Lesson 1a Algebra: Exploring area on a grid

# Episode 1

Reasoning Resources: Isometric dotty grid on board or OHT, Worksheet 1
Whole class preparation
Area as covering of flat space, measured in any convenient unit, e.g. triangular tiles. Display an isometric dotty grid on the board or OHP. Discuss the smallest of the shapes that
can be drawn on the grid: triangle, rhombus, parallelogram and trapezium. Conduct a
discussion about meaning of area as covering the flat space measured in some units,
standard and non-standard. The discussion may lead to comparison with use of square units
as a standard for area and how the triangle is equivalent to half of a square unit. Generalised number. Focus on the parallelogram and model an example on the board. Using one row of
parallelograms of different length base, agree on terms for the base of the shape and the slanted side, e.g. b as the length of the base and s as the length of the side. For one row
s =1.
Pair and small group work
Give out Worksheet 1 for pupils to work in pairs on part A, the parallelograms. They should
work out the areas of the shapes in unit triangles, and investigate the relationship between
the lengths of the sides and the number of unit triangles inside the shape. Ask them to write
down any patterns that they notice and use this information to create a rule. They could try
other parallelograms.
Whole class sharing and discussion
Revisiting relationships in less familiar context than the rectangle. Ask the pupils to contribute their findings, listing rules on the board and valuing all patterns. Accept ‘part rules’ e.g. double the base for parallelograms with side length 1.In symbols, for any length of the base (b), the area will be = 2 x b for side length 1. Agree the rules for finding area in natural language, e.g.‘one row makes twice the number of unit triangles on the base! Some pupils will realise that the area of the rhombus is twice the square of the side, and may discuss why.

Rhombus idea = 252,
analogous to area of square but with twice the number of units.
Discuss rules that are similar and how they are similar

eg. the base times the side doubled ...one side doubled times the other ... 2(base x side)

What if we used a and b for the lengths of the sides? Possible responses: 2ab, (a + a) x b, (b + b) x a. Are these similar/different?
Inverse with a variable. An extension: Supposing the area of a parallelogram is 20 units, What possible sizes can it have?
Give the pupils a few minutes to produce ideas. Accept trial and improvement in drawings,
or in number, and draw out the division as inverse of multiplication. The idea of having a parallelogram without whole unit length may crop up.