Andy's Marbles
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?
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?
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.
Michael from King Henry VIII found a different method for solving the problem:
Andy has x marbles
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!
Andy's Marbles
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?
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!