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Sticky Triangles

I was exploring a puzzle in which headless match sticks had to be moved to make a different number of triangles.

I made one small triangle

Triangle from three matches. $3$ matches

I made it into $4$ small triangles by adding $6$ matches.
Four triangles. $9$ matches

I added another row and counted the number of small triangles and counted the matches.

9 triangles from matches.

I made a table of my results and continued adding rows. I found many patterns.

Have a go and see what patterns you can find. You do not have to use match sticks (or cocktail sticks) - drawing lines will do just as well.

Find a good way to record your results. See if you can predict the numbers for rows of triangles you have not drawn.

When you have done all you can with triangles, see if you get the same sort of results with squares. Then think of other shapes which might make number patterns as they grow.

Why do this problem?

This investigation starts in a very practical way so that all learners can take part. It can lead to several interesting number patterns and is a good context in which pupils can begin to generalise.

Possible approach

You could draw one triangle on the board and indicate that it is made from three matches (or lines or lolly sticks etc). Next to this triangle, draw another identical triangle but this time say that you are going to extend the drawing with another row of triangles. Ask the children how many more matches you have used and therefore how many matches are now used altogether. Invite children to predict how many more matches will be needed for another row in the pattern. Can they make a prediction without drawing? Take some suggestions with reasons for choosing that number, then check how many are needed by drawing the arrangement. Focusing on the different ways the children explain how they visualised the arrangement will help them to build up a pattern of what is happening.

Set the group off on the challenge. You could leave it very open-ended or you could say, for example, that you want them to be able to work out the total number of matches for ten rows of triangles. As they work, stop them at various intervals to share effective ways of recording results, for example in a table. Learners might find it helpful to have matches or sticks, and isometric lined or dotty paper for recording the actual triangles.

To encourage them to look more carefully at the number patterns involved, rather than simply counting matches each time, suggest that you would like them, for example, to be able to work out the total number of matches for $100$ rows of triangles, which of course would take far too long to draw and count.

Key questions

How will you record what you have done?
Can you see why the number of matches increased by that amount when you added that row?
Can you predict how many matches the next row will need? Why?
Can you see a link between the number of rows and the total number of matches?
Can you see a link between the number of small triangles and the total number of matches?

Possible extension

Some learners will be able to express the patterns they have found in terms of words, some might use a letter to stand for the number of rows, for example. In addition, the real challenge here is to explain the patterns found in the numbers.

This investigation can be continued with squares (see Seven Squares ) and even hexagons. Decisions have to be made on how these are to grow which means that variations on the numbers may be found.

Possible support

Some pupils will be able to organise their results without help but others might need the guidance of a table like this: