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 some way towards giving meaning to topics such as collecting like terms and builds in additive and subtractive use of area.
 

Materials

  • 1 set of Tak Tiles
  • 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?
Algebra Through Geometry

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 + Ży
Shape D = 4x - 2y
Shape E = x + 2y
Shape F = x + 1ży
Shape G = 2x + ży
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. If you can find a set of these you will have many more challenges. Mathematics Centre can supply additional sets of Tak Tiles. Contact our Distribution Manager, Ina Koetsier, inak@blackdouglas.com.au
  • The Lancashire Grid for Learning web site provides an excellent Investigation Guide which will considerably extend student understanding of algebraic manipulation through further use of Tak Tiles.
  • The web site of Brighton & Hove City Council's Learning Platform Hub offers a PDF copy of Geoff Giles original work with Tak Tiles using Algebra Through Geometry. Click the image on the right to reach this page, where you will also find a set of Power Point slides prepared by National Centre for Excellence in the Teaching of Mathematics (NCETM).
  • Other shapes could be designed using x and y. Ask the students to invent some and record them on our Recording Sheet.
  • The original Tak Tile 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 Recording Sheet to redraw the pieces and photocopy your drawing.
Note: Copyright law only prevents photocopying or scanning the card(s).

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

Green Line
Follow this link to Task Centre Home page.