Lesson 1a Algebra: Exploring area on a grid

Episode 3

Reasoning Resources
Isometric dotty grid on board, Worksheet 1
Whole class preparation
'Seeing' a shape as combination of two easier shapes. Show the isometric dotty grid on the board or OHP. Focus on the trapezium and model an example on the board. How is it related to the triangle and parallelogram?
Prod for ideas that see the trapezium as ‘made up of bits’ or made by ‘cutting off’ bits.
Pair and small group work
Give out Worksheet 1 for pupils to work in pairs on part C, the trapezia. As before, they
investigate the relationship between the lengths of the sides and the number of unit
triangles inside the shape. They look for rules and express the rules in words or symbols.
Whole class sharing and discussion
Trapezium area as: the difference between two triangles, or combining area of a triangle, or combining area of a triangle and area of parallelogram in two ways.

For areas are grounded with tangible and visible activity.
Ask the pupils to contribute their findings. List rules on the board.
Discuss rules and compare how they relate to each other, e.g.

  • A large triangle on the base less the triangle on the top
  • A parallelogram on the base with triangle subtracted
  • A parallelogram from top side plus a triangle
  • A second trapezium upended and added give a long parallelogram and we take half.
  • Some pupils will go to numeric relationships from tabulating or trialling, e.g. the top plus
    the bottom times the side; the top times the side plus the bottom times the side.
  • Work with variables at the level of quadratics. Algebraic extensions suited for older pupils
    Exploration of formulae produced in different ways, leading to algebraic manipulation. Demonstrate labelling the sides of the trapezium with a, b and c (the base).

  • Look at the rules found in natural language. Try to express the rules using a,b and c. List (Only those pupils proficient with algebra can deal with these formulae. The writing rules can be explored, including placing letters next to each other without the multiplication sign.)
  • The difference between two triangles: c2 - b2. Pupils who have recognised relationships
    between c, a and b, may understand (a + b)2 - b2
  • Triangle plus parallelogram: a2 + 2ab. Parallelogram minus a triangle: 2ca — a2 or
    2a(a + b) - a2
  • Placing two trapezia together: a(b + c) or (ab) + (ac)
  • Older pupils can be given the task of proving through manipulations that these
    formulae are equivalent to each other.
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    Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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