Digital Root Chains

There are a few steps involved with this island that focuses on multiplication. It really exemplifies though that if you just create ‘what if…?’ rules, patterns will follow. Due to the steps required, the Landing Spot requires a bit more work to ensure that the children are confident in proceeding forwards. However, there is lots for them to explore and find out when they have got the hang of them. It also explicitly highlights using an existing multiplication to work out a similar multiplication – a valuable skill.

It will be easier if the children understand how to find the digital root of numbers before you begin. To find the digital root, we find the sum of the digits in a number repeatedly until we have a single-digit number.

14 -> 1 + 4 = 5

38 -> 3 + 8 = 11 -> 1 + 1 = 2

97 -> 9 + 7 = 16 -> 1 + 6 = 7

6 -> 6

5352 -> 5 + 3 + 5 + 2 = 15 -> 1 + 5 = 6

Now we have that established, we are going to multiply two numbers together, one will be a single-digit number.

543 x 4 = 2172

And we will take the digital root of the answer:

2 + 1 + 7 + 2 = 12

1 + 2 = 3

On this island, our goal is a digital root of four as we are multiplying by four. However, we got a digital root of three. Since four is one more than three, what if we add one to our multiplicand and tried again?

What if we keep trying that process until we get a digital root of 4?

CalculationDigital Root of ProductModifier
543 x 4 = 21722 + 1 + 7 + 2 = 12
1 + 2 = 3
+1
544 x 4 = 21762 + 1 + 7 + 6 = 16
1 + 6 = 7
-3
541 x 4 = 21642 + 1 + 6 + 4 = 13
1 + 3 = 4
Target reached!

The number 543 took three steps to reach the target digital root of four.

What if we tried other multiplicands?

We are still multiplying by four in this avenue of exploration. Most numbers require three steps to reach a digital root of four although some are fewer. No numbers appear to take more than three steps.

Hopefully, the children are able to realise that after the initial step, they don’t need to carry out multiplications for further steps as they are either adding or subtracting by multiples of four. Through this, the island is developing a deeper understanding of multiplication and greater fluency. This is a connection that is really worth emphasising in this island. If I’ve worked out 543 x 4, I don’t need to use a formal method to work out 541 x 4.

What if we use different multipliers?

It’s important that they focus on a particular multiplier to be able to identify patterns. As a class, it might be worth focusing on another multiplier together before giving freedom. There is so much to explore.

MultiplierNotes
1It would be easy to dismiss one as a multiplier but it is still a valid exploration. No matter what number is started with, it will take two steps unless the starting number has a digital root of one already in which case it will be one step.
2Unless the original number has a digital root of two, it is impossible to reach a digital root of two. It cycles between two digital roots. For example:

587 x 2 = 1174 -> digital root of 4
585 x 2 = 1170 -> digital root of 9
578 x 2 = 1156 -> digital root of 4
576 x 2 = 1152 -> digital root of 9

There is still a lot to explore though. Are the pairs of digital roots always the same? Does the multiplicand always decrease? Plenty of opportunities for conjectures and investigations.
3With a multiplier of three, whatever the digital root is of the first calculation ends up being the digital root of every calculation in the chain. Those digital roots are either three, six or nine. For example:

602 x 3 = 1806 -> digital root of 6
599 x 3 = 1797 -> digital root of 6
593 x 3 = 1788 -> digital root of 6
4As mentioned above, a digital root of four is obtainable for every number within three steps.
5A digital root of five is never attainable unless obtained from the first step. Instead, the chain goes back to the original number either after 2 or 6 calculations which is curious in itself.
6With a multiplier of six, it functions in the same manner as a digital root of 3. This isn’t surprising with the knowledge of how the digital root is used in the divisibility rules of 3 and 6 (and 9).
7As with a multiplier of four, a digital root of seven is obtainable for any number within three steps.
8A digital root of eight is never attainable unless obtained from the first step. The multiplicand gradually increases (although sometimes a step decreases it). It doesn’t return to the original number.
9A digital root of nine is always obtainable from the first step no matter what the number is.
Strange treasure

Hopefully, you will get a lot of conjectures from what is noticed. It’s nice to look at the similarities and differences between the multipliers. We perhaps wouldn’t have expected four and seven to be similar. The treasure that we uncover here is strange in nature.

It is most likely that the burden of proof here will be from the sheer number of examples that show the patterns that they come up with. In the Isles of What If…?, at primary level, it is appropriate that once an entire class has been convinced that a conjecture is true and there is data to back it up, that is enough. This is quite a difficult one for them to explain independently but understanding is possible. Let’s look at what goes on with a multiplier of five. To show what is going on, I’m going to use an organised list of multiples of five. What can you notice from the table?

CalculationDigital RootModifierNext
Calculation
10 x 5 = 505Target
11 x 5 = 551+415 x 5
12 x 5 = 606-111 x 5
13 x 5 = 652+316 x 5
14 x 5 = 707-212 x 5
15 x 5 = 753+217 x 5
16 x 5 = 808-313 x 5
17 x 5 = 854+118 x 5
18 x 5 = 909-414 x 5
19 x 5 = 955Target

When the ones digit is a zero, the digital root is just the value of the tens digit. With the next multiple, a five replaces the zero in the ones column, giving a digital sum of five more (70 to 75 results in a digital sum going from 7 to 12). As we are finding the digital root and not the digital sum, we add up the digits of 12 and get 3. Looking at it from the point of view of the numbers aligned in a circle as on the left (modular arithmetic), we can see that the digital root goes round by 5, then 6, then 5, then 6. It repeats itself. One cycle of this pattern is shown in the table above.

If we pick an initial calculation with a product that has a digital root that is not five, the next calculations are all unique and if we follow them a cycle is formed and the original number is reached no matter what the starting point is.

Other multipliers can be thought of similarly. Whether or not this is worth delving into would be up to you and dependent on the nature of the class you have. But I thought I would include it for anyone interested. One option might be to look at using a multiplier of one in explaining this as it is easier to see what is going on then.

What if we increased or decreased the tens digit instead of the ones digit?

The nice thing is that it surprisingly makes no difference whatsoever to what happens. No changes is a pattern in itself! It makes sense when you think about that each digit is valued equally.

What if we found the digital product instead?

There is almost as much to explore here as the original island so it might be challenging to fit in. However, why not challenge them to explore this themselves? I often find that children want to go further in their investigations and will carry them on in lunch or at home.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

Combing definitely helps in understanding what is going on. If you can promote this, and get them adjust the numbers they start with by small amounts, they are more likely to see the patterns in the digital roots.

Organise

The children can Arrange and Compare by looking at the different multipliers and what happens. You can also take the opportunity to Classify them into groups (sets). For example, 4 and 7 might be in the 3-or-fewer set.

Discover

So much to make sense of with the different multipliers. This is one where you can expect plenty of conjectures. Most of the conjectures are going to come from them using their Noticing skills.

Investigate
Investigate inquiry reasoning skill in maths

When investigating, think about how you can promote them Expanding their sets of data. Try low numbers, high numbers, different-length digits. When it is harder for them to prove something by explaining it, a breadth of data is important in validating conjectures.

Argue

Explanations of what is going on might be a bit tricky for the children on this island. However, they can make reasonable and age-appropriate attempts. For example, they might think about why using three, six and nine as multipliers produce similar results.

If you get on to adding or subtracting tens instead, their Arguing skills can be used to explain why nothing has changed in the results.

Explore

There is a lot of data required from the initial island to be convinced of what is going on. That might leave less time for further exploration. The tens digit exploration is worth doing though as children are always surprised when nothing changes at all.

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