Formed From Three

Children arrange three similar triangles repeatedly to form polygons with an increasing number of sides. Shape names could be a curriculum focus of the task but more than that will be the use of problem-solving and reasoning skills. Particularly, if they work in pairs and discuss their ideas. There are so many possible directions that each child’s work will be different.

The idea for it came from the fabulous 21st Century Pattern Blocks by Math for Love. Specifically, the Upscale Pattern Blocks found here would be great for this task. They can be found here.

I have made a Polypad version that you can use to drag and rotate shapes here.

We have three similar triangles with a scale factor of three, two and one. We are going to repeatedly use them in the order of largest to smallest. I’m using Polypad to show these

What if we arranged them so that the area they surround forms a triangle?

I have numbered the triangles to show the order that I have placed them. Here is one possibility:

What if we now had three more triangles to form a quadrilateral?

Starting again with the largest triangle and continuing with where we left off, we could do this:

What if we continued and tried to make a pentagon, hexagon…?

I could make my pentagon using only two triangles, seven and eight, I did not need the smallest triangle which would have been my ninth triangle.

For my next shape, my hexagon, I still only have three triangles to form it. However, my next triangle is the smallest triangle so I need to start with that. Whether I use one, two or three triangles to form a shape, the next shape still needs to be formed from three shapes.

How far can we take this? Heptagons? Octagons? Decagons? Even further?

There are so many possibilities that every piece of work is going to be different and there is so much to explore and try out. There are lots of strategies possible. To aid with this, it would help if children had one or two sets of triangles cut out to move around and try different combinations.

Sometimes, as mentioned above, it is possible to create a shape without needing all three triangles. One option might be to start creating the next shape rather than completing the current one to maximise the use of the three triangles allowed. As the number of sides increases, it will become more challenging. This idea might help them go further.

What if we used different shapes?

We could still use triangles. For example, using right-angled isosceles triangles on a square grid. I think they work really well. Or we could use something different. We would still have them as three similar triangles. Math for Love’s upscale blocks gives you some different options.

What if we changed the order we used them?

We could go from smallest to largest instead. Would it change much? That is for the children to decide.

What if we had a fourth triangle with a scale factor of four?

We would now have four triangles to make each shape. Presumably this makes things easier and we could go further in terms of the possible shapes we could create.

What if the shapes could overlap each other?

We would still have to surround an area to form each shape but maybe overlapping them would be a possible approach to go further.

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