Opposite Digit Circles

With this island on addition, it takes a little bit to set it up but once you have done so, it’s fairly easy to follow and the patterns are quite surprising. It’s an example of when maths at first just seems quite magical and mysterious. However, when studied closer, the question of why the same digits are appearing make sense. This allows for verbal reasoning from the children in explaining their findings.

What if we had a circle of numbers and split them into two halves?

We now have the top half (8, 9, 0, 1 and 2) and the bottom half (3, 4, 5, 6 and 7).

Let’s form a number using values from the top half as digits. I would do this several times with the children.

My example is 1,298.

Opposite Digit Circles

Now create another number using the digits vertically opposite the values that were used in the top half. Opposite the one, would be four; opposite the two, would be three; opposite the nine, would be six; and opposite the eight would be seven.

Our second number would be 4,367. You can see how I did this in the image on the left.

What if we added them together?

1,298 + 4,367 = 5,655

Search inquiry reasoning skill in maths

There isn’t an immediate goal of this island. We aren’t seeking something and as such, it is using the Roam skill to see what we can find. Just create some examples and see what you notice.

If you then collect and compare examples that the class come up with, it should be fairly obvious that most of the totals consist of numbers that only have digits of 5 or 6.

1,890 + 4,765 = 6,655

  • 1,290 + 4,365 = 5,655
  • 2,111 + 3,444 = 5,555
  • 1,988 + 4,677 = 6,665

The exception is when the thousands column totals over 10: 8,021 + 7,534 = 15,555.

With this circle, they should be able to explain what is going on. All the pairs add up to either 5 or 15. The two and the three are vertically opposite each other. They make five. In the top half, the one is next to the two and is one fewer. However, the one is opposite the four which is one more than three. The total is maintained. If the total between pairs is 15, then the next column will be a 6 rather than a 5 due to regrouping.

We can look at altering things slightly to explore further.

What if changed the order of the numbers so that zero wasn’t at the top?

What if we had a circle that went up to 6 rather than 9?

In knowing what happened to the original circle and why, we can extend that thinking and reasoning to these further circles. They can try to predict what they think will happen. This linking back is really important for them to do to ensure it isn’t just some addition calculations but instead pattern seeking.

You could get them to Seek through their What if…? questions in finding another circle that has similar outcomes.

What if the digits for a number could come from both halves but you still chose the digits for the second number using those opposite?

For example, 258 for the first number and 307 for the second.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

As there isn’t an immediate goal with this island, we are mainly focusing on Roaming here.

Organise

Comparing plays an important role because it is through sharing the data that we have as a class that we start to see what is going on.

Discover

After we have looked at the initial arrangement of numbers, we can use the data that we have to make conjectures about possible alterations from further ‘what if…?’ questions. It’s important that children think about the impact that their changes will make before they carry out. There should be reasoning behind what they are saying.

Investigate
Investigate inquiry reasoning skill in maths

Once they have conjectured about how they think the changes they make will impact their answers, the Investigation is simply to try out a few examples. It isn’t complex in this example but still worthwhile explicitly talking about.

Argue

There is real potential for them to explain why the answers are as they are. By starting off with a focused case and then expanding, we give them a greater chance of being able to explain what is going on and why.

Explore

The Explore skills are really important in this inquiry to drive it forwards. Their is definite potential for them to use Charting to explore an avenue and they can easily come up with their ideas and explore them further.

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