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)
Develop fluency Worked example 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
t n e t n its o c d e e l o p t t m c Sa bje u S
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 First, draw up a number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Then mark the numbers on the number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
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 40
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4.1 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 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
t n e t n its o c d e e l o p t t m c Sa bje u S
Reason mathematically Worked example
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 a Copy this bank statement. Then fill in the balance column. Date
Details
Debit (£)
31-01-2019 Opening balance 05-02-2019 Energy company 18-02-2019 Credit card
Credit (£)
Balance (£) 187.00
53.62 228.51
b Explain why the closing balance is negative.
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Chapter 4
Solve problems Worked example 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. b c a –17
2
–9
7
–23
–7
t n e t n its o c d e e l o p t t m c Sa bje u S
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
The league table shows the numbers of goals scored by each team and the ✗ goals scored against each team. The goal difference is the difference between the 7 8 9 ÷ 4 5 6 × 1 2 3 – C 0 = +
number of goals scored and the number of goals against. Team Calcusea Adderpool Bracketburn Directington a b c d
Goals for
Goals against
Goal difference
11
7
9
4
5
10
–10
Find Bracketburn’s goal difference. Find the number of goals scored by Adderpool. Find the number of goals scored against Directington. Given that the total number of goals scored by all four teams is the same as the total number of goals against, find Calcusea’s goal difference.
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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
Develop fluency Worked example Use a number line to work out the answers. a 5 – 13 b (–11) + 9 c 6 – 12 – 3 a Starting at zero and ‘jumping’ along the number line to 5 and then back 13 gives an answer of –8. –13
t n e t n its o c d e e l o p t t m c Sa bje u S +5
–8
0
5
b 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
–2
0
c Using two steps this time, 6 –12 – 3 = –9 –3
–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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Chapter 4 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
t n Reason mathematically e t n its o c d e e l o p t t m c Sa bje u S
Worked example
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 two vertical numbers, –1 and 2 the missing top number of the middle column should be –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 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.
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4.2 12 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
Solve problems Worked example In this 4 × 4 magic square, all of the rows, columns and diagonals add to –10. 8
–6
7
–7
–26
t n e t n its o c d e e l o p t t m c Sa bje u S –1
10
–8
Complete the square. Imagine letters in the spaces as shown. e
8
–6
f
7
–7
d
–26
c
10
a
b
g
h
–1 –8
Work out a: –10 = a + 8 + 7 + –26 so a = –10 – (–11) = –10 + 11 = 1 Then b = –10 – (10 + 1 + –8) = –10 – (3) = –10 – 3 = –13 Then c = –10 – (–6 + –7 + –13) = –10 – (–26) = –10 + 26 = 16 Then d = –10 – (–1 + 16 + –26) = –10 – (–11) = –10 + 11 = 1 Then e = –10 – (–8 + 16 + 7) = –10 – (15) = –10 – 15 = –25 Then f = –10 – (–25 + 1 + 10) = –10 – (–14) = –10 + 14 = 4 Then g = –10 – (–25 +8 + –6) = –10 – (–23) = –10 + 23 = 13 Finally h = 10 – (13 + –1 + –8) = –10 – (4) = –10 – 4 = –14
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Chapter 4 13 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
14 In this 4 × 4 magic square, all of the rows, columns and diagonals add to the same number. Copy and complete the square. 0
–26 –10
–14 –18 –20 –24
t n e t n its o c d e e l o p t t m c Sa bje u S
–6
–30
15 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.
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4.3 4.3 Subtraction with negative numbers ●
I can carry out subtractions involving negative numbers
Develop fluency Subtracting a negative number is the same as adding an appropriate positive number.
Worked example 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)
t n e t n its o c d e e l o p t t m c Sa bje u S
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 Worked example 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.
t n e t n its o c d e e l o p t t m c a e j S Solve problems b u S
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?
Worked example
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
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4.4 4.4 Multiplication with negative numbers ●
I can carry out multiplications involving negative numbers
Develop fluency
✗ 7 4 1 C
8 5 2 0
9 6 3 =
÷ × – +
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. To summarise this: (–) × (+) = (–), (+) × (+) = (+) and (–) × (–) = (+)
Worked example Work out the answers. a –12 × 4 a –12 × 4 = –48
t n e t n its o c d e e l o p t t m c Sa bje u S 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 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, …, …, …
Solve problems Worked example 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
t n e t n its o c d e e l o p t t m c Sa bje u S
12 a Julie asked Chris to think of two integers smaller than ten and tell her their product. Chris said: ‘The product is –24.’ Julie said: ‘There are four different possible sets of numbers that give that product.’ Write down the four possible pairs of numbers Chris could have been thinking of. b Chris asked Julie to think of two numbers smaller than ten and tell him their product. Julie said: ‘The product is 12.’ Chris said that there were four different possible sets of numbers with that product. Write down the four possible pairs of numbers Julie could have been thinking of. 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
3
–2
1
–4
3
b Andy said: ‘You will always have a positive number at the top of the brick wall if there are two negative numbers in the bottom layer.’ Is Andy correct? Explain your answer. c What combination of positive and negative numbers do you need on the bottom layer to end up with a negative number at the top?
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4.5 4.5 Division with negative numbers ●
I can carry out divisions involving negative numbers
Develop fluency 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. You can summarise this as: (+) ÷ (+) = (+), (–) ÷ (+) = (–) and (–) ÷ (–) = (+)
Worked example Work out the answers. a –14 ÷ 2 a –14 ÷ 2 = –7
t n e t n its o c d e e l o p t t m c Sa bje u S 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 Worked example 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.
t n e t n its o c d e e l o p t t m c Solve problems Sa bje u S
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
15 What are the six different pairs of integers less than 10 that complete this calculation? ..... ÷ ..... = –3 16 Write down five calculations involving division that give the answer –5.
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Review Review Develop fluency 1 What sign, < or > should go between the numbers? –5 …. –3 2 Which temperature is higher, –11 °C or –8 °C? 3 Use a number line to work out the answers. a –4 + –5 b –3 + –2 c –7 + –1
d –8 + –6
4 Without using a number line, write down the answers. a 2–4 b 4–9 c 8 – 11
d 5–7
5 Without using a number line, write down the answers. a –2 – (–5) b –3 – (–4) c –5 – (– 8)
d –9 – (–7)
t n e t n its o c d e e l o p t t m c Sa bje u S
6 Use a number line to work out the answers. a 7 – (–4) – 4 b 5 – (–2) – 8 c 9 – (–5) – 6 7 Work out the answers. a –2 × 5.2 b 3 × (–3.6)
c –4 × (–8.1)
d 5 × (–6.3)
8 Work out the answers. a –3 × (5 – 8) b –4 × (7 – 9)
c 7 × (1 – 6)
d –5 ×(4 – 8)
9 Work out the answers. a 5.3 ÷ –2 b 5.7 ÷ –3
c 7.2 ÷ –4
d 3.4 ÷ –5
c 7 ÷ (3 – 8)
d –5 ÷ (6 – 8)
10 Work out the answers. a –3 ÷ (5 – 8) b –4 ÷ (6 – 7)
Reason mathematically
11 Dylan was told: “When you halve a number you always get a smaller number.” Explain why this isn’t true. 12 This is Ben’s solution to a problem. 3 – (–1) + (–5) = 3 – 1 – 5 = –1 He has the correct solution but he has made some errors in his calculation. Explain where he has made errors. 13 Amir was told: “When you add two numbers together you always get a larger number.” Is this correct? Explain your answer.
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Chapter 4 14 This is Mimi’s solution to a problem. –12 ÷ (–8 + 5) = 12 ÷ 3 = 4 She has the correct solution but she has made some errors in her calculation. Explain where she has made errors. 15 You are given two numbers, –10 and –2. There are eight different ways of combining these numbers with +, –, ×, and ÷ . You are told that half of these combinations give negative answers. Is this correct? Explain your answer.
Solve problems 16 Harry is on the 29th floor of a skyscraper. He goes up 18 floors and then down 23 floors. He wants to go to the 2nd floor. How many floors does he need to go down now?
t n e t n its o c d e e l o p t t m c Sa bje u S
17 Karen and Geza have parked their car on level –5 of the shopping centre car park. They take the lift to the shops on level +6, then Karen realises she has left her purse in the car. She goes back to the car to get it. Then she returns to the shops and meets Geza on level +4. How many levels has Karen travelled through altogether? 18 The Mariana Trench in the Pacific Ocean has a maximum depth of 10 911 metres below sea level. The summit of Mount Everest is 8848 metres above sea level. a How much higher than the base of the trench is the summit of Mount Everest? b If Mount Everest was set in the deepest part of the trench, how far would its summit be below sea level? 19 In a popular TV programme, each of two teams has to buy three items at an antiques fair. The items are sold at auction and the team that makes more money wins and keeps any profit they make. Copy and complete each team’s score sheet. a Red team Item Silver dish
Buying price (£) 59
Umbrella Toy car Total
Selling price (£)
–18 47
45
Profit (£) +14
55
137
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Review b Blue team Item
Buying price (£)
Necklace Doll
49
Watch
110
Total
197
Selling price (£)
Profit (£)
55
+17
85 –52
c Which team won and by how much? d Teams can go for a bonus buy, chosen by an expert. The Red team bonus buy was bought for £33 and sold for £25. The Blue team bonus buy was bought for £21 and sold for £29. Copy and complete this table to show who would win in each case.
t n e t n its o c d e e l o p t m ct Self-reflectiona S bje u ✓✓✓ S Red team without the bonus buy
Red team with the bonus buy
Blue team without the bonus buy
Blue team with the bonus buy
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) I can carry out additions and subtractions involving negative numbers I can use a number line to calculate with negative numbers I can carry out subtractions involving negative numbers I can carry out multiplications involving negative numbers I can carry out divisions involving negative numbers
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓ ✓ ✓ 55
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