Triangles & Colours
Task 221 ... Years 2 - 10
SummaryThis 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?
IcebergA 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:
That's 10 different triangles altogether for 3 colours.
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.
Not a lot of data on which to predict but some things are certain:
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:
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:
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...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.
Sum of Triangle Numbers1 + 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
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 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.
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:
The Triangles & Colours lesson is an integral part of: