Distinct Digits to 5555

This island focuses on the use of column addition and even though it focuses on only four digits, it promotes a trial and improvement approach that requires a lot of mathematical thinking as well as having the potential to naturally involve decimals. The best thing of all is the twist at the end that truly exemplifies how maths is a creative subject. It’s one of my favourite lessons to teach – it always brings out the best in the children and the sense of astonishment at the end with a score of 9 is so joyful.

Start by asking for any addition that makes 5,555.

And then another…

And another…

At this point, the children should be focused on the target sum of 5,555 and we can introduce the extra rule.

What if we scored a point for every digit that we use only once in the addends?

How many scores are possible with a sum of 5,555? The example I’ve given scores 3 (for 1, 8 and 7). This is a nice starting point for them to try and beat your score.

Search inquiry reasoning skill in maths

Once you have done an example together, the children can explore which scores are possible. This provides the opportunity for the use of all 3 Search skills. They can Roam initially, Comb by making adjustments to their existing addends to try and score different number of points and Seek to try and target specific scores. It’s not just about the highest score, it is all the scores that are possible.

Seek in particular might be used to try and obtain the highest score which you can expect a conjecture to claim as 8. There are 48 ways this can be done (for example, 2097 + 3458). Examples of scores of 0 to 8 are given below.

01234 + 4321
One number is a reversal of the other in this solution.
12667 + 2888
Forced to think about how you can get away from pairs of numbers through regrouping. It takes a bit of thought to realise the importance of both addends being in the two thousands. Also possible to solve with decimals.
21113 + 4442
32383 + 3172
41103 + 4452
4 points is the most common number of points. Something you might highlight with a frequency chart from their solutions. This brings in the Organise skills.
52026 + 3529
62180 + 3375
74582 + 973
This is only possible with a 4-digit and a 3-digit number.
82097 + 3458
The nice thing about this solution is that no two columns total the same amount which is somewhat surprising.
Argue inquiry reasoning skill in maths

To beat a score of 8, the obvious answer is to use decimals. It really brings out the determination in the children and an understanding of needing to try and make pairs to different totals. But no matter how they arrange them, a score of 9 or 10 just seems impossible. The likely conjecture of 8 being the highest score seems to hold true. Most of the time, the teacher’s role is not to provide the ‘answers’. There is no point usually in the children making all sorts of discoveries and then at the end of it, the teacher saying, actually, you were wrong. However, in this example, the potential shift in mindset about maths is too good an opportunity to miss. I ask them if they think we’d have to be really creative to get a score of 9. I then ask them how creative it would be if we could get a score of 9 without using decimals. It brings in the Argue skills beautifully because the children can explain why that just seems impossible. There are only 4 columns, so how can you have more than 8 digits…

What if we used 3 addends?

3498 + 2056 + 1

This is why I love maths! The seemingly impossible achieved!

If you have time, you could search for a score of ten using this route. I don’t think it is possible. The closes I’ve managed to 5,555 is by making 5,553 with 1 + 2 + 8 + 9 + 73 + 5460 which is pretty nice in itself.

Explore inquiry reasoning skill in maths

What if we tried 4,444 or 6,666?

This is the obvious way we could use the Reorient skill to Explore further. Normally, I find that there isn’t enough time for this in a lesson but you could look at it in a further lesson.

What if we used subtraction?

A possible new island altogether through Voyaging is with subtraction. Just remember that a score of 8 is possible without regrouping (9,876 – 4,321) so I normally add a new rule that their calculations must involve regrouping in at least one column as I want to build their fluency in column subtraction.

What if we had to have a decimal component in our addends?

Which scores are possible now? Would that change if we tried to make 555.5 instead? A nice link to place value with the latter and opportunities for them to use the Argue skills.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

An important aspect of this island is to encourage Seeking and Combing. Get the children to keep track of which scores they are creating. This will encourage them to try and make the scores that they haven’t made. It isn’t too hard for them to make a score of 4 lots of times but some of the scores require more creative thinking and problem solving.

Organise

The main aim here is to keep track of the scores that they have made as this will encourage them to create the scores that they haven’t managed to make.

Discover

The commentary on the various scores above provide some insight into what sort of conjectures you might expect. The nice thing is that with this island, there are conjectures made that prove to be false. For example, that 8 is the highest score possible.

Investigate
Investigate inquiry reasoning skill in maths

Because there are so many possible combinations of digits, it’s one where you should really emphasise the need to Investigate conjectures thoroughly. It’s impossible for them to try every combination so they have to try and consider what they can do to try and think of counter-examples.

Argue

There is so much potential here. For example, how the digits have to be arranged for a score of 1 – one of the pairs in a column needs to be reused in another column which means you have to use regrouping to ensure that different totals in those two columns are made. What does a score of 7 require and why? I’ve had some brilliant explanations and ah-ha moments with this one.

Explore

This isn’t necessarily an island that needs lots of new directions to be successful. The creativity is within the solutions that they come up with for the initial island. That’s ok though, if the depth is there, there is no need to go too much further. It doesn’t mean you couldn’t though and children can still come up with some great ideas.

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