Shape Algebra

Task 65 ... Years 4 - 8


Shape Algebra blows away the traditional opening line of an algebra lesson, viz: "Let x stand for any number". In this task x is a shape! Well actually it is the number that stands for the area of the shape, but the exciting feature of the task is that the algebra becomes geometry. The task focuses on creating and manipulating algebraic terms, but all stages of the symbol manipulation are grounded in the concrete material.


  • Shape Algebra tiles
  • Investigation board, marker and cloth
  • Recording sheet - when printing from Adobe Acrobat set Page Scaling to none.


  • area
  • use of pronumerals
  • manipulating algebraic symbols
  • like and unlike terms
  • concrete representation of algebraic addition, subtraction and multiplication by a whole number
Shape Algebra


A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

This task is a simplified version of Task 71, Algebra Through Geometry, a concept attributed to the great Scottish educator, Geoff Giles.

The areas of the shapes in the puzzle are:

Shape A = 2x + 2y
Shape B = 2x + y
Shape C = x + y
Shape D = 4x - y
Shape E = 2x
Shape F = 3x - y
Shape G = 2x + 2y
Summing these gives a total of 16x + 4y. That is, the total area of the 7 pieces is 16x + 4y. We can also see the same result by realising that the whole puzzle is made up of 16 squares plus 4 rounded pieces in the corners, ie: also 16x + 4y.

A second, albeit more complex, way to check the total is to see that all the pieces fit into a 5 by 4 frame of x shapes (=20x) from which a piece has been rounded out of each corner.

So if we knew how big this rounded section was we could take it away four times from 20x and it should be the same as the total area of the 7 pieces. Examining any corner shows the 'removed rounded' piece must be x - y. So the total is also:

20x - 4(x - y) = 20x - 4x + 4y = 16x + 4y

If the middle step is a little confusing, think of it as starting with a 5 by 4, taking out 4 corner x pieces, then adding 4 y pieces into the corners to replace them.

  • Other shapes could be designed using x and y. Ask the students to invent some and record them on Shape Algebra Paper.
  • The original Shape Algebra pieces fit together to make a spatial puzzle in a 'rectangle'. So the second level of extra challenge is to create other pieces which also fit into this frame (or a frame of the student's own design).

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

The best way to turn this task into a whole class investigation is to purchase some extra copies of the task and set up a work station. Over a number of days all students can try the task. However, if this is not possible, use the Shape Algebra Paper to redraw the pieces and photocopy your drawing.
Note: Copyright law only prevents photocopying or scanning the card(s).

At this stage, Shape Algebra does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Shape Algebra task is an integral part of:

  • MWA Pattern & Algebra Years 5 & 6
  • MWA Pattern & Algebra Years 7 & 8

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Follow this link to Task Centre Home page.