Inside Triangles

'Inside Triangles' printed from https://nrich.maths.org/

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Inside Triangles

Here is a four by four dotty grid:

I have joined three dots on the grid to make a triangle which has one dot inside it:

triangle with one dot inside

How many different triangles with one dot in the middle can you draw?

How do you know have found them all?

You may like to experiment with the pegboard interactivity below, or you could print off this page of dotty grids to work on.

Why do this problem?

This low threshold high ceiling activity is accessible to all pupils but has the scope to be extended in many directions. In order to make a start, children will need to be familiar with properties of triangles, but drawing triangles on the grid will help them to clarify for themselves what they understand by the term "triangle". In order to find all the possible triangles, pupils will need to work in a systematic way.

This task also offers the chance to focus in particular on reasoning, problem solving and developing a positive attitude to mathematics, three of the five key ingredients that characterise successful mathematicians.

Possible approach

You may want to begin this task with the whole class and, this way, the notion of "different" will come up quite quickly. How is the group going to define "different"? This is a great discussion point and one where there isn't a right or a wrong answer. You could decide to count triangles which could be picked up and placed exactly on top of another triangle as the same. Or, you could decide that they are different if they are in a different orientation on the grid. The former suggestion makes a more manageable number to count!

It might help to suggest working in pairs on this activity so that children are checking they haven't duplicated triangles. Learners might find it helpful to use a pegboard and/or to draw their triangles on this sheet of grids.

The interactivity, if projected or used on an IWB, allows findings to be shared easily. In addition to checking that the triangles are indeed all different, a plenary could focus on how the children know that they have found them all, which is quite a challenge. Listen out for learners who have a 'system' of some description which they follow to make sure they don't miss any out. Alternatively, you could ask children to draw each triangle on a different grid and try to group the triangles that have been found. That way, the imposed method of grouping will help to identify any that have been omitted.

Key questions

Tell me about the way you're working.
How will you remember which triangles you've found?
How do you know that your triangles are all different from each other?
How do you know that you have found them all?

Possible support

Having a range of different equipment available for children to use to tackle this problem (e.g. pegboards, grids on paper, the interactivity) will help everyone get started.

If learners are finding it difficult to work systematically, you could offer them 6 Beads and Three Ball Line Up first, which might be a more familiar context.

Possible extension

In order to extend the problem, pupils could be asked to find triangles with three spots inside them or no spots inside... Differently sized grids could be drawn and compared. You could sort the triangles across differently sized grids, for example all right-angled triangles together, or all triangles which are the same shape but different sizes together.