

27 65

2.03 Square roots

Understanding

17 Answer may look like:

1 The square root of 64 is 8. 2 Answer may look like: They are inverse operations. That is, if you take the square root of a number and then square the result, you get the original number. 3 Answer may look like: If the number is a perfect square, the square root is a whole number, and if it is not a perfect square, the square root is not a whole number. 4 a Yes

b No

c No

d Yes

e Yes

f No

g Yes

h No

5 Answer may look like: Since the garden is square, the lengths of the sides of the garden will be the square root of the area.

One way to estimate the value of a square root is to ﬁnd the perfect square that is closest to the number and then estimate the square root based on its distance from that perfect square. For example, the square root of 53 is between the perfect squares of 49 and 64, so we can estimate it to be between 7 and 8. 18 Answer may look like: , Ansha correctly identiﬁed that but her mistake was saying the square root is is closer approximately 6. This is because to 7, so it should be approximated as 7. is closer to 7 because We know 49 − 43 = 6 but 43 − 36 = 7. 19 a D b Answer may look like: and is closer to 3 because 11 − 9 = 2 while 16 − 11 = 5. Because the is difference between 11 and 9 is smaller, closer to

Fluency 6 5 7 B 8 a 2

b 4

c 7

d 12

Problem-solving

9 a 20

b 8

c 60

d 2

20 a 10

e 11

f 10

g 5

h 1

21 24 m

i 242

j 139

k 21

l 5

10 9 m

d 12

22 48 cm

24 a b

13 a 3 and 4

b 2 and 3

c 5 and 6

d 9 and 10

e 7 and 8

f 11 and 12

g 15 and 16

h 20 and 21

14 4

824

c Yes

23 1600 cm

11 125 m by 125 m 12 a 16 < 22 < 25

b 10

Mathspace Victorian Curriculum – Year 7 mathspace.co

Shape number (n)

Total number of tiles (T )

b The total number of tiles (T ) is equal to the square of the shape number (n) that is added to 1.

Shape number (n)

Number of light purple border tiles (B)

g 2×2×2×2×2×2×2×2 h 7×7×7×7×7×7 10 a 5 × 5

d The number of light purple border tiles (B) is equal to 4 times the shape number (n).

b 7×7×7×7×7×6×6×6×6

e Shape number 6.

f 100 tiles

d 77 × 77 × 77 × 77 × 77 × 13 × 13

g 64 tiles

h 32 tiles

c 4×4×4×4×4×4×9×9 11 a 5 × 5 × 5

2.04 Powers of whole numbers

Understanding

b 125

128

256

512

1 9

13 a 1, 3, 9, 27, 81, 243, 729

2 7

b 1, 5, 25, 125, 625

3 a The value of the expression increases.

Reasoning

b The value of the expression decreases.

14 As the base is 4 and the power is 5, you multiply 4 by itself 5 times. So 45 = 4 × 4 × 4 × 4 × 4

4 C

15 a Fluency 5 a 22

b 33

c 52

d 23 Index notation

5×5×5×5 ×5×5

6×6×6×6

Eight cubed

8×8×8

Eleven to the power of three

11 × 11 × 11

113

Twenty three to the power of five

23 × 23 × 23 × 23 × 23

235

7×7

15 × 15 × 15 × 15

154

Five to the power of six Six to the power of four

Seven squared Fifteen to the power of four 7 a 92

b 93

c 94

b 1 when raised to any power will always be 1.

Expanded notation

Words

d 95

16 a The answer is halved. c i

iii

b 1

125

625

100

101

102

103

104

100

1000

10 000

d Every non-zero number raised to the power of zero equals one. Problem-solving 17 a 0

b 5

c 3

d 1

e 4

f 4

g 9

h 6

8 a 53

b 16

c 34

d 24

18 a 64

b 1

c 116

d 243

e 4

f 7

e 128

f 9

g 68

h 130

g 122 × 93

h 112 × 73

i 140

j 98

k 129

l −37

i 123 × 152

j 22 × 42

19 a >

b =

c >

d =

l 22 × 33 × 52

20 81

k 24 × 12 9 a 9×9

b 9×9×9

c 9×9×9×9

d 9×9×9×9×9

e 3×3×3×3×3

f 5×5×5×3

Answers mathspace.co

825

numbers as whole numbers, we get 90 000 and 200 000. This shows 9 × 104 is smaller.

2.05 Powers of ten Understanding

13 Answer may look like:

1 12 zeros

Multiplying by 10 increases the value of a number by a factor of 10, while raising to the power of 10 multiplies the number by itself 10 times.

2 D 3 105

For example, 2 × 10 = 20 but 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2× = 1024

Fluency 4

101

102

14 Answer may look like:

100

We can use powers of ten to compare numbers by writing them in expanded form, then comparing the powers on the bases of ten. If the powers on the bases of ten are the same, then we can compare the digits that are multiplied to the powers of ten.

103

1000

104

10 000

105

100 000

106

1 000 000

107

10 000 000

108

100 000 000

Problem-solving

5 6

15 384 000 kilometres

6 a 200 000

b 300

c 7000

d 10 000

e 80

f 6 000 000

g 3

h 50 000 000

7 a 102 e 6 × 103

16 171 300 000 kilograms

b 103

c 4 × 102

d 9 × 102

f 7 × 103

g 8 × 104

h 3 × 104

8 500 000 000 kilometres

b 41 400 000 km c The distance between Venus and Earth

2.06 Factors and multiples Understanding

9 a 2 × 103 + 2 × 102

1 Answer may look like:

b 8 × 103 + 3 × 102 c 2 × 103 + 4 × 102 + 8 × 101 d 7 × 103 + 2 × 101 e 6 × 103 + 2 × 102 + 6 × 101 f 9 × 104 + 2 × 103 g 3 × 104 + 3 × 103 + 5 × 101 h 6 × 104 + 2 × 103 + 4 × 102 + 8 × 101 10 a 37 000

b 1040

c 410

d 7640

e 60 305

f 32 095

g 324 020

h 22 776

Reasoning 11 a > b 12 9 × 104 because it only has 4 place values while 2 × 105 has 5 place values. If we write both

826

17 a 50 300 000 km

Mathspace Victorian Curriculum – Year 7 mathspace.co

A factor of a number is a whole number that divides that number exactly without any remainder. 2 Answer may look like: A factor of a number divides the number exactly, whereas a multiple of a number is obtained by multiplying the number by another whole number. Factors will be smaller or equal to the number itself, whereas multiples will be equal or larger than the number itself. 3 1 4 2 5 A number is a multiple of another number if that other number divides into it evenly.

Fluency

Reasoning

7 a Yes

b No

c No

d Yes

16 For a number to have exactly three factors, it must be a square of a prime number. This is because a prime number has only two factors (1 and itself), so the only way for a number to have exactly three factors is for it to be the product of two identical prime numbers.

e Yes

f No

g Yes

h No

17 Answer may be like:

Factor pairs of 50 (1, 50) (2, 25) (5, 10)

b 1, 19

Yes, the number itself is both a factor and a multiple of itself.

c 1, 3, 9

18 This is possible because 35 is a factor of 280.

8 a 1, 2, 7, 14

d 1, 3, 11, 33

19 a 6

e 1, 2, 4, 7, 14, 28 f 1, 2, 4, 8 g 1, 2, 3, 6, 7, 14, 21, 42

20 a 1, 2, 4, 8, 16

h 1, 2, 4, 8, 16

b The number of ﬂowers that can be in each row are the factors of 16.

i 1, 3, 5, 9, 15, 45 j 1, 7, 49 k 1, 2, 3, 6, 9, 18, 27, 54

Problem-solving

l 1, 2, 3, 6, 8, 9, 12, 24, 36, 72

21 108 eggs

9 a 1, 5, 23, 115 b 1, 2, 7, 14, 49, 98

22 3

c 1, 3, 7, 11, 21, 33, 77, 231

23 72

d 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312

24 Mindt

10 a 4, 8, 12, 16, 20

b 3, 6, 9, 12, 15

25 a 36 ﬁsh

c 7, 14, 21, 28, 35

d 9, 18, 27, 36, 45

26 a 15 rooms

e 10, 20, 30, 40, 50

f 11, 22, 33, 44, 55

27 16 m by 24 m

g 12, 24, 36, 48, 60

h 15, 30, 45, 60, 75

11 a Yes

b No

c No

d Yes

b 20 ﬁsh

c Wally

b 18 weeks

2.07 Prime factorisations Understanding

12 32

1 a Factor of a number

13 80

b Prime number

c Prime factorisation of a number

14 6 15 a

b 4

c There are marbles left over because 6 is not a factor of 40.

2 a Prime: 3, 5 Week

Total earnings (in dollars)

b Prime: 7, 13, 11

Composite: 14, 15

120

150

180

b $150

Composite: 4, 6, 10, 12

c Prime: 19, 17 Composite: 1, 21, 27 3 Factors and prime factors are the same in that they both divide the number evenly. They are different because factors can be prime

Answers mathspace.co

827

numbers, composite numbers, or neither, such as 1. Prime factors are always prime numbers.

ii 24 × 32

f i 144

Fluency 12

36 3 2

18 2 2

ii 23 × 52

g i 3

200

5 45 10

b 22 × 32 × 5 × 7

6 a 1260 7 a i

ii 2 × 3

b i

ii 22 × 7

ii 23 × 32 × 5

h i 360

4 10 2

c i

ii 2 × 3 × 7

2 7

d i

8 a 2×2×3×3

ii 34

e 2×2×2×3×3

f 2×2×2×2×2×3

g 2 × 2 × 3 × 11

h 2×3×5×5

9 a 3 ×5×2

b 52 × 23

c 22 × 32 × 5

d 52 × 32 f 24 × 52

e 2 × 13 3

g 7×5×2

ii 2 × 32 × 7 126 6

828

21 3

h 23 × 32 × 7

Reasoning

e i

b 2×3×7 d 3×2×2×2×2

c 2×3×3×3

Mathspace Victorian Curriculum – Year 7 mathspace.co

10 Answer may look like: No, two different numbers cannot have the same prime factorisation. This is because the product of the prime factors will result in the same number. We cannot multiply the same numbers together and get two different answers.

11 Rae and Josie are both correct. 2 and 5 are both prime factors of 30, so they will need to divide by both numbers at some point in their work. It does not matter which one is divided ﬁrst as either way will lead to the same ﬁnal result. 12 The prime factors of 35 are 5 and 7. If both 5 and 7 divide 420 evenly, then 35 is a factor of 420. Problem-solving 13 a No, 175 = 5 × 5 × 7 b Yes c No, 286 = 2 × 11 × 13 d No, 27 = 3 × 3 × 3 e No, 20 = 2 × 2 × 5 f No, 189 = 3 × 3 × 3 × 7 14 a No, 56 = 23 × 7 b No, 495 = 3 × 3 × 5 × 11 c Yes d Yes e No, 504 = 23 × 32 × 7 f No, 300 = 22 × 3 × 52

b Least common multiple is the smallest positive integer that is a multiple of two or more integers. 2 a True b False. If the HCF and LCM of two numbers are equal, then the numbers are equal. c True d False. The HCF of two numbers cannot be larger than either of the two numbers because a factor of a number cannot be greater than the number itself. e False. The LCM of two numbers cannot be less than either of the two numbers because a multiple of a number must be greater than or equal to the number itself. Fluency 3 a i 1, 2, 3, 6 1, 2, 3, 4, 6, 12 ii 6 b i 1, 3, 9 1, 2, 3, 4, 6, 8, 12, 24 ii 3 c i 1, 2, 7, 14

15 36

1, 2, 4, 8, 16, 32

16 a The ﬁnal number is the same as the original number.

ii 2

b All six-digit numbers where the ﬁrst three digits are the same as the last three digits have 7, 11 and 13 as prime factors. c 1001 = 1000 + 1 d If we multiply 1001 by any three-digit number, it is the same as multiplying by 1000 + 1. For example, 123(1000 + 1) = 123 000 + 123 = 123 123. In other words, multiplying it by 1000 makes the ﬁrst 3 digits the same as the original number, then multiplying by 1 makes the last 3 digits the same as the original number.

d i 1, 2, 4, 7, 14, 28 1, 2, 3, 6, 7, 14, 21, 42 ii 14 4 60 5 28 6 350 7 a 8

b 6

c 3

d 12

e 6

f 36

g 11

h 10

i 16

j 24

8 a 2940

b 8820

9 a i 6, 12, 18, 24, 30, 36, 42, 48

2.08 HCF and LCM Understanding 1 a Highest common factor (HCF) is the largest positive integer that divides two or more integers without a remainder.

10, 20, 30, 40, 50, 60, 70, 80 ii 30 b i 8, 16, 24, 32, 40, 48, 56, 64 14, 28, 42, 56, 70, 84, 98, 112 ii 56

Answers mathspace.co

829

b 324

c 676

d 6400

42, 84, 126, 168, 210, 252, 294, 336

c i 24, 48, 72, 96, 120, 144, 168, 192

e 14 500

f 27

g 80

h 231

ii 168

i 32

j 867

k 7

l 13

5 a 9

b 13

c 5

d 11

e 1

f 147

g 21

h 4

6 a 64 < 70 < 81

b 8<

7 a 8 and 9

b 9 and 10

c 11 and 12

d 17 and 18

d i 15, 30, 45, 60, 75, 90, 105, 120 30, 60, 90, 120, 150, 180, 210, 240 ii 30 10 108 11 3780 12 a 42 e 1872

b 126

c 168

d 78

f 5850

g 72

h 378

2 a 225

3 625 chairs 4 46

13 a 75

b 34 650

14 a 17 640

b 1050

8 36 m

15 a 4 centimetres

b 25 strips

9 4000 m

16 a Every 76 years

b Every 60 years

10 a 64

17 a Every 126 minutes

b A

b 152 × 73

c 163 × 202

d 22 × 33 × 52

11 a 4 × 4 × 4 × 4 × 4 Reasoning

b 5×5×5×5

18 The student is correct because all numbers have 1 as a factor. If the numbers have no other factors in common, they will still have the factor of 1 in common. Any two numbers can be multiplied together to create a common multiple.

c 2×2×2×2×2×2×2×2

19 The product of the prime factors that the two numbers have in common will be the highest factor the numbers have in common. 20 a Sasha is not correct. Although the highest common factor of 8 and 12 is 4, the lowest common multiple is 24. b Suri is correct because both numbers have 4 as a factor. Since 4 is a factor, both numbers will be multiples of 4. c 12, 16 or 4, 48

d 7×7×7×7×7×7 e 7×7×7×7×7×6×6×6×6 f 4 × 4 × 4 × 4 × 4 × 4 × 9 × 9 g 77 × 77 × 77 × 77 × 13 × 13 h 120 × 120 × 120 × 120 × 120 × 12 × 12 12 a 12 × 12 × 12 × 12

b 20 736

13 a 729

b 1024

c 207

d 432

e 704

f 512

g 6689

h 473

14 192 rabbits 15 a 300 000

b 800 000

c 7 000 000 3

d 90 000 000 2

16 a 2 × 10 + 2 × 10 c 5 × 104

b 8 × 103 + 4 × 102 d 6 × 104 + 9 × 103

Problem-solving

17 a 530

21 28 cm

18 2826

22 Year 2214

19 a 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

23 a 18 seconds

b 8 red balls

24 6 minutes

830

c 32 073

d 93 874

b 1, 2, 4, 8, 16, 32, 64 c 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 d 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Chapter 2 review 1 a 27, 18

b 8620

b 52, 24

Mathspace Victorian Curriculum – Year 7 mathspace.co

20 60

21 a 2 × 3 × 3 × 3 c 2×3×3×5

b 2×2×3×5

9 a i We will GO TO Step 3.

d 2×3×5×5

ii We will GO TO Step 5. iii We will GO TO Step 7.

22 24

iv 758

23 a 45

b 20 790

24 a 1260

b 5250

25 30 cm

b i W e must regroup into tens and units, then GO TO Step 3. ii We will GO TO Step 5. iii We will GO TO Step 7.

3.01 Solve problems using an algorithm Understanding 1 C

iv 795 10 a i GO TO Step 2.

ii 1

iii 60

iv 61

b i R egroup the tens from the ﬁrst number to give a ten to the units, then GO TO Step 2.

2 B 3 a Yes

b Yes

d Yes

e Yes

c Yes

b Yes

c Numbers with a units digit from 7 to 9. 5 a Yes

b No

c Numbers with a units digit from 3 to 9. 6 a No

iii 40

b No

c Numbers with a units digit from 0 to 2. 7 a i We will add 1 and 8. ii We will not need to regroup.

iv 45

11 a i 35 = 30 + 5

ii 5 × 6 = 30

iii 30 × 6 = 180

iv 180 + 30 = 210

b i 12 = 10 + 2

Fluency 4 a No

ii 5

iii 10 × 82 = 820

ii 2 × 82 = 164 iv 164 + 820 = 984

Reasoning 12 No this will not always work because this algorithm does not consider any regrouping of tens when adding the units. 13 Add a step to subtract the hundreds place. 14 Answers may vary. Adding 45 and 37

iii We will add 40 and 30. b i We will add 8 and 7. ii We will need to regroup into tens and units. iii We will add 30, 20 and 10.

Problem-solving 15 55 16 1 time 17 35

8 a i 3 and 6

ii GO TO Step 3.

iii GO TO Step 5.

iv 99

18 C

ii GO TO Step 3.

19 2

b i 2 and 3

iii First regroup into hundreds and tens, then GO TO Step 5. iv 155 c i F irst regroup into tens and units, then GO TO Step 3.

3.02 Following algorithms Understanding 1 Example answer: Making a cake.

ii First regroup into hundreds and tens, then GO TO Step 5.

2 Measure 250ml of water.

iii 154

3 Step 4: Make sure both socks are from the same pair.

Answers mathspace.co

831

Fluency

10 a For 5:

4 Painting a room and changing a car tyre.

• Multiply 5 by itself. 5 × 5 = 25

5 a Choose wrapping paper.

• Subtract 5 from 25:

b Decorate with a bow.

25 − 5 = 20

6 a Put soap on your hands.

• Divide 20 by 5:

b Turn off the tap. 7 Place the board on a ﬂat surface.

For 6:

8 C

• Multiply 6 by itself.

9 a

832

Step

Instruction

Output

Start with the number 100.

100

Subtract 22 from 100.

Round the result to the nearest ten.

Step

Instruction

Output

Start with the number 70.

Double 70.

140

Add 21.

161

Step

Instruction

Output 16

Start with the number 16.

Add 5 to 16.

Double the result.

Step

Instruction

Output

Start with the number 2.

Add 2 to 2.

Round your answer to the nearest ten.

Step

Instruction

Output

Start with the number 36.

Halve 36.

Add 3 to the result.

Step

Instruction

Output

Start with the number 12.

Multiply 12 by 10.

120

Add 5 to the result.

125

Step

Instruction

Output

Start with the number 6.

Subtract 1 from 6.

Double the result.

Mathspace Victorian Curriculum – Year 7 mathspace.co

6 × 6 = 36 • Subtract 6 from 36: 36 − 6 = 30 • Divide 30 by 6:

For 7: • Multiply 7 by itself. 7 × 7 = 49 • Subtract 7 from 49: 49 − 7 = 42 • Divide 42 by 7:

b The pattern is that the final answer is always one less than the original number. c 121 d The magic trick works because of the mathematical properties of the operations used. When you square a number (multiply it by itself) and then subtract the original number, you’re left with a multiple of the original number. When you then divide by the original number, you’re left with the original number minus one. This will always be the case, no matter what number you start with.

Reasoning 11

Step

Problem-solving Instruction

Justiﬁcation

Before creating an email account, you must ﬁrst decide which email Choose an email service service provider you want to use. This is the provider. foundation of your email experience.

To create an email account, you need to provide some basic Fill out the signup form information to the service with personal provider. This typically includes personal details details. like name, age, and location.

Your email username and password will be used Select a to uniquely identify and unique email secure your account. username and They are essential for password. logging in and keeping your emails private. Email service providers need to ensure that the email address provided Verify your email account during the signup process is valid and controlled by through a confirmation the user. Therefore, they send a confirmation email email. with a link or code for verification purposes. Once your account is Start sending verified and active, you can start using it for its emails to primary purpose: sending friends. and receiving emails.

12 Adding tea leaves before boiling the water can result in the tea steeping at a lower temperature than desired. This can affect the extraction of ﬂavors from the tea leaves, leading to a brew that might be weaker or different in taste than intended. 13 Tasks involving emotions, judgment, intuition, or creativity can be challenging to break down into algorithms. For instance, making a subjective judgment, understanding art, or interpreting emotions in a complex social scenario often require nuanced understanding and cannot be reduced to a simple set of instructions.

14 $15.70 15 53 points 16 $25.50

3.03 Branching Understanding 1 Branching in algorithms is used to decide what steps to follow next based on certain rules (conditions). 2 A conditional in coding refers to a decisionmaking point where the algorithm decides which step or path to take next based on certain conditions. 3 a No branching c Branching

b Branching d No branching

Fluency 4 a 5

b 5

5 a 24

b 25

6 a 49

b 54

7 a 27

b 32

8 a 30

b 77

9 a 250

b 982

10 a 7

b 4

c 8

d 7

e 25

f 26

Reasoning 11 No, a number cannot remain unchanged after completing the algorithm. In the case of the number 30, it’s neither bigger than 30 nor less than 30, so it would not ﬁt any of the conditions. However, the algorithm doesn’t specify what to do when the number is exactly 30, leading to ambiguity.

Answers mathspace.co

833

Step

Instruction

Result

b i

Step

Instruction

Result

Start with the number 54

Start with the number 13

Add 6 to this number

Add 13 to this number

Repeat Step 2 three more times

Round this number to the nearest 10

Step

Instruction

Result

72 78

Start with the number 35

Add 2 to this number

Repeat Step 2 two more times

Half the number

20.5

Instruction

Result

Step 1

Start with the number 40 000

40 000

Divide this number by 10

4000 400

Repeat Step 2 three more times

40 4

Step

Instruction

Result

Start with the number 5

Double this number

10 20

Repeat Step 2 until the number is greater than 40.

40 80

b 7+7+7+7+7

8 a 5 c 35

d 35

9 a i 32

ii 6

b i 17

ii 5

10 a i

52 65

ii 5 × 13 Reasoning 11 Step 3 would have to be changed to say repeat Step 2, 11 more times. If it is only repeated 6 more times it will only show up to (7 × 7 = 49). 12 Limiting the iterations to 3 would end the times table at (4 × 3 = 12). The listing would be (0, 4, 8, and 12). Numbers 16 to 24 would be absent. 13 The algorithm is designed for batches of ﬁve hens. This suggests that the farm has optimized its egg collection process for groups of ﬁve. For larger quantities, like 25 hens, the farm can still manage by repeating the steps, but there will always be a repetitive cycle based on batches of 5. It implies that the collection process and storage containers are likely optimized for 5 hens at a time. Problem-Solving 14 a 15 b The mistake in Step 3 is repeating Step 2 three times, which only adds 4 three times. To reach 19, repeat Step 2 four times instead. c

Step

Instruction

Start with the number 3.

Add 4 to the number.

Repeat step 2 four times.

15 The iteration runs 7 times: (5, 10, 20, 40, 80, 160, 320, 640)

Step

Instruction

Result

Start with the number 6

Add 6 to this number

Repeat Step 2 three more times

18 24

16 Tom will need to refill the can 2 times. He waters 3 plants on the first fill, refills and waters the next 3 on the second fill, then refills again to water the remaining 2 plants. 17 Steps 2 to 4 will be repeated 5 times.

30 ii 5× 6

Answers mathspace.co

835

18 The results after each iteration will be: 37, 145, 577. The ﬁnal result after applying the algorithm will be 577.

15 a

Start SET n = number of steps

3.05 Flow charts

Is n = 10 000?

Understanding

Count steps

Yes Celebratory Notification

1 Branching chooses different actions based on conditions (like using if statements). Looping repeats actions multiple times until a condition is met (like using for and while loops). 2 You would use a diamond to ask the question about the number’s nature (whether it’s even or odd).

End

b You would need to take 1 more step. 16 a

START Input two numbers a and b, ensuring a > b

3 “PRINT” in a ﬂow chart context means to display or showcase the result or answer. This could be on a computer screen, on a piece of paper, or having someone write the answer down in their book.

Divide a by b to get the quotient and remainder r

4 The variable “number” would have the value of 4. 5 a Shape D

SET n = n + 1

b Shape C

Set a = b and b = r

Is r = 0?

Yes

HCF = b

c Shape B

Fluency

END

b i 35

6 33

ii 20

iii 3

3.06 Algorithms in code

7 77 Understanding

8 50

1 To assign the value of 5 to the variable x.

9 4 10 a Flow chart C

b Flow chart C

11 a Flow chart D

b Flow chart E

Reasoning 12 No, Mary is incorrect. If f is 7, the ﬂow chart will take the ‘No’ branch and PRINT “f is 7 or greater”.

2 When a decision needs to be made. 3 Outputs a value or message. 4 To represent the logic of an algorithm that can be followed by humans and not language speciﬁc for computers. Fluency

13 Yes, Susan is correct. The values of h will be decreased in the sequence: 6, 4, 2, 0, −2. When h becomes −2, the message will be printed.

5 a 100

b 286

d 6

e 31

6 a C

b 120 m

Problem Solving

7 a D

b 32 m

14 Your score would decrease by 3 points.

8 a is odd

836

Mathspace Victorian Curriculum – Year 7 mathspace.co

c 8 f 151 2

Reasoning 9 Ivana is right as the order should be E, A, C, D, B. 10 Division by zero is undeﬁned in mathematics. If b is set to 0, the pseudocode will attempt to divide by zero, leading to an error or undeﬁned behavior. 11 The multiplication by 2 accounts for both sides of the rectangle. A rectangle has two widths (W) and two lengths (L), hence to calculate the perimeter, both dimensions are multiplied by 2. Problem-solving

15 1 2 3 4 5 6 7 8

START SET a=[some value] SET b=[some value] SET temp=a SET a=b SET b=temp PRINT a, b END

3.07 Loops Understanding 1 To repeat a block of code until a certain condition is met.

12 1 2 3 4 5 6

START SET x=5 SET y=10 SET z=(x+y)/2 PRINT z END

2 a The loop used in the provided pseudocode is a “WHILE” loop. b The loop will run until the condition a < 9 is no longer true. c This line adds the current value of a to the running total in the “sum” value. Fluency

1 2 3 4 5

START SET n=[some value] SET square=n × n PRINT square END

14 Area: 1 2 3 4 5 6

START SET b=[some value] SET h=[some value] SET area=1/2 * b * h PRINT area END

3 a a=5

b sum = 35

c 5 times 4 a a=9

b a=3

c OUTPUT = 39 5 a i a=3

ii OUTPUT = 52

b i a=3

ii OUTPUT = 7

6 n=2 7 a 5 c 1, 2, 4, 8, 16

b 5, 6 d 12, 10, 8, 6, 4

e 5, 1, 0.2, 0.04, 0.008 8 D, A, C, E, B

Perimeter: 1 2 3 4 5 6 7

START SET a=[some value] SET b=[some value] SET c=[some value] SET perimeter= a + b + c PRINT perimeter END

9 G, B, C, E, A, F, D 10 a 10

b F, C, A, G, D, E, B

11 a 720

b F, D, A, E, B, G, C

Answers mathspace.co

837

Reasoning 12 The condition a > 5 ensures that the loop continues to iterate as long as the value of a is greater than 5. Once a becomes less than or equal to 5, the loop terminates. This condition determines how many times the loop will iterate based on the starting value of a. 13 The value of a is incremented by 1 in each iteration to progress through each number from the initial value of a up to 8. This allows the loop to add each consecutive integer to the sum. 14 If the condition were changed to a ≤ 14, then the number 14 would also be included in the sum and the count. This would increase the sum and the count by one, thereby changing the ﬁnal average output. Problem-solving 15 The initial value of the product variable should be 1. Multiplication will replace the addition. The modiﬁed pseudocode: START SET a = 7 SET product = 1 WHILE a > 3 product = product × a a = a − 1 END WHILE PRINT product END 16 Introduce a new variable for the sum of squares. In each iteration of the loop, square the current number and add it to this new variable. The pseudocode: START SET a = 9 SET sum = 0 SET sumOfSquares = 0 WHILE a > 3 sum = sum + a sumofSquares = sumofSquares + a2 a = a − 1 END WHILE PRINT sum, sumOfSquares END

17 Use variables to track the smallest and largest numbers. Update these variables inside the loop based on conditions. The pseudocode: START SET a = 1 SET sum = 0 SET smallest = ∞ SET largest = −∞ WHILE a ≤ n sum = sum + a IF a < smallest smallest = a END IF IF a > largest largest = a END IF a = a + 1 END WHILE PRINT sum, smallest, largest END

Chapter 3 review 1 Example answer: Planting a ﬂower. 2 The step has been executed 7 times during the 7 iterations. 3 The variable “number” would have the value of 8. 4 a The loop will run until the condition a < 15 is no longer true. b This line adds the current value of a to the running total in the “sum” value. 5 a Yes b No c Numbers with a units digit from 8 to 9. 6 a i W e must regroup into tens and units, then GO TO Step 3. ii We will GO TO Step 5. iii We will GO TO Step 7. iv 571 b i We will GO TO Step 3. ii We must regroup into hundreds and tens, then GO TO Step 5. iii We will GO TO Step 7. iv 773 7 a No

838

Mathspace Victorian Curriculum – Year 7 mathspace.co

b No

c Yes

d No

8 a

Step

Instruction

Output

Start with the number 120.

120

Subtract 25 from 120.

Round the result to the nearest ten.

Step

Instruction

Output

Start with the number 18.

Add 6 to 18.

Double the result.

Step

Instruction

Output

Start with the number 40.

Halve 40.

Add 4 to the result.

10 275 Step

Instruction

Result

Start with the number 10

Add 5 to this number

Repeat Step 2 three more times

25 30

c 20

d 140

b 40 m

16 a D 17 a is even 18 a i a = 4

ii OUTPUT = 72

b i a=4

ii OUTPUT = 8

19 a 8

b 2, 6, 18

c 7, 8, 9

d 4, 12, 36

20 The problem with the algorithm is that there are overlapping conditions. Using the number 42, since it’s even, it would become (42 + 4) × 12 = 552. Given that 552 is greater than 40, it would further change to

= 185. Clear non-

26 Your score would decrease by 10 points.

27 Area:

Start with the number 50

Add 7 to this number

Repeat Step 2 three more times

Round this number to the nearest 10

Instruction

Result

Start with the number 30 000

30 000

Divide this number by 5

6000

Repeat Step 2 three more times

240 48

12 a 5 c 45

b 9+9+9+9+9 d 45

24 The book is returned 2 days after the 10-day period. Thus, the fee is $5 plus 2 days at $0.50 each, making a total fee of $6.

Result

23 Price = $17 + 8 MB × $0.15 per MB = $17 + $1.20 = $18.20

Instruction

Step

22 1 time

25 Lisa will need to prepare the ﬁsh food 3 times. She feeds 2 tanks on the ﬁrst preparation, prepares again and feeds the next 2 on the second preparation, then prepares again to feed the remaining 1 tank.

Step

1200

13 107

b 27

21 69

15 a 85

overlapping conditions are essential for consistent results in an algorithm.

9 53

11 a

14 6

1 2 3 4 5 6

START SET l=[some value] SET b=[some value] SET area=l × b PRINT area END

Perimeter: 1 2 3 4 5 6

START SET l=[some value] SET b=[some value] SET perimeter=2 × (l + b) PRINT perimeter END

Answers mathspace.co

839

Fluency 1 2 3 4 5 6 7 8

START INPUT X INPUT Y SET temp=X SET X=Y SET Y=temp PRINT X, Y END

6 a 2

b −3

c 2

d 9

e −2

f −9

g 4

h −1

i 2

j −6

7 a

START SET b = 1 SET product = 1 SET smallest = ∞ SET largest = −∞ WHILE b ≤ n product = product * b IF b < smallest smallest = b END IF IF b > largest largest = b END IF b= b+ 1 END WHILE PRINT product, smallest, largest END

4.01 Integers and the number line Understanding

−10

−5

−10

−5

−10

−5

−10

−5

−10

−5

g −15 −10

−5

−15 −10

−5

h 8 9 9 −1 10 −8 11 −2 12 −1

b Integer

c Non-integer

d Non-integer

e Non-integer

f Integer

g Integer

h Non-integer

2 a To the right of 0

b To the left of 0

3 a 1, 2, 3

b −1, −2, −3

4 Neither 5 Integers that are the same distance from 0 on the number line, but are on different sides of 0. Example: The integers 3 and −3 are opposite integers.

840

−5

29 Use variables to track the smallest and largest numbers. Update these variables inside the loop based on conditions. The pseudocode:

1 a Integer

−10

Mathspace Victorian Curriculum – Year 7 mathspace.co

13 7 14 a 9

b 9

e −1

f 7

c −8 ii −2

15 a i 2 b i −12

ii 12

c i 14

ii −14

d i −5

ii 5

16 a −7 e 10

d −5

b 8

c 5

d −27

f −402

g 88

h −123

4.02 Compare integers

17 a −10

−5

Understanding b −10

−5

18 16

1 a 6

b −4

c 6

2 a −4

b −6

c −4

3 The number on the right hand side is the larger. Reasoning 19 False, as when the integer is positive, the opposite will be negative and hence less than the number itself. 20 0 serves as the centre point of the number line, and hence the starting point of the positive and negative integers. 21 Integers are opposites if their locations on the number line are the same distance from 0. If we start at 0 we also end up at 0. 22 8 is an integer value as it is a whole number. If north is considered positive, then 8 km due north can be represented by 8 and 8 km due south can be represented by −8. The integers 8 and −8 are opposite. 23 Some possible answers: Early civilisations believed only countable numbers were real. There was no need for place holders as numbers were not large. 24 No, infinity is not an integer. It is not a number that can be expressed as a fixed value on the number line. Infinity is a concept used to represent a quantity that is unbounded or limitless. Problem-solving 25 a i No

4 a greater than b less than c Order from smallest to largest d Order from largest to smallest 5 Ascending order Fluency 6 a i −3, 9, 14

ii 14

b i −18, −3, 0

ii 0

7 a i 1, −4, −18

ii −18

b i 10, 4, −1

ii −1

8 a 13

b −7

c 8

d 35

9 a −21

b −24

c −1

d −36

10 a 8 < 10

b 2 > −4

c −5 < −1

d −3 > −8

11 a i Negative seven is less than negative two ii True b i Eight is greater than negative six ii True c i Zero is greater than negative one ii True d i T wenty three is less than negative forty seven ii False

ii Yes

e i Seven is greater than and equal to four

b −400

ii False

26 a Arriving 14 minutes early

b −14

27 a Ascending 2 floors is represented by 2.

f i Three is less than and equal to two ii False

b “A weight loss of 11 kg” is represented by −11.

12 a >

b <

c >

d >

c “A temperature of 10 °C above freezing point” is represented by 10.

e >

f >

g <

h >

28 a 21

b −21

29 a −5100

b Being at sea level

13 a −19 °C < 4 °C or 4 °C > −19 °C b −17 °C < −3 °C or −3 °C > −17 °C 14 a −15 < −2 or −2 > −15 b −10 < 0 or 0 > −10

Answers mathspace.co

841

b −4

c 0

d −26

e −73

f −17

g −22

h 1

i −2

j 25

k −30

l 0

10 a 9

9 a 9

Reasoning 15 Gian is correct as −5 is to the right of −7, therefore −5 is the greater integer.

b −15

c 5

d 12

b I would rather be $225 in the black because it means a profit, whereas $784 in the red means a loss. Profit is preferable to loss.

e 0

f 3

g 81

h 2

i 14

j −14

k −27

l −14

m −3

n 7

o −14

p 5

17 Krypton has the lowest boiling point. −153 °C is further to the left of zero than −62 °C.

11 a 32

b 1

c 11

d 13

16 a $8000 profit

Reasoning Problem-solving

12 a i 0

18 −15 °C, −11 °C, −9 °C, 2 °C, 30 °C

ii 0

b The result is always 0 because you move the same distance left and right from zero.

19 −25 < −11 < 10 < 16 < 27

13 0, because it is a neutral number and when you add or subtract zero you do not move left or right on the number line.

20 15 > 11 > 3 > −9 > −13 21 a 2, 4, 6, 8, 10, 12 b 2, 1, 0, −1, −2, −3

14 Yes, the result will be negative if the magnitude of the negative integer is larger than the magnitude of the positive integer. For example, −6 + 5 will be negative because the magnitude of −6 is larger than the magnitude of 5.

c −16, −12, −8, −4, 0, 4 d −4, −1, 2, 5, 8, 11 e 6, 3, 0, −3, −6, −9 f −17, −12, −7, −2, 3, 8

15 Negative. For example, −5 + −8 will be negative because we move further left on the number line.

4.03 Add and subtract integers Understanding 1 Right

Problem-solving

2 Left

16 a D

b 8 km

3 Left

17 a 3

b −4

c −4

4 Right 5 a Start with 3 and move 12 units left. b Start with −5 and move 9 units right. c Start with −2 and move 8 units left. d Start with −4 and move 11 units right. Fluency

d 1 m below the surface

18 18 °C 19 a −6

b 6m

20 185 m 21 525 metres 22 14 people

6 a 14

b 5

c −3

d 0

e 19

f 7

g 4

h 7

7 a 21

b 8

c −9

d 0

e −9

f 0

g 58

h −11

i 0

j −3

k 40

l 32

26 a i 254 mL

ii 243 mL iv 251 mL

23 −210° C 24 56 points 25 a 1

b −2

8 a 3

b −2

c 0

d −18

iii 250 mL

e 1

f 1

g −3

h −19

b 2499 mL

842

Mathspace Victorian Curriculum – Year 7 mathspace.co

c 1

d 2

27 a i 5

ii −13

5.01 Identify, order and compare fractions

iii 11

b 3

Chapter 4 review 1 a Integer

Understanding b Non-integer

1 a i 8

ii 9

b i 3

ii 7

c Integer

d Integer

e Non-integer

f Non-integer

c i 22

ii 50

g Non-integer

h Integer

d i 10

ii 3

2 The number on the left hand side is the smaller. 3 a 3

b 5

c −2

d 0

4 a 7

b 11

c −6

d −6

e 3

f −2

5 a i −5, 11, 17

ii 17

b i −20, −1, 3

ii 3

2 a One half

b Three quarters

c Five hundredths d Three and six eighteenths 3 a

6 a −29

b −28

c −3

d −40

7 a <

b >

c >

d >

e >

f >

g <

h <

8 a 27

b −13

c −18

d 0

e 50

f −3

g 30

h 42

5 a

9 a 13

b −25

c −4

d −58

f 0

g −28

h 0

b −21

c 0

d 14

f −4

g −20

h 4

b 1

c 16

d 19

e 23 10 a 18 e −20 11 a 49

12 a Descending 18 metres

b −18

Fluency 4 a

6 a i

b i

c i

d i

e i

f i

13 −20m, −13m, −8m, 5m, 25m 14 −18 < −5 < 8 < 13 < 22 15 a 3, 6, 9, 12, 15, 18 c −5, −2, 1, 4, 7, 10

b −14, −10, −6, −2, 2, 6 d 8, 5, 2, −1, −4, −7

16 17 °C 17 90 m 18 25 boxes 19 a i 506 g ii 495 g

iii 500 g

iv 503 g

7 a

b 5005 g

Answers mathspace.co

843

15 To compare these fractions, we can convert them to have a common denominator. The smallest common denominator is 10, so we can write:

8 a 0

b Since

is greater than

, we know that

the greater fraction. Therefore,

16 If the number line from 0 to 2 is divided into 8

equal parts, each part would represent a

distance of

from 0 to 2.

The point that is halfway between 1 and 2 would

be at a distance of

from 1 on the number line.

To ﬁnd this point, we can start by identifying the

position of 1 on the number line. Since the distance between 0 and 2 is divided into 8

equal parts, 1 would be located at the fourth tick mark

Next, we need to ﬁnd the point that is halfway 9 a

10 a

between 1 and 2. Since each part represents of the total distance from 0 to 2, we can count two ticks to the right of 1 to reach the desired point. 0

Therefore, the fraction represented by the point that is halfway between 1 and 2 on a number

11 a

line from 0 to 2 divided into 8 equal parts is

12 a False

b True

or . c True

d False

Reasoning 13 One half 14 One ﬁfth

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Mathspace Victorian Curriculum – Year 7 mathspace.co

17 First, we can draw two number lines of equal size and label them 0 and 1 on either end. Then, we can divide the line into equal parts to represent the denominators of the two fractions. In this case, we need to divide the line into 4 equal parts for

and 10 equal parts for

Next, we can plot the fractions on the number

5.02 Negative fractions

line by locating their numerator values. For , Understanding we mark a point at the third tick mark from 0, since

is three-quarters of the way from 0 to 1

on the number line. For

, we mark a point at

the seventh tick mark from 0, since

2 Each tick represents 3 Answer may look like:

and 0

Then, we can see that the fraction to the right of

would be to the right of zero, while

on the

number line.

Fluency 4 a

5 a i −1 − 6 − 5 − 4 − 3 − 2 − 1 0 7 7 7 7 7 7

Visually, the midpoint between these points (equally distant) is

would

be to the left.

on the number line, and

are located at the same distance from

zero on a number line, but on opposite sides.

is located

on the number line, meaning

that it is larger. Therefore,

and − .

fractions in the given options are

seven-tenths of the way from 0 to 1 on the number line. 0

1 Answer may look like: The negative fractions have a negative sign (−) in front of them. Therefore, the negative

1 7

2 7

3 7

4 7

5 7

6 7

Problem-solving

b i

19 a

20 a

−2

−1

ii c i

−1

ii d i

21 Suma

−1

−

3 4

−

1 2

−

1 4

22 B

23 mango smoothie, apple smoothie, strawberry smoothie

6 a

7 a

1 4

1 2

3 4

8 9

Answers mathspace.co

845

10 a

11 a

−1

−

−1

When we talk about direction in terms of distance, we use positive and negative numbers to represent movements in different directions. In this case, we are considering a person who is hiking both south and north. Let’s imagine that the starting point is 0 km. km due south, it can

When the person hikes

be represented by the fraction

3 4

5 6

−

5 8

2 3

−

1 2

−

1 2

3 8

−

1 4

1 3

−

1 8

1 6

Problem-solving 15 Answers may look like: 1 8

−1

−

7 8

1 8

−

3 4

1 8

−

1 8

5 8

−

1 2

Since we have moved from

km due north, it can be represented by the

we have moved right

1 8

−

3 8

−

1 4

−

1 8

and

(positive fraction).

16 Answers may look like:

Since positive and negative numbers are on opposite sides of 0 on a number line, the and fractions opposite sides of 0.

−

We can use the number lines to arrange the fractions from smallest (furthest from 0) to largest (closest to 0). In ascending order:

On the other hand, when the person hikes

fraction

−

Reasoning 12 Answer may look like:

7 8

would also be on

200 m

1 of 600 = 200 m 3

400 m

In summary, the fractions representing hiking due north and due south are on opposite sides of 0 because they represent movements in opposite directions, with one being positive and the other being negative.

600 m 800 m

2 3

1000 m

13 Answers may look like:

1000 m below sea level

We can use a number line to compare negative fractions.

5.03 Equivalent and simplified fractions

−1

−

8 9

−

7 9

−

2 3

−

5 9

−

4 9

−

1 3

−

2 9

−

1 9

Understanding 1

−1

and

−

5 6

−

2 3

−

1 2

−

1 3

−

1 6

are located at the same spot on the

Simply compare their numerators. The fraction with the larger numerator is the larger fraction.

number line because they are equivalent

3 Answer may look like:

fractions, meaning they represent the same value.

Write both fractions with a common denominator first.

As we can see from the number line, closest fraction to zero.

is the

14 Answers may look like: We can use a number line to compare negative fractions.

846

2 Answer may look like:

Mathspace Victorian Curriculum – Year 7 mathspace.co

4 5 C, D

Fluency

Reasoning

6 a

15 Answers may look like: Yes,

and

are equivalent fractions. To see

why, we can simplify each fraction to lowest terms.

is already in lowest terms, while

can be simpliﬁed by dividing both the 7 a

numerator and denominator by 4 to get . Therefore, the two fractions are equivalent.

8 a

9 a Yes

b No

c No

d Yes

10 a

11 a

13 a i 35 ii iii b i 32 ii iii 14 a i 55 ii iii b i 65 ii iii

with a

denominator of 12, we need to multiply both the

, which is 3, by 4 to get 12. To keep the

and

12 a

To ﬁnd an equivalent fraction to

numerator and denominator by the same factor. We can start by multiplying the denominator of

16 Answers may look like:

fraction equivalent, we also need to multiply the numerator by 4. Therefore, an equivalent fraction to

with a denominator of 12 is

17 Answers may look like: To determine which pairs are equivalent, we need to simplify each fraction to its lowest terms. For , the only factor that can be divided out is 1, so it is already in lowest terms. For , we can simplify by dividing both the numerator and denominator by 2, giving us . Therefore,

and

are equivalent fractions.

For , the only factor that can be divided out is 1, so it is already in lowest terms. We can say the same for . Therefore,

and

are not equivalent fractions. For , we can simplify by dividing both the numerator and denominator by 3, giving us . For

, we can simplify by dividing both the

numerator and denominator by 4, giving us . Therefore,

and

are equivalent fractions.

In conclusion, the pairs of fractions that are equivalent are

and , and,

and

Answers mathspace.co

847

18 Answers may look like:

3 Answer may look like:

Yes a fraction with prime numbers in numerator and denominator is in fully simplified form as there is no common factor that the numerator and denominator can be divided by, other than 1.

Write both fractions with the same denominator.

Problem-solving

4 Answer may look like: Convert the mixed numbers to improper fractions, write with a common denominator and then add as usual. Fluency

5 a 6

b 36

c 20

d 12

6 a

4 8 4 8

Rewrite with a common denominator

7 a

Blue

Green

Red

Did not answer

Evaluate the sum

Example answer:

ewrite with a common R denominator

4 students like blue, 4 students like green, and 8 students like red.

Evaluate the sum

8 students did not answer.

Rewrite as a mixed number

20 Answers may look like: If 3 out of 8 pizza slices have been eaten, then 5 slices remain. Therefore, the fraction of the

ewrite with a common R denominator

Evaluate the difference

pizza that remains uneaten is . An equivalent

denominator of 16 is

21 Answers may look like:

Rewrite with a common

fraction for the remaining fraction with a

denominator

Evaluate the difference

Rewrite as a mixed number

has the simpliﬁed fraction

(divide the

8 a

numerator and denominator by 5). Multiply both numerator and denominator by 7 (equivalent fraction). So, 7 students

to have

roup the whole number G parts and fraction parts

will like blue in the class next door

5.04 Add and subtract fractions Understanding 1

is correct.

2 Answer may look like: Simply add or subtract the numerators and keep the same denominator.

848

Mathspace Victorian Curriculum – Year 7 mathspace.co

Add the whole number parts and fraction parts

10 a 7

Evaluate the sum

Then we can shade in 2 of those equal parts to

Problem-solving

represent the

of the quantity.

20 21

hours Counting the number of shaded parts and

cups

gram

multiplying by 10, we can ﬁnd that

of 50 is:

Multiplying fractions:

5.05 Fractions of a quantity

To ﬁnd of 50 using multiplication, we can multiply the fraction by the quantity:

Understanding 1 Answers may look like: Yes,

is equivalent to

This is

Therefore, we have found that

by using a diagram and by multiplying fractions.

because of the commutative property of multiplication, which states that the order of factors does not affect the result of a multiplication operation. 2 Answers may look like: Let’s say we want to ﬁnd

3 Fluency 4 a 7

of 50.

of 50 is 20, both

b 32

c 3

d 27

h 40

d 90

e 10

f 7

g 24

h 45

i 80

j 14

k 40

l 180

6 a 390 L

b 200 m

c 1100 kg d 414 m2

e 135 m

f 165 kg

g 225 L

e 56

Using a diagram: The diagram below represents the total quantity of 50

5 a

7 a 20 cents Divide the rectangle into 5 equal parts, since the denominator of the fraction is 5,

850

Mathspace Victorian Curriculum – Year 7 mathspace.co

h 380 cm

b 72 seconds

c 16 days

d 5400 mL

e 42 g

f 36 mm

g 100 cm

h 9000 mg

i 24 minutes

j 7200 m

k 1150 cents

l 1.05 L

m 1320 minutes

n 1980 hours

o 336 days

p 1.1 kg

14 12

Reasoning 8 a

b 360

c Georgia

d In part (a), we know the total and the quantity handed out, so we are finding the fraction using the two quantities. In part (b), we know the fraction handed out so we have to find the fraction of the total to find the quantity handed out. They are effectively the reverse of each other, in terms of working. 9 Answers may look like: There are 180 students in the drama club. To explain our reasoning, we used multiplication to find

of the total number of students in the

school. We then simpliﬁed the resulting fraction to get the ﬁnal answer of 180 students in the drama club.

15 Sister has $16 000 so he would have $24 000 16 $320 17 a A fair coin should land on tails 50% of the

which is . b 25 c 20 times assuming the coin is fair, we can expect that it has an equal chance of landing on heads or tails on any given ﬂip. 18 Fe scored 35 points, Jamie scored 22 points, Sarah scored 20 points 19 a 1159 L is used in total

b 551 L

10 Answers may look like:

5.06 Multiply fractions

If of the marbles in the jar are blue, then the remaining fraction of marbles that are red is

Understanding 1 a To multiply

To ﬁnd how many red marbles are in the jar, we can multiply the total number of marbles by the fraction that are red: So there are 300 red marbles in the jar. If not done previously, to ﬁnd the fraction of marbles that are red, we can divide the number of red marbles by the total number of marbles: Therefore,

of the marbles in the jar are red

or . This coin is

time. That is equivalent to

landing on tails a bit more than expected,

and

using shading:

We can draw two rectangles, one with four equal parts and one with three equal parts, and shade in one part of the first rectangle and two parts of the second rectangle. Then, we can combine the two rectangles and count the number of shaded parts in the combined rectangle to get the product. b To multiply

and

using multiplication:

You multiply the two numerators together and the two denominators together. The resulting product is your answer.

11 Answers may look like:

2 Answer may look like:

The row with the error is the second row. The correct working is below:

is equivalent to 1, so multiplying it with

Number of apples

Number of pears

Number of oranges

another fraction will result in the same fraction that we started with. This is because any number multiplied by 1 is equal to itself. 3 a Yes

b No

c Yes

d No

Therefore, there is only 1 orange in the fruit bowl.

Fluency

Problem-solving

4 a

b 7

h 1

12 $4 13 270 pages

Answers mathspace.co

851

5 a e 1

We can multiply

cup by 6 to find the total

amount of juice in the pack. Amount of juice in the pack

6 a

cup × 6

boxes = 4 cups So, there will be 4 cups of juice in the pack. 17 Answers may look like:

7 a 1

To find the area of the rectangular garden, we need to multiply its length by its width. The length of the garden is

metres, which

can be written as an improper fraction: 8 9

The width of the garden is

metres, which

can also be written as an improper fraction:

10 11 The farmer will use planting. 12

years

cups

of an acre of the land for

of an acre remains.

Now we can multiply the length and width to find the area:

We can convert the improper fraction to mixed fraction.

Reasoning Therefore, the area of the rectangular garden is

15 Answers may look like: If the recipe calls for

square metres. 10 square metres of turf will

cup of sugar, and you

cover it.

only want to make one-third of the recipe, then you need to find one-third of To find one-third of by

cup.

cup, we need to multiply

18 19

So, you only need

cup of sugar to make

one-third of the recipe. 16 Answers may look like: Cynthia is incorrect because if each juice box contains

cup of juice, then a pack of 6 juice

boxes will contain 6 times that amount of juice.

852

Problem-solving

Mathspace Victorian Curriculum – Year 7 mathspace.co

20 a

cups

cups of sugar is needed to make 5 cakes b

cups

c $2.50 ÷ 10 = $0.25 per cup $0.25 per cup ×

cups = $0.625

It would cost approximately $0.63 to buy the sugar needed to make 5 cakes.

Reasoning

5.07 Divide fractions

To divide

15 Answers may look like: by , you need to invert the second

fraction and multiply it with the ﬁrst fraction.

Understanding

So, the correct fraction to invert is . When you

1 Answer may look like: Dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 2 a

invert a fraction, you ﬂip it upside down. So, inverting

gives you

or simply 4. Then,

you can multiply the ﬁrst fraction

by the

inverted second fraction (4) as follows:

Therefore,

4 Answer may look like:

16 Answers may look like:

Yes, you can divide a fraction by itself, and the result is always equal to 1.

We would divide the 8 slices by :

Fluency

divided by

is equal to .

So, we could feed 10 people (not 11 as whole person). We can check this as:

5 a 33

6 a

17 Answers may look like:

e 1

h 7

If it takes

7 a

This is just under the 8 slices.

of an hour to read

9 16 glasses

of a book, we

can use reasoning and proportion to ﬁnd out how long it will take to read the entire book. If it takes

of an hour to read , then to read 1 third

would be half this time - divide

8 8 days

is not a

by 2 to get

of an hour. To read the whole book, we would take 3 times this long, which is

of an

hour.

10 11 30 days

Therefore, Sam is correct.

12 5 presents

Problem-solving

13 7 loaves

18 The tank needs to be ﬁlled with an additional

14 9 bows

412.5 litres of water to reach 19 a i 12 letters ii 9 letters b

cm and

of its capacity. iii 6 letters

cm wide

Answers mathspace.co

853

5.08 Order of operations with fractions Understanding

1 • Evaluate expressions inside the brackets • Evaluate multiplication or division from left to right • Evaluate addition or subtraction from left to right 2 Answers may look like: The brackets in the expression mean that instead of doing the division first, we perform the subtraction inside the brackets first, before dividing

by the

result. 3 a Multiply c Divide e Add

9 a

and

b Divide

d Multiply

and

g Subtract

from

by and

f Multiply

and

h Multiply

and

f 15

Reasoning 10 Answers may look like: To simplify the expression

, we must

follow the order of operations, which is a set of rules that tells us which operation to perform ﬁrst when there are multiple operations in an expression. The order of operations is as follows: • Perform any calculations inside parentheses ﬁrst. • Multiply or divide from left to right. • Add or subtract from left to right. Using the order of operations, we can simplify the expression

by multiplying ﬁrst.

11 Answers may look like: To solve the problem, we need to ﬁnd the total number of students going on the trail walk, which is the sum of the students in the ﬁrst scout group and the students from the second scout group. , which means

Riley’s expression is

Fluency

he ﬁrst calculates the number of students from the ﬁrst scout group who are going on the trail

4 a

5 a

expression is

walk

and then adds the 20 students

from the second scout group. Riley’s calculation for the ﬁrst part of the , which means 9

students from the ﬁrst scout group are going on

the trail walk. Adding the 20 students from the second scout group gives us a total of 29 students on the trail walk.

6 a

7 a

8 a

Reese’s expression is

, which means

he ﬁrst adds the number of students in both

854

Mathspace Victorian Curriculum – Year 7 mathspace.co

scout groups (12 + 20) and then multiplies by . Reece’s calculation for the ﬁrst part of the expression is (12 + 20) = 32, which means there are 32 students in total. Multiplying 32 by gives us 24 students going on the trail walk. So, Riley and Reese do not get the same result. Riley gets 29 and Reese gets 24. This is because they are calculating different things. Riley is

finding the number of students from the first scout group going on the trail walk and adding the students from the second scout group. Reese is finding the total number of students going on the trail walk, including both scout groups. 12 Answers may look like: To find the area of the composite shape, we need to add the areas of the two rectangles together. Ruby uses the expression to find the area, which simplifies to (9 + 6.5) = 15.5 square units. Jade, on the other hand, uses the expression to find the area. This simplifies to

d 9

5 a False 6 a i

b False

c True

−1

d True

ii b i

−2

−1

ii 7 a

8 a

b Yes

c No

d Yes

square units.

So, both Ruby and Jade get the same result for the total area of the composite shape, which is 15.5 square units. Therefore, both methods are correct and will give the same answer.

9 a Yes

Problem-solving

iii

c 11 a i 12

b i 70

13 720 people were surveyed

hours, or 1 hour and 24 minutes.

Chapter 5 review 1 a

iii

12 a

b 1

c 8

f 1

2 a i

13 a 7

b i

3 a

b 7

15 16 a

4 a 0

10 a

However, Jade’s method involves ﬁnding the area of the composite shape as the difference between the area of a larger rectangle and the area of a smaller rectangle, whereas Ruby’s method involves adding the areas of two separate rectangles.

17 15 cups

c 5

Answers mathspace.co

855

18 a

f 1

servings

b Two places because there are two zeros in 100.

kilometres

c 97.3

22 7 batches 23 a

6.01 Decimals and place value

1 10 tenths 2 a 100 hundredths 3 a iii

b 10 hundredths

b i

c ii

Fluency 4 a 7 hundredths c 7 ones

b 7 thousandths d 7 tenths

b 0.4

c 1.605

d 4.9999

6 a 8

b 2

c 9

d 5

c 0.39 > 0.389

b 20.9999 > 20.9998 d 0.005 < 0.01

e 0.483 > 0.4

16 a Tuesday b Wednesday and Friday 17 a Daniel, Dylan, Nicky, Bernice b 2nd 18 9.633

8 a 0.0024, 0.024, 0.204, 0.24, 2.4

6.02 Add and subtract decimals

b 3.068, 3.1, 3.68, 3.681, 3.86

Understanding

9 a 0.54, 0.504, 0.5, 0.054, 0.0054 b 7.82, 7.8, 7.28, 7.168, 7.078

1 Answer may look like: When adding decimals we have to make sure that the decimal points are lined up.

b 52

c 35

d 674

e 648.9

f 5761

g 3.784

h 0.03

i 930

j 5943

k 40

l 8

2 Answers may look like: The decimal points are not lined up.

11 a 3.4

856

For example, in 0.8, the digit in the tenths place is 8, while in 0.672 33, the digit in the tenths place is 6. Since 8 is bigger than 6, we can say that 0.8 is larger than 0.672 33, even though it has fewer digits after the decimal point. Problem-solving

5 a 0.71

7 a 2.305 < 2.309

14 Multiplying and dividing decimal numbers by powers of 10 is different from other numbers because powers of 10 uniquely affect place values, shifting the decimal point left or right. This shifting does not occur when multiplying or dividing by other numbers, which requires more complex arithmetic. 15 Just because a number has more digits after the decimal point doesn’t always mean it’s larger. To compare decimals properly, look at the value of each digit based on its place.

Understanding

10 a 5.6

12 Since both digits in the ones place are the same, we compare the digits in the tenths place. In 3.45, the tenths digit is 4. In 3.045, the tenths digit is 0. Since 0 is smaller than 4, we can conclude that 3.045 is smaller than 3.45. 13 a Right

20 8 cakes 21

Reasoning

b 0.0832

c 0.675

d 0.048

e 9.535

f 75.835

g 0.057

h 0.39

i 0.7966

j 0.423

k 0.4233

l 0.0022

Mathspace Victorian Curriculum – Year 7 mathspace.co

23 0.54 kg

24 a The processes from lowest to highest defect rate: • Process C has a 0.0225 defect rate • Process A has a 0.023 defect rate • Process D has a 0.025 defect rate • Process B has a 0.032 defect rate b The smallest difference is between Process C and Process A with a difference of 0.0005.

6.03 Convert between fractions and decimals Understanding 1

Ones

Tenths

Hundredths

2 Convert the denominator to a power of 10 such as 10, 100, 1000. 3 To convert decimals to fractions, we can start by counting the number of decimal places, that is, the number of digits after the decimal point. 4 If there are 2 decimal places then our fraction will have a denominator of 100. Fluency 5 0.53

Fraction Decimal

Decimal

0.3

0.14

b 0.4

c 0.46

d 0.76

e 0.35

f 0.73

g 0.716

h 2.25

i 1.5

j 1.25

k 1.24

l 8.68

12 a 0.08

b 0.375

c 0.156

d 0.313

e 0.175

f 0.163

g 0.219

h 0.225

13 a 2.75

b 5.5

c 1.875

d 4.4

e 3.3

f 6.833

g 2.714

h 8.667

14 a

e f 15 a

d 0.3

7 a

b 0.67

d 0.33

8 a 0.1

b 0.2

c 0.7

d 0.01

e 0.27

f 0.93

g 0.001

h 0.006

i 0.068

j 0.111

k 0.747

l 0.954

m 1.3

n 9.4

o 6.98

p 5.527

18 0.375 cups

9 a 1.1

b 9.3

c 9.65

d 7.12

19 3.75 litres

16 a 0.8

17 a 0.28

Reasoning

Fraction 0.1

0.01

0.001

20 a True, because the shaded portion is larger than the unshaded portion. b False, because the unshaded portion is less than half of the total rectangle.

b Fraction

858

0.4

11 a 0.24

Decimal

0.02

Fraction

b 0.7

Decimal

0.05

6 a

10 a

0.2

0.5

0.25

0.75

Mathspace Victorian Curriculum – Year 7 mathspace.co

c True. From the image, it appears that the shaded portion is more than 0.6 but less than 0.8

21 Answer may look like:

2 a Greater than both numbers.

Yes, a fraction can have a decimal equivalent that is greater than 1 if the numerator is greater than the denominator.

b In between the two numbers.

22 a

e In between the two numbers. Fraction

c Less than both numbers. d Greater than both numbers. f Less than both numbers.

Decimal

0.5 0.25 0.20 0.125

0.10

g Greater than both numbers. h In between the two numbers.

b Answer may look like: As the denominator of a fraction increases, the decimal value of the fraction becomes smaller. This is because the larger denominator means that the whole is divided into more parts, resulting in smaller individual parts and a smaller decimal value when the fraction is converted to a decimal. Problem-solving 23 a b c 24 a b c 25 0.945 hectares 26 The third friend paid $45. 27 The baker needs 3 cups of flour, 1.5 cups of sugar and

cup of milk.

6.04 Multiply decimals Understanding 1 Answer may look like: When you multiply two decimals that are less than 1, the product will be smaller than the decimals being multiplied. For example, 0.5 × 0.3 = 0.15, which is smaller than both 0.5 and 0.3.

i In between the two numbers. 3 a When only one of the numbers is less than 1. b When both numbers are less than 1. Fluency 4 a i 2

ii 0.15

b i 3

ii 0.108

c i 3

ii 0.006

d i 3

ii 0.040 = 0.04

e i 4

ii 0.0015

f i 6

ii 0.000 042

g i 4

ii 0.0090 = 0.009

h i 4

ii 0.0030 = 0.003

i i 3

ii 0.036

j i 3

ii 0.064

k i 2

ii 1.26

l i 3

ii 0.385

5 a 20

b 200

c 2

d 0.2

e 0.18

f 8.8

g 3.84

h 8.46

i 38.4

j 595.2

k 27.23

l 19.04

6 a 2 0.3 0.04

b 11.70

1.5

7 a 40 6 6

240

0.20 0.2

1.2

8 a 4

0.9

0.02

0.6

0.54

0.012

2.4

9 a 2

0.3

0.9

0.1

0.2

0.03

10 a 0.8 0.07 7

5.6

0.49

0.8

0.64

0.056

b 277.2

b 2.952

b 7.13

b 6.786

Answers mathspace.co

859

numerator and denominator by the same non-zero number, you get an equivalent fraction. Therefore, the value of the fraction does not change. In this case, Jo and Isla multiplied both the numerator and denominator by different numbers, but the result is still the same, because they multiplied by equivalent numbers (10 and 1000, respectively). Therefore, they will get the same answer, which is 4. 14 a i 5 cm ii He has increased the area which would have the effect of increasing the width and also decreased the divisor which will have the same effect. b i 5 cm ii He has decreased the area which would have the effect of decreasing the width and also increased the divisor which will have the same effect. c i 5 cm ii The changes he has made have opposite effects. Increasing the area would tend to increase the width, but he has also increased the divisor which would tend to decrease the width. Problem-solving 15 Juliet got better value, paying approximately $3.91 per kilogram. Luna’s bananas were $4.04 per kilogram. 16 Oliver will need 33 complete bottles and a third of the 34th. 17 160 cups 18 a i 7.5 m

ii 13 frames with 8.3 m metal left over. iii 8 of each type of frame with 2.8 m metal left over.

Understanding

b 6

5 a B

b 25

Fluency 6 a 3

b 6

c 24

d 34

e 1

f 0

g 1

h 90

7 a 0.1

b 7.4

c 3.6

d 9.0

e 27.9

f 289.4

g 0.7

h 10.0

8 a 2.90

b 125.46

c 14.29

d 67.00

e 0.01

f 13.00

g 100.00

h 131.10

9 a 0.4

b 8.4

c 19.4

d 0.3

e 3.6

f 20.0

g 13.9

h 1.0

10 a 2.70

b 393.59

c 0.44

d 7.36

e 80.00

f 2.00

g 149.96

h 0.10

11 a 11.848

b 2.160

c 4.504

d 1.200

e 90.034

f 0.599

g 41.501

h 10.000

12 a 5.6596

b 5.6010

c 12.0067 d 40.0000

13 a C

b Lower

c 108 km

14 a 70 pages

b Lower

c 71 pages Reasoning 15 Opie needs to calculate the decimal value up to four decimal places. The number on the fourth decimal place is the basis whether to round up or round down the number in the third decimal place. 16 a She should only use the 4th decimal place to round to three decimal places. b i 0.321 5

ii 10.9 m

b i 20 frames with no metal left over.

6.06 Round decimals

4 a C

ii 0.322

17 a 4.45 b Go back to the original number. 4.449 is closer to 4.4 than 4.5, so should round to 4.4. Once rounded to two decimal places 4.45 rounds to 4.5, which is further from the original number. 18 It is a convention (an agreed upon standard) and is consistent with the check digit being 5.

1 11

Problem-solving

2 39

19 a 30 cm

3 B and D

Answers mathspace.co

861

b Overestimate, both length and width are rounded up. c 28.88 cm 20 a $50

d 1.1 cm

16 $88.45

b Yes.

17 1.8 cups of cocoa powder and 1.2 cups of milk.

c No, because the exact total cost is $50.60 d No, by doing this Tommy has estimated the cost as lower than it actually is. 21 a 12 g and 16 g c Overestimate

b 4g

4 a 1.8 e 5.33

c 5

7.01 Fractions, decimals and percentages Understanding

f 79.465 b 3.1

c 6.7

d 0.23

1 a Multiply

f 6.87

g 0.427

h 53.21

2 100

c 0.54

d 0.45

e 0.768

f 1.75

g 1.28

h 0.25

i 3.74

j 6.75 b

9 a 0.28

d 0.040 = 0.04

e 3.96

f 13.05

g 622.5

h 19.26

ii 0.2, 0.4, 0.6, 0.8, 1.0 d i 1 2.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%, 100% ii 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0

10 a 9.09

b 5.08

c 9.76

d 80.12

e 60.18

f 80.1

g 96.55

h 170.1

b 4

c 600

d 102.5

f 900

g 3000

h 800

12 22 days. b 7.8313 d 22.9995

14 1.18 kg 15 a

862

b i

c i 20%, 40%, 60%, 80%, 100%

b 0.0012

c 0.0054

c 10.0569

5 a i 25%, 50%, 75%, 100% ii 0.25, 0.5, 0.75, 1.0

ii 0.33..., 0.66..., 1.0

8 0.84 cm

13 a 3.2457

4 45% Fluency

7 a

e 4000

b Divide

3 50% b 0.32

11 a 70

c No

d 1000

5 $5.50 6 a 0.17

b Yes

d No, by doing this Ally has estimated the cost as lower than it actually is.

2 3.7, 0.37, 0.073, 0.0073, 0.00 073 e 1.71

19 Sally will need 42 complete containers and two-thirds of the 43rd.

21 a $420

1 1.351

b 0.74

18 $7.00

20 120 packets

d 3.4 g

Chapter 6 review

3 a 0.7

Mathspace Victorian Curriculum – Year 7 mathspace.co

e i 1 0%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100% ii 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 6 a 0.18

b 0.57

c 0.234

d 0.055

e 1.2

f 3.03

g 5.00

h 68.97

7 a 7%

b 85.7%

c 530%

d 2400%

f 4.1%

g 0.35%

h 137.5% d 80%

e 1304% 8 a 47%

b 68%

c 55%

e 16.67%

f 61.11%

g 63.89% h 53.33%

i 650%

j 235%

k 720%

l 332%

9 a 20%

b 25%

c 62%

d 25%

10 a 0.485

b 0.082

c 0.068

d 0.026

e 0.043

f 0.135

g 0.222

h 0.618

11 a

b 0.2

21 a

c 20%

22 203 marbles are not red 23 Test 1, 64%. Test 2, 67.5%. Juliet did better on Test 2. 24 a 80%

b 60%

12 a

7.02 Compare percentages

Understanding

1 a

b 50%

70%

2 a

b 40%

0.53

53%

0.02

3 Convert all three numbers to the same format in order to compare them. So all percentages, or all fractions or all decimals.

0.006

0.6%

Fluency

1.23

123%

4 a i 50%, 100%, 30%

4.3

430%

b i 20%, 70%, 35%

2.8

280%

c i 37.5%, 20%, 80% ii

11.25

1125%

d i 20%, 45%, 27%

Decimal

Percentage

0.7

Fraction

Reasoning

5 a 13%

14 a Yes

c For example: Can get a more than 100% pay raise. 15 16 a C

b A

c D

17 For example:

18 a

b Same

c Same

b No. Test score maximum is 100%. You can’t score more than 100% of the test.

b 100

d 46%

6 a 69% > 0.65

b 71% > 0.59

c 51% < 0.87

d 0.052 > 4.9%

7 a True

b True

c True

d True

e True

f False

g True

h False

8 a False

b True

c False

d False

9 a

c 92%, 60.1%, 0.36, 0.1 Problem-solving 19 15% 20 70%

d Reasoning 10 For example: 88%, 88.5%, 89%

Answers mathspace.co

863

11 For example: 0.6930, 0.6935, 0.6937

4 a i

ii 42%

b i

ii 80%

13 For example:

c i

ii 5%

Problem-solving

d i

ii 50%

14 40

e i

ii 175%

f i

ii 40%

g i

ii 160%

h i

ii 12%

12 For example:

15 Shop C 16 Stella

7.03 Express a quantity as a percentage Understanding 1 a 30% is the same as 30 out of 100 and 3 out of 10.

5 a i 40%

b 15% is the same as 15 out of 100 and 3 out of 20. c 75% is the same as 75 out of 100 and 3 out of 4. d 80% is the same as 80 out of 100 and 4 out of 20. e 50% is the same as 50 out of 100 and 1 out of 1. f 6 2% is the same as 62 out of 100 and 31 out of 50.

b i 75%

ii 35%

c i 220%

ii 350%

6 a 18%

d 25%

7 a 20%

b 54%

e 10%

8 9 a 20 students b 37.5%

c 62.5%

10 40%

2 C Fluency

11 a 49.8%

b 20%

c 1.6%

d 3.5%

e 70%

f 30%

g 15%

h 33%

3 a i

b i

12 61.40%

c i

13 a

d i

14 Sale 2

e i

f i

g i

h i

Reasoning

Problem-solving 16 55% b 40%

c 40%

d 20%

18 9% 19 a 80%

Mathspace Victorian Curriculum – Year 7 mathspace.co

15 a

17 a 20

864

iv 20%

iii

b 76%

c History

20 a 30%

b 60%

c Georgia

7.04 Percentages of a quantity Understanding 1 Fraction Percentage 100% 50% 25% 20% 12.5% 10% 5% 4% 2% 1%

b When ﬁnding a percentage of 82 that is below 100%, the result will be smaller than 82. This is because a percentage below 100% means that you are considering less than the original value (82 in this case). For example, 50% of 82 would be half the original value, so it would be smaller than 82. c i Method 1: Using fractions 60 % can be

b 2

2 a

For example, 200% of 82 would be twice the original value, so it would be larger than 82.

4 It means that the percentage is more than the original quantity.

multiply this fraction by Method 2: Using decimals 60% can be expressed as a decimal:

Fluency 5 a i 3

ii 7

iii 21

b i 14

ii 3.5

iii 49

c i 7

ii 9

iii 63

d i 7

ii 3

iii 21

6 a 0.9 × 20 = 18 c 0.6 × 60 = 36

d 0.21 × 400 = 84 b

. Now

multiply this decimal by 82: 0.6 × 82 = 49.2 So, 60% of 82 is 49.2. ii Method 1: Using fractions 120% can be expressed as a fraction:

b 2.2 × 10 = 22

7 a

. Now

multiply this fraction by Method 2: Using decimals 120% can be expressed as a decimal:

. Now

multiply this decimal by 82: 1.2 × 82 = 94.4 So, 120% of 82 is 94.4

8 a 9

b 4

c 9

d 8

e 55

f 98

g 18

h 72

13 90 minutes 14 a 28

b 28

i 26

j 27

k 32

l 160

c They give the same result.

9 a 8g

b $12

c $60

d 6 mL

d Yes.

f 14 km

g 30 L

h $18

e 72 kg

. Now

expressed as a fraction:

3 Divide by ten

. The two expressions

are both equal to

10 a 62% of 46

b 27% of 230

c 250% of 18

d 45% of 180

Problem-solving

e 29% of 13

f 36% of 48

15 16.67%

Reasoning

16 38 minutes and 20 seconds

11 Some possibilities, student responses may vary:

17 152 18 a 2910

b 2340

19 a 3500 people

12 a When ﬁnding a percentage of 82 that is above 100%, the result will be larger than 82.

20 2625 mg

This is because a percentage above 100% means that you are considering more than the original value (82 in this case).

21 a $3840

c 5250 b 385 children

b $99 840 c $1920

d 4.17%

Answers mathspace.co

865

7.05 Percentages with technology Understanding 1 a 45 + % or 45 + Shift + % b =

b Multiply 40% by 12

c Multiply 130.8% by 9 d Divide 17 by 80% Fluency 3 a 187.5%

b 83%

e 68.75% f 124%

c 60%

d 52.5%

g 1843.75% h 83.3%

4 a 5.2

b 2.6875

c 4.0625

d 9.6875

5 a 13.24

b 0.093

c 0.4505

d 431.9

e 33.21

f 2.8

g 49.395

h 0.24

6 a d 109.872 7 a 61

b 89.1268

c 9.3515

e 2.755

f 0.834 372

b 65.9

c 40

c The manager

d Yes

e $80 870

f $80 000

Problem-solving

c S⇔D

2 a Divide 29 by 60

b 1.08 × 1.04 = 1.1232

13 a No

d 78

14 a 17.5 kg

b 67.5 kg

15 a $7.50

b $22.50

16 a 8

b 80

17 a 60%

b 40%

Chapter 7 review 1 a 0.10

b 0.93

c 0.234

d 0.045

e 1.6

f 4.09

g 7

h 78.94

2 a 3%

b 2.1%

c 3500%

d 45.7%

e 620%

f 1504%

g 0.35%

h 332.5%

3 a 87.5%

b 32.14%

c 62%

d 14.5%

Fraction

8 128

Decimal

Percentage

0.07

1.51

151%

9 3 780 000 megalitres 10 66

0.625

Reasoning 11 Possible answers include:

and 0.2

and 2.25

and 0.1666

and 0.625

6 a i 40%, 80%, 20%

b i 20%, 45%, 27%

5 a

• When ﬁnding 1%, 10% or 100% of a quantity. • When converting easy fractions or decimals like

to percentages

12 a A 25% discount leaves us with 75% to pay, and a 12% discount leaves us with 88% to pay b To verify that Betty is correct, we can multiply 0.75 by 0.88: 0.75 × 0.88 = 0.66 Since the result is 0.66, Betty is indeed correct. Bernie could have just multiplied the price of the jacket by 0.66 to get the final price he needs to pay. c Yes, Bernie would get the same result if he applied the two discounts in the reverse order. This is because the multiplication is associative, meaning that the order of the discounts does not matter.

866

Mathspace Victorian Curriculum – Year 7 mathspace.co

c i 12

%, 60%, 30% ii

, 0.30, 70%

d i 62.5%, 30%, 70% ii 30%, , 0.7 > 0.40%

b 0.75 < 80%

< 57%

d 0.095 <

7 a c

8 a True

b True

c True

d True

e True

f False

g False

h False

9 a 12%

d 65%

10 a 22%

b 36%

c 60%

d 54%

11 a 35%

b 40%

c 25%

d 15%

e 68.75% f 25% 12 35% 13 a 20

b 35%

c 35%

d 30%

14 35% 15 7.5% b 2

16 a

6 a 5:7

b 47 : 43

c 13 : 17

d 37 : 17

e 5 : 26

f 13 : 29

g 15 : 76

h 130 : 171

7 a 9:8

b 100 : 31

c 200 : 409

d 21 : 120

e 360 : 659

f 43 : 52

g 37 : 24

h 13 : 120

8 a i 3:2

ii 4 : 2

b i 2 : 12

ii 4 : 12

17 a 4

b 55

c 98

d 18

9 a 11 : 6

b 12 : 5

c 5:8

e 72

f 26

g 27

h 32

d 3:2

e 2:1

f 50 : 1

18 a 250% of 14

b 450% of 16

10 a 5 : 2

c 29% of 13

d 36% of 48

Reasoning

19 a 20

b 20

c They give the same result. d Yes.

× s.

× r.

The two expressions are both equal to 20 a 2370

b 2280

c 4650

21 1875 mg 22 a $4250

b $89 250 c $1716.35 d 5.88%

23 a 6.725

b 0.01 025 c 0.370

e 65.5668 f 2.8 24 a 43

b 43

d 2.635

g 49.395

h 0.24

c 40

d 53

8.01 Identify ratios Understanding

b 6:2

12 Both Lucidia and Annalee are correct. The order in which they state the length : width and the width : length are stated correctly. 13 First, we need to ﬁnd the total length of the bridge by adding all of the three lengths. So, we have the total length of the bridge is 12 m. Next, we need to ﬁnd the length of PR by adding the lengths of PQ and QR. So PR = 5 m. Finally, we can express the ratio of PR to the total length of the bridge as 5 m : 12 m. So, the ratio of PR to the total length of the bridge is 5 : 12.

b Answers vary.

Problem-solving

c Answers vary.

d Answers vary.

14 3 : 8

b The number of dogs being compared. c 3:5

d 7 : 20

11 The ratio of the number of oranges to the total number of fruits in the salad can be written as “oranges : total fruits”. Since the total number of fruits is 4 + 6 + 8, the ratio can be written as 6 : 18.

1 a Answers vary.

2 a The number of cats being compared.

c 5:6

15 a 0.08 litres 16 a 1 : 4

b 0.025 litres b 4 : 45

c 45 : 16 : 4

17 a 3 : 1

Fluency 3 a 2 : 15

b 5:6

4 a i 15 : 5

ii 15 : 20

b i 7 : 13

ii 7 : 20

5 a i 11 : 4

ii 11 : 15

b i 14 : 2

ii 14 : 16

c i 15 : 1

ii 15 : 16

b The length will also be doubled to remain in the proportion of 3 : 1, and it will still be 3 times as large as the width. c The ratio of the area to the width is 6 : 1. This is different from part (a) where the ratio was 3 : 1. This is because the area increases with the square of the increase in dimensions, hence when the dimensions were doubled, the area became four times bigger.

Answers mathspace.co

867

8.02 Equivalent and simplified ratios Understanding 1 a 2:1=4:2 c 3 : 9 = 6 : 18 e 2 : 7 = 4 : 14 g 5 : 20 = 1 : 4

b 2 : 5 = 4 : 10 d 8 : 4 = 12 : 6 f 6 : 4 = 18 : 12 h 4 : 16 = 8 : 32

2 6 Fluency 3 $54 4 24 5 a 2:5 e 1 : 20 i 2:3

b 1:2 f 2:1 j 9:7

c 1:2 g 3:8 k 9 : 11

d 5:1 h 3:4 l 9:7

6 a Example answer: 6 : 2, 9 : 3, 30 : 10 b Example answer: 3 : 5, 9 : 15, 12 : 20 c Example answer: 4 : 10, 6 : 15, 8 : 20 d Example answer: 1 : 10, 2 : 20, 5 : 50 7 a 1:6

b 4:3

c 2 : 45

d 9:5

e 33 : 28

f 9:7

g 8 : 15

h 9:4

i 9 : 10

j 33 : 35

k 11 : 27

l 21 : 17

8 a 10 : 9

b 5:8

c 2:9

d 8 : 13

e 7:4

f 5 : 26

g 3 : 10

h 52 : 95

9 a 41 : 58

b 90 : 7

c 4 : 17

d 33 : 7

e 4:7 Reasoning 10 We can increase 1 by the same multiple to ﬁnd the number of security guards. We will divide 8400 spectators by 120 spectators to get the multiple. The multiple is 70, so multiplying 1 by 70, the number of security guards required is 70. 11 First, we will convert one of the amounts so that the units match. Let’s convert $4.00 to cents. Next, we will write the ratio and decide if we need to simplify it. 620 : 400 and the greatest common factor is 20. So we can divide both sides by 20 to simplify the ratio: 620 ÷ 20 : 400 ÷ 20. Finally, the simpliﬁed ratio is 31 : 20. This is the smallest integer valued ratio.

868

Mathspace Victorian Curriculum – Year 7 mathspace.co

12 a This is true. In order for the ratio to be equivalent, increasing apples also requires us to increase oranges. b This is false. When we increase one number in a ratio, the other number must increase by the same amount. c This is true. Both numbers in a ratio increase by the same amount in order to keep the ratio equivalent. d This is false. While it is true that there cannot be any apples if there are no oranges, the opposite is not necessarily true. It is possible to have oranges without apples. 13 a The units are not the same. b

c 6:7

d 12 metres

14 a 6 : 9, because it simpliﬁes to 2 : 3, while all other ratios simplify to 3 : 4 and it could be rewritten as 6 : 8 to make it equivalent to the others. b 14 : 30, because it simpliﬁes to 7 : 15, while all other ratios simplify to 1 : 2 and it could be rewritten as 14 : 28 to make it equivalent to the others. Problem-solving 15 a 16 litres

b 20 litres

c The painter will be able to paint 2 full rooms, with paint left over to paint almost half of a third room. 16 a 16

b 128

c No

17 Bristlecone: 600 Aleppo: 540

8.03 Find part or whole measures Understanding 1 The total parts of the ratio is the sum of the ratio parts. 2 4:3:7 3 We can write a part to whole ratio and ﬁnd what multiple of the whole will reach the total number of blocks. Then, we can multiply each part of the ratio in order to create an equivalent ratio. The equivalent ratio will show us how many blocks Mohamad and Isabelle will receive.

Fluency

Problem-solving

4 35 melons

13 a 7 parts

5 a 80 adults

b 280 children

14 27 litres

6 a $160

b $720

15 a 3 cm2

7 a 1:3

b $40

8 87 km/hr Reasoning 9 a 1:2 b 9 is 3 times of the original parts of blue paint, so multiply also 6 by 3 to get 18. 10 a 11 : 9 b To find the number of metres of green material, Avril can divide 33 metres of red material by the part of the simplified ratio representing red: 33 ÷ 11 = 3. Then, Avril multiply both sides of the ratio by 3 to find the number of metres of green material needed.

b i 36 cm2

b 12 cm

c 9 cm

ii 21 cm2

iii 93 cm2

16 72 minutes

Chapter 8 review 1 a 24:42

b 18:54

c 90:150

2 a 3:2

b 3:1

c 7:5

3 a 3:20

b 11:12

c 3:1

4 a 1:2

b 4:3

c 4:11

5 a 12:1

b 4:1

c 5:7

6 $7000 7 21 8 2:5 9 a 9 litres b 15 litres

So Avril needs 27 metres of green material. 11 a The student is making a mistake in interpreting the given ratio. The ratio of Kathleen’s investment to Luke’s investment is 3 : 10, which means for every 3 parts that Kathleen invests, Luke invests 10 parts. The student incorrectly assumes that Kathleen’s investment of $3000 corresponds to 3 parts and Luke’s investment of $7000 corresponds to 10 parts, but this doesn’t match the given ratio. b Kathleen and Luke invest money in a business in the ratio 3 : 7. 12 One possible way to ﬁnd the number of Gumtree trees is by writing the part to whole ratio of Gumtree trees to the total, which is 8 : 17. Then,

c The artist will be able to paint 3 full studios, with paint left over to paint a portion of a fourth studio. 10 a 36

b 132

c No

11 Gala: 550 Honeycrisp: 660 12 18 oranges 13 $42 14 27 km/hr 15 a 15 cm

b 12 cm

16 25 litres 17 25

multiply this by the total trees:

18 4 cups

Gumtree trees. Another way is by ﬁnding the

9.01 Introduction to variables

number of trees in one part, that is by dividing 340 by 17 total parts, followed by multiplying 20 trees in each part by 8 parts for the Gumtree trees, which gives 160 Gumtree trees.

Understanding 1 A variable is used to represent an unknown number.

Answers mathspace.co

869

Reasoning

9.03 Substitution

11 a i 48

Understanding

ii 48

iii 48

iv 48

b The apple is red.

b The answer is always 48 because the numbers in each expression multiply to 24 which is then multiplied by p.

c The ﬂag is red and blue.

c 48

d My favourite colour is red.

d The result is also 48 because p + p = 2p and 24 × 2 = 48.

1 a A red pencil.

2 a Emma is 8 years old.

12 Any three expressions that are equal to 23 such as 4x + y

b I have 8 pens. c The result of 10 − 2 is 8. d 8 is one more than seven. 3 a 5×3

b 3 + 17

c 1−3

4 a x=1

b x=7

c 19

13 The smallest values of a and b is 1, therefore 10a + b = 11. The largest values of a and b is 9, therefore 10a + b = 99. Therefore 10a + b is always a two digit number.

b 12 × 4 − 8

Problem-solving

14 12

5 a 7×6+5 c

15 −15

Fluency 6 a 11

b 12

c 2

d 2

7 a 3

b 20

d −9

8 a 17

b 29

c 3

d 36

e −30

f 2

g 9

h 17

i 6

j 17

k 51

l 7.2

n 22

o 13

p 5

q 38

r 58

9 a 19

b 28

e 83

f 138

10 a

872

d −10

c 27

p+7

6p + 4

34 244

2x + 4

5p – 3

11p + 5

60 126

Mathspace Victorian Curriculum – Year 7 mathspace.co

16 a i Increases

ii Decreases

b i Decreases

ii Increases

17 a i Halves

ii Doubles

b i Doubles

ii Halves

18 a i Doubles b i Multiplied by 4

ii Multiplied by 3 ii Multiplied by 9

19 a 5y

b 150

20 a dp

b 3p

21 a k + m

b 20

9.04 Collect like terms Understanding 1 Like terms have the same variables raised to the same power 2 A 3 B Fluency 4 a Like

b Like

c Like

d Unlike

e Like

f Unlike

g Like

h Like

5 a 10 m and 8 m, 2 and 3 b 9s and −6s c k and 4k

d 10y and −y, 4 and 3 e 9a and −11a, −7b and −15b, 9 and 20

16 When x = 5, 6 × 5 + 2 × 5 − 5 = 35 and 7 × 5 = 35. Therefore 6x + 2x − x = 7x When x = 100, 6 × 100 + 2 × 100 − 100 = 700 and 7 × 100 = 700. Therefore 6x + 2x − x = 7x

f Does not contain like terms g 11 mn and −8 mn h −17ab and −5ab, −3cd and 7cd 6 a Yes

b No

c Yes

d No

7 a Yes

b No

c Yes

d Yes

8 a No

b Yes

c No

d No

When x = 0.5, 6 × 0.5 + 2 × 0.5 − 0.5 = 3.5 and 7 × 0.5 = 3.5. Therefore 6x + 2x − x = 7x 17 a i 15x

iv 1.5k

b 7 × 5 m means 7 lots of 5 m. You can multiply the numbers together to get 35 m.

ii 13u + 15

Problem-solving

b i 7u and −7u

ii 2v

c i 4x and 5x

ii 9x − 6

18 a 2x + 6x = 8x

d i 12 and −8

ii 6x + 7y + 4

9 a i 5u and 8u, 7 and 8

iii −12c

ii 20p

b 3y + 3 + 5y + 5 = 8y + 8 c 7x − 5x + 10 − 9 = 2x + 1

b 5b

d 9x + 4y + 2y = 9x + 6y

d 9 + 7m − 2p + 3q

e 7x + 4x + 10y − 4y = 11x + 6y

e 9y + 1

f 11x − 5

f 8ab + 2b + 9b + 6ab = 14ab + 11b

g 14c

h 7x

i b

j 6n

k 12x

l 5xy

10 a 7a c 7 + 6a

19 a 3x + 9 + 10x − 3 + x + 6 20 12a metres

m 5ab

n 8b + 11

21 15y metres

o 10m + 7

p 16x + 5

22 a 8x

b 8q

23 a 15h

11 a 25m + 8n c 10m + 10n

d 3x + 19y

e 18a + 8b

f 5m + 12n

g 3ab + 2c

h 6s + 2t

i 21xy + 17

j 11a + 9b − 4c

k 4x + 4y + 5

l 15m + 19n + 7

m 25x + 2y + 11

n 12y + 2

o 8x − 7y + 4z + 13

p 23j + 2k + 5

12 a 4x

b 4n

b 10x

c 18x

b 20h

c 35h

d $70h

d 14 hours

e $560 24 a

Chapter 9 review 1 a 3 is multiplied by y

c 6m

d 8k

e 2y

f 9l

g 15h

h 21x

i 6z

j 6a

k 4b

l 8c

13 a 13x + y + 10z + 6

b 14x + 12

b 10xy + 2yz + 6xz + 3

c 16x + 2y + 41

d 11 mn + 2 m

e st + 7s + 7t + 5

f 19ab + 3bc − 9b

b 4 is added to z c The difference of a and 5, divided by 7 2 a 3p + 8

3 a

b zy + 3

4 15 + 0.75p 5 2x − 6

Reasoning

6 60 − r

14 1 × y is just y therefore the 1 is not needed.

7 a 19

b 9

c 35

15 5abc and −9acb are like terms, because they have the same variables raised to the same power. The order of multiplication does not matter.

8 a 7

b 34

c 42

9 a

r+5

Answers mathspace.co

873

5r + 3

178

6 a 4

b 10

c 17

d 7

e 4

f 9

g 4

h 4

7 a True b False. The equation could be: 57 = 15 + 6 × 7 57 − 4 = 11 + 6 × 7

10 10

c False, the equation could be:

11 −9

5 × 7 − 19 = 16

12 4k

5 × 7 − 22 = 13

13 a 3 + 8b

b 10xz

14 a 16x + 2

b 4p + 11q − 6

15 a 52x

b 13.5a

16 a ru + 9r + 9u + 5

c 10cd

c 15c b 19cd + 6de − 10d

17 16x metres

d True 8 a x = 12

b x = 12

c x=7

d x = 27

e m = 13

f s = 23

g u=6

h m = 48

9 Answers may vary a The product of eight and x is equal to thirty two.

18 15y

b The quotient of sixty three and x is equal to nine.

19 $95k

c The product of ﬁve and y is equal to sixty. b 30 hours

20 a

d The quotient of seventy two and p is equal to eight.

10.01 Introduction to equations

e The product of six and x plus four is equal to sixty minus twenty.

Understanding

f T he product of eight and x plus seven is equal to seventy two minus twenty ﬁve.

1 An operator joins two equations together using an equals sign. An equation is true if the expression on the left hand side is equal to the expression on the right hand side. 2 Answer may look like: An equation has an equals sign but an expression doesn’t. 3 Answer may look like:

b m – 16 = 26

10 a c q – 12 = 25

d 7y = 25 – x

f 5q = 29 – p

g 65 – 7p = 12

h 72 – 12r = 36

11 a Yes

b No

c Yes

d No

To solve an equation, you need to ﬁnd the values of any variables that will make the equation true.

e Yes

f Yes

g No

h Yes

12 a No

b Yes

c No

d No

e No

f No

g Yes

4 Answer may look like:

13 a No

b No

c Yes

d No

e No

f Yes

14 a Yes

b No

c No

d No

e No

f No

g Yes

h Yes

He uses substitution where he puts x = 6 into the equation and checks the equation is true. Fluency 5 a Equation

874

b Expression

c Expression

d Equation

e Equation

f Expression

Mathspace Victorian Curriculum – Year 7 mathspace.co

15 a

m−17

b m = 29

16 a

4×k

96 100

c The number of matchsticks is equal to 3 times the number of squares plus 1.

d M = 3F + 1

b k = 24

6 a N = 2F + 1

Reasoning

b 25 tiles

17 a Left-hand side

7 a T = 3W−2

b Try a number larger than 10.

b T = 298 tiles

c Right-hand side d The solution to the equation lies between 10 and 20. 18 a Right-hand side b Try a number larger than 30.

8 a i y increases by 1

ii y = x + 8

b i y increases by 1

ii y = x − 5

c i y increases by 7

ii y = 7x

d i y increases by 5

ii y = 5x + 2

e i y decreases by 3 ii y = −3x + 17

c Left-hand side d The solution to the equation lies between 30 and 40. Problem-solving 19 C

f i y increases by 2 9 a y=x−3

ii y = 2x − 5 b y = 5x

c y=x+7

d y = 2x + 2

e y = 3x − 13

20 C = 20 + 12h

Reasoning

21 10p = 30

10 Method 1: continue the pattern of adding 4 to the b values and the missing number is 14 + 4 = 18.

22 b + 16 = 42

Method 2: Find the rule for the table b = 4a − 10 and use it when a = 7 to ﬁnd b = 4 × 7 − 10 = 18.

10.02 Write rules for relationships

11 Yes the rules are the same as it doesn’t matter what order you do addition in.

Understanding 1 a Increasing by 1

b Decreasing by 2

c Increasing by 7

d Decreasing by 5

12 a Each triangle has 3 sides therefore we multiply by three.

2 a The coeﬃcient is 3

b The coeﬃcient is −2

b M = 4S, where s is the number of squares.

c The coeﬃcient is 3

d The coeﬃcient is −4

c M = 6H, where H is the number of hexagons. Problem-solving

Fluency 3 a

Figure number (F )

Number of matchsticks (M)

4 a

Figure number (F )

No. matchsticks (M)

b Increasing by 1

No. of bunches (x)

No. of bananas (y)

b The number of bananas is equal to 4 times the number of bunches.

c M = 3F

b 3

13 a

c M=F+4

5 a 3 b The number of matchsticks increases by three

c y = 4x 14 a

No. of decks (d)

No. of cards (c)

104

156

208

b The number of cards is equal to 52 times the number of decks. c c = 52d

Answers mathspace.co

875

15 a The number of marbles will be equal to 4 times the number of bags plus his original 5. b M = 4b + 5 16 a

Week (W)

Account $200 $215 $230 $245 $260 total (A) b A = 15W + 200 17 a

c i p − 20 = 5

ii p = 25

d i p − 13 = −15

ii p = −2

e i p + 39 = 83

ii p = 44

f i 75 = p + 42

ii p = 33

g i p − 17 = 25

ii p = 42

h i

ii p = 96

i i 10p = 60

ii p = 6

Figure number

j i

ii p = 72

No. of tiles used

k i 3p = 18

ii p = 6

l i 12p = 84

ii p = 7

ii p = 121

n i 7p = 84

ii p = 12

b The number of tiles used are the square numbers. c N=F2

d 49 tiles

10.03 One-step equations

Reasoning 11 Answer may look like:

Understanding 1 a Addition

b Multiplication

2 a Divide by 8

b Subtract 8

c Multiply by 8

d Add 8

Fluency

I would subtract 5 from both sides of the equation. This is because the goal is to isolate the variable (x) on one side of the equation, and subtracting 5 from both sides will eliminate the constant term 5 from the left side. 12 Answer may look like:

3 a Divide by 7

b Add 2

c Subtract 3

d Add 4.5

e Divide by 4.1

f Add 1.39

g Divide by 4

h Divide by 7.3

Yes, it is possible to solve this equation by adding 8 to both sides. Since the constant term on the right side is negative, adding 8 to both sides will eliminate it from the left side.

i Subtract

j Add

13 a 30h = 300

k Subtract

l Add

There could be several reasons why Samantha might charge more or less per hour for her freelance work compared to Lisa. It could be based on factors such as their experience, skills, expertise, location, demand for their services, etc. Samantha may have more experience or expertise in a particular area, or she may be in higher demand than Lisa. Alternatively, Lisa may live in a location where the cost of living is lower, allowing her to charge less per hour.

b 10 hours

c 4 hours

d Answer may look like:

4 a x = 50

b y=9

c b = 35

d k=7

5 a x = 50

b y=9

c b = 62

d k = −18

6 a x = 30

b d = 12

c x=9

d a=7

7 a x = 24

b y = 36

c m = 200 d x = 0

8 a q=2

b s = 21

c r = 47

d u = −1

e v = 48

f u = 26

g k=5

h k=7

i r = 20

j q = 42

k y = −11

l p = 35

mx=0

n n=5

o x = 78

p m = −5

9 a x = 1.8

b x = 6.6

e x=0

f b = 12

10 a i 10 + p = 12

ii p = 2

b i −9 + p = 21

ii p = 30

876

Mathspace Victorian Curriculum – Year 7 mathspace.co

Problem-solving 14 a 10n = 80, n = 8

c m − 25 = 44, m = 69 d h + 56 = 80, h = 24

15 a The company needs to hire 50 new under 35 year-old employees.

b The percentage of the total workforce that will be under 35 years old after the new hires will be 45.45%.

10.04 Two-step equations

Understanding 28

1 a Take shoes off then take socks off. b Turn lamp on then get out of bed. d Finish eating lunch then put lunch box back in bag. b ÷6 then + 8

c +2 then × 5

d ×9 then − 5

6 a i y

b +7

c ÷2

d ×5

10 5y 10

ii Determine the inverse operations.

3 a −3

c Stop watching movie then turn off TV.

2 a −11 then ÷ 3

5y 10

Fluency 2

4 a 3 2 4

Solve using the inverse operations. 3

b (6 3) 2

18 3

b i 8

q 3

q 4

5(q 3)

q 3

5(q 3)

ii Determine the inverse operations.

5 a

Solve using the inverse operations.

Answers mathspace.co

877

c i

s 9

Solve using the inverse operations.

6 6(s 9)

ii Determine the inverse operations.

6 s 9

6(s 9)

Solve using the inverse operations.

f i

9 8y 9

ii Determine the inverse operations.

y 8

d i

9 8y 9

5 9

n 8

Solve using the inverse operations.

ii Determine the inverse operations.

n 8

Solve using the inverse operations.

7 a 2n + 6

b 2n + 6 = 12

c Subtract 6 from both sides then divide both sides by 2. 8 a

c Multiply both sides by 5 then subtract 8 from both sides. 9 a e i

c Subtract 5 from both sides then multiply both sides by 2. 10 a w = 14

b x=4

ii Determine the inverse operations.

878

Mathspace Victorian Curriculum – Year 7 mathspace.co

c m=3

Chapter 10 review

d y=5

b Yes

c No

2 a Yes

b No

c Yes

3 C

1 a No

4 b + 12 = 39

e x=2

f t=6

g m=1

h p = 10

i x = 40

j x = 15

k p = 66

l n = 72

5 a y = 3x + 2

b y = 8x − 3

11 a u = 3

b q = 34

c m=9

d t = 37

6 a Divide by 3

b Subtract 8.5 c Add

12 a t = 3

b p = 15

c x=1

d x=4

d Divide by 4

f s = 23

g p=3

h s = 12

e n = 22

7 a x = 30

13 a t = 6

b z=5

c x = –12

d p=0

e x = 42

f n = 32

g t = 45

h s=3

i s=2

8 a x = 43

l x = 21

d 14

j x=5 14 a

k x=4

9 a

b 18 Reasoning

b x = 20

e 16

f 56

b x = −24

c x = 9.4

e 13

f 18

c x=0

10 a i 15n = 225

ii n = 15

b i n − 19 = 76

ii n = 95

c i

ii n = 288 b

c b − 20 = 30, b = 50

16 Answers may vary 17 Subtract B from both sides then divide both sides by A 18 a Divide both sides by C then add D to both sides b

12 a a = 3

b b=4

c c=3

13 a x = 4

b m=3

c n=3

14 a x = 50

b y = 27

c z = 25

15 a a = 0

b b=3

c c=0

16 a x = 4

b y=9

c z = 13

17 a The number of circles in the next step is obtained by adding 4 circles to the previous step.

Problem-solving 19 44 kilometres

b 17

20 12 hours

c C = 4S − 3

21 38 months

18 6

22 40 weeks

19 35 kilometres

23 8 months b width = 5 m

25 c = 10 cm 26 a F = 95

f Subtract 7

11 a 6c = 90, c = 15

15 Answers may vary

24 a A = 27 m2

d 17

e Add 5

b C = 100

20 q = 12 m 21 a

Figure number

No. of Circles

b Increasing by 4

c C = 4F − 1

22 w = 13.2 cm

Answers mathspace.co

879

12.02 Parallel lines and transversals

12 a 5, 9 and 14

b 8, 12 and 15

c 9 and 14, 10 and 13, 11 and 16, 12 and 15 Understanding

d 112°

1 a Parallel lines are indicated using arrows or ‘chevrons’. b Example answer:

e 105°

f 75° 13 A; vertically opposite to the corresponding angle of ∠NBP 14 C; alternate angles Reasoning

2 a ‘C’ shape

b ‘Z’ shape

3 a Equal

b Supplementary

c Equal Fluency 4 a Non-parallel

b Parallel

c Non-parallel

d Non-parallel

5 a Alternate angles

b Co-interior angles

6 a x = 116: The angles are co-interior angles, so they are supplementary. This means they add to 180°, so, 64 + x = 180 64 + x − 64 = 180 − 64 x = 116 b x = 107: The angles are corresponding, so their measures are the same. c x = 96: The angles are alternate angles, so their measures are the same. 7 Corresponding angles 8 a i ∠EPQ

ii ∠APQ

iii ∠APF

b i ∠CQP

ii ∠EQP

iii ∠DQE

9 a x = 110° since these are corresponding angles b x = 75° since these are alternate angles c x = 55° since these are co-interior angles 10 a Angle 5

b Angle 4

c Angle 4

d Angles 1, 4, 5, 7

e 105°

f 85°

g 140° 11 a Angle 4 b Angle 3 c 124°

890

d 113°

Mathspace Victorian Curriculum – Year 7 mathspace.co

15 a The diagram contains parallel lines. The marked angles are supplementary and co-interior so we have a transversal of parallel lines. b The diagram does not contain parallel lines. The marked angles are not supplementary so they are not co-interior. We therefore do not have a transversal of parallel lines. c The diagram contains parallel lines. The marked angles are equal and corresponding so we have a transversal of parallel lines. d The diagram does not contain parallel lines. The marked angles are not equal so they cannot be corresponding. We therefore do not have a transversal of parallel lines. e The diagram contains parallel lines. The marked angles are equal and alternating so we have a transversal of parallel lines. f The diagram contains parallel lines. The marked angles are not equal so they cannot be alternating. We therefore do not have a transversal of parallel lines. g The diagram does not contain parallel lines. The marked angles are not equal so they cannot be corresponding. We therefore do not have a transversal of parallel lines. h The diagram contains parallel lines. The marked angles are supplementary and co-interior so we have a transversal of parallel lines. 16 It is not possible to say if the lines are parallel as there is not a matching pair of alternate or corresponding angles, or a pair of supplementary co-interior angles.

12.03 Triangles

17 a p = 64° b Example answer: Angle p and the angle marked 26° are adjacent angles. Together, they form a right angle. This means p + 26 = 90 p + 26 − 26 = 90 − 26 p = 64° c q = 64°

Understanding 1 180° 2 a Its interior angles are all acute or less than 90°. b One of its interior angles is obtuse or more than 90°. c One of its interior angles is a right angle or exactly 90°.

d Example answer: The line at the bottom of the diagram is a transversal that passes through the two parallel lines shown. This means angles p and q are corresponding angles, so their measures are the same. Problem-solving 18 a They are alternate angles, so they are also equal.

3 a All of its interior angles have different sizes and all its sides have different lengths. b At least two of its interior angles are the same size and at least two of its sides are the same length. c All of its interior angles are the same size and all its sides are the same length. This implies all its interior angles are 60°. Note that an equilateral triangle is also isosceles.

b They form a straight angle, so they are also supplementary.

4 a ∠ABF = ∠BCD + ∠CDB

c 96°

∠BDE = ∠DBC + ∠DCB

19 a They are vertically opposite angles, so they are also equal. b They are co-interior angles, so they are also supplementary. c 73° 20 a They are co-interior angles, so they are also supplementary.

∠CDF = ∠DBC + ∠DCB b Exterior angles CDF and BDE are both exterior angles of angle BDC. They are also vertically opposite angles, which means they are equal. Fluency 5 a Right-angled, isosceles triangle

b They form a straight angle, so they are also supplementary.

b Acute, isosceles triangle

c 97°

d Obtuse, scalene triangle

21 x = 39

e Acute, isosceles triangle

22 a x = 114°, y = 54

b x = 40, y = 50

23 a x = 117°

b y = 117°

24 a x = 33°

b y = 57°

c 180° 25 a x = 29° 26 x = 80°, y = 47°, z = 100°

c Acute, equilateral triangle

b y=

f Right-angled, scalene triangle g Acute, scalene triangle h Obtuse, isosceles triangle 6 a Isosceles

b Scalene

c Equilateral

d Scalene

e Equilateral

f Isosceles

g Isosceles

h Isosceles

7 a Acute

b Obtuse

c Acute

d Right-angled

e Obtuse

f Obtuse

8 a 50°

b 90°

c 27°

d 50°

Answers mathspace.co

891

9 a i x = 70

c False, as seen in any acute triangle. The sum of the measures of the interior angles of a triangle is 180°.

ii Acute, scalene triangle b i x = 110 ii Obtuse, scalene triangle 10 a x = 55

b x = 40

e x = 126

f x = 68

c x = 69

d x = 56

Reasoning 11 A different way to find x is by using the exterior angle to find the measure of the third interior angle. Since the third interior and the exterior angle form a straight angle, they are supplementary:

180 − 108 = 72

d False, as seen in any right triangle that is not an isosceles right triangle. In a right triangle, the right angle is 90° and the remaining two angles are always complementary. They are only equal in an isosceles right triangle. Problem-solving 16 a x = 67

b x = 62

c x = 49

17 a x = 74, y = 43

b a = 61, b = 29

c x = 61, m = 138

d a = 60, b = 13

18 a x = 64, y = 32

b x = 38, y = 32

c x = 59, y = 121

d x = 60, y = 65

Then, we can set the sum of the three interior angles equal to 180: x + 52 + 72 = 180 x + 124 = 180 x + 124 − 124 = 180 − 124 x = 56

20 a They are corresponding angles.

12 The sum of the angle measures exceeds 180°.

21 a They are alternate interior angles.

13 Yes, this is true. In the diagram, the interior angles ∠1 and ∠2 have a sum that is equal to two different exterior angles, ∠3 and ∠4. The exterior angles are vertically opposite angles, so they are also equal. 1

2 3

d x = 44

19 Yes, Jena has pitched her tent correctly. b They are vertically opposite angles. c x = 41 b x = 24

12.04 Quadrilaterals Understanding 1 a False

b True

c False

d False

e True

f True

g True

h False

2 Yes, a trapezium. 3 CD 4 a Concave b Convex c Concave d Convex

14 The smallest exterior angle has a measure of 95°. The third internal angle has a measure of 180 − 37 − 85 = 58 and we know the measure of an exterior angle of a triangle is equal to the sum of the interior angles on the opposite side. The smallest exterior angle will therefore be the sum of the two smallest interior angles which gives us 58 + 37 = 95°. 15 a False, as seen by finding the sum of any non-right triangle, such as an equiangular triangle. The interior angles of any triangle sum to 180°. b True

892

Mathspace Victorian Curriculum – Year 7 mathspace.co

e Convex Fluency 5 a x = 30

b x = 110

c x = 230 d x = 106

e x = 66

f x = 65

g x = 63

6 a Kite b Rectangle, parallelogram c Rhombus, kite, parallelogram d Kite e Rectangle, square, rhombus, kite, parallelogram f Kite g Rectangle, parallelogram

h x = 87

h Parallelogram i Trapezium 7 a Trapezium

b Neither

c Parallelogram

d Trapezium

8 a Any parallelogram that doesn’t have any right angles, and consecutive sides are different lengths. For example: B

15 100°

12 80°

b Any figure that is a rhombus. For example: 30°

30°

c Any figure that is a rectangle (including squares). For example: D

10 A

16 a The quadrilateral is a square. A quadrilateral with one right angle and four congruent sides is a square by definition. b The quadrilateral is a rhombus. A quadrilateral with four congruent sides is a rhombus by definition. c The quadrilateral is a parallelogram. A quadrilateral with opposite sides and angles congruent is a parallelogram. d The quadrilateral is a rhombus. A quadrilateral with opposite sides parallel and consecutive sides congruent is a rhombus.

8 cm 8 cm

15 All parallelograms have opposite sides that are equal in length. Since we are not told anything about the angles or the exact lengths of all four sides, parallelogram is the most precise classification.

17 It is because some quadrilaterals share the same properties.

Problem-solving

18 a x = 19

19 ∠CDE = 24°, use the fact that ∠CDA = 90° so ∠EDA = 90° - 24° = 66°.

b x = 104

c x = 135

d x=5

9 AB and CD

Then, use the fact that the adjacent angles of a rectangle are equal so:

10 a ∠a = 77° ∠b = 103° ∠c = 103°

b ∠a = 69° ∠b = 69° ∠c = 111°

66°

c ∠a = 120° ∠b = 120° ∠c = 60° d m∠a = 105° m∠b = 75° m∠c = 75° 11 a x = 99, y = 107 b a = 63, b = 63, c = 117 c a = 79, b = 101 d p = 5, q = 9 Reasoning

B 24° 66°

24° E

66° 24°

24°

66° C

20 x = 2 m

Chapter 12 review

12 1, because otherwise the angle sum would be larger than 360.

1 a Parallel

13 3, because otherwise the angle sum would be larger than 360.

2 a False

b False

c True

d False

14 4, this is the maximum number of angles a quadrilateral can have, and it would give the correct angle sum. Squares and rectangles have four right-angles.

e False

c Non-parallel

b Non-parallel d Parallel

3 a i ∠BQD

ii ∠FQB

iii ∠FQE

b i ∠DPB

ii ∠DPF

iii ∠FPA

b x = 74

c x = 78

4 a x = 125

Answers mathspace.co

893

5 x = 38

6 a 24 000 mm

6 a x = 34

b y = 56

7 a x = 21

b y = 17

b 3.484 m

c 600 000 cm

d 8.668 km

e 9 000 000 mm

f 1.5284 km

g 9030 mm

h 0.15 m

8 a Right-angled, isosceles triangle

i 743 000 cm

j 0.453 km

b Obtuse, isosceles triangle

k 6 710 000 mm

l 0.32 km

c Acute, equilateral triangle

7 5.068 km

d Obtuse, scalene triangle

8 12 500 cm

e Acute, isosceles triangle f Right-angled, scalene triangle

9 Nadia is shorter.

g Acute, scalene triangle

10 a Haonan

h Obtuse, isosceles triangle

11 The whiteboard is longer.

9 a x = 30

b x = 37

c x = 70

d x = 128

b x = 210

c x = 77

d x = 56

e x = 70 10 a x = 95

11 a x = 81, y = 73

b Sandy

c Ellie

12 Plant 1 13 a

Name

Height Height (metres) (kilometres)

b a = 61, b = 61, c = 119

One World Trade Center

541.3

0.5413

Shanghai Tower

632

0.632

13 a x = 73, y = 43

b x = 61, m = 137

451.9

0.4519

14 a x = 60, y = 37

b x = 65, y = 59

Petronas Twin Towers Zifeng Tower

450

0.45

15 a x = 103

b x = 137

Burj Khalifa

828

0.828

12 x = 24

b Burj Khalifa

14 a Desk

13.01 Units of length

b The desk is 19 cm longer then the rug.

Understanding

Reasoning

1 a m

b km

e cm

f m

c cm

d mm

2 a 1000 m = 1 km

b 100 cm = 1 m

c 1 cm = 10 mm

d 1000 mm = 1 m

3 a Divide

b Multiply

4 a 100

b 10

e 100

f 10

c 1000

d 1000

Fluency

15 So that there is no confusion as everyone is using the same unit of measurement. 16 It is easier to convert by multiplying or dividing by 10, 100, 1000. The Imperial system has inconsistent and difficult conversion factors. 17 a The distances are too large in space. b Light-years, which is the distance light travels from the sun in one year. Problem-solving

5 a 30 mm

b 6 cm

c 600 cm

d 8m

e 6000 m

f 9 km

g 8.82 cm

h 733 cm

20 2.22 m

i 0.149 m

j 12.226 km

21 5.34 m

k 46 mm

l 5690 m

894

Mathspace Victorian Curriculum – Year 7 mathspace.co

18 495 000 mm, 5.23 m, 499 cm 19 0.19 km, 19 150 cm, 192 m

22 356 cm

Reasoning

23 0.24 m

17 Method 1: P = 2 × (8 + 5)

24 0.74 cm

= 2 × 13

25 0.7 mm

= 26 cm

26 60 cm

Method 2:

27 a 1 26 000 cm

=2×8+2×5

b 1.26 km

= 16 + 10

28 666.67 years

= 26 cm

29 20 years

Another method: P=8+8+5+5

30 2 m from each long side and 4 m from the short sides.

= 16 + 10 = 26 cm

13.02 Perimeter

18 No. The side length of the square is 6 cm. If you increase it by 4 cm, the new side length will be 10 cm, resulting in a new perimeter of 40 cm. 40 cm is not 4 times the original perimeter of 24 cm.

Understanding 1 a 16 units

b 16 units

2 a 16 units

b 22 units

c 20 units

d 42 units

19 Yes. The side length of the square is 10 cm. If you increase it by 4 times the new side length will be 40 cm, resulting in a new perimeter of 160 cm. 160 cm is 4 times the original perimeter of 40 cm.

3 P = 55 + 73 + 48 = 176 cm Fluency

6 a 28 cm

b 72 m

c 20 cm

d 16 cm

20 Yes. the perimeter will be doubled. Consider a rectangle with a length of 5 units and a width of 3 units. Its perimeter is 2(5) + 2(3) = 16 units. Doubling the length of each side will result to a new perimeter of 2(10) + 2(6) = 32 units which is double the original perimeter.

7 a 27 mm

b 38 cm

c 51 cm

d 28 cm

21 4a cm

4 a 30 m

b 54 m

d 88 m

e 74 mm

5 a 60 m

c 26 cm b 30 m

e 30 mm 8 a 48 mm

22 6p mm b 12 m

c 30 mm

9 3 mm

Problem-solving

10 48 mm

24 8 m

11 a 28 cm

b 30 mm

c 48 m

12 a k = 8 m

b n = 2 cm

c x = 21 cm

13 4 cm 14 a y = 12 m

b n = 23 cm

c d = 14 cm

d k = 6 cm

15 a 4 m 16 a 11 cm

23 10x mm

26 70 m 27 11 cm 28 12 cm 29 6 mm

b 4 cm b 7m

25 9 cm

c 15 mm

30 15 m by 5 m 31 38 cm 32 $4800

Answers mathspace.co

895

33 $10 080

Problem-solving

34 a 52 cm

b 69 cm

35 a y + 3 + y + 3 + 2x + 2x

c 121 cm

15 a 387 m

b 4x + 2y + 6

16 a length = 26 m, width = 16 m

36 a 4y – 1 + 3x − 2 + 3x − 2 + 3y − 2 + 3y − 2 b 6x + 10y − 9

b 84 m 17 a 82 m

13.03 Perimeter of composite shapes

b $13 738.50

b 40 m

18 54 m 19 a 144 m

Understanding 1 Add together the length of the sides around the outside of the shape. 2 A and B 3 a 5 sides

b False

4 P = 10 + 7 + 9 + 11 + 9 + 8 P = 54 cm 5 P = 4 + 6 + 3 + 4 + 14 + 4 + 7 + 6 P = 48 cm

c 106 m

b $2700

20 a 8.8 m b The trim is added to the existing edges of the tablecloth. So, for each edge, we have 2 metres of trim and another 0.1 metres in each corner. Therefore, the total length of the trim is (2 + 0.1) × 4 = 8.4 m. Since the cost of the trim is $5.40 per metre, the total cost of the trim for the tablecloth is 8.4 m × $5.40 = $45.36.

13.04 Area of rectangles and squares

Fluency

Understanding

6 a 30 cm

b 120 m

c 68 cm

7 48 mm

1 a 16 square units

b 15 square units

2 cm2

8 a 58 m

b 60 mm

3 a i 4 square units

9 66 m

ii 9 square units

iii Shape A

10 a 54 cm

b i 25 square units

b 36 mm

11 a 58 cm

b 58 m

e 18 cm

f 76 cm

c 52 cm

d 170 m

Reasoning 12 a 60 cm b 60 cm c Yes, the answers are the same. The shape can be redrawn so that it becomes a rectangle with sides 17 cm and 13 cm. 13 P = 5 + 4 + 8 + 5 + 13 + 9 = 44 m or P = 2 × (9 + 13)

ii 16 square units

iii Shape B 4 a i 10 square units

ii 9 square units

iii Shape A b i 9 square units

ii 10 square units

iii Shape B c i 16 square units

ii 9 square units

iii Shape A d i 16 square units

ii 25 square units

iii Shape B e i 12 square units

ii 8 square units

iii Shape A f i 18 square units

ii 20 square units

= 44 m

iii Shape B

14 Adding sides gives 2 + 6 + 1 + 2 + 5 + 6 = 22 mm. Rectangle method gives 2 × (6 + 5) = 22 mm which is the same.

5 Shape A, Shape C, Shape B

896

Mathspace Victorian Curriculum – Year 7 mathspace.co

6 Shape A 7 a i Length = 6 units, width = 2 units ii 12 units2

b i Length = 12 units, width = 9 units

c The table from part (b) shows that since area is done through multiplication while perimeter is done through addition and multiplication, having the same area cannot guarantee of having the same perimeter.

ii 108 units2 c i Length = 3 units, width = 8 units ii 24 units2 d i Length = 5 units, width = 5 units ii 25 units2

Problem-solving

Fluency

22 24 m2

8 a 24 cm2

b 30 m2

c 50 m2

d 27 mm2

e 138 mm2 9 a 81 mm

24 $22 500

b 16 cm

c 324 m

ii 24 cm

b i 20 cm

ii 24 cm2

c i 14 cm

ii 10 cm2

d i 20 cm

ii 21 cm2

11 a 49 cm2

b 30 cm2

10 a i 22 cm

27 144 cm2 28 10 boxes

c 60 cm2

13.05 Area of triangles Understanding

b 64 cm2

13 a 8 cm

1 a

14 6 mm 15 a 9 m

b 25 cm2

c 20 cm2

2 a b = 6 cm, h = 2.7 cm

b 2m

b b = 36 mm, h = 15 mm

Reasoning

c b = 12 cm, h = 6.4 cm

16 a 49 cm

b 196 cm

d b = 19.5 cm, h = 12.5 cm

c No

17 Doubling the side lengths does not double the area. 18 a 4 times the original b Each side is multiplied by 2 and the area is found using side × side. As 2 × 2 = 4 the scale factor is 4. c Factor of 25 times the original 2

d Factor of x times the original 19 One possible answer is a square with side = 4 units

Length

Width

Area

50 m2

Length 25 5

Width 2 10

Area

Fluency 3 a i 50 cm2

ii 25 cm2

b i 30 cm2

ii 15 cm2

ii 11 cm2

c i 22 cm 2

ii 40 m2

e i 24 cm2

ii 12 cm2

f i 64 m

ii 32 m2

4 a 21 cm2

b 30 m2

d i 80 m

d 1320 cm2

c 24 cm

20 Find the area of the rectangle and halve it.

25 $150 26 150 tiles

12 5 mm

21 a

23 16 cm2

5 a 36 cm2

b 12 m2

c 28 mm2 d 13.5 cm2

e 60 cm2 f 5.2 m2 6 a Yes

b No

c Yes

d Yes

7 b = 8 mm 8 a Base = 4 cm Perimeter

b Base = 8 cm

c Base = 2.5 mm

d Base = 20 mm

50 m

54 m

9 a Height = 9 cm

b Height = 6 cm

50 m

30 m

c Height = 15 mm

d Height = 10 mm

Answers mathspace.co

897

Base (m)

2 a Base 2 cm, height = 3.6 cm

Height (m)

b Base = 10 cm, height = 3 cm

Area (m2)

c Base = 3.6 cm, height = 1.8 cm d Base = 5.5 mm, height = 8 mm

Reasoning 11 The rectangle can be cut into two equal right triangles. Find the area of the rectangle and halve it to find the area of the triangle.

Fluency 3 a

12 a The area is also halved.

6 cm

b The area is also doubled. 13 1 cm × 48 cm, 2 cm × 24 cm, 3 cm × 16 cm, 4 cm × 12 cm 14 No. There are multiple triangles of different dimensions that have the same area. For example, a triangle with the base of 4 cm and height of 10 cm has the same area as a triangle with the base of 8 cm and height of 5 cm, but the triangles are a different shape. b 50 mm

15 a

16 a

12 cm

b 72 cm2 4 a i 10 cm 5 a Base = 8 m

20 a 300 m2

e 1500 mm

f 360 cm2

g 96 cm2

h 266 cm2

7 a 10 mm2

b 105 cm2

b $7500

13.06 Area of parallelograms Understanding b 6 cm

c The area of the paralellogram is 9 × 6 = 54 cm2 while the area of the rectangle is 9 × 6 = 54 cm2. So the area of the rectangle is equal to the area of parallelogram.

Base (cm)

Height (cm)

Reasoning 13 The formula is A = b × h. If the base is the same then the larger height will determine the larger area. 14 a No, the area is not increased by 2 cm2. b No, the area is not doubled. c The area has increased by a factor of 4. d Each side is multiplied by 2 and 4 is equal to 2 × 2. 15 2 cm × 16 cm, 1 cm × 32 cm

898

Mathspace Victorian Curriculum – Year 7 mathspace.co

d No

b 7m

Area (cm )

24 ℎ = 8 cm

c Yes

10 12 cm

22 5 litres b 146 tiles

b Yes

9 a 12 cm

11 11 m

1 a 9 cm

d 270 mm2 2

21 $600

23 a 132 tiles

b 117 cm2

c 50 mm

8 a Yes b 0.2 m

d 40 m2

6 a 88 mm2

18 150 m2 19 a 2000 cm

b Height = 5 m

c 90° 2

17 48 m2

iii 60 cm2

ii 6 cm

b 60 cm2

Problem-solving

c 72 cm2

b 168 units2

16 a A = 6kp 17 a

The length of rectangle 1 can be found by adding the two horizontal distances: 6 + 7 = 13 m. And the width of rectangle 1 is 2 m, so the area is 13 × 2 = 26 m2.

7 cm

The length and width of rectangle 2 are 6 m and 13 m, so the area is 6 × 13 = 78 m2.

5 cm

So by adding the areas of the two shapes, the area of the composite shapes is 104 m2.

The rectangle will have the largest area as the height of the parallelogram must be less than 5 cm as the hypotenuse of a right triangle is always the longest side.

Fluency

Problem-solving

6 a 6 cm by 3 cm

b 18 cm2

c 42 cm2

d 60 cm2

19 9 m2 20 a 160 m2 21 a 800 000 cm 22 a 7500 m

b i 24 mm2 2

b 52 500 m

23 $3000

13.07 Area of composite shapes

c 280 cm2

5 56 m2

iii 48 cm2

b $40 000

b 224 cm2

7 a i 68 cm2

b $1600 2

4 a 56 cm2

ii 8 cm by 6 cm iv 116 cm2 ii 4 mm by 4 mm

iii 16 mm2

iv 40 mm2

8 a 96 cm2

b 112 m2

c 78 mm2

d 143 cm2

f 120 cm2

e 37 cm

g 70 cm2

Understanding

9 a 117 cm2 b 15 cm2

1 a Rectangle and parallelogram b Example answer: Add the area of the rectangle and parallelogram together. 2 Example answer: We breakup the shape into a rectangle and a triangle and add their areas. 3 a We breakup the shape into two rectangles and add their areas. b We calculate the area of the large rectangle 6 m × 15 m and subtract the area of the smaller 7 m × 13 m rectangle. Example answer: We can breakup the shape into two rectangles, as shown, and add their areas. 1

2m 7m

13 m

c 102 cm2

10 a 264 m2 b 136 cm2 c 80 m2

d 120 cm2

Reasoning 11 a 5 × 9 + 5 × 8 = 85 m2 b 9 × 13 − 8 × 4 = 85 m2 12 35 × 39 − 9 × 11 = 1266 m2. This is the most efficient method because it only involves two rectangle area calculations. If we used addition we would need to calculate three rectangle areas. 13 Find the area of the interior rectangle shaded in figure 2. Find the area of the corner rectangle shaded in figure 3. Find the area of the entire rectangle shaded in figure 4. Calculate the area of the shaded region as shown in figure 1 by subtracting the interior rectangle and two times the corner rectangle from the entire rectangle of figure 4.

Answers mathspace.co

899

Problem-solving 14 a 70 m

b 40.84 cm

c 25.76 cm

d 29.85 cm

11 a 119.38 cm

b 141.37 cm

b $700

15 264 cm2 16 a 414 m2

b 96 m2

c 318 m2

d $8268

17 a 416 m2

b 184 m2

c 169.65 cm

c $2308.80

18 $4481.25 19 450 cm2 20 a 109 m2

b $1308

21 a 22 603 m2

b 33 904.5 kg

13.08 Circumference of a circle

1 Circumference 2 a Radius

b Diameter

c Circumference 3 a False

b True

e False

f True

d Centre c True

d False

b 6.68 m

13 a 8.28 cm

b 16.55 cm

14 a 2.23 cm

b 2.86 cm

c 2.37 cm

d 15.60 cm

15 a 11.46 cm

b 5.73 cm

16 a 11.78 cm

b 14.01 cm

c 22.28 cm

d 7.80 cm

Radius (cm)

b π ≈ 3.1416

c π ≈ 3.1

d π ≈ 3.142

5 a C = 2πr

b C = πd

Fluency b 9 cm

12.57

50.27

257.61

170

340

1068.14

Reasoning

19 Sample answer: In fields such as engineering or architecture, where precision is crucial, knowing the exact circumference of a circle can be essential for determining accurate measurements for various constructions.

Diameter (cm)

Radius (cm)

Problem-solving 22 100.5 cm

110

132

Diameter (cm)

Radius (cm)

21 The circumference is also doubled.

23 106.8 cm 24 25.1 m

Circumference (cm)

20 They will get the same result because the diameter is always equal to twice the radius, so πd and 2πr are equivalent formulas.

7 2x

25 94.2 mm

26 311.0 cm

13.5

6.8

3.4

27 22.1 laps

15.6

7.8

9 40 cm

28 39 788 revolutions 29 345.6 m 30 163.4 m

900

Diameter (cm)

18 For each circle: C ÷ d = π

4 a π ≈ 3.14

6 a 7 cm

d 11.31 cm

12 a 13.37 m

Understanding

8 a

10 a 113.10 cm

Mathspace Victorian Curriculum – Year 7 mathspace.co

31 8

Fluency

32 231.0 km

3 a Triangular prism

Chapter 13 review 1 a 88 cm2

b 49 mm2

2 a 8m

b 11 m

3 a 20 cm2

b 42 cm2

4 a Base = 8 cm

b Base = 10 cm

c Base = 6 mm

d Base = 12 mm

5 a 63 mm2 6

b Rectangular pyramid

c Pentagonal pyramid d Octagonal prism 4 a

b 132 cm2

Base (cm)

Height (cm)

Area (cm2)

7 a 135 cm2

b 99 cm2

8 a 116.5 cm2

b 89 m2

9 a 106.81 cm

b 65.97cm

10 a 4.46 cm

b 22.28 cm

11 2 cm × 25 cm, 1 cm × 50 cm

7 Rectangle

14 $750

8 Example answers:

15 $90 000

• Rectangular Pyramid • Rectangular prism • Triangular Prism

16 $1100 17 $742.00 b $1716

Reasoning 9 In order to find the surface area of a solid when using its net, calculate the area of each 2D face/base and then add up each area to get the total surface area.

19 1244.1 cm 20 20.5 laps

14.01 Nets of solids Understanding

c Octagonal pyramid

b Square prism

c Pentagonal pyramid

13 180 m2

1 a Triangular prism

b Pentagonal pyramid

c Hexagonal pyramid d Octagonal prism 6 a Triangular prism

12 54 cm2

18 a 143 m2

5 a Square pyramid

b Rectangular prism d Octagonal prism

2 A net of a solid is a 2-dimensional representation of a 3-dimensional solid.

10 A prism and a pyramid can be classified by their base, and the number of faces a prism or a pyramid will have is the same as the number of sides of its base. A prism will have two bases and a pyramid will only have one base. 11 The sketch and reasoning of the net should look proportionate to the given length, width, and height of the rectangular prism described.

Answers mathspace.co

901

Problem-solving

e i

12 a Answers will vary.

b Answers will vary.

c Answers will vary.

d Answers will vary.

13 Answer may look like:

iii

The number of faces in a net is equal to the number of faces in its corresponding 3-dimensional solid. The number of edges and vertices may be different due to the folding and joining of the faces

4 a D

14.02 Visualise solids

5 A

b C

6 a Cube

Understanding 1 A 2D drawing of a 3D shape. 2 A perspective drawing is more realistic.

c B

d Cylinder

e Hexagonal prism

f Pentagonal prism

7 a Hexagonal prism or pyramid c Cylinder or cone or sphere

9 B 10 B 11 D 12 a i b i

iii iii

b i c i

ii iii

iii c i

d i

ii iii

iii

902

Mathspace Victorian Curriculum – Year 7 mathspace.co

c Triangular prism

8 A

iii

b Rectangular prism

b Trapezoidal prism or pyramid

Fluency 3 a i

d i

j i

iii iii

e i

13 D. By process of elimination, we can see that option A has a hexagon as its top view while options B and C each have two rectangles as its top view. The only solid with three rectangles would be option D.

iii f i

iii

g i

Reasoning

14 Yashi’s observation is accurate. Both the first and second solids share an identical ‘U’-shaped front view. However, their side views differ, as the first solid comprises just three squares, in contrast to the five distinct squares forming the second solid. Moreover, their top views are not the same; the first solid is composed of four squares, while the second solid features six squares. 15 The front view has a hole through the middle that is missed by the side view, and the top view has cubes cut out such that they would be missed by the side view.

iii

Problem-solving h i

16 Answers will vary. 17 Answers will vary.

14.03 Volume using unit cubes Understanding

iii

1 a It is the amount of space that is contained within that shape. b A cube with side length 1 unit. i i

2 a One possible solution:

iii b One possible solution:

Answers mathspace.co

903

c One possible solution:

d 16 rectangular prisms 1 × 1 × 120 1 × 2 × 60 1 × 3 × 40 1 × 4 × 30

d One possible solution:

1 × 5 × 24 1 × 6 × 20 1 × 8 × 15 1 × 10 × 12

3 a 18 unit cubes

b 16 unit cubes

c 36 unit cubes

d 99 unit cubes

e 140 unit cubes 4 a 4 rectangular prisms 1 × 1 × 12 1×2×6 1×3×4 2×2×3 b 9 rectangular prisms 1 × 1 × 48 1 × 2 × 24 1 × 3 × 16

2 × 3 × 20 2 × 4 × 15 2 × 5 × 12 2 × 6 × 10 3 × 4 × 10 3×5×8 4×5×6 Fluency 5 a 4 cm3

b 9 cm3

e 10 cm

f 20 cm

c 8 cm3 3

g 27 cm

d 45 cm3 3

i 48 cm

1 × 4 × 12

Reasoning

1×6×8

6 No, the volume will not change because no unit cubes are added or taken away from the rectangular prism simply by turning it.

2 × 2 × 12 2×3×8 2×4×6 3×4×4 c 12 rectangular prisms 1 × 1 × 72 1 × 2 × 36 1 × 3 × 24 1 × 4 × 18 1 × 6 × 12 1×8×9 2 × 2 × 18 2 × 3 × 12 2×4×9 2×6×6

904

2 × 2 × 30

7 Since 1 metre is 100 centimetres, and the edge of a 1-centimetre cube is 1 centimetre, we can stack 100 layers of a square area that is 100 cubes long and wide. 8 We know that if we lay one row of unit cubes the length and width of the rectangular prism, we can calculate the area of the top of the layer using the area formula, A = l × w. Then, we can stack how ever many layers of the unit cubes that are needed to reach the height of the rectangular prism. The area of the first layer multiplied by the number of layers in the rectangular prism is equal to the volume of the rectangular prism, l × w × h.

3×3×8

Problem-solving

3×4×6

9 98 cubic units

Mathspace Victorian Curriculum – Year 7 mathspace.co

14.04 Units of volume

14 a One possible answer:

Understanding

The books on the shelves could be distributed to fill two of the shelves, so we

1 Answer may look like: Volume is a measure of how much space an object occupies for example how much space bottles take on a supermarket shelf. Capacity is a specific type of volume, it measures the amount of liquid or gas that a container can hold, such as how much liquid is in a bottle. 2 a Answers vary. One possible solution: A small seed. b Answers vary. One possible solution: A sugar cube. c Answers vary. One possible solution: A washing machine. d Answers vary. One possible solution: A drop of liquid from an eyedropper. e Answers vary. One possible solution: A bottle of soda. f A nswers vary. One possible solution: A large shipping container.

could multiply the volume by so that the estimated volume of the books is 0.32 m3. b One possible answer: Looking at the books on the shelves, it seems that the books could take up about of one shelf, so multiplying the volume by , the books may have a volume of 0.0267 m3 Problem-solving 15 200 days. 16 1200 mL. 17 a The 2 L bottle. b Buying 2 bottles of 500 mL and a 2 L soda for a total of $12.80.

14.05 Volume of right prisms

g Answers vary. One possible solution: A lake. Understanding Fluency 3 a No

b No

c Yes

4 a m3

b m3

c cm3

1 One possible answer: d m3

5 Litres and millilitres 6 mL

To determine the volume of a composite solid, you need to calculate the volume of each individual simple shape that forms the composite solid, and subsequently sum up these volumes.

11 Approximately 2.7 L

2 One reason that every cube is a rectangular prism and not every rectangular prism is a cube follows: we can use the formula for the volume of a rectangular prism when calculating the volume of any cube, but we cannot use the formula for the volume of a cube when calculating the volume of any rectangular prism.

12 Approximately 1.2 L

Fluency

Reasoning

3 a 720 mm3

7 mL 8 C 9 a cm3

b Approximately 56 cm3

10 Approximately 44 250 cm3

13 1200 mL. The liquid fills half of the container, so half of the total amount of liquid that could fill it is the amount in the container.

b 2250 cm3 3

c 645.975 m 3

d 288 m3

e 32 cm

f 88 mm3

4 a 1728 cm3

b 216 cm3

5 2744 m3 6 3 cm

Answers mathspace.co

905

7 90 000 cm3 8 a 4900 cm

b We could calculate the volume of the solid by breaking it into four rectangular prisms and finding the sum of their volumes.

b i 16 537.5 cm3

ii No

9 68 600 cm3

Problem-solving

10 45 000 cm3

17 288 000 L

11 a 252 cm3

b 588 cm3

c 840 cm3

12 a 168 cm3

b 112 cm3

c 280 cm3

13 a 1344 units3

b 88 cm3

c 68 cm3

d 384 cm3

Reasoning 14 The volume is 3136 cm3. One way to calculate the volume is by breaking the composite solid into four identical rectangular prisms. Then, one of the rectangular prisms is 16 cm × 7 cm × 7 cm = 784 cm3 so the four that make up the composite solid is four times that volume. Another way to calculate the volume is by breaking the solid into two rectangular prisms: the smaller one on top and the larger 21 cm × 16 cm × 7 cm prism. The volume of each prism is 784 cm3 and 2352 cm3, respectively. The sum of the volumes is the volume of the composite solid. We could argue that calculating the volume of one rectangular prism and recognizing that four prism make up the composite solid leads to more efficient calculation.

18 a 37.5 litres. b One possible answer: The aquarium should not be filled to the top in case of spilling over, so we could assume that the water could be filled with 2 centimetres of space from the surface of the water to the top of the tank. Then the volume of the aquarium is 35 000 cm3, meaning that a reasonable amount of water to fill the aquarium is 35 litres. 19 10 units 20 7.81 cm 21 2.72 m3

14.06 Attributes in measurement formulas Understanding 1 a Constant

b Varying

c Varying

d Constant

2 a Distance

b 120

c Time

d Speed

3 a Base, height, area

15 First, convert 2 m to centimetres by multiplying it by

Then, multiply the length, width and height of the box to find the cubic centimetres: 200 cm × 30 cm × 40 cm = 240 000 cm3. 16 a In order to calculate the volume of the solid, we need to calculate the volume of the entire solid with the outermost dimensions and the volume of the rectangular prism that was cut out. Then, we will subtract the volume of the rectangular prism that has been cut out from the volume of the entire solid. The volume of the solid shown is 1080 cm3.

906

Mathspace Victorian Curriculum – Year 7 mathspace.co

b Base

c 12

d Height and area

e None

f A=6×h

Fluency 4

Length (cm)

Width (cm)

5 a

Length 4 4 4

Width 1 2 3

9 10

Area b A = 4w

4 8 12 16 20 24 28 32 36 40

d Linear

c Area 40

9 a

35 25

varying width, length = 4 cm

9 10

Speed (km/h) 300

Width 1 2 3 4 5 6 7 8 9 10

240 180

Length

Area

6 12 18 24 30 36 42 48 54 60

b A = 6l c Area 60

120

9 10

d Linear

60 Time (hours) 1 2 3 4 5 6 7 8 9 10

c No

d 24 km/hr

Reasoning 10 Constant means “always the same” and a constant term never changes its value. Variable means “changeable” and a variable term such as w is a term that can take on many different values.

varying length, width = 6 cm

Length 1 2 3 4 5 6 7 8 9 10

7 a 8 m3 d

360

55 50 45 40 35 30 25 20 15 10 5

Speed 360 180 120 90 72 60 45 40 36 (km/h)

6 a

Time (hours)

b 16 m3

Height

11 The sides of a square are equal. Therefore, the length must be a constant of 12 cm.

c 24 m3 5

9 10

Problem-solving

Volume 8 16 24 32 40 48 56 64 72 80

12 a True

b True

c False

d False

f V = 8h

13 a True

b False

c True

d True

14 a True

b False

c True

d False

e Volume (cm)3 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5

15 Double the base 16 Double his speed

Chapter 14 review 1 a Triangular pyramid

b Rectangular prism

c Pentagonal pyramid d Pentagonal prism 2 a Square pyramid

b Pentagonal pyramid

c Hexagonal pyramid d Octagonal prism

Lenght (cm) 1 2 3 4 5 6 7 8 9 10

3 a

8 a 30 cm3 b

Width

Length

Height

Volume 40 50 60 70 80 90

100

c V = 10h

d V = 350 cm3

Answers mathspace.co

907

19 a 90 cm3

4 a i

Width

Length

Height

Volume 108 126 144 162 180 198 216

iii

d V = 450cm3

c V = 18h b i

iii

b 6

6 a m3

c cm3

7 a 280 mm

d False

Understanding 1 A transformation is when an object or point is moved, turned, flipped, or changed in shape or size.

b 210 cm

c 2145 mm 8 43 750 cm

d cm3 3

b True

c False

15.01 Transformation of shapes

5 a 4 b m3

20 a True

d 280 cm

2 a Yes

9 a 4807 cm3 b i 16 223.625 cm3

d No

b It turns the shape over.

b 304.75 cm3

c 150 mm3

c No

3 a It slides the shape.

ii No

10 a 32 cm3

b No

c It spins the shape.

d 432 cm3

Fluency

11 k = 11.5 12 y = 10.1 13 a 76 cm3

b 184 cm3 3

d 452 cm3

c 1024 cm

4 a No

b No

e No

f Yes

c Yes

d No

5 a No

b No

c Yes

d No

6 a Yes

b No

c No

d No

e No

f Yes

e Yes

14 792 m3 15 7 units 16 7.5 cm

7 a Translation

b Rotation

17 a Constant

b Varying

c Translation

d Reflection

c Varying

d Constant

e Translation

f Reflection

8 a Translation

b Rotation

c Reflection

d Reflection

18 a Length 6

9 10

Width

Area

6 12 18 24 30 36 42 48 54 60

b A = 6w c Area 40

d Linear

35 30 25 20

varying width, length = 6 cm

15 10 5

908

Width 1 2 3 4 5 6 7 8 9 10

Mathspace Victorian Curriculum – Year 7 mathspace.co

Reasoning 9 Reflection and translation. Flag B is a mirror image of flag A but on a higher position. This means flag B is a transformation of flag A using reflection and translation. 10 We cannot use reflection because flag C is not a mirror image of flag A. 11 No. Block E and block N have the same vertical distance to the top of the tower, but different horizontal distances.

Fluency

20 The order is 2, 3, 1, 4.

4 a A, B, C

1. Half the cards are black, so this has an even chance.

b 1, 2, 3, 4, 5, 6

5 6, 7, 8, 9, 10, 11, 12 6 a Unlikely

2. There are only 4 sevens, so this is unlikely. b Impossible

c Even chance

d Impossible

e Unlikely

f Certain

7 a Impossible

b Even chance

c Certain 8 a Unlikely

b Even chance

c Unlikely

d Impossible

9 a Likely

b Impossible

10 a No

b No

c Yes

11 a Apple

b Unlikely c C

d D

12 a Yes

b Yes

d No

c No

3. There are 13 diamonds, so this is unlikely, but more likely than drawing a seven. 4. There are 9 numbers from 2–10, and there are 4 of each number, so there are 9 × 4 = 36 cards numbered 2–10. This is more than half the deck, so this is likely. 21 An example is a bag containing 3 blue marbles, 3 red marbles, and 3 green marbles. Drawing any colour is equally likely as the next since there are 3 of each colour, but the colours do not have an even chance of being drawn. Even chance refers to one event having half the total number of outcomes. Equally likely means all the events have the same number of outcomes.

13 a Green

b Blue

14 a Blue

b Likely

15 a Impossible

b Likely

16 a Even chance

b Unlikely

22 Marie is not correct. Even if there are fewer Fords than Holdens, there is still a chance for the Fords to win. It can be said that it is unlikely, but not impossible, for Fords to win.

17 a Certain

b Even chance

Problem-solving

c Impossible

d Unlikely

e Certain Reasoning 18 Unlikely. Since there are 12 sides on the die, there are 12 outcomes total. Rolling an 8 or higher has 5 outcomes: 8, 9, 10, 11, 12. Since 5 outcomes is less than half the total number of outcomes, rolling an 8 or higher is unlikely. 19 C The prime numbers between 1 and 6 are 2, 3, and 5. Rolling a prime number has an even chance. The even numbers on a standard die are 2, 4, and 6, so rolling an even number also has an even chance. The numbers less than 3 are 1 and 2. There are 6 total outcomes when rolling a die, so 2 outcomes is less than half, which means rolling a number less than 3 is unlikely.

23 a True

b False

c True

d False

24 Lisa 25 a 4 b Yes, there are 13 of each suit, so each suit has an equal chance of being drawn. c Drawing red or drawing black

16.02 Probability as a number Understanding 1 Probability = 2 Probability = Fluency 3 a Unlikely

b Certain

c Impossible

d Likely

e Even chance

f Likely

g Unlikely

h Unlikely

Answers mathspace.co

921

i Likely

j Likely

k Unlikely

l Even chance

Problem-solving

4 a

5 a

d 0

6 a

b 0

b 75%

c 25%

d 50%

18 a

19 Here is an example answer: A spinner with four colours, where red and blue are likely to occur, and yellow and green are unlikely to occur.

e 7 a 100% 8

20 a

21 a

9 0.8

10 a 35% 11 a

b 60% b

16.03 Theoretical probability d Understanding

1 7

12 a 27, 28, 29, 30, 31, 32, 33, 34 b

2 a True

13 a

c False

d 10

3 4 a

b 60%

5 a 822, 827, 828, 872, 877, 878, 882, 887, 888 b

15 If 4 marbles are black, then 23 – 4 = 19 marbles are not black. This means there is a probability of

that Paul will pick a marble that is not

black.

7 3 8 a 6

b 3

9 a

10 a

17 Spinning a red or a yellow

Reasoning

There are 8 total sections on the spinner. 2 are red and 2 are yellow, so each have a probability . Since they have the same probability

of occurring, they are equally likely.

6 12

16 No, the bag only contains red, blue, green, and orange marbles. This is because the probabilities of the colours listed sum to 100%, which means these colours represent all the marbles in the bag.

Mathspace Victorian Curriculum – Year 7 mathspace.co

11 35

12 4 times. To find the expected number of outcomes, we multiply the probability of rolling a 1 by the number of trials. The probability of rolling a 1 on a standard die is , and

922

d True

Fluency

Reasoning 14 a 4

b False

× 24 = 4.

13 Example answer: No, probability helps us predict outcomes, but it does not guarantee that the outcomes will happen. Although it is expected that the coin will land on heads 5 out of 10 times, there is still a chance that it will not land on heads exactly 5 times.

6 a

7 a

8 a 157 9 a 110 b i

14 a Lose the game b Yes, it is possible because there is still a chance of rolling a 1. The probability is unlikely, but not impossible. c No, Elsie is only expected to win money 1 out of the 3 times. However, she will need to pay a total of $3 to play those 3 times. Even if she won $2 once, it would not make up for the $3 she paid to play.

iii

b 270

10 a 11 1265 12 a 7

13 a 5

14 15 a 4

b No

15 a

b 20%

Reasoning Problem-solving 16 a 145

b 50

c 155

d 300

18 a Twenty-sided die

b Four-sided die

c Ten-sided die b 130

20 a 0

b 2

c 60

d 5

16.04 Experimental probability Understanding 1 Experimental probability = 2 Experimental probability is the probability of an event occurring based on actual observations or experiments, while theoretical probability is the probability of an event occurring based on the mathematical likelihood of the outcomes. 3 a Experimental

b Experimental

c Theoretical Fluency 4 a 90 5 6%

d No, because an experiment does not always go the way we expect.

17 a 0

19 a 65

b 10

16 a

17 A deck of cards has the same number of red and black cards. A deck of cards also has the same number of cards in each suit. Table 1 shows almost the same number of red and black cards while Table 2 shows same number of Spade and Diamond cards, but there is a big difference from the numbers of Heart and Club cards. This means that Table 1 has the closest results to the expected outcome. 18 a Theoretically, each number should occur once. Experimentally, we see the numbers occurred once, twice, or no times. Since these are close to the theoretical probabilities, the program appears to be fair. b Theoretically, each number should occur 10 times. Experimentally, we see each of the numbers occurred close to 10 times. Since these are close to the theoretical probabilities, the program still appears to be fair. Problem-solving

19 a 76% 20 a

b Tuesday b

c 85%

Answers mathspace.co

923

21 a 200

d 20

Understanding

Chapter 16 review

1 Categorical and numerical

1 C 2 a Impossible

b Likely

c Likely

d Unlikely b

3 a 4 a 0%

2 a ii

7 a Yes

3 a Numerical

b Categorical

c Categorical

d Categorical

b 50%

e Categorical

f Categorical

g Numerical

h Numerical

i Categorical

j Categorical

k Numerical

l Categorical

c No

d Yes

e No 8 a O

9 a

10 a

c iii

b b Yes

b i

Fluency

5 6 a 4

17.01 Types of data

4 a Discrete

b Discrete

c Discrete

d Continuous

e Continuous

f Continuous

g Discrete

h Continuous

i Continuous

j Continuous

k Discrete

l Discrete

m Continuous

n Discrete

o Discrete

p Continuous

5 a Continuous numerical 11

b Discrete numerical c Discrete numerical

12 68

d Categorical

13 2

e Continuous numerical f Discrete numerical

b No

14 a

g Categorical h Categorical

15 32

i Continuous numerical 16 a

j Discrete numerical

17 a 58%

b 42%

k Continuous numerical l Categorical

18 3%

Reasoning

19 0.9990 or 1

6 a Continuous: the amount of liquid is measured b 10

20 a

d No, because an experiment does not always go the way we expect. 21 a 157

b 33%

c 77%

22 a 70%

b 96%

c 86%

924

Mathspace Victorian Curriculum – Year 7 mathspace.co

b Discrete: the number of cans is counted and must be a whole number c Discrete: the number of views is counted and must be a whole number d Continuous: the amount of time is measured 7 a Continuous numerical: the amount of time is a number that is measured

b Categorical: the types of pets are words, not numbers c Discrete numerical: the number of pets is counted and must be a whole number

9 a 25 people 10 a

b 12 students 11

9 The question could say: “What is the brand of the shoes you are wearing?” If it were written like this, the answer to the question would be a word.

c 32 people

Number of bedrooms Number of students

d Continuous numerical: the weight is a number that is measured 8 The question could say: “How many students scored higher than 70 in your most recent science test?” If it were written like this, the answer to the question would be a number.

b 11 people

c 4 bedrooms

Animal

Number of friends

Cat

Dog

Fish

Problem-solving 10 a Continuous data

Reasoning

b 45.5 km

11 a Categorical b Comedy 12 a Discrete numerical

c 20 people

b 2 people

c 9 hours

d 9 hours

13 a Categorical

b 6 cats

c 30 pets

d One dog

17.02 Frequency tables

12 To find the total number of apples sold, you would multiply each number of apples by its corresponding frequency, and then add up the results. The calculation would be (10 × 5) + (15 × 8) + (20 × 12) + (25 × 6) = 560. This means that the total number of apples sold by the vendor in a week was 560. 13 a

Understanding 1 a Mode 2 a False

b Frequency b True

c False

d True

Fluency 3 a 13

b 2, 5

c 23

4 7 5 a

Sport

Number of friends

Football

Rugby

Netball

Swimming

Cricket

b 15 friends

c Rugby

6 a 36 classmates

b 1 hour

c 30 classmates 7 a 149.9 cents 8 a 107 people

Colour

Frequency

Red

Blue

Green

Yellow

Purple

b The favourite colour among the students in the class is blue, with a frequency of 9 as it has the largest number. 14 Add all the frequency for vowels A, E, and I, 22 + 28 + 20 = 70. Next, subtract 70 from the total vowel letters on a piece of text (320 − 70 = 250) to determine the remaining vowels for O and U. Since O + U = 250 and O = 1.5U, we can determine the value of U by 1.5U + U = 250. Simplifying the equation, we have U = 100 and O = 150.

d 6 classmates b 17 days

b 50 people

c 42 people

Answers mathspace.co

925

7 a • Red Rockets

Problem-solving 15 a

Hair colour

Frequency

Brown

Blonde

Black

Red

b 44 students

b If the Blue Blazers’ first game score changes from 70 to 90, their mean increases, their median increases, and their mode remains unchanged (no mode).

c 4.5% d

Sick days

Frequency

b 2 students 18 a

• Blue Blazers Mean: 82.4, Median: 85, Mode: None

16 a 15 trains b 2 trains c 17 a

Mean: 84, Median: 85, Mode: None

Mean: 60, Median: 60, Mode: None • Sarah Mean: 57, Median: 60, Mode: 60 b The mean, median, and mode all decrease for both sales representatives, with Sarah’s mode also changing to 45 and 60.

c 4 days

Weight in kilograms

Frequency

2 0

10 a 11

11 12

• Café Delight Mean: 8.6, Median: 9, Mode: 9

c $9

1 It measures the average of the values. 2 It measures the middle value of the set of values. b False

c True

d False

e False

c 9

12 a i 14

ii 18

b i 19

ii 18

b i 4 minutes

ii 3 minutes

iii 2 minutes

iv 11 minutes

14 a 9 tests b i 86

ii 87

iii 87

iv 10

15 Set A 16 Set D Reasoning

Fluency b 9.25

c 6

d 8.75

f 13

g 12.6

h 12.2

6 a 5

b 17.5

c 12.5

d 5

e 5

f 12

g 4.7

h 33.2

5 92

926

b 2

13 a 10 calls

Understanding

e 12.6

b • Café Delight Mean: 8.4, Median: 8, Mode: 8

17.03 Summarise data

4 a 6.5

9 a Bistro Bliss could have a slightly higher mean rating.

b 2 passengers

3 a False

8 a • Jacob

Mathspace Victorian Curriculum – Year 7 mathspace.co

17 a Sometimes. If the data set has an odd number of values, the median is the middle value. If the data set has an even number of values, the median is between the two middle values and may not be in the data set. b Never. The mean will always be less than or equal to the maximum value in the set.

c Always. The range represents the difference between the largest and smallest values, and the difference tells us how far apart the values are. 18 The range would not change. 19 a The difference between the most points scored last season and the lowest points scored last season was 43 points. b The range shows that there is a big difference between the most points scored and the lowest points scored during the season. This means the team is not consistent with the number of goals they score, since sometimes they score a lot and other times they score less. If the coach can figure out why they didn’t score as many points in the lower-scoring games, the coach may be able to help them play better and score more points more often. 20 a These are the means of each season:

25 56 26 85 27 4, 5, 6, 7, 8 28 Example answers: • 9, 11, 13, 15 • 1, 12, 12, 20 • 2, 10, 14, 22 29 Example answer: 13, 14, 14, 16

17.04 Outliers Understanding 1 An outlier is a data point that varies significantly from the body of the data. An outlier will be a value that is either significantly larger or smaller than other observations. 2 a The mode does not change. b The range always decreases.

• Season 1 is 8.1

3 a Larger

• Season 2 is 8.45

Fluency

• Season 3 is 8.15 Season 2 has the greatest mean rating. b These are the medians of each season: • Season 1 is 8.10 • Season 2 is 8.40 • Season 3 is 9.05 Season 3 has the greatest median rating. c The answer to part (a) would change but the answer to part (b) would not. With a rating of 6.3 for Episode 6, the mean of Season 3 would be about 8.48, which is the highest mean of the three seasons. With a rating of 6.3 for Episode 6, the median would still be 9.05, which is still the highest median of the three seasons. Problem-solving 21 11 22 12th score 23 17th and 18th scores 24 15 goals

b Smaller c Larger

d Smaller

4 a 1

b 99

c 275

d 5.1

5 a 18 kg

b 14 kg

c 18 kg

d The mode are the same before and after removing the outlier. 6 a 18 kg

b 13.3 kg

c Lower

7 a Increase

b Decrease

c Decrease

d Increase

8 a Yes

b No

9 a i

c Yes

Mean:

Median:

Mode:

Range:

102

Mean:

d No

ii 126 iii

Median:

Mode:

Range:

iv The mean, median and range of the data decreased, but the mode stayed the same.

Answers mathspace.co

927

b i

Mean:

1.92

Median:

1.7

Mode:

0.9

Range:

3.8

Mean:

1.61

Median:

1.5

Mode:

0.9

Range:

1.9

ii 4.7 iii

iv The mean, median and range decreased, but the mode stayed the same. c i

Mean:

4540

Median:

4750

Mode:

4700

Range:

3800

ii 1500 iii

Mean:

4877.8

Median:

4800

Mode:

4700

Range:

800

iv The mean and median increased, the range decreased, but the mode stayed the same. 10 a 12.5 kg c 13.875 kg

b 14.2 kg d 14.25 kg

Reasoning 11 The mean will be higher, but the median will stay the same. 12 a 15

b 13.29

c The median because it is not affected by the outlier. 13 She should use the mean since it is the higher measure of average because it is affected by the outlier. 14 Median, because it is the least affected by the outlier. 15 a Example answer: The mode. Since most people get paid $58 000, the owner might think it is the most representative of the company’s annual pay rate.

928

Mathspace Victorian Curriculum – Year 7 mathspace.co

b Example answer: The mean. If Ted wants to get a raise, the mean will show the largest difference between the average pay rate and his pay rate. c Example answer: The median. A mathematician would recognise that $400 000 is an outlier and know that outliers affect the mean but do not always affect the median. Problem-solving 16 a 280

b A

17 a $462 000

b $527 000

d $929 000

e $446 600

c Median

18 x = 6 or x = 9 19 a The mean will be lower since the outlier is larger than the rest of the values. b The median will be unchanged since the middle of a large group of data points is mostly unchanged when extreme values are removed. c The mode of the data set is unchanged. d The range will decrease since the largest data value is being removed.

17.05 Dot plots and column graphs Understanding 1 By counting the number of dots above a particular category 2 The mode is the highest column of dots on a dot plot or the highest column on a column graph. 3 Column graph Fluency 4 a 4

b i 15

9 10 11 12 13 14 15 16 17 18 19

ii 15

iii 15

c 8 people 5 a The dots represent how many times she worked a particular number of shifts per week. b 3 weeks

c 6

d The distribution of the number of shifts per week is approximately symmetric.

6 a False

b True

7 a 2009 8

c True

d True

b 2006

Sport

12 a The distribution of ratings given by the judges to the winning contestant is positively skewed. b 7 judges

Number of People

Cricket

Basketball

Soccer

Tennis

Netball

13 a 1 goal

b 2 goals

14 a 15 people

b 16 names

c 10 names

d 6 names

b Underlook and Elf Fortress c 32 students 16 a Rating 4 d

Number of websites visited daily

Frequency

b The clusters in the number of websites visited daily are 2 and 4 websites, where the frequencies are relatively higher. c Yes, there is a gap in the distribution between the values 0 and 2, as there are no individuals who visited exactly 1 website daily. b 23 people

18 Marbles

12 10 8 6 4 2 Black

18 a

Blue Yellow White Colour

Red

Sydney Opera House visitors 11

c 3 people

d The distribution of band members across different sections is negatively skewed, with the highest number of members in Strings and the lowest in Percussion.

14 Number of marbles

d Mean: 41.05, Median: 40

c 4 people

Frequency

Number of people (in millions)

c The distribution will become less skewed and more symmetric, as the outlier was pulling the mean towards the higher end of the scale. The median will remain unchanged, as it is not affected by the outlier.

b 18 ratings

Hotel Rating

b Mean: 49, Median: 40

11 a Percussion

d 10 judges

15 a Bash Sisters

9 a The scores have higher-frequency columns on the left (the lower scores), so the distribution is positively skewed. There appears to be an outlier as there is one value far to the right of the others. The data set also shows a large gap between 75 and 200, and clustering around 40.

10 a

c 16 judges

2015 2016 2017 2018 2019 Year

b Yes

Answers mathspace.co

929

15 a

2 If the scale is too small, small changes in the data can appear to be significant changes, which can be misleading. If the scale is too large, important changes or trends in the data may seem small or may get overlooked.

Month

Number of employees

November

December

January

February

4 a 32 °C

March

5 a 2300 visitors

Angle size

3 a Pie chart

b Line graph

c Dot plot

Fluency b 35 °C

d 33 °C

b 2650 visitors

c Month 5 and 6 b Holidays taken over summer

c 34 °C d No

Vegetables eaten

December

Number of vegetables

7 January

80° 100° 60° 40° November 80° February March

6 5 4 3 2 1

16 a

ManufacResponses turer Apple

2051

Samsung

967

Huawei

531

Angle size

Sun Mon Tue Wed Thur Fri

Sat

Day

10 9 8

451

Smartphone market share

Frequency

Other

7 6 5 4 3 2 Apple

Samsung

Huawei

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Other

8 a 2013

17.08 Line graphs Understanding 1 A line graph clearly shows how one value increases or decreases as another value changes.

Score

b 2016

9 a Sunday

c 50

d 25

b Monday and Tuesday

c 7 °C warmer 10 a May c July to August

b October d February to March

e November to December

Answers mathspace.co

933

11 a 0.2 km, 0.9 km and 1.2 km

21 a They travelled 4 km closer to home.

b 0.6 km and 1 km c 0.2 km

d 0.5 m/s

c They travelled 4 km closer to home.

12 a 16 L

d They spent 1 minute not moving at all.

b The driver filled the tank. c 16 L

d 12 p.m.

13 a Increasing b $200 000 c June ‘13, September ‘13, and June ‘12 14 a Berlin

b 30 mm

c 23 mm

d London

e December 15 a $1.50

b December and June

c $2.75

d April and September

e $312.50

22 a The line graph only shows the data for the half of the year. Showing only some parts of the data makes the graph bias or misleading. b The y-axis starts at 0 which makes the line smaller showing small changes in each period. Problem-solving 23 a Apollo left home at 2:00 p.m., while Artemis left at 6:00 p.m. b They were the same distance from home sometime between 8:00 p.m. and 9:00 p.m. c Artemis

b 12:00 p.m.

d No

c 0.6 L

d 4 hours

24 a 20 km

e 1L

f 0L

16 a 1.6 L

g 2.4 L

25 a 4 km

b The first 4 minutes b 5 km

c 3 times d 9 km

e Between 4 and 5 hours

17 a 2.5 km

b 3:00 p.m.

c 3.5 km

d 4 hrs

Reasoning

26 a 2 hours into the journey

b 400 km

c The final 3 hours and the first 2 hours 27 a 4 km

b 3 km

c 9 hours into shift

18 D

d Pollux’s speed in his last two hours

Water

20 18 16 14 12 10 8 6 4 2 0

Chapter 17 review 1 a Discrete

b Nominal

c Continuous

d Ordinal

2 a 9 tests b i 90.8 1

3 a Yes

Time

ii 91

iii 92

iv 10

b No

c Yes

d No

4 a The number of days exercised in one week b 3 weeks

1080

5 a True

960

Water

840

b False

c True

d False

b Green and red

c 44 vehicles

480

7 39

240 120 0

c 3

6 a White

720 600 360

Time

934

b They travelled 1 km closer to home, but slower than before.

Mathspace Victorian Curriculum – Year 7 mathspace.co

8 a 17 m

b 27 m

c 22.72 m

9 a Red

b Blue

c 112 fish