Algebra Through Geometry
Task 71 ... Years 7 - 10
SummaryIn 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.
IcebergA 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 + 2ySumming 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 + 4yExtensions
Whole Class InvestigationTasks 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: