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Sticky Triangles

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Tom is putting a great deal of effort and hard work into investigations.

The first thing that Tom did is an important, helpful strategy whenever you are problem solving, he used a tool. In this case it was matchsticks. Being able to 'build' the problem and see how it 'looks' and then being able to move pieces of the problem around is one of the most useful and vital ideas in solving problems.

Organising your findings into an easy to read table is also important. This way patterns get revealed making it easier to predict what happens next.

Here's Tom's ideas for the Sticky Triangles investigation.

I used matchsticks to work out the answers for the one, two, three and four rows of triangles. I put my answers in a table and saw there was a pattern and I worked out what I thought the answer would be for five rows. I checked it with the matchsticks and it was right.

Matches Triangles Rows
3 1 1
9 4 2
18 9 3
30 16 4
45 25 5

This can be explained using equations. Letters can be used to represent the different pieces of information. This saves writing out the full words each time. Tom used:

m = number of matches
t = number of triangles and
r = number of rows

By looking at the table above, Tom was able to work out that:

t = r 2

(which is the same as saying that the number of triangles is the same as the number you get when you square number of rows - meaning, you multiply by the number of the rows by the same number)

and

m = t + 2 for the first row
m = t + 2+3 for the second row
m = t + 2+3+4 for the third row
m = t + 2+3+4+5 for the fourth row
m = t + 2+3+4+5+6 for the fifth row
and so on.

I didn't know how to write this as an equation so I went to "Ask Nrich" and they told me it was:

m = t + r*(r+3)/2

Using "Ask Nrich" is using your initiative Tom. Well done!
Do you understand the equation our mathematical helpers gave you, Tom?