Product Parting

The Product Parting island results in a lot of multiplying of numbers. In creating an optimal solution, there are lots of possibilities and perhaps some surprising ones. How many can the children find?

Let’s start with a multiplication: 485 x 3 = 1455

What if we split the last digit from the rest of the digits to create two new numbers and found the product?

We take 1455 and form 145 and 5. We now have 145 x 5 = 725.

What if we kept going? What will happen?

485 x 3 = 1455
145 x 5 = 725
72 x 5 = 360
36 x 0 = 0

We managed four multiplications before reaching zero and that is our end point.

What if we tried other three-digit and one-digit numbers? Can it be beaten?

Search inquiry reasoning skill in maths

There are a lot of numbers to choose from across this island and so it is about looking at trends. Initially, the children will likely Roam and just try other examples. I think it is a good idea to get them to focus on using the same number as a multiplier each time to encourage Combing. This can then allow us, by comparing results, to think more in a Seek mindset by looking at what works best.

We want to avoid zeros appearing in the ones column as that results in the end of the sequence.

You could also arrange the starting numbers they have into 3 lists with the total number of calculations next to each one. One list is for even x even, one for odd x odd and one for odd x even. Sometimes, an intervention like this is important as it will then encourage greater mathematical thinking than otherwise. At the beginning, there are hundreds of numbers to choose from, we need a strategy for narrowing things down. Whilst not guaranteed, it’s likely that they might use the Notice skill to see that odd x odd starting points generally allow more multiplications. Thinking about why, we can improve our search. Odd x odd = odd. This makes it less likely that we produce a number that has a zero as the ones digit.

Starting with a five as the multiplier is an interesting case to look at to bring out their Arguing skills as well. A five is perhaps a problematic number to have because as soon as it is multiplied by an even number, there will be no more multiplications. However, when multiplied by an odd number, it will produce five again as the multiplier. A nice link to the multiples of five.

911 x 5 = 4555
455 x 5 = 2275
227 x 5 = 1135
113 x 5 = 565
56 x 5 = 280
28 x 0 = 0

They might conjecture that larger numbers will produce a greater number of multiplications. Yet 999 x 9 produces an underwhelming five multiplications.

999 x 9 = 8991
899 x 1 = 899
89 x 9 = 801
80 x 1 = 80
8 x 0 = 0

Two numbers with nines in the ones columns results in a product with a one in the ones column. Not very helpful for our problem! But once again, an accessible explanation of why it isn’t a Search area of promise.

I’ve not found a way of getting more than eight multiplications. Let them convince each other of it though than telling them. It is possible using a range of numbers but I like 117 x 9 as a starting point because it is the opposite of the larger numbers conjecture.

117 x 9 = 1053
105 x 3 = 315
31 x 5 = 155
15 x 5 = 75
7 x 5 = 35
3 x 5 = 15
1 x 5 = 5
0 x 5 = 0

It’s interesting to look at because it brings us back to using five as a multiplier and how the five persists for so long.

I think there is enough in the initial island itself to build a deep inquiry. However, there is an interesting ‘what if…?’ question lurking:

What if we use two two-digit numbers and split the final two digits from the answer each time?

Some children might think there will be more multiplications possible, some children think there will be fewer. The result is still eight! This is definitely some interesting and possibly strange treasure.

81 x 49 = 3969
396 x 9 = 3564
356 x 4 = 1424
142 x 4 = 568
56 x 8 = 448
44 x 8 = 352
35 x 2 = 70
7 x 0 = 0

What if we ranked the solutions by the number of different multipliers used?

As there are so many optimal solutions, I like this as a way of separating them. It doesn’t need to take long but could provoke some extra exploration.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

This is an island where the Search skills are important and they come from all the other skills. If the children just Roam all lesson, they won’t get much out of it. It is likely that you will need to do a bit of leading to ensure that that isn’t the case.

Organise

It’s most likely that patterns are spotted on this island through the use of the Organising skill as mentioned above. They can definitely do comparisons of multipliers if different children focus on different multipliers.

Discover

There is a lot of potential for conjectures here and some will surprise them such as the presumption that bigger numbers always produce more multiplications.

Investigate
Investigate inquiry reasoning skill in maths

Lots of possible conjectures can lead to lots of Investigating. They have to combine their Investigating with their Search skills to be focused on thinking about which numbers will produce counter examples. That is often the case when there are many options in choosing numbers. Sometimes, the conjectures might act as error-checking mechanisms. If they have suddenly got a new highest total, do they need to check it for mistakes?

Argue

As mentioned above, there are nice links to multiples of numbers and the parity of numbers that the children can use to form arguments to explain what is going on. Organising the results effectively will be key to this.

Explore

This isn’t necessarily an island that promotes lots of different directions coming from the children. However, it is important for them to focus on different multipliers to enable coverage of the whole island in a timely fashion.

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