Add the Ones, Double or Halve

Chains are always a good source of exploration and practice. Particularly so if you can make some sort of class version. Doubling and halving are such fundamental aspects of being confident with playing with numbers. I have a few chains on doubling and halving numbers under 100. This one results in some intriguing discoveries.

Start with a two-digit number. Add the ones digit to the number. If it is greater than or equal to 50, halve it, otherwise, double it.

37

Add 7 to get 44. 44 is less than 50 so double it.

88

Add 8 to get 96. 96 is greater than 50 so halve it.

48

Add 8 to get 56. 56is greater than 50 so halve it.

28

Add 8 to get 36. 36 is less than 50 so double it.

72

Add 2 to get 74. 74 is greater than 50 so halve it.

37

We have reached our original number. Does this happen to all numbers?

As can be seen in the examples, all starting numbers end up in a loop although not necessarily involving their original number.

23
+3
x2
52
+2
÷2
27
+7
x2
68
+8
÷2
38
+8
x2
92
+2
÷2
47
+7
÷2
27
55
+5
÷2
30
+0
x2
60
+0
÷2
30
17
+7
x2
48
+8
÷2
28
+8
x2
72
+2
÷2
37
+7
x2
88
+8
÷2
48

There are a few possible loops:

A) 44, 96, 51, 26, 64, 34, 76, 41, 84

B) 68, 38, 92, 47, 27

C) 28, 72, 37, 88, 48

D) 40, 80

E) 30, 60

These loops consist of just 25 numbers in total. Interestingly, which loops numbers appear in also is a pattern. Starting with the number ten and going up each number appears in the loops as follows:

ACBAD ACBAD ACBAE ABCAD ACBAD ACBAD ACBAE ACBAE ACBAE ACBAE ACBAD ACBAD ACBAD ACBAD ACBAD ACBAD ACBAE ACBAE ACBAE

So essentially, we have this pattern of always having the A, C, B and A loops with either D or E and then repeating.

Some loops contain the starting number but not many:

27, 28, 30, 34, 37, 40, 48, 51, 60, 64, 72, 76, 84, 88, 96

It’s curious how most of these are multiples of 4.

The obvious thing to change is the rule that we are using between numbers. They could try and find other looping rules.

What if we used a number other than 50?

This is the simplest change that can be made. Generally, larger numbers result in longer chains and lower numbers result in shorter chains.

What if we added the tens digit instead?

With this, you need to avoid decimals and so if halving, an additional rule of adding one is required if there is an odd number. This does eventually loop but they do not start looping you have 25 or more numbers so be prepared for long chains if you go down this route.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

This is not an island that really promotes Searching as a skill as it is less about choosing numbers and more about following the numbers.

Organise

Organising plays a big role if patterns are to be spotted. It would be easy to miss the pattern in the way the loops are predictable for different numbers if there is no organisation in terms of the outcome from consecutive numbers.

Discover

This is a an island that promotes the use of noticing as their is not a problem-solving element to it. It is more about generating the chains and then spotting the patterns. Once they have been spotted, conjectures can be made about the results of untried numbers.

Investigate
Investigate inquiry reasoning skill in maths

There will be a large part of investigating as essentially, the children will be seeing if all numbers loop and whether starting numbers follow a loop that has been predicted of them.

Argue

To highlight this skill, it would be worth focusing on the 40, 80 and 30, 60 loops as they are the most easily explained.

Explore

This is going to be more challenging with this island. That is because making changes to the rules will not necessarily result in looping patterns. However it is something that could be explored. I have a few other similar looping chains involving doubling and halving (that I will write about when I get a chance) and so it might be after they have experienced several that they could explore their own idea.

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