Lesson 7 Which offer shall I take

# Episode 3

Reasoning Resources: Worksheet 1 on board or OHT, and for pupils
Pair/group work on drawing graphs

The variables only have whole-number values, so bar charts will be valid representations. However we can join the points up with a dashed line which will indicate the linear pattern of the discrete data, as distinct from a continuous linear function. The fact the intermediate points are meaningless need not be emphasised. Ask pupils what they notice about the values, so they realise that the values will ‘cross each other‘ This is the appropriate time to ‘see’ the club offers in a different way, as graphs, which may help them to make greater sense of things.

Explain how to draw the graphs (using different colours) on the same axes, and allow pupils time to do this. They can join the dots to see what happens and should label each line with its equation. As they draw their graphs, draw your own ready to show on the OHP or the board.
Whole class sharing/discussion

Step and rise visual comparisons are translated to numbers for the gradient (slope/slant).

Intercept is experienced as start-up/beginning value.
This episode centres around how the graphs help us to compare the two Club offers by seeing the gradient and intercept in the formula in a meaningful way and using natural language. Key questions are: What do you see/notice? What is similar/different?

The slope can be represented visually by the amount of ‘rise’ for each ‘step’ The step can be made as one month, and the rise then can be read numerically as either 2 or 3. Ask pupils to draw the step and rise at different months. For the ‘intercept’ pupils could use ‘starting value’ or the ‘value at zero!

Typical comparisons go along the lines of:

• ‘That one starts lower but ends higher’
• ‘You get more in the end with the second club’
• ‘That's where it starts from!’
• ‘That one goes up in threes.

Allow time for pupils to match features of the graphs to terms in the formula and to natural language.
• Reflection
Comparing representations

Linking to other similar situations.
• Ask for ideas on which description of the pattern for each club is best, accepting that for ordinary conversation full sentences are adequate but that for comparisons you need tables or graphs. Discuss the role of algebra and how it can be translated into meaning.
• Ask which other situations produce similar patterns, prodding for recognition that services like telephone, electricity and so on have a standing charge plus prices for units used, or have a discounted start plus a price per unit. Pose the challenge of how to compare such situations. Some pupils will recognise the need for two lines to compare the patterns without necessarily being able to show them correctly.
• Extension: imagining and adding new clubs to the graphs

A form of peer assessment, which ends the lesson with all of the pupils engaged.
Tell the class in groups to invent a new club just like the two they have been looking at today. They have to draw the line on a new sheet of squared paper and pass it to another group.Their job is to write the formula for the line, add the values to the table and describe the pattern in a sentence.
End of lesson Reflection Why are equations a useful way to describe relationships? How do graphs help us to see things differently?

## License

Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.