# Shape Algebra

### Task 65 ... Years 4 - 8

#### Summary

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.

#### Materials

• 1 set of foam Shape Algebra tiles or use this set of Shape Algebra pieces.
• Investigation board, marker and cloth
• Recording sheet - when printing from Adobe Acrobat set Page Scaling to none.

#### Content

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

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 equation is a little confusing, think of it as starting with a 5 by 4, taking out 4 corner x pieces, then adding four y pieces into the corners to replace them and hence create the rounded corner effect.

Extension
• Other shapes could be designed using x and y. Ask students to invent some and record their outline and symbol expressions on the Recording Sheet. Encourage new 'subtraction' shapes as well as 'addition' shapes.
• 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 ideas on the card and the Extensions above provide plenty of material for a whole class investigation. You will need one set of Shape Algebra pieces for each pair. The investigation could be with all the students at once, or as a work station in an algebra unit that included three to five algebra investigations. See the Task Cameo Content Finder for other algebra based investigations.

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 