Top-Three-Digit Times Tables

The memorisation of times tables can often be prioritised over reliable strategies that will work with any mental multiplication. In part, this is because memorisation requires lots of practice and the easiest way to do that is to just have a list of calculations. With this isle, it gives the opportunity for repeatable practice but also promotes connections between multiplication facts and strategies for efficiently calculating them. This is not designed for just one lesson but instead over say, a term. Set up a number grid on your wall and cross a number off any time that score has been achieved. The children love crossing off a new number.

If you are doing Maths Week England this year (2023), it links in nicely with the theme of three although I actually designed it before I realised that and have been using it with my class (successfully!). However, you could have submissions from the whole school and see how many totals you are able to cross off. It would be fun to see how many totals everyone can make and who could get the highest total (and what it is). My class have loved trying to achieve this.

To do this, we start with a times table. We are going to calculate five in total. I am going to start with:

6 x 5 = 30

With each multiplication, I look at all the digits that have been used in the multiplicand, multiplier and product. I total the three highest digits.

MultiplicationThree Highest DigitsScore
6 x 5 = 306, 5, 314

In determining my next multiplication fact, I use a link to the previous one. My link could be via:

  • Adding a multiple – in this case, I could add 1 x 5 to make 7 x 5 or 6 x 2 to make 6 x 7 for example.
  • Subtracting a multiple – the opposite of adding a multiple
  • Scaling – I can scale one of the multiples. For example I could calculate 12 x 5 or 6 x 10 next.
  • Halving – I could halve one of the multiples. For example, 3 x 5 would be possible with this link.
  • Equivalent – I could have one multiple and double the other. 3 x 10 would be possible in this case.

In a set of 5 multiplications, each link can only be used once and the same multiplication cannot be repeated. This is to encourage a range of strategies used.

A set of five multiplications might look like this then:

MultiplicationThree Highest DigitsLinkScore
6 x 5 = 306, 5, 314
3 x 5 = 153, 5, 5Halve13
4 x 5 = 204, 5, 2Add multiple11
8 x 5 = 408, 5, 4Scale by 217
8 x 4 = 328, 4, 3Subtract multiple15
Total: 70

Once the rules are established, it is about seeing what scores are possible. This could happen over weeks and weeks and weeks. And that is great as far as I am concerned. If you have a class number grid up to 130, then you should have all the possible scores covered. Children can then write their name on a number when they achieve that score. Initially, lots of new scores will be created. When a score over one hundred is created, it could be celebrated. Over time, it will be more challenging to create new scores. Sheets for each score could be retained in folders so that they can make adjustments thoughtfully to make uncreated scores using existing data. There is a lot of scope for mathematical thinking whilst practicing their times tables. I’ve blurred out names, but this is how we are keeping track of scores attained:

The totals for the greatest three digits in each multiplication does not have a massive amount of overlap which is nice. Some interesting patterns are noticeable but I would not provide this to children or there is no need for them to do the multiplication. I have just provided it here for interest.

x123456789101112
135791113151719335
2581114810131619568
371115913171215197912
4914914111419151991216
511813111514171719111513
6131017141418172220131815
7151312191717232122152119
8171615151722212224172423
9191919191920222426192719
1035791113151719335
11369121518212427347
1258121613151923195710

We also have fairly low numbers to add at the end. Notice that in each set, not all links are used. I think that is ok.

One thing to consider is whether you want to limit the size of the multiples. Personally, I am happy for the children to go over 12 x 12 if they want to as I don’t see what is being lost really.

I think the highest total possible is just under 130. I could be wrong though!

Over time, you could offer the option of changing the rules to see if it helps them create hard-to-reach scores. Alternatively, they could be used to freshen things up after a while or like a bonus round one week.

What if we could select the three digits to total rather than using the highest ones?

This might be a nice possibility. I do not do this straight from the beginning as I think it creates too many variables initially. However, once they are used to the system, giving them this option might lead to new scores.

What if we used ten multiplications?

Higher scores would obviously be possible. But how high? Would we reach a number above 200?

What if we used the highest four digits?

Again, scores would be higher but how high?

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

There is going to inevitably be a lot of Roaming to start with. However Combing and Seeking will come into it the more the children have a go. Ensuring they do not use the same link or multiplication twice ensures that they will never just completely Roaming and somewhat thoughtful in their selections.

Organise

A class number grid that the children can come back to (as described above) and use to write their names on when they achieve a score will ensure there is a purpose to their activities and reflection on what scores need to be made.

Discover

There are plenty of opportunities to Conjecture. They might initially Conjecture that a score over one hundred is impossible. This would make a nice early challenge. To draw out conjectures, they need time to reflect as well as do.

Investigate
Investigate inquiry reasoning skill in maths

As mentioned in the Discover section, conjectures on the maximum score possible can result in Digging for counter examples. Going beyond 12 x 12 might be a way of Expanding data to stretch the limits of conjectures.

Argue

There is a lot of scope for children to think about what starting times table will help them achieve certain scores. This will likely emerge once they have had a few goes at it.

Explore

Since this is an island that occurs over many sessions, Exploration might not happen for a long time. However, the children could consider alternatives to keep things fresh and interesting. If they are struggling to construct sets that achieve particular scores, they could consider what sort of rule change might make it more attainable.

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