Lesson 1 Algebra: Roofs – isometric trapezia

# 1 Algebra

Roofs - isometric trapezia
Overview Resources
Pupils explore the mathematics of four-sided shapes drawn on the isometric grid.They relate the area and perimeter of shapes to the lengths of their sides.

Challenges range from recognition of one-step and multi-step relationships to algebraic symbolisation and proof. The greatest benefit for pupils at each step is in generating ideas, verbalising and sifting through them and testing patterns. Advanced mathematical ideas are treated implicitly and lightly, and the mathematical terms are only used in passing.

Optional tasks illustrate counter-examples, enumerate combinations within rules, and extend geometric reasoning to several steps.
Random dots on board or projector
Isometric dotty grid on board or projector
Isometric dotty paper
Worksheet 1
• Focus on the features of the triangular/isometric grid and the quadrilaterals possible on it
• Explore the perimeters of shapes in relation to lengths of sides
• Develop some generalised rules and the relationship between them.
• 2D shapes
• Number patterns
• Algebra
• Reasoning and justification.
• EPISODE 1 (Optional)
Isometric grid and four-sided shapes
An episode suited to many classes where challenges are confined to exploring the grid, following rules for drawing shapes, giving descriptions of sets, and seeing the rhombus as a subset of the parallelogram. It ends with using number to describe sides.

For some classes the episode can be a short whole class preparation for the second episode.
EPISODE 2
How the sides of a trapezium relate to each other
This activity is based on a familiar investigation for the middle school years that can be handled at a very wide range of levels and could benefit from repeat handling. It leads to symbolising and the notion of a variable in a visual context. Some pupils in early secondary classes may have missed these steps in early algebra.

Pupils investigate how the sides of a trapezium relate to each other.They find partial or full rules, use the notion of counter-example, and explore symbolisation in algebra, including understanding an expression as dependent on the value of the symbol within it.
EPISODE 3 (Extension)
Exploration with proof 3
Pupils may move to looking at one or more of the following logical tasks, each allowing a chain of reasoning. The first being the easiest:

• all the possible trapezia with a given base, i.e. listing all the combinations
• the possible sets of least necessary data to describe a trapezium on the isometric grid
• the reasons why the relationships are what they are.