Lesson 29 Straight line graphs

Episode 2

Reasoning Resources: Worksheet 2 for pupils, Worksheet 2 on A2/A3 or OHT
Whole class preparation
The gradient and intercept notions of linear relations are explored with or without actual algebraic notations. It is left to the teacher to introduce these to classes who will benefit from them. Display the coordinate grid from Worksheet 1, Plot two points and join them with a line. How do the ‘across’ and ‘up’ values change as you move along the line? Repeat for other pairs of points.

Show how any two points on the grid lie on a line in which the ‘across’ and ‘up’ values change regularly as you move along the line.
Pair work
Refer the class to TM23: Rates of change where the step and rise are used. Ask pairs of pupils to work on Worksheet 2. Guide pupils’ struggles to describe lines using both the axes. They can imagine a slanting line as a horizontal line rotating around the point where it crosses the y-axis. Therefore they need two values: a value for the crossing point at the y-axis, and a value for the slope. Acknowledge and emphasise words such as ‘slope; ‘incline, ‘slant; ‘steepness’ or similar as everyday language equivalents for ‘gradient! Similarly, accept ‘crossover, ‘crossing points’ ‘cutting; ‘starting from’ and such phrases for ‘intercept’.
Whole class sharing and discussion
The format for the equation of a straight line e.g. y=2x+1 is just a convention. This could equally be y= 1 + 2x or L =2K+1

In fact, it seems nearer to pupils’ experience at this stage to start the expression for the line with the y-intercept value followed by the slope then the x value,
since the crossover point is a more concrete concept than a slope of countless lines.
Pupils should see that the slope itself is a relation of two values: the amount a point on the line goes up or down for each step it goes across. Use the term ‘gradient’ relating to gradients in steep roads they may have noticed such as 1:8, and that it can be positive or negative.

Depending on the class, different issues and misconceptions may crop up during the work. You will have to decide which are the most fruitful discussions to pursue and which might be interesting but inappropriate diversions.