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Andy's Marbles


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

 

Bag of Marbles


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

Bag of Marbles with marbles falling out

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.

Pile of marbles


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?


 

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:
"I bought some apples at the market. After I had given half of them to my sister I had $7$ left. How many did I buy?"
"Tom gave a quarter of his bag of sweets to Ben and ate half of them himself. He had $6$ left. How many sweets were there in the bag to begin with?"
Encourage learners to talk to each other about how they solved each of the above - they may not have used exactly the same method.

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

Where shall we start?
How many marbles did Andy and Sam rescue?
How might this help you to work out the number of marbles Andy had before the bag split?

Possible extension

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


Possible support

It might help some learners to start slowly from the end asking questions such as:
"How many marbles did Andy have at the end?"
"How many did he give to Sam? And how many did he keep for himself?"
"Can you work out from that how many Andy and Sam picked up altogether?"