Lesson 1 Algebra: Roofs – isometric trapezia

# Episode 3 Exploration with proof (20-25 mins)

Reasoning Resources Isometric dotty grid on board, projector or visualiser, isometric dotty paper
1 What does fixing only one variable do?
Ask: Suppose we only know that the base of the roof-shape is 4. What possible values can the other sides have? Allow some time for pupils to generate ideas, then give them other values for the 4th side, e.g. 6, and 8 and 3.

Ask them to generalise in words how to find the number of possibilities when the 4th side is 20 or 100, and come up with an expression for the number of possibilities for any value.
Some pupils will come up with the idea that, 'if the base is n there are n - 1 possibilities' and with the reasoning that you have to cut off the top'.
2 Which two variables can be independent?
Enumeration, combinations, recognising exceptions. Remind the class that they were free to use any numbers for the 1st and 2nd sides, but that the 3rd and 4th were dependent on them.
Ask: Is it possible to make different sides/numbers independent? Give a few minutes to pairs and small groups to produce ideas, including on how many different pairs of independent variables we can have for roof-shapes.

Some pupils would recognise that any pair is possible except the two equal sides.
Symmetry. 3 Proof:
Recognising mathematical properties of the grid. Ask: Why are these rules necessary for making a roof-shape? Why do they always work, and why are they sufficient? Give several minutes for this work. The equality of the 1st and 3rd sides are easiest to explain, including by 'you must go down as much as you went up', or by recognising symmetry. But the fact that the 4th must be the sum of the 1st and 2nd (or 2nd and 3rd) is challenging.
Multi-step reasoning proof You could keep up the challenge when pupils vaguely recognise it to be related to the nature of the grid, (Teachers resort to trigonometry to solve this!) Typical pupils' ideas revolve around the fact that 'when you go up and down you go across half as much too. Some may go as far as representing it on the diagram as two right-angled triangles, or splitting the trapezium into a parallelogram and an equilateral triangle. At this stage the class may be using one specific roof-shape as a generic example, occasionally checking that what they are doing applies to all others 