Algebra Through Geometry

Task 71 ... Years 7 - 10

Summary

In a cunning twist algebra becomes geometry (or is it the other way around?). Whatever, this approach is not new in the sense that the majority of the original investigation into algebra by the ancient Greek scholars was from a geometric starting point. They didn't even try to separate the two. Perhaps it is a pity that school algebra has been so disposed to symbolic algebra without consideration of visual and kinaesthetic learners. This task goes a long way towards giving meaning to topics such as collecting like terms and builds in additive and subtractive use of area.

Originally designed to be used with the Plastazote shapes shown which were known as Tak Tiles (currently difficult to obtain), the investigation can still be explored in full by preparing the equivalent pieces linked under materials. Print the page on thin card, laminate and cut out a set of puzzle pieces and frame. When printing from Adobe Acrobat set Page Scaling to none. You only need one set for one task, but you might want to make a class set while you are on the job.

Materials

1 set of original Tak Tiles or use this set of Algebra Through Geometry pieces.
Investigation board, marker and cloth
Recording Sheet - when printing from Adobe Acrobat set Page Scaling to none.

Content

area measurement
use of pronumerals
collecting, summing and subtracting like (and unlike) terms
manipulating algebraic symbols
concrete representation of algebraic addition, subtraction and multiplication, including examples with fractions
problem solving
application of the question Can I check this another way?

Iceberg

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

Tak Tiles were designed by Geoff Giles for the DIME (Developments in Mathematics Education) Project. This task is a more sophisticated partner to Task 65, Shape Algebra.

The traditional x and y of algebra are given immediate physical presence in the task and the intriguing foam shapes allow students to see and touch algebraic procedures related to operations on like and unlike terms. The context invites students to apply visual and tactile intelligences, and even the fact that it includes algebraic fraction operations related to halves and quarters is not a barrier to most.

The areas of the shapes in the puzzle are:

Shape A = x + 2y
Shape B = 2x
Shape C = 2x + ¹/₂y
Shape D = 4x - 2y
Shape E = x + 2y
Shape F = x + 1³/₄y
Shape G = 2x + ³/₄y
Shape H = 3x - y

Summing these gives a total of 16x + 4y. That is, the total area of the 8 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.

Extensions

Look in your cupboards. Some schools will not only have sets of Tak Tiles, but will also have the DIME workbooks that were available with them, or the later version, the cover of which is shown here. If you can find a set of these you will have many more challenges.
The original Tak Tile pieces fit together to make a spatial puzzle in a 'rectangle' as on the book cover and in the Pieces master above. 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).
This Investigation Guide was originally freely distributed through the Lancashire Grid for Learning (LGfL) web site. This is the Mathematics Centre version, produced when the LGfL went off line. Designed to work with pairs, or a whole class, this guide will considerably extend student understanding of algebraic manipulation - like terms, adding and subtracting like terms, some factorisation - through further use of Tak Tiles. It will also give you ideas for creating similar challenges. Perhaps it should be your students who create the additional challenges?
Other shapes could be designed using x and y. Ask the students to invent some and record them on the Recording Sheet with their symbolic expressions. Encourage new 'subtraction' shapes as well as 'addition' shapes.

The card tells us to assume the small arc section has an area of ¹/₄y. The small arc is clear in the image above where the bottom right and bottom centre pieces join. Assuming, rather than proving, ¹/₄y is sensible in the context because the focus remains on manipulating the 'algebra', however at another time it is a nice piece of maths in itself to calculate that the small arc section is truly ¹/₄y.

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 Tak Tile 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, Algebra Through Geometry 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 Algebra Through Geometry task is an integral part of:

MWA Pattern & Algebra Years 7 & 8

Follow this link to Task Centre Home page.