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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

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.

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.

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.

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

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

Year 7: Negative numbers Scheme of learning

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?

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

ÂŠ HarperCollinsPublishers Ltd 2019

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

_________________

c –4.2, –3.8, –4.9

_________________

b –2.6, 3.4, –4.8

_________________

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

_________________

b –5 + 4

_________________

c –2 – 7

_________________

d –5 – 3

_________________

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

_________________

c 2 – (–7) – 13

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b 3 + (–8) – 2

_________________

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

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

_________________

b 5 × (–8)

_________________

c –6 × (–3)

_________________

d 7 × (–9)

_________________

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)

_________________

c –32 ÷ (–9 + 1)) _________________

d 40 ÷ (–5 – 3)

_________________

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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