The Shanghai Maths Project Practice Book 5B Sample Unit

Chapter 7 Introduction to positive and negative numbers

Unit 7.1 Positive and negative numbers (1) Conceptual context This chapter introduces negative numbers. Negative numbers help to describe values less than zero. Due to their abstract nature, they should be introduced in context.

Negative numbers represent opposites. Positive numbers represent values greater than a given benchmark (reference point), while negative numbers represent values less than the benchmark (the benchmark is often zero). If positive represents a movement to the right, negative represents a movement to the left.

Pupils will already have met negative numbers in everyday life. They may have seen negative floor numbers in lifts, corresponding to basement floors below ground level. They will be familiar with temperatures expressed as negative numbers. Temperatures above zero are positive while temperatures below zero are negative. Domestic freezers are generally set to −18 °C and the fridge temperature varies between 3 and 5 °C. There is often a live display of these values on a control panel.

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Positive numbers are usually written without a ‘+’. The + is implied. Thus, positive ten (or ‘plus ten’) can be written as +10 or, more commonly, simply as 10. Negative numbers are always written with a − sign, thus negative ten (or ‘minus ten’) is written as −10.

Learning pupils will have achieved at the end of the unit ●

Temperatures below 0 °C will have been read as negative numbers (Q1, Q5)

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Pupils will have practised reading positive and negative numbers (Q2, Q3, Q4, Q5, Q6)

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The concept that positive and negative numbers are opposites will have been explored (Q2, Q3, Q4, Q6)

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Pupils will have learned that negative numbers must always have − in front of the number (Q2, Q3, Q4)

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The understanding will have been established that positive numbers may be prefaced by + but if there is no + it is inferred (Q2, Q3, Q4)

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Pupils will have developed and extended their knowledge and understanding of specific, well-known temperatures (Q5)

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Pupils will have consolidated their understanding that on the Celsius scale temperatures below zero (the freezing point of water) are represented by negative numbers (Q5)

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The concept of a benchmark (reference point) where numbers greater than the benchmark are positive and numbers less than the benchmark are negative will have been introduced (Q6)

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Chapter 7 Introduction to positive and negative numbers ●

Unit 7.1 Practice Book 5B, pages 29–31

Pupils will have explored how a negative number and its corresponding positive number are ‘opposite’ each other across the benchmark (reference mark); they are both the same distance from the benchmark but in opposite directions (Q6)

Resources −10 to 110 °C thermometers; diagrams of −10 to 110 °C thermometers; fever scan thermometer and/or classroom thermometer if available; Resource 5.7.1a Matching temperatures; Resource 5.7.1b Thermometers; Resource 5.7.1c Positive and negative numbers; Resource 5.7.1d Identifying common temperatures; Resource 5.7.1e Test results 1; Resource 5.7.1f Test results 2

Vocabulary positive number, negative number, benchmark, temperature, ° Celsius, thermometer

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Chapter 7 Introduction to positive and negative numbers

Question 1

Unit 7.1 Practice Book 5B, pages 29–31

numbers using the word ‘negative’, not ‘minus’. Strictly speaking, ‘negative 10’ is a number while ‘minus 10’ is a mathematical operation that can be done to another number. However, you should refer to negative numbers as, for example, ‘negative 3’ and ‘minus 3’ (also ‘positive 3’ and ‘plus 3’) interchangeably so pupils become familiar with both. ●

Pupils should complete Question 1 in the Practice Book.

Same-day intervention ●

Give pupil pairs Resource 5.7.1a Matching temperatures.

What learning will pupils have achieved at the conclusion of Question 1? ●

Temperatures below 0 °C will have been read as negative numbers.

Activities for whole-class instruction ●

Display a large diagram of a blank −10 to 110 °C thermometer and examine the scale with pupils. Find zero and read the increasing scale together, noting increments of 10 °C. Now look at the scale below zero and agree that the next interval below zero is also marked 10. Ask if pupils know how the two 10s are differentiated – they may already have used thermometers of this type in their Science lessons. If possible, allow pupils to see and handle real thermometers as well as looking at the diagram.

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Explain that the numbers above zero are positive temperatures and the numbers below are negative ones.

Remind pupils that the higher the height of the column in the thermometer, the higher the temperature. Negative numbers represent cold temperatures; in fact, negative temperatures are below freezing because 0 °C is the freezing point of water.

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Show different temperatures on the thermometer diagram, both negative and positive values, and ask pupils to read the temperatures.

Answers: 1. D; 2. A; 3. C; 4. B; 5. F; 6. E

Same-day enrichment ●

Give pupil pairs Resource 5.7.1b Thermometers.

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Negative numbers are correctly read as ‘negative (number)’ thus −10 is ‘negative ten’. In the past, the word minus was used, thus −10 was read as ‘minus 10’.

Pupils will still hear this used today, for example in television weather forecasts. Explain that while to nonmathematicians ‘minus 10’ does mean −10, they should read

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Chapter 7 Introduction to positive and negative numbers

Questions 2, 3 and 4

Unit 7.1 Practice Book 5B, pages 29–31

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Display 14 and +14 and invite pupils to read these numbers. They may read them as ‘fourteen’ and ‘positive fourteen’.

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Ask: Do these two numbers have the same value?

Confirm that they have the same value. Explain that we do not usually write + in front of positive numbers because numbers are usually positive. When you see a number, you can assume it is positive and so the + does not need to be written or said. However, if we want to emphasise that a number is positive, for example when we are comparing it to a negative number, we can put in the + without changing the number’s value.

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Display the following numbers and invite individual pupils to read them: 15 −4 +12 −11 −200 34.2 −2.6 12

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Ask: Which two numbers are the same?

Twelve and positive twelve (or ‘plus twelve’) have the same meaning and value.

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What learning will pupils have achieved at the conclusion of Questions 2, 3 and 4? ●

Pupils will have practised reading positive and negative numbers.

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The concept that positive and negative numbers are opposites will have been explored.

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Pupils will have learned that negative numbers must always have − in front of the number.

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The understanding will have been established that positive numbers may be prefaced by + but if there is no + it is inferred.

IN

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Remind pupils that in Question 1, temperatures below zero were written with a negative sign. Display −5 and ask pupils to read the number. Agree it is ‘negative 5’ (or ‘minus 5’).

Repeat with more negative numbers, including decimal numbers and fractions, for example: −14, −3.4, −25, −34, −273.

Negative numbers always have a negative sign in front of the number.

DOWN

SPEND

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Agree that the opposites are OUT, UP and SAVE. Remind pupils that negative numbers mirror positive numbers and are opposite to them. Negative numbers are often used in the context of money. Adding money is positive and taking money away is negative.

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Display the following words and say: Here are some words that describe money transactions. Discuss whether they describe a positive movement of money or a negative one.

DEPOSIT

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SPEND

SELL

WIN

RECEIVE

Establish the following positive/negative descriptions and agree the following opposites: DEPOSIT (positive) – WITHDRAW (negative)

Activities for whole-class instruction ●

Display the following words and ask pupils to tell you the opposite.

SPEND (negative) – SAVE (positive) SELL (negative) – BUY (positive) WIN (positive) – LOSE (negative) RECEIVE (positive) – GIVE (negative)

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Zero is neither positive nor negative. An integer can either be greater than zero, called positive, or less than zero, called negative. ●

Pupils should complete Questions 2, 3 and 4 in the Practice Book.

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Chapter 7 Introduction to positive and negative numbers

Same-day intervention ●

Unit 7.1 Practice Book 5B, pages 29–31

Same-day enrichment

Give pupils Resource 5.7.1c Positive and negative numbers to complete.

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Present pupils with the following problem:

Answers: Number in words

Numeral Positive or negative? (delete as necessary)

Sally had £100 in the bank.

On Monday, she withdrew £40.

On Tuesday, she deposited £50.

On Wednesday, she withdrew £20.

negative ten

−10

positive/negative

On Thursday, she deposited £30.

three point five

3.5

positive/negative

On Friday, she spent £10.

negative five point six

−5.6

positive/negative

four hundred and nine point two

409.2

positive/negative

Complete the balance sheet, showing deposits as positive entries and withdrawals as negative ones, to work out the final balance.

positive eleven

11

positive/negative

Deposits/withdrawals Balance

negative eighty-five

−85

positive/negative

Starting balance

positive seven eighths

7 8

positive/negative

Monday

negative twelve point four

−12.4

positive/negative

twelve point four

12.4

positive/negative

Thursday

negative thirty-four point nine

−34.9

positive/negative

Friday

negative two point zero three

−2.03

£100 −£40

£60

Tuesday Wednesday

positive/negative

How much was in Sally’s bank account on Saturday morning? ________________________ Answer: Sally had £110 in her bank account on Saturday.

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Chapter 7 Introduction to positive and negative numbers

Question 5

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Unit 7.1 Practice Book 5B, pages 29–31 Display the temperatures that have been discussed, increasing from the left: −18 °C

What learning will pupils have achieved at the conclusion of Question 5? ●

Temperatures below 0 °C will have been read as negative numbers.

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Pupils will have practised reading positive and negative numbers.

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Pupils will have developed and extended their knowledge and understanding of specific, wellknown temperatures.

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Explain to pupils that there are a few other temperatures that are useful to learn, if possible. These are: –

Room temperature is about 21 °C. Pupils can look at a thermometer in the classroom to confirm this.

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Body temperature is 37 °C. If your temperature is higher, then you have a fever and by the time it reaches 40 °C, you feel really ill. Use a forehead thermometer to take the temperature of a volunteer pupil.

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Fridges are set at between 3 and 5 °C. This is just a few degrees above freezing so as to stop germs multiplying but not have frozen food.

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Freezers are set at −18 °C. Suggest pupils look at the fridges and freezers in school and at home. Many have the temperatures displayed on them.

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The temperature on the surface of most planets varies hugely from very hot (several hundred degrees Celsius) to very, very cold (large negative temperature values). Pupils can explore this on the internet if they are interested.

21 °C

37 °C

100 °C

Invite pupil pairs to read and identify these temperatures and then choose pupils to share their answers.

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Pupils should complete Question 5 in the Practice Book.

Same-day intervention ●

Display the following table and ask pupil pairs to copy and complete it, circling the correct answers.

Temperature of …

Pupils will have consolidated their understanding that on the Celsius scale, temperatures below zero (the freezing point of water) are represented by negative numbers.

Lead a discussion about temperatures in everyday life.

Remind pupils that the temperature at which water freezes is 0 °C and that it boils at 100 °C. These are temperatures that pupils should commit to memory.

3–5 °C

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Activities for whole-class instruction ●

0 °C

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1. … boiling water

100 °C −100 °C /+100 °C

2. … the human body

73 °C

37 °C/ +37 °C

3. … a fridge

3 °C or 4 °C

0 °C

4. … the classroom

21 °C or +21 °C

40 °C

5. … water freezing

0 °C

10 °C

6. … a freezer

−18 °C

18 °C/ +18 °C

Ask pupils to discuss their answers with another pair.

Answers: 1. 100 °C/ +100 °C; 2. 37 °C/+ 37 °C; 3. 3 °C or 4 °C; 4. 21 °C/+21 °C; 5. 0 °C; 6. −18 °C.

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Chapter 7 Introduction to positive and negative numbers

Activities for whole-class instruction

Same-day enrichment ●

Give pupil pairs Resource 5.7.1d Identifying common temperatures.

Unit 7.1 Practice Book 5B, pages 29–31

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Ask pupils if they know how much they weighed when they were born – or if they know the birth weight of a younger brother or sister. Tell them that the weight of babies born at full term is usually between 2.7 kg and 4.1 kg. So, a newborn baby typically weighs about 3.5 kg.

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Display the weights of eight babies born in a hospital during one week: 3.9 kg 4.0 kg

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Answers: 1. 100 °C and +100 °C; 2. 37 °C; 3. 3 °C and 4 °C; 4. 21 °C and +21 °C; 5. 0 °C; 6. −18 °C; 7. +75 °C; 8. −196 °C

Question 6

3.0 kg 3.6 kg

3.5 kg 3.7kg

Ask pupils to arrange them in increasing order: 2.8 kg 3.6 kg

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3.2 kg 2.8 kg 3.0 kg 3.7 kg

3.2 kg 3.9 kg

3.5 kg 4.0 kg

Remind them that 3.5 kg is the typical weight. Ask: –

Were any of the babies’ birthweights exactly 3.5 kg?

–

How many babies were below the typical birthweight?

–

How many babies were above the typical birthweight?

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Explain that sometimes, mathematicians use a reference point, called a benchmark, against which the other values can be compared. Values below the benchmark are negative, and values above the benchmark are positive.

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Work together to complete the differences between the 3.5 kg benchmark and the other values.

2.8 kg

−0.7

3.0 kg

−0.5

3.2 kg

−0.3

3.5 kg 0 Benchmark

What learning will pupils have achieved at the conclusion of Question 6? ●

Pupils will have practised reading positive and negative numbers.

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The concept that positive and negative numbers are opposites will have been explored.

●

The concept of a benchmark (reference point) where numbers greater than the benchmark are positive and numbers less than the benchmark are negative will have been introduced.

●

Pupils will have explored how a negative number and its corresponding positive number are ‘opposite’ each other across the benchmark (reference mark); they are both the same distance from the benchmark but in opposite directions.

3.6 kg

+0.1

3.7 kg

+0.2

3.9 kg

+0.4

4.0 kg

+0.5

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Ask: Which two weights are exactly the same distance from the 3.5 kg benchmark? Agree that the babies weighing 4.0 kg and 3.0 kg are both the same distance, 0.5 kg, from the benchmark but in opposite directions, one positive, +0.5, and one negative −0.5.

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The smallest baby at 2.8 kg was −0.7 from the benchmark.

Ask: What would the weight be of the baby the same distance from the benchmark in a positive direction? Agree it would be 4.2 kg.

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Ask individual pupils to tell you: –

the weights of the two babies 0.6 kg either side of the benchmark

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Chapter 7 Introduction to positive and negative numbers

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the difference between the benchmark for some other weights, for example 2.9 kg, 4.1 kg, 3.3 kg, 3.8 kg

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the weight of a baby whose benchmark value is +0.3 (−0.1, −0.8 and so on).

Pupils should complete Question 6 in the Practice Book.

Unit 7.1 Practice Book 5B, pages 29–31

Challenge and extension questions

Question 7

Same-day intervention ●

Give pupil pairs Resource 5.7.1e Test results 1.

In this question, pupils explore an income and expenditure problem, where income is a positive amount and expenses are negative numbers, which are subtracted.

Question 8

Answers: Table: +2; −5; +3; −1; +10; −2; +6; 0; −6; 1.

Harry; 2. 4; 3. 4; 4. Negative five; 5. Ella (+10)

The problem in this question considers the elevation of places above and below sea level. Pupils could be invited to draw a sketch to clarify their answer.

Same-day enrichment ●

Give pupil pairs Resource 5.7.1f Test results 2.

Answers: Table: +2; −5; +3; 79; 90; −2; 86; 0; 74; 75; 100; 1. Harry; 2. 5; 3. 5; 4. Negative five; 5. Isla (−6); 6.

Jamie and Ben 7. Kaia; 8. Kaia

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