Triangles & Colours

Task 221 ... Years 2 - 10

Summary

This wonderful, easy to start investigation, simply involves making triangles with coloured sticks the same length. If only one colour is used, only one triangle can be made. If only two colours are used, there are four possible triangles. One all Colour A, one all Colour B, one 2A and 1B, one 1A and 2B. But what happens if we have three colours to use? The task card leads the students on to make all the possibilities for four colours, then, based on the data gathered so far, they are asked to predict for five colours.

But why stop there? If I tell you any number of colours can you tell me the number of triangles that can be made?

 

Materials

  • Coloured pop sticks - at least 15 in each of four colours

Content

  • algebra, concept of a variable / function
  • algebra, cubic
  • algebra, factorisation
  • algebra, generalisation in words & symbols
  • algebra, linear
  • algebra, quadratic
  • combination theory / ordered arrangements
  • counting
  • numbers, triangle
  • patterns, colour
  • patterns, number
  • patterns, visual
  • recording mathematics
  • sorting, classifying, ordering
Triangles & Colours

Iceberg

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

A triangle only has three sides so colouring those sides can only be with:

  • all the same colour (triples)
  • two of the same colour (doubles)
  • all different colours (singles)
The card shows there is only one triangle if there is one colour and it's a triple. The card also show that with two colours there must be 2 triples and 2 doubles - why are there no singles? Things begin to get interesting with 3 colours.

Triples

= 3

Doubles

= 6

Singles

= 1

That's 10 different triangles altogether for 3 colours.
For 4 colours...

Triples

= 4

Doubles

= 12
...because once the double colour is chosen each of the 3 remaining colours become the third side
...and there are 4 choices for the double colour.

Singles

= 4

The card suggests there is now enough data to make a prediction for 5 colours. What would a mathematician do with this data to help them find a pattern on which to build this prediction (or hypothesis)? Make a list or a tables seems like a good strategy to try.

No. Colours Triples Doubles Singles Total
1 1 1
2 2 2 4
3 3 6 1 10
4 4 12 4 20

Not a lot of data on which to predict but some things are certain:

  • The number of triples will always be the same as the number of colours. Let's call the number of colours n.
  • The number of doubles will always be the product of the number of colours with one less than the number of colours. This is n(n - 1).
Singles is a bit trickier, but the 5 colour line can be partially completed:

No. Colours Triples Doubles Singles Total
1 1 1
2 2 2 4
3 3 6 1 10
4 4 12 4 20
5 5 20

There is more data in the Total column than in the Singles so it seems sensible to look first for a pattern there. The differences between these terms are 3, 6, 10, which isn't much data, but we might notice that the differences between those differences are 3 and 4. Therefore it's not unreasonable to guess that the next one is 5, which, working backwards, means that the next difference in the Total column might be 15 and hence the Total number for 5 colours might be 35.

If it was then the number of singles would have to be 10. This can then be checked either with sticks or using a colour code like R, Y, B, G, P(ink) and trying all possible cases. Testing will confirm the hypothesis and the table becomes:

No. Colours Triples Doubles Singles Total
1 1 1
2 2 2 4
3 3 6 1 10
4 4 12 4 20
5 5 20 10 35

Some might notice that the Totals column is the sum of the triangles numbers. That's exciting!

The task card has now opened the door to:

  • What happens if there are 6 colours?...
  • What happens if there are 10 colours?...
  • What happens if there are n colours?
We can see from the table (and common sense confirms) that the number of triples will always be the same as the number of colours. So if there are n colours, there will be n triples. The column for doubles is more challenging, but it is likely several in the class will soon find a link between the number of colours and the number of doubles. (Finding a 'common sense' reason for the pattern is a different challenge.) If a hint is needed to find the pattern you could encourage students to look at the factors of each of the doubles numbers.

However, finding the total for any number of colours is going to be easier if we can reason out how to find the Singles. The following might make make sense to students:

If order mattered, the number of arrangements would always be n(n - 1)(n - 2). But that would mean that these...

= 6

...are all different but in the context of this problem, they are not.

So, if we take order into account we get six times the number of answers than is correct, so dividing the 'order' answer by 6 should give the answer for Singles. That is, Singles = n(n - 1)(n - 2) / 6. This can be checked against the tabulated results so far.

A consequence of investigating Triangles & Colours for any number of colours is we have now found two ways to calculate any value in the Total column.
  • Looking down the column to any cell, the value can be calculated by adding the Triangle Numbers: 1 + 3 + 6 + 10 + 15 + ... + nth Triangle Number
  • Looking across the row to the same table cell, the Total can be calculated by adding the numbers of triples, doubles and singles: n + n(n - 1) + [n(n - 1)(n - 2)] / 6
Therefore it must be true that:

Sum of Triangle Numbers

1 + 3 + 6 + 10 + 15 + ... + nth Triangle Number
= n + n(n - 1) + [n(n - 1)(n - 2)] / 6

Especially interesting if you are familiar with Task 101, Pyramid Puzzle, through which we discover that:

1 + 3 + 6 + 10 + 15 + ... + nth Triangle Number
= n(n + 1)(n + 2) / 6

We are looking at equivalent algebraic expressions which are cubic!

Hmm... both derivations make sense in their context. Perhaps it all depends on how you look at things in mathematics.

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 develop a whole class investigation from this task, you will need plenty of pop sticks in multiple colours. An alternative is to use drinking straws cut in three, or coloured plastic sticks which are available in some schools.

Begin the lesson at a central table cleared of all objects. Explore 1 and 2 colours together and set pairs to work on exploring 3. Return to the central table at various times through the lesson as a gathering point for class discussion of information discovered as partners work through the growing challenges. Guide the investigation using the exposition above.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 154, Triangles & Colours, which develops in more detail the relatively complex cubic governing the investigation. Paradoxically, this relates more clearly to the context of the problem when left in its non-simplified form.

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 Triangles & Colours task is an integral part of:

  • MWA Pattern & Algebra Years 5 & 6
  • MWA Pattern & Algebra Years 9 & 10

The Triangles & Colours lesson is an integral part of:

  • MWA Pattern & Algebra Years 9 & 10

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