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Lesson 7: �Understanding straight line graphs

Objectives

Understand how linear relationships are represented by straight line graphs
Understand gradient as steepness and rate
Interpret the y-intercept as a constant
Use graphs to identify information about a relationship
Make connections between equations of a straight line and their graphical representations

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Bike and scooter hire

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Pedal bike

Debbie

E-bike

E-scooter

Edwin

What are some different ways to charge for a hire?

Tell students that Debbie and Edwin are starting up a bike and scooter hire business in the city where they live. They have pedal bikes, electric bikes (e-bikes) and electric scooters (e-scooters) for hire. They borrowed some money to buy the bikes and scooters. They want to be able to pay off the loan and start making some money as soon as possible.

Debbie and Edwin need to decide on a pricing model. Ask students to suggest some different ways that Debbie and Edwin could charge for a hire. You may like to discuss things such as an hourly rate, or charging to hire for the day. Encourage all students to contribute to the class discussion, providing an opportunity for students to explain their thinking to each other and promoting a collaborative culture (Key Principle 5).

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Hire charges

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Debbie

Edwin

So, the hire cost is 15 times the number of minutes … I’ll write an equation.

Let’s charge 15p per minute.

Pedal bike

h = 15t

Debbie and Edwin decide to charge by the minute, as city hires can often be quite short. They discuss how much to charge per minute for hiring a pedal bike. They want to make sure they are charging a price that attracts customers and isn’t too expensive, but that also means that they will start making a profit as soon as possible.

Debbie suggests they charge 15p per minute.
Edwin writes the equation h = 15t

Ask students what h and t represent in Edwin’s equation and establish that h represents the hire charge amount per minute in pence. Now ask them to write on their mini whiteboards what the charges would be for a 2-minute hire and a 10-minute hire, for example, to check their understanding of substitution. Most students should be able to do this, and the activity provides an opportunity for you to value, and build on, what they already know (Key Principle 2).

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Hire charges

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Pedal bike

h = 15 × 20

h = 300

What will the charge be for a 20-minute hire?

Debbie

Edwin

h = 15t

We need to get at least £3.50 for a 20-minute hire.

Let’s charge 15p per minute.

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Pedal bike hire

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Debbie

Let’s draw�a graph.

Edwin

We could charge a fee to unlock the bike and then charge 15p per minute.

h = 15 + (15 × 20) ✗

h = 20 + (15 × 20) ✗

h = 25 + (15 × 20) ✗

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Constructing the graph

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DISCUSS

Pedal bike hire

Hire charge in pence (h)

Length of hire in minutes (t)

(0,0)

(20,300)

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Plotting the line

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DISCUSS

Hire charge in pence (h)

Length of hire in minutes (t)

(0,0)

(20,300)

Edwin

Will it be a straight line?

Debbie

Yes. The charge increases 15p every minute.

Pedal bike hire

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Adding vertical and horizontal lines

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DISCUSS

Pedal bike hire

Hire charge in pence (h)

Length of hire in minutes (t)

(0,0)

(20,300)

What are the vertical and horizontal lines for?

t = 20

h = 350

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£3.50 for a 20-minute hire

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DISCUSS

Pedal bike hire

Hire charge in pence (h)

Length of hire in minutes (t)

(0,0)

(20,300)

How could they reach the target?

t = 20

h = 350

Target

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Unlocking charge

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Use the ‘Pedal bike hire’ app: https://www.geogebra.org/m/adyxa7fq

DISCUSS

Pedal bike hire

Hire charge in pence (h)

Length of hire in minutes (t)

(0,0)

(20,300)

t = 20

h = 350

What should the unlocking charge be?

Use the ‘Pedal bike app’, which can be found at https://www.geogebra.org/m/adyxa7fq to explore changes in the graph.

Ask students to predict what will happen to the graph if a fee is charged to unlock the bike. Keep the ‘Charge’ slider at 15 and increase the ‘Unlock’ slider to a number less than 50. Discuss what happens to the graph. Check that students recognise that the line is parallel to the original line and that it crosses the y-axis at the same value as the unlocking fee amount. Establish that the two straight line graphs have the same steepness/gradient/are parallel because the price per minute (15p) has stayed the same and is identifiable from the change in cost, h, (‘change in y’) divided by the change in length of hire, t, (‘change in x’).

Ask students what the unlock fee needs to be to ensure that Debbie and Edwin are charging £3.50 for a 20-minute hire (50p).

Return the ‘Unlock’ slider to 0. Ask students to predict what Debbie and Edwin would need to increase the per minute charge to if they did not include a fee to unlock the bike. Establish that 17p per minute is just below £3.50 and 18p per minute is just above. Ask students how they could work out the per minute charge exactly (350 ÷ 20 = 17.5). Establish that it is not possible to charge 17.5p per minute so the charge would have to be 18p per minute, resulting in a £3.60 charge for a 20-minute hire.

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Equation of the line

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Edwin

h = 15t

means 15p per minute.

Write an equation for�‘50p to unlock &�15p per minute’

Hire charge in pence (h)

Length of hire in minutes (t)

Pedal bike hire

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E-scooter hire

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Debbie

Edwin

We need at least £4�for a 20-minute hire.

What is the unlocking fee?

How can we find the charge per minute?

DISCUSS

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

Explain to students that Debbie and Edwin need to agree on the prices for hiring an e-scooter and an e-bike. They first decide between two different pricing options for hiring an e-scooter. They want to make sure that they are not charging too much, as this could result in them not getting many hires if people think the charges are too expensive. They are aiming to achieve at least £4 for an average length (20 minute) e-scooter hire.

Tell students that the two options have been represented on a graph.

Ask students to look at the scale on the y-axis and check that they can identify the unlocking fee for the two options from the graph (25p).

Now direct their attention to the x-axis and discuss how they could identify the charge per minute (for example, reading off the hire charge for a 5-minute hire or a 10-minute hire). If the charge for a 5- or 10-minute hire and the unlocking fee are known, subtracting the fee and dividing by the length of hire gives the charge per minute.

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Missing equations/graph

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YOUR TURN

Handout�available

Complete the missing descriptions, equations and graph.
Which options are best?

We need £4 for�20 minutes.

We need £4.50�for 20 minutes.

Give copies of the ‘Pricing’ handout to each pair of students. You may want to cut and give the two sections out separately or ask students to fold the handout so that they can focus on the e-scooter hire first, before asking them to work on the e-bike hire.

Remind students that Debbie and Edwin need at least £4 for an average length e-scooter hire (20 minutes). Tell students that Debbie and Edwin would like at least £4.50 for an average length e-bike hire (20 minutes).

Ask students to complete the missing descriptions, equations and graph on the handout and identify which option Debbie and Edwin should choose for each type of hire. Tell students that they should use a sharp pencil when completing the graph for the e-bike hire.

Once students have had sufficient time to complete the task, hold a class discussion.

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E-scooter hire charges

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OPTION 1 �25p to unlock &

10p per minute

OPTION 2

25p to unlock &

20p per minute

We need £4�for 20 minutes.

Hire charge in pence (h)

Length of hire in mins (t)

E-scooter hire

REVIEW

Option 1

Option 2

OPTION 1: h = 25 + 10t

OPTION 2: h = 25 + 20t

Use Slides 14 – 17 to support a class discussion.

Check that students have correctly identified Option 1 as ‘25p to unlock and 10p per minute’ and have labelled the graphical representations of the two options correctly.

Check what students have written as an equation for Option 2 and ask them to explain how they identified it. If students have written h = 20 + 25t rather than h = 25 + 20t, ask them to use the description for Option 2 to calculate the charge for a 10-minute hire and compare this with substituting t = 10 into equation h = 20 + 25t. Check that students understand and can explain why the missing equation is h = 25 + 20t and check that students understand why this is the case.

Ask students to comment on what‘s the same and what’s different about the two straight line graphs and how this relates to the two equations. Highlight that both graphs start at (0, 25), as the fee to unlock the scooter is 25p for both options. The charge per minute for Option 2 (20p) is more expensive than the charge per minute for Option 1 (10p), so the graphical representation is steeper for Option 2 (gradient = 20) than for Option 1 (gradient = 10). Confirm that charging ‘25p to unlock and 20p per minute’ (Option 2) will ensure that an average e-scooter hire will bring in the desired amount of money. Ask students to use the graph to determine what the charge for a 20-minute hire would be for Option 1 (£2.25).

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E-bike hire charges

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REVIEW

Hire charge in pence (h)

Length of hire in mins (t)

E-bike hire

Option 3

OPTION 3: h = 100 + 10t

OPTION 4: h = 50 + 20t

Option 4

OPTION 3 �£1 to unlock &

10p per minute

OPTION 4

50p to unlock &

20p per minute

We need £4.50�for 20 minutes.

Check that students have correctly identified that Option 3 has already been represented and have drawn the graph for Option 4 correctly. Students may have struggled when drawing the graph so spend time asking a couple of students to explain how they identified where the graph crosses the y-axis (0, 50) and its gradient (20).

Discuss the features of the two e-bike graphs, focusing on the differences in gradient and y-intercept. Ask students to interpret what the point of intersection of the two graphs means in terms of e-bike hire (the cost of a 5-minute hire would be the same for both of the options).

Check that students can identify the 100p in the equation and on the graph for Option 3 as a £1 unlocking fee (they may have described it as ‘100p to unlock’). Confirm that charging ‘50p to unlock and 20p per minute’ (Option 4) will ensure that an average length e-bike hire will bring in the desired amount of money. Ask students to explain how they could determine what the length of hire would need to be for Option 3 for the charge to be £4.50 (35 minutes). Check whether students can explain how to do this using both the equation (450 = 100 + 10t) and the graph.

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Graphical representations

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REVIEW

A:

B:

C:

1:

2:

3:

There is no unlocking charge

The unlocking charges �are the same

The charge per minute �is the same

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Features of the straight line graphs

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REVIEW

If there is no unlocking charge, �the straight line graph starts at the origin

When the unlocking charges are the same, �the straight line graphs have �the same starting position

When the charge per minute is the same, �the straight line graphs are parallel

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Placing the cards

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YOUR TURN

Work in pairs to match the cards to the empty cells.
Take turns giving a reason for your choice.
Complete the blank graphs and any missing ‘hire charges’ and ‘equations’.

Handout�available

Give each pair of students a copy of the ‘Hire charges’ handout, the ‘Cards’ and some scissors (if the cards have not already been cut up).

Explain to students that Debbie and Edwin collect information about two other city hire companies (Company A and Company B) so that they can see how competitive their proposed charges are. The charges are described in writing, algebraically and graphically, but some of the information is missing. Ask students to work in pairs to place the cards in the correct empty cells on the handout. They also need to fill in any empty cells and draw three graphs. You might want to work through one of the graphs they need to draw with them, because although they have already drawn a graph in Explore 1, some of them may still need help.

Encourage students to take turns, explaining their thinking to their partner. Explain that they need to complete the three blank graphs on the handout, making sure they use a sharp pencil so that they can draw accurately.

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Company A: Pedal bike/E-scooter

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REVIEW

10p per minute

20p per minute

h = 10t

E3 h = 20t

Why do the graphs start at

the origin?

G2

Pedal bike hire

Hire charge in pence (h)

Length of hire in mins (t)

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

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Writing equations

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REVIEW

h = 20t +10

D2 10p to unlock &

20p per minute

How is

h = 20t +10 different to

h = 50 + 20t?

OPTION 4

50p to unlock &

20p per minute

h = 50 + 20t

E-bike hire

Hire charge in pence (h)

Length of hire in mins (t)

For the e-bike, the correct card is D2. Students need to draw the graph. Check that their straight line graph starts at the point (0, 10) and has a gradient of 20. Ask students to explain how they drew their graphs.

Remind students of the second option (Option 4) that Debbie and Edwin were considering charging for an e-bike hire earlier in the lesson. Ask them what they notice about the equation for this (h = 50 + 20t) compared with h = 20t + 10. In h = 50 + 20t the unlocking fee (the constant term) has been written as the first part of the equation, whereas for h = 20t + 10 the equation has been given in the more conventional ‘y = mx + c’ form and so the variable term comes first.

You may not need to hold this discussion, but it is important that students understand the structure of the equation of a straight line so that they are able to identify the charge per minute (gradient) and unlocking fee (y-intercept) when it is written in different ways.

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Company B: Pedal bike/E-scooter

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REVIEW

h = 10t + 10

D1 10p to unlock &

10p per minute

E2 h = 10t + 20

20p to unlock &

10p per minute

Pedal bike hire

Hire charge in pence (h)

Length of hire in mins (t)

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

For Company B, for the pedal bike, students need to draw the graph using the information given in the equation and choose Card D1 for the hire charge.

For the e-scooter, the unlocking fee and rate need to be read off the graph, giving the equation h = 10t + 20 (card E2).

After checking that students understood what to do and could do it, you may want to discuss how these graphs are the same and how they are different, pointing out that the gradients (rate per minute) are the same and the y-intercepts (unlocking fee) are different. Highlight that if two graphs have the same gradient, the change in y will be the same for a given change in x, irrespective of the y-intercepts of the two graphs. The graph for the e-scooter can be described as the straight line graph for the pedal bike hire translated 10 units in the positive y-direction.

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Company B: E-bike

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REVIEW

E1 h = 20t – 100

5 minutes free then

20p per minute

G1

E-bike hire

Hire charge in pence (h)

Length of hire in mins (t)

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Comparing to Companies A and B

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REVIEW

h = 10t

h = 20t

h = 20t + 10

h = 10t + 10

h = 10t + 20

h = 20t – 100

h = 15t + 50

h = 20t + 25

h = 20t + 50

Company A

Company B

Debbie & Edwin

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h = 20t

h = 10t + 20

Constant term and the y-intercept

DISCUSS

No�constant�term (it is 0)

Goes�through�(0,0)

Constant�term = 20

Intersects with the y-axis at�(0,20)

Company A

Company B

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

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25

h = 20t

h = 10t + 20

Variable term and the gradient

DISCUSS

Variable term t,

coefficient 20

Gradient = 20

For every 5 across, �it is 100 up

Variable term t,

coefficient 10

Gradient = 10

For every 5 across, �it is 50 up

5

100

50

5

Company A

Company B

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

E-scooter hire

Hire charge in pence (h)

Length of hire in mins (t)

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Practice question (1)

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REVIEW

Handout�available

The line L is shown on the grid.

Find an equation for L.

y = 3x – 6 (3)

Q22 from June 2018, 1MA1/2F

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Practice question (2)

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REVIEW

Handout�available

Tom uses his lorry to deliver bricks.

You can use this graph to find the delivery cost for different distances.

For each delivery, there is a fixed charge plus a charge for the distance.

How much is the fixed charge?

£10 (1)

Tom makes two deliveries of bricks.�The distance of one delivery is 20 miles more than the distance of the other delivery.

(b) Work out the difference between the two delivery costs.

£30 (2)

Q12 from November 2018, 1MA1/1F

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Lesson review: �Understanding straight line graphs

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Suggested further steps/areas to work on

Interpret graphs of quadratic functions

Objectives

Understand how linear relationships are represented by straight line graphs
Understand gradient as steepness and rate
Interpret the y-intercept as a constant
Use graphs to identify information about a relationship
Make connections between equations of a straight line and their graphical representations

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Lesson 7: �Credits

Photo acknowledgements

Shutterstock.com: Keport, MVelishchuk, Perla Berant Wilder, Vectorfair.com

Text acknowledgements�2022 GeoGebra

Pearson Education Ltd: Pearson Edexcel GCSE (9-1) In Mathematics (1MA1) Foundation (Calculator) Paper 2F

Pearson Education Ltd: Pearson Edexcel GCSE (9-1) In Mathematics (1MA1) Foundation (Non-Calculator) Paper 1F

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