Learn and Practice Book

Develop your skills to master maths

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. a 6 ■ 10 b 5 ■ –2 c –1 ■ 9

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 54

4.1 8 Put these numbers into order, starting with the highest. a –4, –1, –6, –3 b –2, 2, 0, –1 c –4, –4.4, –2.5, 1 9 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

10 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. 11 Carla was told 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. 12 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 13 Explain how you know that –5.1 is higher than –5.9. 14 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

15 Work out the number that is halfway between the numbers in each pair. a b c –17

2

–9

7

–23

–7

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Chapter 4 16 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? 17 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?

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. b (–11) + 9

a 5 – 13 a

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 4–6 c 2–3

d 1–9

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

d –8 + –7

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

d –2 + 6

4 Use a number line to work out the answers. a –3 – 1 b –3 – 3 c –1 – 4

d –3 – 8

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4.2 5 Use a number line to work out the answers. a –2 + –3 b –1 + –4 c –3 + –5

d –1 + –6

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

d 4–6

7 Without using a number line, write down the answers. a –9 + 3 b –8 + 4 c –7 + 5

d –3 + 8

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

d –6 – 7

9 Use a number line to work out the answers. a 1–5+4 b 2–7+3 c 3–9+2 10 Use a number line to work out the answers. a –3 + 1 – 5 b –6 + 8 – 4 c –8 – 2 + 5 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.

11 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

12 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. 57

Chapter 4 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. 13 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. 14 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

15 A hotel has 24 floors. Phil get in the lift at floor at floor 18. The lift takes him down nine floors, then the lift goes up twelve floors before going down twenty two floors where he gets out. What floor does Phil end up at?

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4.3 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)

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)

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Chapter 4 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? d Dipesh’s first roll is a 2 and he wins the game in seven rolls. What possible final scores could he get? 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.

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4.4 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 –2 × (–4) b –3 × (–6)

c –6 × (–8)

d –4 ×(–9)

4 Work these out. a –1 × 3

c –7 × (–8)

d 4 × (–6)

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

c –0.5 × 8

d –1.5 × 7

6 Work out the answers. a 2.5 × (–3) b 3.1 × (–4)

c 6.2 × (–3)

d 4.3 × (–4)

7 Work out the answers. a –2.5 × (–3) b –3.5 × (–4)

c –1.5 × (–8)

d –4.1 × (–7)

8 Work out the answers. a –1.4 × 4 b 3.2 × (–4)

c –7.1 × (–9)

d 4.3 × (–5)

9 Work out the answers. a –3 × 4.3 b 2 × (–4.7)

c –5 × (–9.1)

d 3 × (–5.4)

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

c 6 × (2 – 7)

d –4 × (5 – 9)

b 3 × (–7)

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Chapter 4 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 11 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, …, …, … 12 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 12 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. 13 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.

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4.5 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 Work out the answers. a –6 ÷ 2 b –8 ÷ 4

c –16 ÷ 4

d –21 ÷ 7

2 Work out the answers. a 12 ÷ (–3) b 35 ÷ (–5)

c 60 ÷ (–5)

d 28 ÷ (–7)

3 Work out the answers. a –12 ÷ (–4) b –18 ÷ (–6)

c –16 ÷ (–8)

d –45 ÷(–9)

4 Work out the answers. a –15 ÷ 3 b 35 ÷ (–7)

c –27 ÷ (–9)

d 42 ÷ (–6)

5 Work out the answers. a –2.6 ÷ 2 b –4.4 ÷ 4

c –8.4 ÷ 2

d –1.5 ÷ 3

6 Work out the answers. a 4.5 ÷ (–3) b 12.8 ÷ (–4)

c 6.9 ÷ (–3)

d 4.8 ÷ (–4)

7 Work out the answers. a –2.5 ÷ (–5) b –12.4 ÷ (–4)

c –4.5 ÷ (–5)

d –2.1 ÷ (–7)

8 Work out the answers. a –8.4 ÷ 4 b 3.2 ÷ (–2)

c –8.1 ÷ (–9)

d 6.5 ÷ (–5)

9 Work out the answers. a (–4 × –6) ÷ 8 b 28 ÷ (–4 × –1)

c –35 ÷ (–9 + 2))

d 36 ÷ (–5 – 4))

c 60 ÷ (2 – 7)

d –42 ÷ (5 – 7)

10 Work out the answers. a –12 ÷ (4 – 7) b –45 ÷ (6 – 9)

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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 64

Teacher Handbook

Prepare todayʼs pupils for GCSE (9–1)

Chapter 4 Negative numbers Concept comprehension Ordering positive and negative numbers Q1 Which is the lowest temperature, 2 ⁰C or –9 ⁰C?

Teacher guidance:

Ans: –9 ⁰C

Check pupils are clear that even though 9 is higher than 2, –9 is on the other side of the number line to 0.

Q2 Explain how you know that 5 is higher than –6 Ans: Use a number line showing –6 to the left of zero and 5 is to the right Q3 Work out the number that is halfway between –4 and 2 Ans: Draw a number line showing both –4 and 2. Count 6 spaces between –4 and 2, so halfway is 3 spaces from the –4, which can be seen on the number line to be –1

Use a number line to help with each of these questions, frequently asking pupils to identify the smaller or higher number on the line. You want pupils to see the number line in their heads, but this comes from concretely using it on paper themselves. Get them to use the line first, then to see the line in their own head.

Adding negative numbers Q1 What is the answer to –2 + 1 – 3

Teacher guidance;

Ans: –4

Get pupils to use the number line in their heads if possible or on paper.

Q2 Grannie has £25 in the bank. She writes a cheque for £35. Saying that she now has £10 in the bank. Is she correct? Explain your answer Ans: No, she is not correct, as 25 – 35 is -10, so she has – £10 in the bank which is what we call an overdraft. Q3 Solve this magic square where each row and column adds up to –3.

In Q1 ask pupils to explain how they found the answer, some will look at it as 1 – 5 = –4, others will work from left to right, –1 – 3 = –4 Q2 may need overdraft explaining it’s what we call negative money in the bank. What we are looking for is a recognition that taking a larger amount away from a smaller amount will give a negative answer. Q3 Ask the question without (a), (b) etc – that is the order to be looking at for the answers at first.

Ans: a 3, b –1, c 2, d 0, e 1, f –3

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Year 7 Negative numbers Concept Comprehension

Chapter 4 Negative numbers Concept comprehension Subtracting negative numbers Q1 What is the answer to 3 – (-4)

Teacher guidance:

Ans: 7

We want pupils to recognise immediately that minus a minus is adding, hence –(–4) is simply +4

Q2 This is Ben’s solution to a problem (–5 + 2) – (1 – 3) = 7–2 = 5 Ben has made some errors in his calculation. Explain where he has made errors. Ans: the (–5 + 2) should be –3, the (1 – 3) should be –2 giving the solution –3 – (–2) = –3 + 2 = –1 Q3 –5 6 –8 7 –4 9 By subtracting two of these numbers, what is the largest possible answer and what is the smallest? Ans: Largest is 9 – (–8) = 9 + 8 = 17; Smallest is –8 – 9 = –17

Pupils will still see 1 – 3 as a subtraction but 1 from 3 giving 2. Keep reinforcing the fact that subtracting larger from smaller will always give a negative answer. Q3 could first be set as what possible answers can you get by subtracting two of the numbers, then look for the extreme values. Again seeing that subtracting a large negative will make a larger addition.

Multiplication with negative numbers Q1 Work out: a 2 × (–8) Ans: a –16

Teacher guidance: b –5 × (–3) b 15

Q2 What are the next three numbers in the sequence –5, 10, –20, 40, …, …, ….

Pupils need to recognise that multiplication of a negative with a positive will always give a negative answer, while multiplying a negative with a negative will give a positive answer. Take care not to say “two negatives give a positive” as this can result in pupils thinking this applies when adding or subtracting as well.

Ans: –80, 160, –320 Q3 Give two numbers that have a product of –12 Ans: there are six: –3×4, 3×–4, –2×6, 2×–6, –12×1, 12×–1

In Q2 do get pupils to explain why they are suggesting their answers term by term.

Division with negative numbers Q1 Work out: a –12 ÷ 3 Ans: a –4

Teacher guidance: b 30 ÷ (–5) b –6

c –10 ÷ (–2)

Pupils need to see the same rules apply to division as with multiplication of negative numbers.

c5

Q2 This is Theo’s solution to a problem (11 – 35) ÷ (–5 + 2) = 24 ÷ 3 = 8 Theo has the right answer but has made some errors in his calculation. Explain where he has made errors.

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Signs the same gives a positive answer; different signs give a negative answer. Reinforce that this is only for multiplication and division, as adding and subtracting can give different results depending on the size of the numbers. Year 7 Negative numbers Concept Comprehension

Chapter 4 Negative numbers Concept comprehension Ans: (11 – 35) should be –24, (–5 + 2) should be –3 Giving the answer as –24 ÷ –3 = 8

In Q2 get pupils to explain the errors as these are common errors made by pupils.

Q3 Give me two numbers that will divide to give a negative answer. Will this always be one positive and one negative Ans: Many different answers, all with one positive and one negative.

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Year 7 Negative numbers Concept Comprehension

Chapter 4 Negative numbers National Curriculum links

Contexts and connections

Understand and use place value for decimals, measures and integers of any size

4.1 The number line

Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥ Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

Understand and use place value for decimals, measures and integers of any size Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the

Context may include working with temperatures. There are connections with using negative numbers in Algebra.

Learning outcomes To use a number line to order positive and negative numbers, including decimals To understand and use the symbols < (less than) and > (greater than) Assessment links Year 7: Fractions Core, Fractions Extended and Percentages and statistics Core Year 8: Baseline Test and Algebraic expressions Core

Misconceptions –5 is not greater than –3 –3 is not “bigger than” than –5

Support Use horizontal and vertical number lines (including thermometers) to show progression through zero.

Learning Sequence Comparing two numbers

Extension through depth

Sequencing two numbers

Students could be encouraged to look at a thermometer and use it as a number line, or look at a number line that includes negative.

Sequencing three numbers in context One step word questions Two step word questions

Find five numbers that have a mean of –5. Then find five numbers that have a mean of 0.

Misconceptions 2 – 3 is not the same as 3 –2 Subtracting from a negative number increases the digit value. For example –3 – 2 = –5, rather than –1 or 1

Support Using a thermometer with varying intervals solve similar questioning that require careful reading and understanding of the question, such as: If the temperature is 5 degrees and it

Pictorial questions

Key Vocabulary Greater than, less than, positive, negative, sequencing 4.2 Arithmetic with negative numbers Learning outcomes To carry out additions and subtractions involving negative numbers To use a number line to calculate with negative numbers

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Year 7: Negative numbers Scheme of learning

real numbers; use the symbols =, ≠, <, >, ≤, ≥ Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

Assessment links Year 7: Baseline test Year 8: Algebraic expressions Core Key Vocabulary positive, negative, diagonal, addition, subtraction

Learning Sequence Subtraction beyond zero Addition of two negative numbers Addition of a positive and a negative number Subtraction of a positive from a negative number Addition of a positive and two negative numbers One step word questions Two step word questions

Understand and use place value for decimals, measures and integers of any size Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥ Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

drops overnight by 7 degrees. What is the new temperature? If the temperature is –5 degrees and it drops overnight by 7 degrees. What is the new temperature? Support Extension through depth Practice moving around zero. Write equation involving 2 operations where the answer is 0 e.g. 5 – 7 + 2 Use all four operations to construct an equation where the answer is 0

4.3 Subtraction with negative numbers Learning outcomes To carry out subtractions involving negative numbers

Misconceptions –2 + 1 is not –3 20 – (–12) is not 8

Assessment links Year 7: Algebra Extended Year 8: Algebraic expressions Core and Algebraic expressions Extended

Learning Sequence

Key Vocabulary positive, negative, subtraction

Subtracting a negative number from a negative number

Support Use a calculator to show that 2 – (–3) =1 Model that the two negatives create a positive, in the same way a double negative phrase is positive: “I am not doing no homework”, for example, means that the homework is going to be done. Focus on the language and that 2 – 3 = –1, moving left on a number line, decreasing but that 2 – (–3) would require moving right or positively.

Mixed addition and subtraction involving three positive and negative numbers

Extension through depth

Subtracting a negative number from a positive number

One step word questions Two step word questions

Start on –12. What could the calculation be if the solution is 5? Use 2, then 3 subtraction calculations

Open ended enquiry questions Multiple calculations in a question © HarperCollinsPublishers Ltd 2019

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Year 7: Negative numbers Scheme of learning

Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥ Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

4.4 Multiplication with negative numbers Learning outcomes To carry out multiplications involving negative numbers

Misconceptions

−3 × 2 = 6 −2 × (–3) = –6

Assessment links Year 8: Algebraic expressions Core

Learning Sequence

Key Vocabulary positive, negative, multiplication, decimal, sequence, integer

Multiplying a positive integer by a negative integer

Multiplying a negative integer by a positive integer

Multiplying a negative decimal by a positive integer

Support Use the bar model method to show that – 4 × 3 = –12. This will pictorially demonstrate that you have 3 lots of –4 and therefore –4 + –4 + –4 = –12 rather than 12. Extension through depth If the solution is –10 what could the question be if you multiply a positive and negative number?

Multiplying a positive decimal by a negative integer Sequencing through repeated multiplication One step word questions Two step word questions Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

4.5 Division with negative numbers Learning outcomes To carry out divisions involving negative numbers

Misconceptions –8 divided by 2 = 4 –45 divided by (–5) = –9

Assessment links Year 8: Algebraic expressions Extended

Learning Sequence

Key Vocabulary positive, negative, division, decimal, integer © HarperCollinsPublishers Ltd 2019

Support Use the bar model method to show that – 4 ÷ 2 = –2. This will pictorially demonstrate that a whole –4 can be divided into 2 parts of –2

Dividing a negative integer by a positive integer

Support and extension through depth

Dividing a positive integer by a negative integer

The solution is –12 what could the division calculation be?

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Year 7: Negative numbers Scheme of learning

Dividing a negative decimal by a positive integer Dividing a positive decimal by a negative integer One step word questions Two step word questions

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Year 7: Negative numbers Scheme of learning

Chapter 4 Negative Numbers Topic Review Tick the statement when you feel confident with the skill.

Now I can… use a number line to order positive and negative numbers, including decimals Now I can… understand and use the symbols < (less than) and > (greater than) Now I can… carry out additions and subtractions involving negative numbers Now I can… use a number line to calculate with negative numbers Now I can… carry out subtractions involving negative numbers Now I can… carry out multiplications involving negative numbers Now I can… carry out divisions involving negative numbers

The number line 4.1.1 Put these numbers into order, starting with the lowest. a –1.7, 1.6, –2.3

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c –4.2, –3.8, –4.9

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b –2.6, 3.4, –4.8

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4.1.2 Mae said –8 is larger than –3 Hann said she was wrong. How could she explain to Mae why she was wrong? _________________________________________________________________________ _________________________________________________________________________ 4.1.3 Chris got into a lift on floor 3. He went up four floors to a meeting room. Then he took the lift down five floors to call in his office before taking the lift eight floors up to the library. Finally he took the lift down three floors to the sales department. On which floor is the sales department? _________________

Arithmetic with negative numbers 4.2.1 Without using a number line write down the answers. a –2 + 4

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b –5 + 4

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c –2 – 7

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d –5 – 3

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Chapter 4 Negative Numbers Topic Review 4.2.2 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. _________________________________________________________ _________________________________________________________ _________________________________________________________ 4.2.3 In a quiz, 4 points are awarded for each correct answer but 2 points are taken off for each incorrect answer. If a question is not answered at all then it scores –1 point. There are eight questions in the quiz. a What is the lowest possible total score? _________________ b Joy got five correct answers and left out two questions. How many points did she score? _________________ c Nadia scored 27 points. How many questions did she get right? _________________ d Explain why it is impossible to score 23 points. _________________________________________________________________________

Subtraction with negative numbers 4.3.1 Without using a number line write down the answers. a 2 – (–6) + 3

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c 2 – (–7) – 13

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b 3 + (–8) – 2

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4.3.2 This is Sophia’s solution to a problem (–12 + 4) – (2 – 6) = 16 – 4 = 12 Sophia has made some errors in her calculation. Explain where she has made errors. _________________________________________________________________________ 4.3.3 Choose a number from each list and subtract one from the other. Repeat for at least three pairs of numbers. What are the biggest and smallest answers you can find? A B

12 –21

–15 7

–23 –16

13 19

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Chapter 4 Negative Numbers Topic Review Multiplication with negative numbers 4.4.1 Work out: a –2 × 4

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b 5 × (–8)

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c –6 × (–3)

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d 7 × (–9)

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4.4.2 Alan was asked to write down the next three numbers in the number sequence. –1, 3, –9, 27, …, …, … He wrote 81, –243, 729. Is Alan correct? Explain your answer. _________________________________________________________________________ 4.4.3 Dave asked Helen to think of two numbers smaller than ten and tell him their product. Helen said: ‘The product is –8.’ Dave said: ‘There are four different possible sets of numbers that give that product.’ Write down the four possible pairs of numbers Helen could have been thinking of. _________________________________________________________________________

Division with negative numbers 4.5.1 Work out the answers a (–3 × -8) ÷ 4 _________________

b 30 ÷ (–5 × –1)

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c –32 ÷ (–9 + 1)) _________________

d 40 ÷ (–5 – 3)

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4.5.2 This is Ben’s solution to a problem (7 – 43) ÷ (–6 + 3) = 36 ÷ –9 = –4 Toni has made some errors in his calculation. Explain where he has made errors. _________________________________________________________________________ 4.5.3 What are the six different pairs of integers less than 13 that complete the calculation. ÷

= –4

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