# Linked Chains

Here are lots of small chains, each with a certain number of links. The chains that we will use have these numbers of links: 2, 3, 4, 5, 6, 8, 9, 10 and 12. There are many of each size available.

Suppose we want to make a twenty-link chain from the sizes above, then we could use two of the ten-link small chains. Each ten would just be $\frac{1}{2}$ of the 20.

In this case we could write:

where the twenty-link chain is the ONE (1)

What other selection of chains could you put together to make the twenty-link chain, using only the sizes that you see at the top?

Try writing the fractions down in different ways.

Find as many different ways of making a twenty-link chain as you can.

Here is an example using a twelve-, an eight- and a four-link chain:

$\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1$

where the twenty-four-link chain is the ONE (1)

Or, we could write:

a half plus a third plus a sixth makes a one

How many ways can you find to make a twenty-four-link chain?

Could you replace any of your chains with two or three other small chains to make a new combination?

I recommend you look at this Word file Westridge school .doc which shows these pupils' excellent work in an easy format to read and discuss.

Also Sofie from Westridge, sent in the following,

For this problem, I first started with the example the problem showed: 1/2+1/2=1, and split each half into half, which resulted in quarters. Then I split each half, which is a chain of 5, into chains of 2 and 3. When I figured that out, I split chains of 10 (1/2) into chains of 4,6,2, and 8. Then I split those numbers other chains, like a factor tree. If you continue with this pattern, you get many results. But don't forget, there are no 1,7,or 11 chains!

Elliot and Lilly from Husbourne Crawley Lower School wrote briefly,

We found 45 soloutions but we think there will be many more as we only had a maths lesson 2+8+10=20 6+6+4+4=20

Four from Thomas's Academy sent these solutions in;

Yaqub

My way of finding out the equation is to find the equivalent fraction, so for example for finding the 24 chain link, I did 1/3+4/6 (which is the same as 8+16) because 2/6 is equal to 1/3 so 4/6 is equal to 2/3 so the equation really was 1/3+2/3 and for the other equations in the 24 chain link I did the same [I don't mean I copied the same equation I did the same method to solve the over equations]

Hafsa

First I worked out the fractions for the 24 chain then tried a few fraction equations.1/3 + 2/3 =1.

Then I tried some more:

6/12 + 6/12 =1

6/12 =12 so 2 of those would make 24

2/3 + 2/6 =1

2/3 =16 and 2/6 =8 and 16 + 8 =24

Rihanna

First I worked out the fractions for twenty four and used used a linked chain.

6/12+6/12=1.How I did that was 6/12 is 12 and another 6/12 is 12, so 12+12=24. 24 is one. 2/3+2/6=1

Tyra-leighrose

First, I did 2/5+2/4+1/10= 20. Then,I did 4/8+1/2=20. Next,I did 1/2+2/4=20. After,I did 8/10+1/5=20. Finally,I did 4/5+2/10=20.[I used equivalent fractions and different strategies to solve the equations.]

To get to 24.

First, I did 6/12+3/6=24. Then,I did 4/8+1/3+1/6=24. Next, I did 2/6+1/3+4/12=24. Finally, I did 3/6+1/3=24. [I used different methods to work this out.]

Thank you all those excellent solutions which have come in for this new activity.

Why do this problem?

This activity is an unusual context in which pupils can consolidate recognising, finding, naming and writing fractions. The rich environment also gives them the opportunity to identify, name and write equivalent fractions of a given fraction, represented visually as a chain. Furthermore, learners will be adding and subtracting fractions with the same denominators and denominators which are multiples of the same number.

### Possible approach

Having some practical resource available for children to use if they wish is key to making it accessible. You may be able to borrow links that can be joined to make a chain (perhaps from your Early Years setting), or you could make decorations from loops of paper. Alternatively, you could use a rod of each number of connecting cubes that are the same colour for each length.

Have many examples of each size of chain and give the group the challenge of making a twenty-link chain. Depending on their experience of naming fractions, you could just use tens and fives to begin with to help the pupils have a feel for them being halves and quarters of the twenty.

Once they are comfortable about referring to the small chains as fractions of the twenty-link chain then you can invite them to explore the many ways of making the twenty-link chain as a whole.

The first part of the activity focusing on the twenty-link chain could be a 'simmering task', so that learners consider it out of lesson time and contribute their ideas to a working Maths wall, for example. You can then revisit that first part, perhaps even several times, before moving on to the twenty-four-link chain and the twenty-seven-link chain.

### Key questions

Tell me about the fractions you've decided to use.

Could you replace your half (*or other suitable fraction*) with two or three other small chains?

### Possible extension

Encourage learners to take three or four random small chains and put them together. What fraction of the twenty/twenty-four/twenty-seven do they make?

For example, if you chose the 4, 6, 8 as $\frac{1}{5} + \frac{3}{10} + \frac{4}{10}$ you would have $\frac{9}{10}$ when twenty links is the whole 1.

The same three would be $\frac{1}{6} + \frac{1}{4} + \frac{1}{3}$ and you would have $\frac{3}{4}$ when twenty-four links is the whole 1.

### Possible support

Children who have difficulties with fine motor skills may need help manipulating the small chains.