Partitioning Polygons

This island on 2d shapes feels like it should be easy to do but actually requires quite a lot of thought. We draw lines across polygons, partitioning them so that each polygon that is created from the segments of the lines each have a different number of sides. Equally though, the reason it is challenging is explainable to children. It extends really nicely to another island about overlapping shapes.

Partitioning Polygons inquiry/island

To start with, let’s just look at how many ways there are to create diagonals (lines between two vertices). Emphasise the shapes that are created from the two segments. In the example above, we have a quadrilateral and a hexagon.

How many ways are there to create two shapes with a different number of sides? What shapes can they be?

What if we tried to add a second diagonal? How many lines and what shapes could we make?

Can we still create segments where each shape has a different number of sides?

A second partitioning line on the octagon.

We could split the shape into a quadrilateral, a pentagon and a triangle.

At this point though, we face a difficulty. The reason for this is that if we try and partition the triangle at the bottom, we will create two triangles. Partitioning the quadrilateral or the pentagon will also create a triangle and we already have one. I haven’t shown all the possible examples below but this is an opportunity for them to prove that it isn’t possible by trying all possibilities (proof by exhaustion). If we look at the two partitioning lines across our octagon, there isn’t enough vertices between their end points to avoid creating another triangle.

Exhausting all possibilities to make a new shape.

What if our starting shape wasn’t an octagon?

What if we didn’t use regular shapes?

These are perhaps the two obvious ways of expanding this further. This is nice and follows and the format of a standard investigation where you could create a table and notice a pattern with different-sized shapes. However, I think that more creative ‘what if…?’ questions that change the rules more significantly provide a more interesting avenue.

What if we didn’t use diagonals (lines from a vertex to a vertex)?

This is where the power of asking ‘what if…?’ questions and creativity comes into maths. If we didn’t use diagonals, we could easily create four shapes using two lines. We can go a step further as well…

What if we didn't use diagonals?

What if the end points of our lines could be on other partitioning lines?

Both of these ‘what if…?’ questions have a purpose. They are seeking to find creative solutions to creating more shapes. As we always try to do with the Isles of What If…?, we started with something simple and have thrown in new ideas to complicate things and encourage deeper exploration.

What if the end points of our lines could be on other partitioning lines?

We want children to believe that maths is an ever-expanding world because it is. Focusing on the octagon alone, allows us to consider creative strategies to maximise possibilities. It might be at this point that you start looking at different-sized shapes or irregular shapes.

What if we intersected shapes instead of partitioning with lines?

This has so much potential that I consider it a whole new archipelago. I’ll have another page on it shortly. The key is that there is so much potential here that you have to specialise again and start simple otherwise the children will become lost in the exploration. I’d start with just intersecting rectangles.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

This is an island where it will start off with Roaming but the children will quickly realise that they need to move beyond Roaming to be successful.

Organise

This isn’t an island that requires a lot of Organising to be successful. You can have a leader board to see how many points are possible.

Discover

The obvious form of the conjectures that will come out is the maximum number of lines that are possible. But the children can also be encouraged to look at the shapes that are being made and whether lines can go across say, a triangle, after one has been made.

Investigate
Investigate inquiry reasoning skill in maths

This is an island where Inspecting can be really useful as there are going to be conjectures that are hard to find counter-examples through specific investigation but that someone else might have already found through chance.

Argue

With this island, it’s perhaps surprising that more lines aren’t possible. The Argue skills come in to explain why there aren’t more.

Explore

It may well be that the ‘what if…?’ questions in this island are driven by the teacher more than the children. But things such as changing the starting shape is certainly possible from the children.

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