White Rose Maths Key Stage 3 - Student Book 3

White Rose Maths Key Stage 3 Student Book 3

Ian Davies, Caroline Hamilton and Sahar Shillabeer

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Contents Introduction

v

Block 1

Straight line graphs

1

1.1 1.2 1.3 1.4

Plotting and reading graphs Gradients and intercepts Equations of lines Further graphs

2 12 20 30

Block 2

Forming and solving equations

39

2.1 2.2 2.3 2.4 2.5

Unknowns on one side Inequalities with negative numbers Unknowns on both sides Solving problems with equations and inequalities Rearranging formulae

40 52 58 68 77

Block 3

Testing conjectures

90

3.1 3.2 3.3 3.4 3.5

Properties of number Looking for truth The journey to proof (1) The journey to proof (2) Searching for pattern

91 97 106 116 128

Block 4

Three-dimensional shapes

135

4.1 4.2 4.3 4.4 4.5

Into three dimensions Nets and other diagrams Surface area (1) Surface area (2) Volume

136 147 160 174 185

Block 5

Constructions and congruency

202

5.1 5.2 5.3 5.4 5.5

Constructions so far Introducing loci Perpendiculars More loci Congruence

203 218 227 237 245

Block 6

Numbers

257

6.1 6.2 6.3 6.4 6.5

Working with numbers Estimation Solving problems with numbers Fractions Standard form

258 267 274 285 297

Block 7

Percentages

305

7.1 7.2 7.3 7.4

Percentage basics Reverse percentages Solving percentage problems Repeated percentage change

306 318 326 335

Block 8

Maths and money

342

8.1 8.2 8.3 8.4

Interpreting bills and statements Interest Taxes Solving problems with money

343 350 357 364

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Contents Block 9

Deduction

375

9.1 9.2 9.3 9.4

Angles review Chains of reasoning Angles and algebra Geometric conjectures

376 385 394 403

Block 10 Rotations and translations

415

10.1 10.2 10.3

416 423 434

Symmetry Rotation Translations and beyond

Block 11 Pythagoras’ theorem

449

11.1 11.2 11.3

450 461 472

Working with right-angled triangles Finding unknown sides Beyond triangles

Block 12 Enlargement and similarity

482

12.1 12.2 12.3 12.4

483 492 504 518

Calculating with enlargement Enlarging a shape (1) Enlarging a shape (2) More similarity

Block 13 Solving ratio and proportion problems

532

13.1 13.2 13.3 13.4

533 545 555 563

Direct proportion review Investigating inverse proportion Ratio revisited Best buys

Block 14 Rates

571

14.1 14.2 14.3

572 582 590

Speed, distance and time Density and units Rates and graphs

Block 15 Probability

600

15.1 15.2 15.3

601 610 620

Probability of one event More than one event Probabilities and tree diagrams

Block 16 Algebraic representations

631

16.1 16.2 16.3

632 643 650

Further graphs Simultaneous equations Graphs of inequalities

Block 17 Getting ready for Key Stage 4

658

17.1 Handling data review 17.2 Sequences review 17.3 Moving on with trigonometry Glossary Answers

659 665 670 677 681

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3 Testing conjectures In this block, I will learn… les and primes

how to work with factors, multip

24 6

4 2

2

3

2

24 = 2 × 2 × 2 × 3

21 = 3 × 7

24 = 23 × 3

21 The factors of 21 are 1, 3, 7 and 7 The prime factors of 21 are 3 and 21 is a multiple of both 3 and 7

how to test conjectu

res

“When you add two primes

the answer is even.”

3+5=8✓ 11 + 2 = 13 ✗ The statement is some

times true.

how to prove conjectures about numbers using diagrams and sym

bols

“The sum of two even numbers

is even.”

how to expand a pair of binomials 2m + 2n = 2(m + n)

how to use algebra to

look for patterns 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 x 51 52 53 54 55 56 57 58 59 60 ? ?

61 62 63 64 65 66 67 68 69 70 ?

(x + 1)(x + 3) ×

x2 + 4x + 3

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3.1 Properties of number Small steps

Key words

■ Explore factors, multiples and primes

Factor – a positive integer that divides exactly into another positive integer Multiple – the result of multiplying a number by a positive integer Prime number – a positive integer with exactly two factors, 1 and itself Prime factor decomposition – writing numbers as a product of their prime factors

Are you ready?

Work out

1

2

a

4×1

b 4×2

c

4×3

d 4×4

e 4×5

a

What mathematical term is used for multiples of 2?

b

How can you tell by looking at a number whether it is a multiple of 2?

Copy and complete these tables of multiplication facts.

3

×

2

4

6

×

5

3

7

15

5

4

8

9

8

Models and representations Array

This array shows

Bar model

This bar model shows

3

12 ÷ 2 = 6

12 = 6 × 2

12 ÷ 6 = 2

Factor tree

12 ÷ 4 = 3

12 3

12 = 2 × 6

3

3

12

12 ÷ 3 = 4 4

3 12 = 2 × 2 × 3

2

2

12 = 22 × 3

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3.1 Properties of number You worked with factors, multiples and primes in Books 1 and 2. In this chapter, you will review your learning as you will be using these concepts in the rest of this block.

Example 1 Here is a list of numbers: 21

22

23

24

a

Write down all the prime numbers in the list.

b

Write down all the multiples of 3 in the list.

c

Write down all the factors of 48 in the list.

a

25

26

27

23 is the only prime number as all the other numbers have factors other than 1 and themselves, for example:

23

21 = 3 × 7 22 = 2 × 11 24 = 4 × 6 25 = 5 × 5 26 = 2 × 13 27 = 3 × 9

b

3 divides exactly into these numbers with no remainder 21 ÷ 3 = 7, 24 ÷ 3 = 8, 27 ÷ 3 = 9

21, 24 and 27

Another way of saying this is all the numbers are multiples of 3 21 = 3 × 7, 24 = 3 × 8, 27 = 3 × 9

c

24 is a factor of 48 as 24 × 2 = 48 (or 48 ÷ 2 = 24)

24

Example 2 Express 54 as a product of its prime factors.

One way is to draw a factor tree.

54

Start by identifying a pair of factors: 54 = 2 × 27 2 is prime so you can circle it; this branch ends here

27

2

27 is not prime, so you need to continue: 27 = 3 × 9 3 is prime so you can circle it; this branch ends here.

9

3

3 54 = 2 × 3 × 3 × 3 54 = 2 × 33

9 is not prime, so you need to continue: 9 = 3 × 3

3

Again 3 is prime, so you can circle both 3s; the factor tree is complete.

Remember: this result is called the prime Now you can write the factor decomposition product of prime factors in of 54 ascending order.

You can also write your answer in index form as 3 × 3 × 3 can be written as 33

Check that you get the same final answer if you start your tree with 54 = 6 × 9

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3.1 Properties of number

Practice 3.1A 1

2

a What multiplication facts do these arrays show?

b

Use your answers to part a to list the factors of 12

a

In how many ways can you arrange 15 counters in an array? List the multiplications that the arrays show.

b

List the factors of 15

c

List the factors of i 20

3

ii 24

a

16

17

b

18

19

20

21

22

23

24

From the list, write down the multiples of i 2

ii 3

iii 4

iv 6

v 8

How can you tell by looking at the list that there are exactly two multiples of 5?

a Find all the ways in which can you arrange 16 counters in an array. b

Find all the ways in which can you arrange 17 counters in an array.

c

Explain how your answers to parts a and b illustrate that 17 is prime but 16 is not.

d

Which of the numbers in this list are prime?

6

5

iv 27

Here is a list of numbers.

15

4

iii 30

7

9

13

18

19

21

25

29

31

39

a Explain how you can use a calculator to find out whether a number is a multiple of another number.

Here is a list of numbers 90 b

91

92

6

94

95

96

97

98

99

From the list, write down all the multiples of i 3

c

93

ii 7

Write down all the prime numbers in the list.

a Copy and complete this factor tree.

24 4 2

2

b

Use your answer to part a to write 24 as a product of its prime factors.

c

Express each of these numbers as a product of its prime factors. i 30

ii 40

iii 66

iv 28

v 81

vi 120

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3.1 Properties of number 7

8

9

Express each of these numbers as the product of two prime factors. a

26

b 38

c 46

d 55

e 65

f 93

a

Write 108 as a product of its prime factors.

b

Use your answer to part a to write 216 as a product of its prime factors.

c

Use you answer to part b to write 2160 as a product of its prime factors.

Another way of finding the prime factor decomposition of a number is called the “ladder method”.

2

180

2

90

a

Discuss with a partner how ladder method works.

3

45

b

Use the ladder method to find the prime factor decomposition of

3

15

5

5

i 500

ii 252

iii 264

1

Do you prefer the factor tree method or the ladder method?

10 a b

180 = 2 × 2 × 3 × 3 × 5

5a3 = 135. Find the value of a

x = 23 × 32 × 5. Calculate the value of 10x

What do you think?

1

Do you agree with Marta?

2

This book was written in the year 2021

If a is a factor of b, then b is a multiple of a

2021 = 43 × 47 Write down the prime factorisations of a 3

4042

b 20 210

c 606 300

Ali factorises the expression 12x + 18y

12x + 18y 3(4x + 6y) How many other ways can you find to factorise the expression?

4

a

What percentage of the integers from 1 to 10 are prime?

b

What percentage of the integers from 1 to 100 are prime?

c

Estimate the percentage of the integers from 1 to 1000 that are prime.

Check how close you are by finding the correct answer from an internet search.

d

Why do you think the percentage of prime numbers decreases as you include more numbers?

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3.1 Properties of number

Consolidate – do you need more?

1

a What multiplication facts does this rectangle show?

b 2

List the factors of a

3

What other rectangles can you draw with the same area?

10

b 28

c 40

Here is a list of numbers.

25 a

26

27

28

32

33

34

Which numbers in the list have ii the fewest factors?

Write down all the prime numbers between a

5

31

ii 4

i the most factors 4

30

From the list, write down the multiples of i 3

b

29

10 and 20

b

35 and 45

Seb is thinking of a number.

My number is odd. It is a multiple of 3 and a factor of 30

What are the possible numbers that Seb could be thinking of?

6

Express each of these numbers as the product of two prime factors. a

7

33

b 34

c 35

d 46

e 51

f 85

Express each of these numbers as a product of its prime factors. a

75

b 48

c 60

d 600

e 96

f 225

95

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3.1 Properties of number

Stretch – can you deepen your learning?

1 The number 2x will have twice as many factors as the number x a

Find a pair of numbers for which Zach’s statement is i correct

2

3

ii incorrect.

b

Find a general rule to decide when Zach’s statement is correct and when it isn’t correct.

a

Substitute n = 1, 2, 3 and 4 into the expression n2 + n + 41

b

What do you notice about your answers?

c

Explain why your discovery will not be true for n = 41

d

Explore substituting other numbers into the expressions n2 + n + 41 and n2 – n + 41

a

Find the five numbers under 100 that have exactly 12 factors.

b

Find the prime factorisations of the numbers from part a.

Explain why they must have 12 factors.

4

You will explore this relationship again in Chapter 3.3

I can tell by looking at the prime factorisation of 1225 that it is a square number. a

Find the prime factorisation of 1225 and explain how Faith knows it is a square number.

b

Use prime factorisation to determine which of these are square numbers.

1764

729

3969

1980

2744

1 000 000

c

Are any of the numbers in part b cube numbers? How do you know?

d

Use your learning to find numbers that are both square and cube numbers.

Reflect Describe how you could find all the factors of a number such as 14 280

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17 Getting ready for Key Stage 4 In this block, I will review and extend my learning about…

Handling data Number of visits

Frequency

Subtotals

0

5

0

1

8

8

2

17

34

3

6

18

4

3

12

5

1

5

40

77

spaper “The words used in the local new words are shorter on average than the used in a national newspaper.” easiest to “Salt and vinegar crisps are the identify in a blind tasting.” five “Goals are more likely in the last n in any minutes of a football match tha .” iod per e nut other five-mi

77 1.925 total Mean = number of items = 40 =

Sequences

How many sticks are needed for the 4th pattern?

The 10th pattern? The nth pattern? The 100th pa ttern?

1 First differences

4 +3

Second differences

9 +5

+2

16 +7

+2

25 +9

+2

Trigonometry sin 25° = DE = CF = BG AE AF AG cos 25° = AD = AC = AB AE AF AG tan 25° = DE = CF = BG AD AC AB

E F G D

E

D

25° adj

A

B F

hyp opp

C

25°

opp

A C

hyp adj

25°

A

G opp

B

hyp 25° adj

A

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17.1 Handling data review Reminders and links Here are some of the statistical diagrams you drew in Books 1 and 2 Bar charts and pie charts show information about one variable, while scatter diagrams look at two variables and the relationship between them.

Bar chart Choice of sport

Pie chart Vegetables people dislike Other

tennis

Sport

swimming

Carrots

Sprouts

cricket football athletics

Parsnips Broccoli

0 2 4 6 8 10 12 14 16 18 20 Number of students

Scatter graph £10000

Value of a car This graph shows that there is a negative correlation between the age of a car and its value.

Value

£8000 £6000 £4000 £2000 £0 0 1 2 3 4 5 6 7 8 9 10 Age, years

You learned how to calculate these measures of location.

Mean – the total of a set of items divided by the number of items Median – the middle value of a set of numbers when they are arranged in order If there is an even number of items, then the median is the mean of the middle pair.

Mode – the most common item in a set of data

If there are n items, the median is in the (n + 1)th position 2

You also calculated this measure of spread.

Range – the difference between the lowest and the highest values in a set of data The greater the range, the more spread out the data set is.

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17.1 Handling data review

Moving into Key Stage 4 You will continue to use the diagrams and measures of location and spread that you used in Key Stage 3. You will also learn how to draw and interpret the diagrams below: Histograms

Frequency diagrams 10

2.5

9 8

2

Frequency

Frequency

7 6 5 4

1.5

1

3 2

0.5

1 0

0 30

35

40

45

50

55

Age

70

90

110

130

150

170

190

210

Mass

Cumulative frequency diagrams 100 Cumulative frequency

90 80 70 60 50 40 30 20 10 0 10

30

50 70 Speed (mph)

90

110

In addition, you will find out about quartiles, which split a distribution into four equal parts; this is similar to the way in which the median splits a distribution into two equal parts. These can be used to construct box plots, which are very useful for comparing sets of data.

Paper 1 scores 0

10 20 30 40 50 60 70 80 90 100

Paper 2 scores

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17.1 Handling data review In this chapter, you will look at data in frequency tables. Some of this content may be familiar to you if you completed all the Higher

steps in Book 2

Example 1 The table shows how many times a group of 40 people visited a dentist in the past year.

Number of visits

Frequency

0

5

1

8

2

17

3

6

4

3

5

1

Work out a

the mode

b the median

c

the mean

number of visits to the dentist made by the group.

a

2

The mode is the data value with the highest frequency, so the modal number of visits was 2

b

Number of visits

Frequency

Cumulative frequency

0

5

5

1

8

5 + 8 = 13

2

17

13 + 17 = 30

3

6

30 + 6 = 36

4

3

36 + 3 = 39

5

1

39 + 1 = 40

The median is the ( Median = 2

40 + 1 th th 2 ) = 20.5 item.

Number of visits

Frequency

0

5

1

8

2

17

3

6

4

3

5

1

You can find the median from a table using the cumulative frequency. This is the total of all frequencies so far in a frequency distribution.

You find the cumulative frequencies by adding each frequency to the total so far.

The median is in the (n + 1)th position 2 This is halfway between the 20th and 21st items.

Both the 20th and 21st items are 2 so the median is 2

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17.1 Handling data review

c

Number of visits 0

Frequency

Subtotals

5

0

1

8

8

2

17

34

3

6

18

4

3

12

5

1

5

40

77

Instead of working out the mean by adding 0 + 0 + 0 + 0 + 0 + 1 + 1 + 1 +… and so on, you can add an extra column to the table and find subtotals You find the subtotals by multiplying. For example, 2 occurs 17 times so these data items add up to 2 × 17 = 34. In the same way, 3 occurs 6 times so these data items add up to 3 × 6 = 18 You add the subtotals together to find the overall total of all 40 items.

total 77 Mean = number of items = 40 = 1.925

Use the formula for the mean. The number of items is the total frequency.

Practice 17.1A 1

2

Find the mean, median, mode and range of each of these sets of data. a

5

7

10

12

12

b

21

23

8

12

50

12

46

30

A group of people were asked how many holidays they had taken over the last three years. The table shows the results.

Number of visits

Frequency

0

37

1

23

2

28

3

18

4

8

5

6

a

How many people were asked altogether?

b

Explain why it would be easy to represent this data on a pie chart.

c

Write down the modal number of holidays taken.

d

Show that the median number of holidays taken was 1.5

e

Work out the mean number of holidays taken.

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17.1 Handling data review 3

For the data in each of the tables, find the i

mode

ii

median

iii mean iv range. b

a

4

5

Number of cars

0

1

2

3

4

Score in test

6

7

8

9

10

Frequency

4

17

8

3

1

Number of students

2

6

9

6

5

The grouped frequency table shows the masses of some plants.

Mass (g)

Frequency

100 < w ⩽ 150

36

150 < w ⩽ 200

42

200 < w ⩽ 250

53

250 < w ⩽ 300

28

Total

159

a

Identify the modal class.

b

In which class does the plant with median mass lie?

The grouped frequency table shows the lengths of time that a group of 150 people spent exercising last week.

The estimates of the subtotals Time (hours) Frequency Midpoint Subtotals are found by multiplying each midpoint by the 63 0.5 31.5 0<t⩽1 corresponding frequency, for 15 2 30 1<t⩽3 example 63 × 0.5 = 31.5 27 5 Each midpoint is the mean of 3<t⩽7 the endpoints of the class, for 24 7 < t ⩽ 10 3+7 example =5 21 2 10 < t ⩽ 15 Total

150

a

Copy and complete the table to find an estimate of the mean time spent exercising last week.

b

Find the class in which the median lies.

Hint: find an estimate of the overall total by adding up the subtotals.

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17.1 Handling data review 6

For the data in each of the tables, find i

the modal class

ii

the class in which the median lies

iii an estimate for the mean. a

b

7

Length (cm)

0 < l ⩽ 10

10 < l ⩽ 30

30 < l ⩽ 60

60 < l ⩽ 75

Frequency

18

35

32

15

Time online (hours)

0<t⩽1

1<t⩽3

3<t⩽5

5 < t ⩽ 10

Frequency

27

10

35

22

The mean height of a class of 30 students is 162 cm. A student who is 180 cm tall leaves the class. Find the new mean height of the class.

Might the other averages have changed? How can you tell?

8

The mean of five numbers is 16. When a sixth number is added, the mean increases to 17. Find the sixth number.

Challenges 1

Assume that the data is distributed evenly across each of the class intervals in Practice questions 4, 5 and 6. How could you use proportional reasoning to work out an estimate of the median?

2

Describe how you could test these hypotheses: “The words used in the local newspaper are shorter on average than the words used in a national newspaper.”

“Salt and vinegar crisps are the easiest to identify in a blind tasting.”

“Goals are more likely to be scored in the last five minutes of a football match than in any other five-minute period.”

What data should you collect? How would you present your findings? What charts or measures would you use?

3

Investigate the relationship between the amount of time a person spends online and the amount of time they spend sleeping. How is this investigation different from those suggested in Challenges question 2?

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