McMahon's Triangles 2

Task 107 ... Years 4 - 10

Summary

The 24 triangles represent every possible combination of mapping 4 colours into 3 positions. The challenge of doing this is the subject of the partner task, Mc Mahon's Triangles 1, Task 148. This task uses the pieces to involve students in some quite difficult spatial logic puzzles.
 

Materials

  • 24 equilateral triangles, each different, but all coloured using only four colours

Content

  • spatial perception
  • problem solving strategies
  • reflection and rotation
  • combination theory
McMahon's Triangles 2

Iceberg

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

This is not a problem that will captivate everyone. It takes considerable time for most people to think through a solution to Questions 3 and 4 in particular. However, there is success for everyone in the earlier questions. Question 1 is easy and it doesn't take long to find ONE hexagon in Question 2 that has matching inner edges and a border all the same colour.

However, choices have to made in constructing one such hexagon. If, for example, you chose to make the border red, there are 13 pieces that have at least one red section. But you only need six, so which of the 13 do you choose. If the challenge is to make only one bordered hexagon, it probably doesn't matter too much. But if you have to make four, each with a different colour border, then the decisions made for red will affect what is available for the others.

One of the critical realisations in making the four hexagons is that since there are only four pieces all the same colour (all red, all yellow...), one of these must be in each hexagon. Allow students to come back to this problem as often as they wish and stay as long as they need. In their journal they should record what has worked, partially worked, not worked and any insights. Eventually a solution will be found:

     

  There is a sense of rotation in this solution. For example, in Hexagon 1, look at the colour of the bottom right rhombus. It is yellow. Now look at Hexagon 2. Yellow is the outside colour. But where does the outside colour of Hexagon 1 appear in Hexagon 2? It moves to the bottom left. This 'pushes' the bottom left colour to the middle left in Hexagon 2 , which pushes the middle left colour of Hexagon 1 to the bottom right in Hexagon 2. Overall a pattern of:
  • Bottom right ---> Outside
  • Outside ---> Bottom left
  • Bottom left ---> Middle Left
  • Middle left ---> Bottom right
Try this sequence between Hexagons 2 & 3 ... 3 & 4 ... 4 & 1. It works!

Be aware that the student solutions may not be oriented as in the diagram and there are variations, for instance:

An additional interest feature of this task is to challenge students to use the pieces to make shapes that are line or rotationally symmetric.

One solution to Question 4 is:

and one variation is:

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.
   

To make a whole class investigation out of this task - at least one where all students are working on the same problem at the same time, you really need lots of pieces the same as those in the set. With computer drawing packages, colour printers and laminators in schools, that is possible. But it is also time consuming. What about using this task, along with say 9 other challenging spatial tasks to create a menu-based unit of work on Shape, Symmetry & Logic. Mini-lessons/tutorials/fish bowl discussions happening at 'point of need' based on the investigations of particular pairs of students, are teaching techniques that can be employed to share learning.

At this stage, McMahon's Triangles 2 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 McMahon's Triangles 2 task is an integral part of:

  • MWA Space & Logic Years 3 & 4
  • MWA Space & Logic Years 9 & 10
This task is also included in the Primary Library Kit. Solutions for tasks in the this kit can be found here.

Green Line
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