Polygons of Many Multiples

This shares similarities with Triangular Threes but focuses on constructing multiples instead. I used squares as that worked well with my Year 4 class when I came up with the idea but you could easily start with hexagons if you want to start with something more challenging. There are a lot of interesting directions to explore.

We start off with a square and the numbers one to four on the corners.

We can split that square into two triangles.

Since, we started with a square, and a square has four sides and angles, we are going to try and make multiples of four within the triangles. What multiples of four can be created?

What if we had a second square around the original one? Can we construct a different multiple of four in each triangle?

In the example above, 3 x (6 + 2) = 24 and (7 + 3) x 4 = 40. There are ten possible triangles, can we create the first ten multiples of four?

This in itself will likely require a decent amount of investigation. Children can strategically use the four in as many different triangles as possible to create multiples.

Some of the multiples can be created in many different ways whereas some of them are much trickier. Children will have to prioritise the trickier to create multiples.

What if we tried to construct the highest set of ten consecutive multiples of four as possible?

Rather than just focusing on the first ten multiples of four, we could try and make a higher set.

What if we had a third square?

This is the natural progression but it doesn’t necessarily mean that it is the best one to go with. Have a look at alternatives below.

If you do look at this, notice how using the two outer squares, there is a difference of four between numbers going diagonally from the centre. This can be a useful strategy for them.

What if we did not have a four to use?

I really like this more challenging variant. It requires a lot more thought as children have to more creative in their methods of constructing multiples of four.

What if we used hexagons?

You could do this with other shapes but I have found that hexagons work nicely and it might be that you choose to start with hexagons rather than squares. When I did this island with Year 4, squares worked well but I imagine with Year 6 or older, you could easily start with hexagons. In this case, rather than multiples of four, we are constructing multiples of six.

You can then do the same thing as with the squares and remove the six from being usable. I am not sure if it is actually possible in this case but that should not stop it from being attempted. I think it is important to give children problems that are not solvable as well to ensure they have that element of doubt.

What if we used a square and a hexagon?

This one is very intriguing in that it is ambiguous in terms of what the problem is. I think it is interesting to present something for them to interpret. Just as we would try and analyse a book. There is no ‘right’ answer to this. Children can follow a path that interests them. It might be:

  • That we have to make the first set of multiples of four and six
    • Within this, we could play about with the common multiples. Perhaps colour those triangles in. Can they all be adjacent to each other? Can they be apart? How many different positions can we put them in?
  • We might have to construct common multiples of four and six (and allow repetitions)

What if we constructed triangles with only two numbers attached?

With this idea, it won’t take very long to realise that it will not be possible in this case. However, even if that is the case, children will likely realise that quickly and it is still worthy of investigation. Rather than us shutting it down, they can explain why it will not work. It most likely will not take them long and can be a basis for mathematical reasoning. They might also find that triangles with only two numbers attached will work in other places! Where though?

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

With this, the children will be required to use trial and error a lot in their search for combinations that work. Some multiples can only be constructed with some numbers and so they need to prioritise their search for those initially.

Organise

It is useful for them to have a table of the multiples that they have already constructed (and how they have done so) to keep track of them.

Discover

The obvious conjectures they can make are about which multiples are possible where. If you go down the route of trying to find the greatest set of consecutive multiples, that can form a basis for conjectures as well.

Investigate
Investigate inquiry reasoning skill in maths

The whole problem is in essence, investigating whether or not it is possible. Some of the ideas mentioned above are not possible. That is fine! It is good to consider why they are not.

Argue

They can use their Arguing skills to justify why certain multiples have to be placed in certain triangles. This is particularly the case with using multiples of six and when you do not have the number that you are trying to find multiples of.

Explore

There are so many possible directions with this task and that is what makes it so enjoyable. I have given plenty above that my Year 4 class came up with but I am sure there are other possibilities.

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