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Pumpkin Pie Problem

Peter, the pumpkin eater, wanted to make two pies for a party. His mother, a professional pie maker, had a recipe for him to use. However, she always made $80$ pies at a time. She used:
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$10$ dozen eggs
$27$ litres of condensed milk
$480$ tablespoons of sugar
$100$ teaspoons of cinnamon
$140$ cups of pumpkin

Peter looked in the cupboard and found:

$4$ cups of pumpkin
$2$ eggs
$1 \frac{1}{2}$ teaspoons of cinnamon
$\frac{2}{3}$ of a litre of condensed milk
$15$ tablespoons of sugar

Did Peter have enough ingredients to make two pumpkin pies for the party or did he need to buy more?

Why do this problem?

Ratio is a notoriously difficult topic and many teachers avoid it! There are many possible ways of solving this problem which make it ideal for class or group discussion, and offer opportunities to assess your children's understanding and misconceptions. You will need to allow time for a reasonable discussion.

Possible approach

Working in pairs gives the children opportunities to clarify their thinking. Give each pair a large piece of paper on which to record - many may want to draw pictures and working on small pieces of paper isn't helpful. Having explained the problem, provide each pair with the two lists (you can download a printable copy here).

Give a little time for the children to 'get into' the problem. When appropriate, draw the group or class together and ask what they have found useful to begin. Give more time for them to use some of these ideas, or continue with their own.

As each pair comes to a solution, ask them to prepare a new piece of paper which will help them to explain their working. Choose a confident pair to start and then invite each pair in turn to explain what they have done. The rest of the group can ask questions if clarification is needed.

Focus the children's attention on the different ways pairs have solved the problem but how all involve scaling amounts up and down - ratio and proportion.

Key questions

Can you paraphrase the question (ask it in a different way) to make it easier to understand?
What do we need to find out?

Possible extension

Most children will have focused on the eggs first as these are easier to calculate than litres, teaspoons etc. The question is then answered as no matter how much of everything else he has, he doesn't have enough eggs. Children who find this quite easy could be challenged to find out what additional other ingredients would have to be bought for two pies. What is the largest number of pies he could make without buying more of each ingredient?

Possible support

This is probably not a suitable question for children who struggle with the idea of doubling, halving and other simple ratio.