KS3 Maths Now Learn and Practice books

4 Negative numbers 4.1 The number line ●

I can use a number line to order positive and negative numbers, including decimals

●

I can understand and use the symbols < (less than) and > (greater than)

Using a number line helps you to order negative and positive numbers.

Develop fluency Which number is greater, −7 or −3?

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

Because −3 is further to the right on the number line than −7 is, it is the larger number.

Dan wrote −6 < 4 His sister said she could write the same thing but with a different sign.

Explain how she could do this.

Swapping the numbers around and using the greater than sign gives 4 > −6, which is the same thing but with a different sign.

Write these temperatures in order, from lowest to highest.

8 °C, −2 °C, 10 °C, −7 °C, −3 °C, 4 °C Draw a number line then mark the numbers on that line.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9

10

You can see that the order is: −7 °C, −3 °C, −2 °C, 4 °C, 8 °C, 10 °C

1 State whether each statement is true or false. a 5 < 10 b 5 > −10 c −3 > 6

d 3<6

2 Copy each statement and put < or > into the ■ to make it true. b 5 ■ −2 c −1 ■ 9 a 6 ■ 10

d −3 ■ −1

3 Write down the lower temperature in each pair. a −1 °C, −8 °C b 2 °C, −9 °C c −1 °C, 9 °C

d −3 °C, −1 °C

4 Put these numbers into order, starting with the lowest. a −0.5, 0, −1 b −1, 2, −5 c −1.1, −1.6, −1.9 5 Write down the greater number in each pair. a −1, −3 b −6.5, −5.6 c −1.2, −2.1 6 Put these numbers into order, starting with the highest. a −3, −2, −5 b −1, 1, 0 c −3, −3.3, −1.3 7 Put these numbers into order, starting with the lowest. a −0.8, 0.5, −1.2 b −1.5, 2.3, −5.9 c −3.3, −2.7, −3.8 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book.

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47

Chapter 4

8 Write down the higher temperature in each pair. a −211 °C, −108 °C b −58 °C, −29 °C c −101 °C, −99 °C

d −73 °C, −61 °C

9 Write down the highest number in each set. a −5.5, −5.3, −5.6 b −7.5, −7.6, −7.9 c −4.2, −4.5, −4.1 Reason mathematically Leo said, ‘−6 is larger than −4’.

His sister said he was wrong.

How could she explain to Leo why he was wrong?

She could draw a number line and place −6 and −4 onto the line, showing that −6 is further to the left than −4 and that the further left a number is on the line, the lower the number is.

10 Carla was told that the temperature outside would drop by 5 °C that night.

She said that it must be freezing outside then.

Explain why she might be incorrect.

11 A submarine is at 40 metres below sea level.

The captain is told to move to 20 metres below sea level.

Does the captain move the submarine down or up? Give a reason for your answer 12 Explain how you know that −5.1 is higher than −5.9.

13 James has a bank balance of £187. He pays his energy bill of £53.62, and his credit card bill of £228.51. Explain why his bank balance is now negative.

Solve problems Work out the number that is halfway between −3 and 2.

Draw a number line showing both −3 and 1.

2 spaces 2 spaces –3 –2 –1

0

1

Count 4 spaces between −3 and 1, so halfway is 2 spaces from the −3, which is −1.

14 Bernie took the lift. He started on floor 2. First he went up two floors to marketing. Then he took the lift down seven floors to the canteen. Then he took the lift up two floors to the boardroom. Finally he took the lift up four floors to the IT department.

On which floor is the IT department?

15 Luke has £137 in the bank, he pays a bill of £155 and also pays £50 into his bank.

Then pays another bill of £40. How much will he have in his bank account now?

48

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4.2 4.2 Arithmetic with negative numbers ●

I can carry out additions and subtractions involving negative numbers

●

I can use a number line to calculate with negative numbers

It can be useful to use the number line to help you add and subtract.

Develop fluency Use a number line to work out the answers. a 5 − 13 a

b (−11) + 9

c 6 − 12 − 3 Starting at zero and ‘jumping’ along the number line to 5 and then back 13 gives an answer of −8.

–13 +5 –8

0

b

5

Similarly, (−11) + 9 = −2 Notice that brackets are sometimes used so that the negative sign is not confused with a subtraction sign.

+9 –11 –11

c

–2

–3

0

Using two steps this time, 6 −12 − 3 = −9

–12 +6

–9

–6

0

6

1 Use a number line to work out the answers. a 1−5 b −2 + −4 c −1 + 3

d −3 − 1

e −2 + −3

2 Use a number line to work out the answers. a 4−6 b −5 + −2 c −4 + 3

d −3 −3

e −1 + −4

3 Without using a number line, write down the answers. a 1−3 b 3−8 c 7−9

d 4−6

4 Without using a number line, write down the answers. a −9 + 3 b −8 + 4 c −2 − 5

d −3 − 2

5 Use a number line to work out the answers. a 1−5+4 b 2−7+3 c −3 + 1 − 5

d −6 + 8 − 4

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49

Chapter 4 Reason mathematically In a magic square, the numbers in any row, column or diagonal add up to give the same answer. Could this be a magic square?

Give a reason for your answer.

−6 2 −1

−3

The diagonal adds up to −5.

So the missing number on the bottom row will be −1.

And the missing top number in the other diagonal will be −4.

From the middle vertical numbers, −1 and 2, the missing top middle number is −6.

From the two numbers in the top row, −6 and −4, the missing top number of the middle column should be 5.

Since −6 is not the same as 5, these rows and columns do not add up to the same number, so is not a magic square.

6 In a magic square, the numbers in any row, column or diagonal add up to give the same answer. Could this be a magic square?

Give a reason for your answer.

−3

−7 4 −5

7 Alf has £124 in the bank. He makes an online payment for £135.

How much has he got in the bank now? Explain your working.

Solve problems In a quiz, two points are awarded for a correct answer, but three points are deducted for an incorrect answer.

Team A answers eight questions correctly, two incorrectly and don’t answer the last two questions. Team B answers all the questions; nine correctly and three incorrectly.

Who wins the quiz?

Team A scores 8 × 2 points, giving 16, but lose 2 × 3 points, leaving them with 10 points.

Team B scores 9 × 2 points, giving 18, but lose 3 × 3 points, leaving them with 9 points.

So Team A wins the quiz.

50

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4.3 8 In a maths competition, 5 points are awarded for each correct answer but 1 point is taken off for each incorrect answer. If a question is not answered then it scores zero points. There are ten questions in the competition. a What is the lowest score possible? b Given that Amy got four correct answers and only left out one question, how many points did she score? c Given that Barakat scored 34 points, how many questions did she get right? d Given that Conor scored 0 points but answered at least one question, how many questions did he not answer correctly? e Explain why it is impossible to score 49 points.

9 In this 4 × 4 magic square, all of the rows, columns and diagonals add to −18.

Copy and complete the square.

−27

15

6

−12 18 3

−3

−15

10 A hotel has 24 floors. Phil get in the lift at floor at floor 18. The lift takes him down 9 floors, then the lift goes up 12 floors before going down 22 floors where he gets out.

What floor does Phil end up at?

4.3 Subtraction with negative numbers ●

I can carry out subtractions involving negative numbers

Subtracting a negative number is the same as adding an appropriate positive number.

Develop fluency a 12 − (−15) a 12 − (−15) = 12 + 15 = 27

b 23 − (−17) b 23 − (−17) = 23 + 17 = 40

1 Use a number line to work these out. a 2 − (−5) b 4 − (−1)

c 7 − (−4)

d 6 − (−8)

2 Use a number line to work these out. a −3 − (−4) b −5 − (−3)

c −6 − (−5)

d −7 − (−3)

3 Without using a number line, write down the answers. a 2 − (−5) b 3 − (−2) c 3 − (−8)

d 6 − (−)7

4 Without using a number line, write down the answers. a −1 − (−6) b −2 − (−3) c −4 − (−7)

d −8 − (−9)

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51

Chapter 4 5 Use a number line to work these out. a 8 − (−5) − 3 b 6 − (−1) − 7

c 8 − (−4) − 5

6 Use a number line to work these out. a −1 − (−3) + 5 b −2 − (−3) + 4

c −9 − (−5) + 3

7 Use a number line to work these out. a 9 − (−2) − (−4) b 5 − (−3) − (−1)

c 4 − (−6) − (−7)

8 Use a number line to work these out. a −2 − (−4) − (−3) b −11 − (−4) − (−2) c −8 − (−3) − (−4) 9 Without using a number line, write down the answers. a 1 − (−5) − 4 b 2 − (−7) − 3 c 3 − (−9) − 2 10 Without using a number line, write down the answers. a −3 − (−1) − (−5) b −6 − (−8) − (−4) c −8 − (−2) − (−5) Reason mathematically The deepest part of the English Channel is 174 metres below sea level.

The top of a mast of a yacht is 6 metres above sea level. a How much higher is the top of the mast than the lowest part of the English Channel? b If the yacht were sat on the bottom of the English Channel, how far would the top of the mast be below sea level? a 174 + 6 = 180 metres b 174 − 6 = 168 metres 11 A fish is 12 m below the surface of the water. A fish eagle is 17 m above the water.

How many metres must the bird descend to get the fish? Explain your working.

12 Alice and Dipesh are playing a game with a dice. The dice has the numbers −4, −3, −2, −1, 1 and 2. They both start with a score of 20 and whatever number they roll is subtracted from their total. They take it in turns to roll the dice. The winner is the first player to score 40 or higher. a Alice rolls −2, −4 and 1. What is her score? b Dipesh rolls 2, 2 and −3 and Alice rolls 1, −1 and −2.

Who is closer to 40 points? c What is the lowest number of rolls a player could roll to win the game?

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4.4 Solve problems This is Laz’s solution to a problem (−15 + 3) − (3 − 5) = 12 − 2 = 10 Laz has made some errors in his calculation.

Explain where he has made errors. −15 + 3 should have been −12, 3 − 5 should have been −2 giving the answer as −12 − −2 = −12 + 2 = −10 13 Choose a number from each list and subtract one from the other. Repeat for at least four pairs of numbers. What are the biggest and smallest answers you can find?

A

14

−17

−25

11

15

B

−23

9

−18

8

−14

14 This is Alia’s solution to a problem: (8 − 10) − (4 − 9) = 2 − 5 = 3 She has the correct answer but made some errors. Explain where she has made errors.

4.4 Multiplication with negative numbers ●

I can carry out multiplications involving negative numbers

Multiplying a positive number by another positive number gives a positive number.

Multiplying a negative number by a positive number gives a negative number.

Multiplying a negative number by a negative number results in a positive number.

Develop fluency

✗ 7 8 9 ÷ 4 5 6 × 1 2 3 – C 0 = +

Work out the answers. a −12 × 4 a −12 × 4 = −48

b −7 × 3. b −7 × 3 = −21

1 Work out the answers. a −2 × 1 b −3 × 4

c −6 × 7

d −1 × 7

2 Work out the answers. a 2 × (−3) b 3 × (−5)

c 6 × (−5)

d 4 × (−7)

3 Work out the answers. a −1 × 3 b 3 × (−7)

c −7 × (−8)

d 4 × (−6)

4 Work out the answers. a −2.5 × 2 b −1.5 × 4

c −0.5 × 8

d −1.5 × 7

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53

Chapter 4 5 Work out the answers. a 2.5 × (−3) b 3.1 × (−4)

c 6.2 × (−3)

d 4.3 × (−4)

6 Work out the answers. a −1.4 × 4 b 3.2 × (−4)

c −7.1 × (−9)

d 4.3 × (−5)

7 Work out the answers. a −3 × 4.3 b 2 × (−4.7)

c −5 × (−9.1)

d 3 × (−5.4)

8 Work out the answers. a −2 × (4 − 7) b −3 × (6 − 8)

c 6 × (2 − 7)

d −4 × (5 − 9)

Reason mathematically Write down the next three numbers in each number sequence. a −1, −2, −4, −8, …, …, …, b −1, 3, −9, 27, …, …, …, a Multiplying by 2 each time to give missing numbers −16, −32, −64 b Multiplying by −3 each time to give missing numbers −81, 243, −729 9 Write down the next three numbers in each number sequence. a 1, −2, 4, −8, …, …, … b −1, −3, −9, −27, …, …, … c −1, 5, −25, 125, …, …, … d 1, −4, 16, −64, …, …, …

10 This is Tim’s solution to a problem: (7 − 11) × (3 − 8) = 4 × 5 = 20 He has the correct answer but made some errors. Explain where he has made errors.

Solve problems Peter asked Kath to think of two integers smaller than eight and tell her their product.

Kath said, ‘The product is −12’.

Peter said, ‘There are four different possible sets of numbers that give that product.’

Write down the four possible pairs of numbers Kath could have been thinking of.

To have a negative product, one number must be positive while the other is negative.

To have a product of 12, you can have 3 × 4 or 2 × 6 only using integers less than eight.

So the four sets are −3 × 4, 3 × −4, −2 × 6, 2 × −6 11 Julie asked Chris to think of two integers less than ten with a product of −24.

Write down the four different possible pairs of numbers Chris could have been thinking of.

54

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4.5 12 a In each brick wall, work out the number to write in an empty brick by multiplying the numbers in the two bricks below it. Copy and complete each brick wall. ii i 4

–2

–1

1

–4

3

b Andy said: ‘You will always have a positive number at the top of the brick wall if there is one negative number in the bottom layer.’

Is Andy correct? Explain your answer.

4.5 Division with negative numbers ●

I can carry out divisions involving negative numbers

Dividing a positive number by another positive number results in a positive number.

Dividing a negative number by a positive number results in a negative number.

Dividing a negative number by a negative number results in a positive number.

Develop fluency Work out the answers. a −14 ÷ 2

a −14 ÷ 2 = −7

b −60 ÷ 5

b

−60 ÷ 5 = −12

1 For questions 1-10, work out the answers. a −6 ÷ 2 b −8 ÷ 4

c −16 ÷ 4

d −21 ÷ 7

2 a 12 ÷ (−3)

b 35 ÷ (−5)

c 60 ÷ (−5)

d 28 ÷ (−7)

3 a −12 ÷ (−4)

b −18 ÷ (−6)

c −16 ÷ (−8)

d −45 ÷(−9)

4 a −15 ÷ 3

b 35 ÷ (−7)

c −27 ÷ (−9)

d 42 ÷ (−6)

5 a −2.6 ÷ 2

b −4.4 ÷ 4

c −8.4 ÷ 2

d −1.5 ÷ 3

6 a 4.5 ÷ (−3)

b 12.8 ÷ (−4)

c 6.9 ÷ (−3)

d 4.8 ÷ (−4)

7 a −2.5 ÷ (−5)

b −12.4 ÷ (−4)

c −4.5 ÷ (−5)

d −2.1 ÷ (−7)

8 a −8.4 ÷ 4

b 3.2 ÷ (−2)

c −8.1 ÷ (−9)

d 6.5 ÷ (−5)

9 a (−4 × −6) ÷ 8

b 28 ÷ (−4 × −1)

c −35 ÷ (−9 + 2))

d 36 ÷ (−5 − 4))

10 a −12 ÷ (4 − 7)

b −45 ÷ (6 − 9)

c 60 ÷ (2 − 7)

d −42 ÷ (5 − 7)

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55

Chapter 4 Reason mathematically This is Lee’s solution to a problem: (5 − 11) ÷ (−6 + 3) = 6 ÷ 3 = 2 Lee has the correct solution but has made some errors in his calculation.

Explain where he has made errors.

(5 − 11) should have been −6, and (−6 + 3) should have been −3 giving −6 ÷ −3 = 2 11 This is Toni’s solution to a problem: (8 − 52) ÷ (−7 + 3) = 46 ÷ 10 = 4.6 Toni has made some errors in his calculation. Explain where he has made errors.

12 Ann said she has thought of two negative numbers that have a product of −8.

Is this possible? Explain your answer.

13 Dean said that if I divide a positive number by a negative number I will always get a negative answer.

Give four examples to illustrate that Dean is correct.

14 Abby was told that 111 × 13 = 1443 She said, ‘So 1443 ÷ −13 will be −111’. a Is Abby correct? Explain your answer. b Give another similar division that gives the answer −111.

Solve problems What are the four different pairs of integers less than 9 that will divide to give −4?

We have −8 ÷ 2 = −4, 8 ÷ −1 = −4, −4 ÷ 1 = −4, 4 ÷ −1 = −4 15 What are the six different pairs of integers less than 10 that will divide to give −3?

16 Write down five calculations involving division that give the answer −5.

17 Work out the answer to (−8 ÷ 4) × (−9 ÷ –3) × (10 ÷ −2) Now I can… use a number line to order positive and negative numbers, including decimals

carry out additions and subtractions involving negative numbers

carry out divisions involving negative numbers

understand and use the symbols < (less than) and > (greater than)

carry out multiplications involving negative numbers

solve problems with negative numbers

use a number line to calculate with negative numbers 56

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