What if we tried to make a sum of 5555 using each digit as few times as possible?

This scenario provides the opportunity to develop fluency in addition whilst analysing a scenario that promotes thinking about creative solutions.

Self Regulation: Numicon can be used as a way of keeping track of which scores have been found to be possible.

As a starting point, the class can explore the number of points that can be scored. The above example scores 6 with 1, 2, 5, 7, 8 and 0 being used once each. The 3 is used twice.

If a new score is made, the equivalent Numicon piece can be added to a pile in the middle of the table. This helps them focus on new scores rather than repeating old ones. In doing this, the scores that are left are generally the ones that are going to be more challenging.

Initial questions

• What is the highest score possible?
  • Is is it possible to use all ten digits only once?
• What is the lowest score possible?
• Which scores are more challenging to create?
  • Are there any scores that are impossible to make?

Children might start with trivial ways of making 5 in each column using pairs of 5 and 0, 4 and 1, and 3 and 2. The nice thing about this scenario is it rewards using grouping and exchanging to increase the number of possibilities. You then have pairs of numbers that make 15 in the ones column but equally pairs of numbers that make either 4 or 14 in the tens and hundreds columns. It emphasises how each column can be impacted on by the previous one.

Highest score possible: 8?

In the example on the right, 8 points are scored using these ideas. Note that no single column has the same total, something that might seem surprising and worthy of observation.

After much exploration, a score of 8 appears to be the maximum possible. More on that below!

A score of one – easy?

The lowest score of zero is pretty trivial to make: 4444 + 1111 for example. This might lead you to think that a score of one is equally forthcoming. Except it isn’t at all. The importance in this is the exploration of why. The children can tackle it and consider what the problem is and it’s a great opportunity for them to build a justification for the challenge – I try not to rob them of it.

Often, new scores can be achieved by manipulating previous scores, a great thing to promote in itself. If we take the example of scoring zero, 4444 + 1111, and then try and have one digit used only once, we come to a problem. As soon as we change one of the fours or one of the ones, it’s pair in the column also needs to change to make the column total 5: 4443 + 1112. This obviously results in a score of 2. Dealing with this provides the a great example of using creativity within the problem. Decimals.

This in turn, requires the problem to be broken down step-by-step.

1. If we have a tenths column, it needs to total 10 so that we make one whole. Let’s say 9 and 1.
2. This means that the total in the ones column must equal 4.
   1. The consideration then in the ones column is between 2 and 2, 1 and 3, and 0 and 4.
   2. I want to minimise the number of points, so using 2 and 2 serves this purpose.
3. If I have 9 and 1 in the tenths column, I need to repeat one of them in the other columns.
4. Repeating 1 is going to be easier, so I can use 4 and 1 in the tens, hundreds and thousands columns.

So I have, 4442.9 + 1112.1

Definitely, a creative solution! The addition itself might not challenge many people, but its construction, and the depth of thinking behind it, requires resiliency and the use of problem-solving strategies. Even if you had the idea of using decimals, simply using trial and improvement would unlikely get you there. It requires exploring the impact of grouping and exchanging in depth. It’s this kind of challenge that makes this scenario fairly limitless in terms of being low threshold, high ceiling.

Revisiting the highest score

The obvious route from this to try and reach a score higher than 8 is through decimals as well.

Try it.

It just doesn’t seem to work. There just never seems to be the right pair left to make all the columns work. It would be easy to give up. But, as I said, this activity is a great one to promote creativity in maths. At this point, I usually go all out for dramatic effect. I explain to them that not only is a score of 9 possible, it doesn’t even require decimals. They can’t imagine it at all, can you? That’s the beauty of it.

A score of 9!

How is it possible?

Definitely, a creative solution.

As always, there are possibilities to ask further ‘what if…?’ questions. For example, if a score of 10 for 5555 isn’t possible, are there any numbers close by for which it is possible? What about 6666? 7777?

For the first of those questions, there is a nice solution using the idea above: 5553 = 1 + 2 + 8 + 9 + 73 + 5460. Yeah, that took a while to work out!

What if we used subtraction?

The obvious expansion would be to use subtraction. In this case, a score of 8 is perhaps a little too easy: 9876 – 4321. This makes it a bit less effective. You could add in an extra rule requiring at least one example of regrouping. The nice thing about the addition problem is that it naturally rewards grouping and exchanging. You will need to play about with the rules to achieve this with subtraction. But hey, that’s what maths is all about, right? Have a go, be creative with the rules and come up with your own ideas to generate a new scenario that is interesting to explore.