Cambridge Lower Secondary Maths Student Book 7

1A Unit 1A Number and calculation

What’s it all about?

• The importance of place value and ordering numbers • Identifying equivalent fractions, decimals and percentages • Improving mental strategies for multiplication and division • Working with negative numbers You will learn about: • • • • •

Ordering positive and negative numbers Recognising and using place value How to use multiplication tables Divisibility tests How to recognise equivalent fractions, decimals and percentages • Comparing fractions You will build your skills in: • Mental maths • Understanding the use of numbers in real life such as measurements

Unit 1A: Number and calculation

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Unit 1A • Chapter 1

Place value and rounding

You will learn how to: • Interpret decimal notation and place value.

• Multiply and divide whole numbers and decimals by 10, 100 or 1000.

• Order decimals including measurements, changing these to the same units.

• Round whole numbers to the nearest 10, 100 or 1000 and decimals, including measurements, to the nearest whole number or one decimal place.

Starting point Do you remember …?

• what each digit represents in numbers with up to 6 digits?

For example, what does each digit represent in the number 724 568?

• what each digit represents in numbers with up to 2 decimal places?

For example, what does each digit represent in the number 0.43?

• how to multiply and divide whole numbers by 10, 100 or 1000 where the answer is a whole number?

For example, calculate 234 × 100 or 7 500 ÷ 10.

• how to round simple whole numbers to the nearest 10, 100 or 1000 and how to round decimals to the nearest whole number or one decimal place?

For example, round 78 to the nearest 10.

• the relationships between different units of measurement?

For example, how many centimetres are in 2 metres?

This will also be helpful when … you learn how to do more complex calculations with decimals, such as multiplying and dividing by decimals.

Hook Here is a game for 2–4 players.

You will need paper, pens and a die.

Aim of the game: to produce the largest number Each player draws 4 boxes like this, with a decimal point as shown:

Player 1 then rolls the die.

All the players must write the number on the die in one of their four boxes.

Player 2 then rolls the die.

All the players must write the number on the die in one of their remaining three boxes.

Play continues twice more, rolling the die and then writing the number in a box.

Once all four boxes are complete, the players compare their numbers and the player with the largest number wins.

2

Unit 1A: Number and calculation

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• If a 6 is rolled, where should you position it?

• What about if a 1 is rolled?

• Is there a way to ensure that you always win?

• Play again – this time, the player with the smallest number wins. How does this change your strategy?

Did you know?

Place value

In some countries the decimal point is shown as a full stop but in other countries a comma is used instead.

Key terms A digit is a numeral used to show a number or part of a number. 237 is a 3-digit number with digits 2, 3 and 7.

You can use a place value grid (below) to show the value of each digit in a number 1 000 000

100 000

10 000

millions

hundred thousands

ten thousands

1000

100

thousands

10

hundreds

tens

1 units

0.1 decimal point

0.01

tenths

0.001

hundredths thousandths

Worked example 1 State the value of the 7 in each of these numbers: a) 437 061

b) 2.57

a) 437 061

HTh 4

The value of the 7 is 7000

TTh

Th

3

H

7

0

T

100 000

U

6

10 000

1000

100

hundred ten thousands hundreds thousands thousands

1

10

1

tens

units

If the number is a whole number, without decimals, then start by putting the units in the units column.

The 7 is 7 thousands.

b) 2.57 The value of the 7 7 is 100 or 0.07

u

t

h

2

5

7

1 units

If the number has a decimal point then line up the decimal points first. The 7 is 7 hundredths or 0.07.

decimal point

0.1

0.01

tenths

hundredths

Exercise 1 1

State the value of the 8 in each of these numbers: a) 872

b) 2816

c) 28 612

d) 3008

e) 2.8

f) 23.468

Chapter 1: Place value and rounding

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2

Jim is working with this number: 543 182 Jim says, “The value of the 4 in this number is ten thousand”.

Do you agree with Jim? Explain your answer.

3

Here is a number 978 654.321 Which statement is false?

Discuss

A:

B:

C:

D:

The 4 digit

The 7 digit is

The 1 digit is

The 3 digit

worth 7000

1 worth 1000

is worth 0.3

is worth 4

What is the largest decimal you can write that is less than 3?

Multiplication and division by powers of 10 Worked example 2 Calculate: a) 167 × 10 a) 167 × 10 = 1670

b) 1.67 × 100 = 167

c) 248 ÷ 10 = 24.8

b) 1.67 × 100

c) 248 ÷ 10

d) 2.4 ÷ 100

When you multiply by 10 you move each digit one place value to the left. If there are empty place vales between numbers and the decimal point you fill those with zeros.

When you multiply by 100 you move each digit two place values to the left. If there are empty place values between numbers and the decimal point you fill them with zeros.

When you divide by 10 you move each digit one place value to the right. If there are empty place values between numbers and the decimal point you fill them with zeros.

d) 2.4 ÷ 100 = 0.024 When you divide by 100 you move each digit two place values to the right. If there are empty place values between numbers and the decimal point you fill them with zeros.

10 000

1000

100

ten thousands hundreds thousands

1 100

10

1

hundreds

tens

units

1

6

tens

units

6

7

6

7

0

decimal point

100

10

1

tens

units

2

4

8

2

4 0.1

decimal point

0.1

0.01

tenths

hundredths

6

7

7

hundreds

1

1

1

1

units

10

tenths

2

4

0

0

decimal point

0.1

0.01

tenths

hundredths

8 0.01

0.001

hundredths thousandths

2

4

Exercise 2 1

4

Calculate: a) 26 × 10

b) 38 × 100

c) 265 × 1000

d) 2156 × 10

e) 350 × 1000

f) 400 × 10

g) 0.05 × 10

h) 0.004 × 100

i) 3.65 × 10

j) 2.8 × 100

k) 0.48 × 1000

l) 0.104 × 100

Unit 1A: Number and calculation

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2

3

Calculate: a) 260 ÷ 10

b) 380 000 ÷ 1000

c) 34 ÷ 10

d) 28 ÷ 100

e) 560 ÷ 100

f) 7 800 ÷ 1000

g) 0.5 ÷ 10

h) 0.4 ÷ 100

i) 1.6 ÷ 10

j) 37.8 ÷ 100

k) 450 ÷ 1000

l) 14.5 ÷ 100

Match each calculation on the left to its answer on the right.

Complete the missing calculation and answer in the empty boxes to produce 8 pairs of calculations and answers.

5.2 × 10

5200

0.52 × 1000

5.2

5.2 × 10

52

52 × 100 0.52 ÷ 10

520 000

520 ÷ 100

520

52 000 000 ÷ 100

52 000 0.052

4

Emily is working out this calculation: 14.3 × 100.

She says, “I just need to add two 0s on to the end of my number, so the answer is 14.300”.

Do you agree with Emily? Explain your answer.

Ordering decimals and measurements Key terms Ordering numbers means to put them in order of size.

You do this by comparing the numbers working from left to right. If the first two numbers on the left are the same you then compare the next numbers along.

41 is bigger than 38 because 4 is bigger than 3.

48 is bigger than 43 because the 4s are the same but 8 is bigger than 3.

This works with decimals too.

2.45 is bigger than 2.37 because the 2s are the same but 4 is bigger than 3.

To compare 2.338 with 2.33 you first write 2.33 as 2.330 so it has the same number of decimal numbers as 2.338.

2.338 is bigger than 2.330 because the 2s and the 3s are the same but 8 is bigger than 0.

If you are comparing more than two numbers it is often easier to write them underneath each other, in a place value table lining up the decimal points.

2.322 second largest 2.318 smallest 2.890 largest

1 units

0.1 decimal point

tenths

0.01

0.001

hundredths thousandths

2

3

2

2

2

3

1

8

2

8

9

0

Chapter 1: Place value and rounding

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Worked example 3 a) Which of these three numbers is the largest?

2.359

2.36

2.357

b) Order these numbers from largest to smallest.

4.294

3.235

2.338

2.329

2.611

c) Order these lengths from shortest to longest.

2.4 m

224 cm

a) 2.36 is the largest as 6 (h) is bigger than 5 (h)

2.234 m

2 m

Write the decimals under each other, lining up the decimal points and zeros where necessary.

2.359 2.360

U

t

h

th

2

3

5

9

2

3

6

2

3

5

7

U

t

h

th

4

2

9

4

1

3

2

3

5

2

2.357 Compare the numbers in each place value starting with the left-hand column. b) 4.294 is the largest as 4 (U) is Write the numbers under each other, larger than 3 or 2 lining up the decimal points and adding zeros where necessary.

3.235 is next as 3 (U) is larger than 2 4.294

2

6

1

1

3.235

2

3

0

0

2.611 2.32 is next as 2 (th) is larger 2.300 than 0 2.320 2.3 is the smallest Compare the numbers in each place value starting with the left-hand column.

2

3

2

0

U

t

h

th

4

2

9

4

1

3

2

3

5

2

2

6

1

1

3

2

3

0

0

5

2

3

2

0

4

U

t

h

th

2

2

3

4

2

0

0

0

U

t

h

th

2

4

0

0

4

2

2

4

0

3

2

2

3

4

2

2

0

0

0

1

2.611 is next as 6 (t) is larger than 3

c) Writing these in metres you have 2.400 2.240 2.234 2.000 2 m is the shortest as 0 (t) is less than 2 or 4 2.234 m is next as 3 (h) is less than 4 2.240 m is next as 2 (t) is less than 4

To compare these distances you must change them all to the same units.

Divide 224 cm by 100 to change it into 2.24 m.

Write the numbers under each other lining up the decimal points and adding zeros where necessary.

Compare the numbers in the place values from left to right, but this time you are looking for the smallest numbers.

2.4 m is the longest In order, smallest first, the lengths are 2 m, 2.234 m, 224 cm and 2.4 m

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Unit 1A: Number and calculation

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Exercise 3 1

2

3

Which is the larger number or quantity in each of these? a) 2.6 or 2.49

b) 36.375 or 36.366

c) 0.116 or 0.123

d) 1.56 m and 1.479 m

e) 2.79 m and 293 cm

f) 96 g and 0.95 kg

Which is the smallest number or quantity in each of these? a) 1.43 or 1.399 or 1.322

b) 1.2 or 1.122 or 1.109

c) 15.198 or 14.991 or 14.910

d) 2.61 m or 2.55 m or 2.49 m

e) 2.31 m or 249 cm or 2.41 m

f) 2.33 kg or 299 g or 2245 g

Put these numbers in order from smallest to largest. a) 1765, 1800, 1650, 6500, 1675

Remember:

b) 13 454, 14 350, 14 503, 13 544, 14 035 4

5

Tip

Put these numbers in order from largest to smallest. a) 12.3, 13.2, 13, 12.19

b) 0.45, 0.4, 0.54, 0.47,

c) 0.601, 0.62, 0.6, 0.621, 0.612

d) 16.59, 16.6, 16.599, 16.95, 16.9

Put these measurements in order from smallest to largest. a) 3.6 m, 3.36 m, 363 cm, 3.603 m

There are 100 centimetres (cm) in 1 metre (m).

There are 1000 metres (m) in 1 kilometres (km).

There are 1000 grams (g) in 1 kilogram (kg).

There are 1000 millilitres (ml) in 1 litre (l).

b) 14.2 kg, 14.25 kg, 14 285 g, 14.258 kg c) 7.1 litres, 701 millilitres, 7000 millilitres, 0.71 litres d) 18.3 km, 1830 m, 17 000 m, 1.38 km, 1.835 km 6

Use each of the digits 0–5 once only to complete the missing digits to make four correct statements.

2 17.

7

mm < 2.1 cm km >

739 m

8 2.

Think about

mm < 9 cm kg > 2

95 g

Write an ordering question with a list of 6 measurements that are difficult to order. Now write the solution.

What makes your list of measurements hard to order?

What steps should you go through to make sure you have got a set of measurements in the right order?

Rounding Key terms If 34 237 people watched a television programme last night you would probably say that over 34 000 people watched the program. You have rounded the number to the nearest thousand.

Rounding a number often makes it less accurate but easier to use.

Worked example 4 Round: a) 17.5 to the nearest 10

b) 3562 to the nearest 10

c) 3562 to the nearest 100

d) 19.6 to the nearest whole number

e) 23.657 to 1 decimal place Chapter 1: Place value and rounding

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a) 27.5 rounded to the nearest 10 is 30

b) 3562 to the nearest 10 is 3560

27.5 lies between 20 and 30 and you need to decide which it is nearer to. You can do this using a number line or by looking at the number in the next place value to the left of the tens. If that number is 5 or more you increase the 2 in the tens place to 3.

3562 lies between 3560 and 3570 and you need to decide which it is nearer to. The number in the units place is 2 so 3562 will be nearer 3560.

20

27.5 T

U

t

2

7

5

3

0

0

3562

3560

3570 Th H

c) 3562 to the nearest 100 is 3600

3562 lies between 3500 and 3600. The number in the tens place is 6 so 3562 is nearer to 3600.

T

U

3

5

6

2

3

5

6

0

3562

3500

3600 Th H

d) 19.6 to the 19.6 lies between two whole numbers 19 nearest whole and 20. The number in the first decimal number is 20 place is 6 so 19.6 is nearer to 20.

e) 23.657 to one decimal place is 23.7

23.657 is between 23.6 and 23.7. The number in the second decimal place is 5 so 23.657 is nearer to 23.7

Exercise 4 1

30

T

U

3

5

6

2

3

6

0

0

19.6

19

20 T

U

t

1

9

6

2

0

0 23.657

23.6

23.7

T

U

t

h

th

2

3

6

5

7

2

3

7

0

0

1â&#x20AC;&#x201C;12

Give the nearest multiples of 10 that these numbers lie between then state the multiple of 10 that the number is closest to. The first one has been done for you. a) 367 is between 360 and 370 and is nearer to ...................... . b) 648 is between 6_0 and 6_0 and is nearer to ...................... . c) 35 781 is between 357_0 and 357_0 and is nearer to ...................... . d) 2476 is between 24_0 and 24_0 and is nearer to ...................... .

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Unit 1A: Number and calculation

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2

Round these numbers to the nearest 10. a) 35

3

b) 361

c) 4564

d) 1453

e) 19 963

Give the nearest multiples of 100 that these numbers lie between then state the multiple of 100 that the number is closest to. The first one has been done for you. a) 5648 lies between 5600 and 5700 and is nearer to 5600. b) 369 lies between _00 and _00 and is nearer to ...................... . c) 6528 lies between 6_00 and 6_00 and is nearer to ...................... . d) 21 477 lies between 21_00 and 21_00 and is nearer to ...................... .

4

Round these numbers to the nearest 100. a) 516

5

b) 8341

c) 3487

d) 67 549

e) 49 655

Give the nearest multiples of 1000 that these numbers lie between then state the multiple of 1000 that the number is closest to. The first one has been done for you. i)

45 678 lies between 45 000 and 46 000 and is nearer to 46 000.

ii) 4176 lies between .........000 and .........000 and is nearer to ...................... . iii) 31 578 lies between 3......... 000 and 3......... 000 and is nearer to ...................... .

6

Round these numbers to the nearest 1000. a) 7813

7

d) 37 688

e) 630

b) 16.149

c) 173.52

d) 49.74

e) 0.35

d) 21.01

e) 7.98

Round these numbers to one decimal place. a) 0.46

9

c) 63 813

Round these numbers to the nearest whole number. a) 1.87

8

b) 6512

b) 6.413

c) 12.389

Here is a number: 142 874.18 a) Which of these statements is correct?

A:

B:

C:

D:

E:

The number rounded to the nearest 1000 is 140 000

The number rounded to the nearest 100 is 142 800

The number rounded to the nearest 10Â is 142 880

The number rounded to the nearest whole number is 142 874

The number rounded to one decimal place is 142 874.1

b) Correct the statements that are wrong.

10 Alma is rounding the number 46.419 to one decimal place. She says, â&#x20AC;&#x153;Because the last number is a nine, I need to round up. So the answer will be 46.5â&#x20AC;?.

Do you agree with Alma?

Explain your answer.

11 Round 12 765.192 metres: a) to the nearest metre b) to the nearest kilometre c) to the nearest centimetre 12 Round 3734.815 grams a) to the nearest gram

b) to the nearest kilogram

Chapter 1: Place value and rounding

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Tec

Value A

Value B

7.3

2.915

4.817

1.956

h

13 Set up a spreadsheet table like this: Value A × Value B

Write a formula to calculate the result of Value A × Value B in the third column.

Does the spreadsheet round the answers? If so, to what degree of accuracy? If not, how can you change the settings so that it does round?

Did you know?

In sports competitions, measurements are often rounded because the equipment used is only accurate to a certain degree.

For example, times in the 100 m race are rounded to 2 decimal places (or the nearest hundredth of a second).

14 A runner completed a 100 m race in a time of 11.39 seconds.

Given an example of a time with 3 decimal places that would be rounded to 11.39 seconds to 2 decimal places. How many times with exactly 3 decimal places are there that round to 11.39 seconds?

15 Vocabulary feature question Complete the text with the words in the box. round ten decimal place decimal point order multiply divide thousand units place value Our number systems is based around ............................... columns. Each column has a value that is ............................... times bigger than the previous one.

When we use a number that is not an exact whole number, we use a ............................... to separate the ones, tens, hundreds etc. from the tenths, hundredth, thousandths etc.

When we ............................... a number by ten, each digit becomes ten times larger, and thus appears in the next column to the left.

When we ............................... a number by ten, each digit becomes ten times smaller, and thus appears in the next column to the right.

We can compare the value of each digit in a number to help us put them in ...............................

Sometimes we need to ............................... a number. This reduces its accuracy but can make it easier to calculate with.

Numbers that have been rounded to the nearest ............................... will end in three 0s.

Numbers that have been rounded to one ............................... will have exactly one number after the decimal point.

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End of chapter reflection You should know that ...

You should be able to ...

Such as ...

You can organise numbers into the following place value system, where each category is ten times larger than the next one:

State the value of any digit of a whole number of decimal.

The value of the 2 in 12 356

1 000 000 millions

100 000

10 000

1000

hundred ten thousands thousands thousands

100

10

1

hundreds

tens

units

0.1 decimal point

tenths

0.01

0.001

hundredths thousandths

You can multiply or divide whole numbers and decimals by 10, 100 and 1000 by moving the place values of their digits

Multiply or divide a) 23.6 Ă&#x2014; 100 a whole number or b) 356.78 á 10 decimal by 10, 100 or 1000.

You can order decimal numbers including measures.

Put a list of decimal These decimals numbers in order from the largest of size. to the smallest 0.34, 0.4, 0.304, 0.43, 0.403

You can round numbers to the nearest 10, 100 or 1000 or to the nearest whole number or to one decimal place (sometimes written 1 d.p.).

Round numbers to a given degree of accuracy.

34 567 to the nearest 100.

Chapter 1: Place value and rounding

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