Making Triangles

Task 36 ... Years 2 - 8

Summary

Students combine sticks of different lengths in an exploration of different types of triangles. The focus is on the lengths (represented by different colours) not the arrangements of those colours. At one level, the task involves sorting and classifying, and at a deeper level it involves investigating how many different triangles are possible and how we know that we have found them all.
 

Materials

  • At least 60 rods in four different lengths which are indicated by different colours

Content

  • sorting and classifying triangles
  • concepts of length, angle and shape
  • properties of triangles
  • sets and representations of them such as Venn Diagrams
  • problem solving strategies such as try every possible case
Making Triangles

Iceberg

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

   

The order of the colours doesn't matter in this task. For example R-B-Y is the same as Y-B-R. Further, using this notation the bold colour is thought of as the side placed horizontally in front of you on the table as a base.

Students may use the criteria suggested by the card to sort and classify ot they may focus on the number of each colour and come back to the triangle types later. Some students begin quite randomly and it is heartening to see them reconsider this approach and try a more systematic strategy. Mathematical conversation is a key feature of the task and one of the high points for teachers is when students realise that some triangles belong to more than one category. For example all equilateral triangles are also isosceles.

The card suggests drawing as a form of recording - a mathematician records notes and diagrams as they investigate a problem - but if they choose to use a code rather than lines and colours that is fine.

One way to approach an organised search is to start with the smallest rod, Red, and make all the possible 3-side, 2-side and 1-side red triangles, then to repeat this approach for Yellow, Green and Blue, being careful not to build any that have already been made. The intellectual discipline required to follow this through precisely parallels the work of a mathematician using the strategy of try every possible case.

Red as Base

  • 3 sides: R-R-R
  • 2 sides: R-R-Y ... R-R-G ... R-R-B
  • 1 side: Y-R-G ... Y-R-B ... G-R-B (other 2-side situations occur below)
Yellow as Base
  • 3 sides: Y-Y-Y
  • 2 sides: Y-Y-R ... Y-Y-G ... Y-Y-B
  • 1 side: G-Y-B (other possibilities have been made)
Green as Base
  • 3 sides: G-G-G
  • 2 sides: G-G-R ... G-G-Y ... G-G-B
  • 1 side: All possibilities have been made.
Blue as Base
  • 3 sides: B-B-B
  • 2 sides: B-B-R ... B-B-Y ... B-B-G
  • 1 side: All possibilities have been made.
There are 20 different triangles in all. Four are Equilateral, 16 are Isosceles (which includes the 4 equilateral) and 4 are Scalene. One of the Scalene triangles (G-R-B) is also Right Angled.

Extensions to the task could include:

  • Can you make a picture for me which shows how the different types of triangles relate?
    Connect the students' response to the way mathematicians use Venn Diagrams as a tool.
  • Exploring the possibilities if using a fifth colour.
  • Exploring the possibilities of making quadrilaterals.
  • Investigating how you know without making the shape whether three sticks will make a triangle.

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.
   

Converting this task to a whole class investigation requires a considerable amount of dowel cutting and painting or dyeing. But it is worth it. Once prepared the materials can be stored as a class set and used year after year. With the co-operation of the woodwork or art & craft department, the students can prepare the sets of materials as part of their studies in those areas. The iceberg information above suggests the directions in which class investigation could go. Teachers might also set particular tasks for particular groups, eg:
Tables A & B explore all the triangles which can be made with red as the base and Tables C & D...

At this stage Making Triangles 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 Making Triangles task is an integral part of:

  • MWA Space & Logic Years 5 & 6
  • MWA Space & Logic Years 7 & 8

Green Line
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