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/www/nrich/html/content/00/11/penta3/
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2011-02-01T00:00:01
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<p>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:<br></br>
<mdo:image align="right" alt="[copyright Microsoft]" src="fig3.gif"></mdo:image></p>
$10$ dozen eggs <br></br>
$27$ litres of condensed milk <br></br>
$480$ tablespoons of sugar <br></br>
$100$ teaspoons of cinnamon <br></br>
$140$ cups of pumpkin <br></br>
<p>Peter looked in the cupboard and found:</p>
$4$ cups of pumpkin<br></br>
$2$ eggs <br></br>
$1 \frac{1}{2}$ teaspoons of cinnamon <br></br>
$\frac{2}{3}$ of a litre of condensed milk <br></br>
$15$ tablespoons of sugar <br></br>
<p>Did Peter have enough ingredients to make two pumpkin pies for
the party or did he need to buy more?</p>
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<p>It looks like Peter is going to have to go shopping!</p>
<p>A way to think about this problem was to find out what
proportion of the original recipe Peter required to make the two
pies. You might have divided and then perhaps multiplied, depending
on your strategy. Using what you know about fractions could have
helped. A number of you sent the answer that Peter was short of
some of the ingredients. There were two ways that the problem was
solved and fully explained. These mathematicians came up with the
following solutions and used their knowledge of decimals.</p>
<p><strong style="font-weight: 400;">Ruth who</strong> lives in
Gloustershire, decided that the answer to the <em>Pumpkin Pie
Problem</em> is, he hasn't got enough ingredients to make two pies.
She got the answer by dividing the ingredients by 80 and
multipliyingby 2. Both <strong><span style="font-weight: 400;">Helena</span></strong> from Tattingstone
School, and <strong><span style="font-weight: 400;">George</span></strong> of Rosebank Primary
School, Leeds, divided mother's recipe by 40. Why do you think they
decided to do this?</p>
<p>Here's <strong style="font-weight: 400;">Ruth</strong>'s
solution, she did an excellent job of setting out her calculations
in a chart.</p>
<table cellpadding="5" border="2">
<tbody>
<tr>
<th>80 pies take</th>
<th>What he needs</th>
<th>What he has got</th>
</tr>
<tr>
<td>120 Eggs</td>
<td>3 eggs</td>
<td>2 eggs</td>
</tr>
<tr>
<td>27L Milk</td>
<td>0.675L Milk</td>
<td>0.66L Milk</td>
</tr>
<tr>
<td>480 tbs. Sugar</td>
<td>12 tbs. Sugar</td>
<td>15 tbs. Sugar</td>
</tr>
<tr>
<td>100 teasp. Cinnamon</td>
<td>2.5 teasp Cinnamon</td>
<td>1.5 teasp Cinnamon</td>
</tr>
<tr>
<td>140 cups of Pumpkin</td>
<td>3.5 cups of Pumpkin</td>
<td>4 cups of Pumpkin</td>
</tr>
</tbody>
</table>
<br></br>
<p><strong style="font-weight: 400;">Helena</strong> and
<strong style="font-weight: 400;">George</strong> both provided
clear explanations. George explained: "To find out if Peter had
enough to make 2 pies I divided each of the ingredients by 40. For
the 10 dozen eggs I first multiplied 10 by 12, because dozen means
12. Then I divided the answer by 40.<br></br>
10 X12 = 120.</p>
<p>120 / 40 = 12 / 4 = 3</p>
<p>Peter had 2 eggs but he needed 3 eggs so he had to buy 1 more
egg.</p>
<p>To find out how many litres of condensed milk was needed, I
first converted the litres of condensed milk into millilitres of
condensed milk. Then I divided the answer by 40 again.<br></br>
27litres X 1000 = 27,000ml</p>
<p>27,000ml / 40 = 675ml</p>
<p>Peter had 666ml but he needed 675ml so he needed to buy 9ml of
condensed milk.</p>
<p>To find out how many tablespoons of sugar Peter needed I just
divided the amount for 80 pies by 40.<br></br>
480 / 40 = 12 tablespoons of sugar.</p>
<p>Peter had 15 tablespoons of sugar so he had 3 more.</p>
<p>To find out how many teaspoons of cinnamon, I divided the
original amount again by 40.<br></br>
100 / 40 = 10 / 4 = 2.5 teaspoons of cinnamon.</p>
<p>Peter had 1.5 teaspoons of cinnamon, so he needed to buy 1 more
teaspoon of cinnamon.</p>
<p>Finally, to find out how many cups of pumpkin were needed I
divided 140 by 40.<br></br>
140 / 40 = 14 / 4 = 3.5 cups of pumpkin.</p>
<p>Peter had 4 cups of pumpkin so he had 0.5 cups of pumpkin more
than he needs".</p>
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<h2>Pumpkin Pie Problem</h2>
<br></br>
<p>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:<br></br>
<mdo:image align="right" alt="[copyright Microsoft]" src="fig3.gif"></mdo:image></p>
$10$ dozen eggs<br></br>
$27$ litres of condensed milk<br></br>
$480$ tablespoons of sugar<br></br>
$100$ teaspoons of cinnamon<br></br>
$140$ cups of pumpkin<br></br>
<p>Peter looked in the cupboard and found:</p>
$4$ cups of pumpkin<br></br>
$2$ eggs<br></br>
$1 \frac{1}{2}$ teaspoons of cinnamon<br></br>
$\frac{2}{3}$ of a litre of condensed milk<br></br>
$15$ tablespoons of sugar<br></br>
<p>Did Peter have enough ingredients to make two pumpkin pies for the party or did he need to buy more?</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>Ratio is a notoriously difficult topic and many teachers avoid it! There are many possible ways of solving <a href="http://nrich.maths.org/public/viewer.php?obj_id=1026&part=index">this problem</a> 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.</div>
<br></br>
<h3>Possible approach</h3>
<div>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 <a href="/content/00/11/penta3/PumpkinPieListsnewlogo.doc">here</a>
).</div>
<br></br>
<div>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.</div>
<br></br>
<div>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.</div>
<br></br>
<div>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.</div>
<br></br>
<h3>Key questions</h3>
<div>Can you paraphrase the question (ask it in a different way) to make it easier to understand?</div>
<div>What do we need to find out?</div>
<br></br>
<h3>Possible extension</h3>
<div>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?</div>
<br></br>
<h3>Possible support</h3>
<div>This is probably not a suitable question for children who struggle with the idea of doubling, halving and other simple ratio.</div>
<br></br>
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How will you find out what you need to make two pumpkin pies?<br></br>
What would you need for ten pies?<br></br></mdoxml>
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Pumpkin Pie Problem
Peter wanted to make two pies for a party. His mother had a recipe
for him to use. However, she always made 80 pies at a time. Did
Peter have enough ingredients to make two pumpkin pies?
Fractions, Decimals, Percentages, Ratio and Proportion
Calculating with ratio & proportion
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Upper primary mapping document