Lesson 10 Rectangle functions

# Episode 2

Reasoning Resources: Worksheet 2, 1/2 cm squared paper, rulers
Whole class preparation
Halving the perimeter allows several pairs of values to produce it.

Plotting some points where perimeter = 12 against values for length and width.
Switching pupils’ attention now to the perimeter and refreshing their minds on ways of finding it, ask: How many rectangles with perimeter 12 could we draw?

Using prior knowledge of how the perimeter relates to the two dimensions of the rectangles (half perimeter = length + width), the pupils would generate a few pairs of values. You can then ask: What sort of picture would we get of the relationship between width and length for all the rectangles that have perimeter 12?

After some initial ideas, give out Worksheet 2 for paired and small group work. Give guidance to any who find difficulty in working out how to get a perimeter of 12. Demonstrate how to plot (width, length) pairs on a grid, with width on the horizontal axis and height on the vertical axis.
Pair and small group work
Pupils use Worksheet 2 in pairs to develop the data and graphical representation for rectangles with perimeters of 12 and 24 units. Having drawn the graphs, the aim is to encourage answers to the questions on the sheet. Each pair should work with another pair to discuss and refine their ideas ready for the whole class discussion.
Whole class sharing and discussion
Limiting case is not continuous with rest of graph and can be shown with the convention of a circle around the point.
Continuity.
Intuitive placement of a graph as a whole in relation to other graphs.

Looking for similarities and differences in situations is aimed at developing this approach in pupils.
The algebraic skills needed to symbolise the two functions (e.g. l = 12 - w and 1 = 12/w) with the domain O < W < 12, then accept the linearity and non-linearity, is above the middle school maths level.
Review the actual width and length values for the perimeter of 12 units, looking at fractional values where the length is increasing from 5 to 6. The height decreases to compensate for increase in length to keep the perimeter =12. Develop this to consider the limiting case where the height approaches zero and length = 6, Pupils would appreciate that the rectangle remains a rectangle however much the length approaches 6, providing it is less than 6. But it will no longer be a rectangle when the length is actually 6. So the straight line graph does not actually reach the axis.

You could also discuss the continuous possible values between any two values linked to similarity with the graph of the area. Pupils would intuitively place the line for Perimeter = 20 between the lines for P = 12 and P = 24. They can draw it accurately using two values, checking that other points fulfil the function.
Whole class reflection
Move on to comparing the two functions for area and perimeter by looking at their graphs. The aim is to allow pupils to describe in their own words the commonalities and differences between the functions. The teacher's role is to help by questioning, indirect hints and occasional use of formal mathematical phrases. Pupils would intuitively appreciate, but may not be able to put in full sentences, why the graph of the perimeter is a straight line, while that of the area is curved.
End of Lesson Reflection