### Tangles

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

### Bryony's Triangle

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

# Chocolate

## Chocolate

This challenge is about chocolate. You have to imagine (if necessary!) that everyone involved in this challenge enjoys chocolate and wants to have as much as possible.

There's a room in your school that has three tables in it with plenty of space for chairs to go round. Table $1$ has one block of chocolate on it, table $2$ has two blocks of chocolate on it and, guess what, table $3$ has three blocks of chocolate on it.

Now ... outside the room is a class of children. Thirty of them all lined up ready to go in and eat the chocolate. These children are allowed to come in one at a time and can enter when the person in front of them has sat down. When a child enters the room they ask themself this question:

"If the chocolate on the table I sit at is to be shared out equally when I sit down, which would be the best table to sit at?"

However, the chocolate is not shared out until all the children are in the room so as each one enters they have to ask themselves the same question.

It is fairly easy for the first few children to decide where to sit, but the question gets harder to answer, e.g.

It maybe that when child $9$ comes into the room they see:

• $2$ people at table $1$
• $3$ people at table $2$
• $3$ people at table $3$
So, child $9$ might think:

"If I go to:
• table $1$ there will be $3$ people altogether, so one block of chocolate would be shared among three and I'll get one third.
• table $2$ there will be $4$ people altogether, so two blocks of chocolate would be shared among four and I'll get one half.
• table $3$, there will be $4$ people altogether, so three blocks of chocolate would be shared among four and I'll get three quarters.
Three quarters is the biggest share, so I'll go to table $3$."

Go ahead and find out how much each child receives as they go to the "best table for them". As you write, draw and suggest ideas, try to keep a note of the different ideas, even if you get rid of some along the way.

THEN when a number of you have done this, talk to each other about what you have done, for example:

A.  Compare different methods and say which you think was best.

B.  Explain why it was the best.

C.  If you were to do another similar challenge, how would you go about it?

### Why do this problem?

This is an excellent problem for helping youngsters to develop their concepts of fractions. It's not so much to do with arithmetical manipulation of fractions but more to do with youngsters exploring and developing their ideas.  By encouraging learners to share their methods, there is an opportunity to discuss which might be the 'best' (this might depend on the individual's preference too).

### Possible approach

Children will need plenty of (the same sized) paper available for folding and tearing in order to explore sizes of fraction.

To begin the activity, you could act out the problem using large sheets of paper to stand for the chocolate bars (or real bars!), placed on tables. The acting could go through the situation with the first six children coming up one by one to the tables. Encourage them to justify their decisions and make sure the whole group agrees with their choice.

Then learners could work in pairs on what happens when further children come to the tables. By listening to their conversations you can get a good insight into the ways that those youngsters think about and visualise fractions. Some surprises are very likely!

After some time, bring everyone together again to talk about their ways of working.  Invite comments about each method and then once all the different ways have been explained, ask pairs to discuss which method they would use now they have seen so many.  You can then suggest they continue working on the problem, choosing any approach (or see the extension below).  It would be interesting to talk to those pairs who have changed the way they tackle the task to find out what it is about their new method that they preferred to the original one.  Some of their reflections could be recorded for display.

### Key questions

What size do you think this is ...?
Why? (In response to the answer to the above.)

### Possible extension

Challenge children to come up with a system or pattern that would help them to solve similar challenges.

### Possible support

You could start off with just two tables and a total of three bars of chocolate.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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