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Lesson 1: �Multiplicative reasoning

Objectives

Understand the multiplicative relationship between two quantities (non-calculator)
Solve multi-step currency or unit conversions problems (calculator)
Understand how to use representations to provide insight into solving problems

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Staying in a hotel

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How much should the hotel owner charge for:

5 nights? 11 nights?

A hotel owner charges the same price per night�regardless of the length of stay.

You can stay any number of nights.

For a 2-night stay, the charge is £50.

Introduce the context of the owner of a hotel charging the same price per night, regardless of how long someone wants to stay (this is an important assumption in this task) and tell students that the charge for a 2-night stay is £50.

Ask half the students to find the cost of a 5-night stay, and ask the other half of the class to find the cost of an 11-night stay. Give students a few minutes on their own to think about how to tackle the problem, before pairing students that have been working on a different number of nights.

Ask pairs of students to explain their approach to each other and agree on an approach for finding the two lengths of stay. They should write their method for finding a solution on their mini whiteboards.

As students work through the problem, identify any students who use one of the four approaches described on Slide 4. These students can be called on when these methods are covered in Discuss 1.

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Class methods for 11-night stay

3

REVIEW

0

2

11

No. of nights stay

0

50

Cost (£)

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Different approaches

4

Ava’s approach

2 nights ➝ £50

1 night ➝ £25

£125

5 nights ➝ £125

1 night ➝ £25

£275

Bahir’s approach

2 nights ➝ £50

1 night ➝ £25

£25 × 11 = £275

Catia’s approach

£50 for 2 nights ➝ £25 per night

11 nights × £25 per night = £275

Danny’s approach

11 nights is 5.5 as many nights as 2 nights

so £50 × 5.5 = £275

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Ava’s approach

5

2 nights ➝ £50

1 night ➝ £25

£125

5 nights ➝ £125

1 night ➝ £25

£275

0

2

11

No. of nights stay

0

50

Cost (£)

0

2

11

No. of nights stay

0

50

Cost (£)

0

2

11

No. of nights stay

0

50

Cost (£)

4

100

0

2

11

No. of nights stay

0

50

Cost (£)

4

5

100

125

0

2

11

No. of nights stay

0

50

Cost (£)

4

5

10

100

125

250

0

2

11

No. of nights stay

0

50

Cost (£)

4

5

10

100

125

250

275

+2

+1

+5

+1

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Bahir’s approach

6

2 nights ➝ £50

1 night ➝ £25

£25 × 11 = £275

0

2

11

No. of nights stay

0

50

Cost (£)

275

×11

25

1

÷2

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Catia’s approach

7

£50 for 2 nights ➝ £25 per night

11 nights × £25 per night = £275

0

2

11

No. of nights stay

0

50

Cost (£)

0

2

11

No. of nights stay

0

50

Cost (£)

×25

0

2

11

No. of nights stay

0

50

Cost (£)

×25

275

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Danny’s approach

8

11 nights is 5.5 as many nights as 2 nights

so £50 × 5.5 = £275

0

2

11

No. of nights stay

0

50

Cost (£)

0

2

11

No. of nights stay

0

50

Cost (£)

× 5.5

0

2

11

No. of nights stay

0

50

Cost (£)

×5.5

275

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Approaches

9

REVIEW

Ava’s approach

2 nights ➝ £50

1 night ➝ £25

£125

5 nights ➝ £125

1 night ➝ £25

£275

Bahir’s approach

2 nights ➝ £50

1 night ➝ £25

£25 × 11 = £275

Catia’s approach

£50 for 2 nights ➝ £25 per night

11 nights × £25 per night = £275

Danny’s approach

11 nights is 5.5 as many nights as 2 nights

so £50 × 5.5 = £275

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Different approaches compared

10

Ava’s approach

Bahir’s approach

Catia’s approach

0

2

11

No. of nights stay

0

50

Cost (£)

×25

275

Danny’s approach

0

2

11

No. of nights stay

0

50

Cost (£)

×5.5

275

Which approach shows the rate per night?

0

2

11

No. of nights stay

0

50

Cost (£)

4

5

10

100

125

250

275

+2

+1

+5

+1

+2

+1

+5

+1

0

2

11

No. of nights stay

0

50

Cost (£)

25

1

275

×11

÷2

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Direct proportion

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KEY IDEA

If two quantities are in direct proportion:

as one value gets bigger, so does the other (by the same multiplier)
as one value gets smaller, so does the other (by the same divisor)
if one value is 0, then the other value is 0.

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Cost of stay

12

YOUR

TURN

Handout�available

Work through a question on your own.
Take turns to explain your method to your partner.
If your partner used the same method they must describe their thinking in their own words.
If they used a different approach they must explain their thinking.

Ask students to work in pairs and give each pair a copy of the ‘Cost of stay’ handout. It is important that students answer the questions in order because they have been carefully designed to guide them to complete the table in a way that explores different relationships in a sequential way. Ask them not to be tempted just to fill in the table.

Make sure that students explain their approach to their partner, working together to agree a solution and develop their own understanding. If you notice that one student is answering all the questions, or they are not working collaboratively, ask a student to explain a question that has been answered by their partner.

Observe students as they complete this task but do not intervene unless necessary. As students work on the task, notice how they illustrate their strategy. As you observe them, identify those with different approaches whom you can call on to explain in the next part of the lesson.

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Ways of working

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REVIEW

0

2

11

No. of nights stay

0

60

Cost in Euros

5

0

2

11

No. of nights stay

0

2400

Cost in South African Rand

5

0

60

Cost in Euros

0

2400

Cost in South African Rand

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Exchange rates

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The exchange rate �describes the relationship �between two currencies.

1 Euro

is worth

16 South African Rand

The relationship between two currencies is �directly proportional.

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Currency conversions

15

YOUR

TURN

5 Euros are worth 16 East Caribbean Dollars.

What are 60 Euros worth in East Caribbean Dollars?

What are 1344 East Caribbean Dollars worth in Euros?

60

0

5

Cost in Euros

0

1344

Cost in East Caribbean Dollars

16

How many nights could you stay at the hotel for�if you paid 1344 East Caribbean Dollars?

Depending on the class, you may want students to continue working in pairs or to work on these questions individually. Either way, they should answer these questions on their mini whiteboards. Once the students have had an opportunity to answer the question, use the diagram to explore their thinking. It is likely that there will be a variety of approaches to the task and it is important not to promote a particular method with students.

If students are struggling with the third question, you may need to remind them to look back at how much per night in Euros the hotel cost. They should then be able to see that 60 Euros is the cost of a 2-night stay, so the equivalent in East Caribbean Dollars (192) gives the cost of a 2-night stay in East Caribbean Dollars.

Answers:

60 Euros are worth 192 East Caribbean Dollars.

1344 East Caribbean Dollars are worth 420 Euros.

1344 East Caribbean Dollars is the amount you would pay for a 14-night stay at the hotel.

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Cost of stay in East Caribbean Dollars

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How could you find the cost in East Caribbean Dollars of

a 5-night stay
an 11-night stay?

	Number of nights stay
	2	5	11
Cost of stay in Euros	60	150	330
Cost of stay in �South African Rand	960	2400	5280
Cost of stay in �East Caribbean Dollars	192

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Getting to the hotel

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Two friends Eli and Fi are travelling by train from their homes to the hotel.

Eli’s journey is 60 km and takes 1.5 hours on the train.

Fi’s train journey takes 4 hours.

How far does Fi travel on the train?

4

0

1.5

Time (hours)

0

Distance (km)

60

Note that it has been assumed that both trains travel at the same average speed.

Depending on the class, you may want students to continue working in pairs or to work on these questions individually. Either way, they should answer these questions on their mini whiteboards. Give students time to work out the distance travelled, and take note of the different approaches that students use. Students may draw their own double number line to illustrate their thinking.

Once most students have answered the problem, discuss the different approaches that students used, using the double number line to illustrate their thinking.

Answer:

Fi travels 160 km on the train.

Deepening understanding Ask students what the rate of 40 kilometres per hour represents. Can they explain why the multiplier gives the speed of the train?

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Multipliers

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Number of nights stay Cost in Pounds�Number of nights stay Cost in Euros�Number of nights stay Cost in Rand

Euros Rand

Euros East Caribbean Dollars

Time Distance

The multiplier is called the constant of proportionality.

×25

×30

×480

×16

×3.2

×40

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Practice question

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REVIEW

Worksheet�available

2.5 kg of apples cost £3.60.

Work out the cost of 3.5 kg of apples.

£…………………………….

(2)

Question 12 from Nov 2017, 1MA1/3F

£5.04

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Lesson review: �Multiplicative reasoning

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Objectives

Understand the multiplicative relationship between two quantities (non-calculator)
Solve multi-step currency or unit conversions problems (calculator)
Understand how to use representations to provide insight into solving problems

Suggested further steps/areas to work on

Interpret and use a conversion graph

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Lesson 1: �Credits

Text acknowledgements

Pearson Education Ltd: Pearson Edexcel �GCSE (9-1) In Mathematics (1MA1) Foundation (Calculator) Paper 3F, November 2017

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