# Andy's Marbles

Andy had a big bag of marbles but unfortunately the bottom of it split and all the marbles spilled out. Use the information to find out how many there were in the bag originally.

## Problem

Andy and his friend Sam were walking along the road together. Andy had a big bag of marbles.

Unfortunately the bottom of the bag split and all the marbles spilled out. Poor Andy!

One third ($\frac{1}{3}$) of the marbles rolled down the slope too quickly for Andy to pick them up. One sixth ($\frac{1}{6}$) of all the marbles disappeared into the rain-water drain.

Andy and Sam picked up all they could but half ($\frac{1}{2}$) of the marbles that remained nearby were picked up by other children who ran off with them.

Andy counted all the marbles he and Sam had rescued.

He gave one third ($\frac{1}{3}$) of these to Sam for helping him pick them up. Andy put his remaining marbles into his pocket. There were $14$ of them.

How many marbles were there in Andy's bag before the bottom split?

What fraction of the total number that had been in the bag had he lost or given away?

## Getting Started

How many marbles did Andy and Sam rescue?

How might this help you to work out the number of marbles which Andy had before the bag split?

## Student Solutions

Several very well reasoned solutions came in for this problem about Andy's marbles. Katherine from Treffos, North Wales, pupils from Wigginton Primary and Stephen from Frisby School all went about the problem in a similar way. Stephen said:

I know that to solve this problem I have to work backwards, beginning with the last part, and ending with the first.

So... Andy walks away with 14 marbles, 2/3 of what was recovered. His friend walks away with 1/3 of what was recovered. 14 is 2/3 so to find out 1/3 I then half 14, getting 7. Sam had 7 and Andy had 14. A total rescued of 14 + 7 = 21

21 is half (1/2) of what was lying around on the ground, the other half having been taken by the children. To find how many were on the ground nearby I have to double 21. I now have an answer of 42.

42 must be doubled to get the final answer, as 1/3 and 1/6 went down drains or hills (1/3 + 1/6 = 1/6 + 2/6 =1/2).

This gives me a final answer of 84 marbles in the bag.

Michael from King Henry VIII found a different method for solving the problem:

Andy has x marbles

1/6 go down the drain, and 1/3 roll down the slope

That comes to 3/6 or 1/2

x/2

1/2 the remaining 1/2 are stolen, leaving 1/4

x/4

He gave 1/3 to his friend, leaving 1/4*2/3 0r 2/12 or 1/6

x/6

That left 14

x/6=14

Times both sides by 6 and you get x=84

We also received two solutions from students at The British School of Brussels, both of whom used a bar modelling approach. Here is Monty's:

And here is Julia's:

Can you see how they have used a bar to represent the total number of marbles that Andy had to begin with? Then they have split this 'whole' into parts according to the fractional amounts stated in the problem.

These are all very clear solutions, well done. Working from the end of a problem can be a useful way of tackling it - something to remember!

## Teachers' Resources

**Why do this problem?**

This problem involves complicated reasoning about fractions that challenges children's understandings of the concepts involved. It is a good example of how fractions relate to multiplication and division.

Possible approach

You could start a lesson with some oral challenges, such as:

You could introduce the problem verbally, as a printed sheet or on an interactive whiteboard. Ask learners to spend a bit of time working on it in pairs and then share ideas on how to get started amongst the whole group. It would be good to encourage children to jot down whatever they find helpful as they tackle the problem. Make it clear that these jottings are purely for pairs themselves and that you do not need to be able to understand them!

Key questions

### Possible extension

Children could be encouraged to make up their own version of a similar problem for a friend to solve.

Possible support