Dividing Shapes

Task 115 ... Years 4 - 8

Summary

Imagine the pictured surface of the Tricube (actually a Trisquare) is a farmer's paddock that has to be divided to separate stock. Animals in each section need equal amounts of feed and space to exercise. The Trisquare is a shape that gives the farmer several options. The object of the task is to find those options.
 

Materials

Content

  • spatial perception
  • fractions
  • rotation
  • area & perimeter
  • growth patterns
Dividing Shapes

Iceberg

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

Dividing Shapes helps students learn to visualise the way shapes are composed. It is an important skill. For example, there is no simple formula for finding the area of a Sphinx...

... but this irregular shape can be partitioned in several ways to make pieces for which area formulae are known. For example:

  • 6 small equilateral triangles
  • 1 large equilateral triangle and a rhombus
  • 1 small equilateral triangle and an isosceles trapezium
Using this approach is applying the mathematician's strategies of breaking a problem into smaller parts and looking for opportunities to apply previous knowledge. Dividing Shapes encourages this way of thinking in the more familiar situation of a square-based shape. The first three questions are 'warm-ups'. Possible answers are:

         

Students might notice a connection between the required shapes and the rotation of one match pinned at the concave angle of the Tricube shape. In other words it would be easy for a farmer with a rope to mark out the fencing line necessary to create the paddock divisions. The solution to the main challenge relates this task back to Sphinx (Task 166):

Just as a Trisquare can be made from four Trisquares, so a Sphinx can be made from four Sphinxes. But the shape formed is a new Trisquare, so 4 of this new size would make the next size Trisquare - and so on.

Consequently, equivalent backwards reasoning reveals that the original farmer's paddock could also be divided into four equal parts, each a mini-Trisquare shape. (Note: Rather than have students break matches to discover this solution, the challenge has been put in terms of four whole Trisquare paddocks.)

The starting point for each section of this problem is one whole Trisquare. In each case this whole is divided into equal parts. Take the opportunity to discuss and record the fraction connections in each solution. Also students could look for other ways to divide the whole paddock into equal parts.

A further extension is to ask for the area and perimeter of the smaller paddocks in each case. Perimeter becomes interesting when the hypotenuse of a triangle is involved.

If you have enough class sets of Tricubes or Trisquares, students could explore:

  • growing Trisquares in 2 dimensions
  • growing Tricubes in 3 dimensions
However each of these challenges is a task in itself. Growing Tricubes is Task 234 and Growing Trisquares is Task 238.

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.
   

A class set of Trisquares or Tricubes is the most effective way to turn this task into a whole class lesson. However, if you don't have these, it is easy to draw up a 2 x 3 rectangle marked with two Trisquares. Print sufficient of these and supply scissors so students can quickly cut out temporary Trisquares.

At this stage, Dividing Shapes does not have a matching lesson on Maths300, however Lesson 25, Sphinx, and Lesson 99, What's It Worth?, involve related mathematics.

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 Dividing Shapes task is an integral part of:

  • MWA Space & Logic Years 7 & 8

Green Line
Follow this link to Task Centre Home page.