Digital Shape Sequences – Multiplication

This isle is an alternative or continuation of Digital Shape Sequences, focusing on times tables rather than single-digit addition. It mostly follows the same rules though and explores the same repeating shape sequence. Coming back to it shows that maths and mathematical problems do not have to be over in a lesson. They can continue for days, months or years.

We have a repeating shape sequence: triangle, square, hexagon.

In the triangle, the times table must result in a product with a three as a digit since it has three sides and angles. In the square, we need a product with a four as a digit and in the hexagon, a product with a six as a digit.

To transition to the next shape, we can change either the multiplicand or the multiplier (up to 12 x 12). So for example, these next steps are two of many possibilities. The first involves changing the multiplicand and the second, the multiplier.

A product cannot be repeated across multiple shapes. If we start with 8 x 4 = 32, how long can you make the chain?

Here is a chain of length 9 as an example:

Everyone starting with 32 provides a common basis for discussion. Here are all the possible second steps:

  • 1 x 4 = 4
  • 6 x 4 = 24
  • 10 x 4 = 40
  • 11 x 4 = 44
  • 8 x 3 = 24
  • 8 x 5 = 40
  • 8 x 6 = 48
  • 8 x 8 = 64

Such an organised list provides a basis for discussing strategies that can be used which will then play a role in them choosing their own start point. It might be, for instance, that they aim to avoid 24 and 48 early on as they are multiples of 2, 3, 4, 6, 8 and 12 which makes them easily reachable later on. This is not a lesson on factors and multiples but by investigating strategies, we are also providing opportunities for making connections between concepts – such a fundamental aspect of understanding.

What if we started from a different multiplication?

Once they are confident that they have investigated starting with 8 x 4 = 32 fully, they can attempt to see if a different starting point leads to a longer chain. This comparison back is important as it promotes the thinking of why some starting numbers may be better than others. The first twelve multiples of one for instance, only have 3, 4 and 6 that are usable in our chain. That might make them a good starting point.

What if we alternated between changing the multiplicand and multiplier?

This restriction means that we cannot change either the multiplicand or the multiplier twice in a row. We have to alternate. Adding the restriction in after they are comfortable with the initial rules adds an extra challenge to what is possible. Here is an example of following these rules with a chain length of 12:

What if we arranged the chain in a circle?

How big could a circular chain be? Here I have a circular chain that has nine numbers. What would be the largest chain possible now?

This perhaps does not dramatically change the situation but it opens up other possibilities. Showing them directions such as these can simply be to promote them coming up with similar interesting directions in the future.

What if the triangle could not contain a multiple of three, the square could not contain a multiple of four and the hexagon could not contain a multiple of six?

This makes things more challenging. It makes starting with 8 x 4 not an option as no matter which I change, it is guaranteed that the second number will be a multiple of four. In fact, it is challenging to get a chain going any length at all with these rules. This is not a bad thing though, it gives the opportunity for explaining why that is. The fact there is only two numbers possible in a triangle: 32 and 35 make it impossible to go any length. This could be followed up by using three different shapes, perhaps a hexagon, octagon and square might work well as composite numbers.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

This is going to be particularly important when selecting a starting number. As the chain gets longer, there are fewer choices as to what the next number could be. The children’s initial Roaming, through discussion, could lead to a more logical approach later.

Organise

The main use of organising would be to make a list of all the possible multiplications that could go in a triangle, a square and a hexagon. By looking at how frequently the digits 3, 4 and 6 appear in each times table, decisions can be made as to a good starting number.

Discover

Conjectures are largely going to be based around strategies. Is it better to start with a number that provides many options for a second step or one that provides fewer options? They can also Conjecture as to what they believe the longest chain possible is. As always, as the teacher, it is not our job to give a definitive answer to this but to allow them the opportunity and guide them towards investigating it. Once 32 has been investigated, the children can Conjecture as to whether they think starting numbers will offer a longer chain or not.

Investigate
Investigate inquiry reasoning skill in maths

If you give explicit attention to it, there is plenty of scope for investigating the best possible starting number. This island is one that works well if you give different groups in the class a different starting number to focus on. Once they have done this, they can select the starting number that they think will be most effective and carry on.

Argue

Since there are so many possibilities for different strategies, there is lots of potential for children attempting to convince each other of what would work best. Just remember that it is entirely possible that you will need to start this debate by bringing the whole class back. Working in pairs for this task might give more opportunities for them to use these skills.

Explore

Even though this island itself is already a Departure from a previous island, there is so much more potential to go even further with it. The children can definitely come up with their own ‘What if…?’ questions to take things in their own direction. I would love to hear what they come up with.

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