6883
/www/nrich/html/content/id/6883/
1
2011-02-01T00:00:01
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<p>On Tuesdays Green Class sometimes do baking.<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="" height="115" src="Bix1.png" width="167"></mdo:image></div>
<br></br>
<p>One Tuesday Miss King said that William's group were going to make biscuits. There were four children in the group besides William. They are Ali, Jess, Karni and Danny.<br></br>
<br></br>
William, Jess, and Karni were going to work together to make biscuits with chocolate chips in them.<br></br>
<br></br>
Ali and Danny were going to make ginger cookies.<br></br>
<br></br>
When Miss King took Ali and Danny's tray of biscuits from the oven they looked like this:<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="12 biscuits" height="211" src="Bix2.png" width="279"></mdo:image></div>
<br></br>
<p>How many biscuits did they make?<br></br>
<br></br>
"Let's share them out!" cried Danny, "There are lots of biscuits! We'll get lots each, won't we?"<br></br>
<br></br>
If they shared them between just Ali and Danny, how many would they get each?<br></br>
<br></br>
"What about Will, Jess, and Karni?" objected Ali, "They should get some too."<br></br>
<br></br>
Just then Miss King took the other tray of biscuits from the oven. They looked like this:<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="8 biscuits" height="210" src="bix3.png" width="278"></mdo:image></div>
<br></br>
<p>How many biscuits did the three of them make?<br></br>
<br></br>
"Ooh! They smell good!" exclaimed William.<br></br>
"Can we eat them now?" asked Karni.<br></br>
"We will have to cut some of them up, if we are going to divide them between us fairly," said Jess thoughtfully.<br></br>
<br></br>
If they shared them between the three of them, how many would they get each?<br></br>
<br></br>
Ali looked at both trays. He counted all the biscuits. How many were there altogether?<br></br>
<br></br>
They decided to share all the biscuits between all five of them. How many did they get each?<br></br>
<br></br>
"What about Miss King?" Danny asked suddenly. "She might like some biscuits too."<br></br>
Can they share all the biscuits equally between the five children and Miss King?<br></br>
How many more biscuits could they bake so that they could share them fairly without cutting any of them up? </p></mdoxml>
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<p><span class="editorial">Finlay from Stalyhill Junior School
answered the final part of this problem, which asked how many more
biscuits the children would need to bake so that they could share
them equally with Miss King too. Finlay said:</span></p>
<br></br>
They would need to make $4$ more biscuits. They would then get $4$
biscuits each as $24$ divided by $6$ is $4$.<br></br>
<br></br>
<p><span class="editorial">Well done, Finlay. Emerson and
Hamish from St Peter's added a bit more detail:</span></p>
<br></br>
If you add all the biscuits up you get $20$. There are six people
so you need to find the multiple of $6$ that is closest to $20$ but
that is also above it. The answer is therefore $24$.
$24-20=4$ Therefore they need to bake $4$ more biscuits.
<br></br>
<br></br>
<p><span class="editorial">Ed from St Peter's sent in answers
to all parts of the problem. He said:</span></p>
Ali and Danny baked $12$ biscuits. If the biscuits were shared
between Ali and Danny, they would receive $6$ each.<br></br>
Will, Jess and Karni baked $8$ biscuits. If the biscuits were
shared between Will, Jess and Karni they would receive $2
\frac{2}{3}$ each. <br></br>
There were $20$ biscuits baked all together. If they shared all the
biscuits between the $5$ children , each child would receive $4$
bisuits.<br></br>
Yes, they can share all the biscuits between all the children and
Miss King. Each person would receive $3\frac{1}{3}$ biscuits. The
children would have to bake $4$ more biscuits to make $24$ in total
and therefore each person receiving $4$ biscuits each.<br></br>
<br></br>
<p class="editorial">Thank you, Ed. It's a shame you didn't
tell us how you worked out this solution.</p>
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<h2>Lots of Biscuits!</h2>
<br></br>
<p>On Tuesdays Green Class sometimes do baking.<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="" height="115" src="Bix1.png" width="167"></mdo:image></div>
<br></br>
<p>One Tuesday Miss King said that William's group were going to make biscuits. There were four children in the group besides William. They are Ali, Jess, Karni and Danny.<br></br>
<br></br>
William, Jess, and Karni were going to work together to make biscuits with chocolate chips in them.<br></br>
<br></br>
Ali and Danny were going to make ginger cookies.<br></br>
<br></br>
When Miss King took Ali and Danny's tray of biscuits from the oven they looked like this:<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="12 biscuits" height="211" src="Bix2.png" width="279"></mdo:image></div>
<br></br>
<p>How many biscuits did they make?<br></br>
<br></br>
"Let's share them out!" cried Danny, "There are lots of biscuits! We'll get lots each, won't we?"<br></br>
<br></br>
If they shared them between just Ali and Danny, how many would they get each?<br></br>
<br></br>
"What about Will, Jess, and Karni?" objected Ali, "They should get some too."<br></br>
<br></br>
Just then Miss King took the other tray of biscuits from the oven. They looked like this:<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="8 biscuits" height="210" src="bix3.png" width="278"></mdo:image></div>
<br></br>
<p>How many biscuits did the three of them make?<br></br>
<br></br>
"Ooh! They smell good!" exclaimed William.<br></br>
"Can we eat them now?" asked Karni.<br></br>
"We will have to cut some of them up, if we are going to divide them between us fairly," said Jess thoughtfully.<br></br>
<br></br>
If they shared them between the three of them, how many would they get each?<br></br>
<br></br>
Ali looked at both trays. He counted all the biscuits. How many were there altogether?<br></br>
<br></br>
They decided to share all the biscuits between all five of them. How many did they get each?<br></br>
<br></br>
"What about Miss King?" Danny asked suddenly. "She might like some biscuits too."<br></br>
Can they share all the biscuits equally between the five children and Miss King?<br></br>
How many more biscuits could they bake so that they could share them fairly without cutting any of them up? </p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/6883&part=">This activity</a> is an opportunity for young children to solve a practical problem that involves sharing into groups of two and five, and even six. By introducing the problem within the context of baking, children might surprise you with the level of mathematics they can handle. Having high expectations of learners is key and this challenge
allows children to work at their own level. Some might be keen to discuss cutting up biscuits, whereas others might be happy to have some 'left over', or some for another day, for example. You may be able to bake biscuits with the children so that they have the 'real thing', alternatively you could make salt dough cookies or cookies out of plasticene/play dough. Or, you could simply introduce the
context as a story and allow the children to create their own representations.<br></br>
<h3>Possible approach</h3>
<div>You could read the book 'The Doorbell Rang' by Pat Hutchins before tackling this problem.</div>
<div> </div>
<div>The problem might be best done with a small group of children so that the story about William's group and Miss King can be told gradually. At each question, give plenty of time for the pupils to try to solve it. You might encourage them to work all together, or in pairs, depending on how many you have in the group. You can spend time talking about their solutions as a whole group
before continuing with the story.</div>
<div> </div>
<div>As the children talk together, you have the opportunity to observe and assess. How are they approaching the task? Do they make the biscuits and physically share them? Are they happy to use something else to represent the biscuits? Perhaps they choose to draw a picture or jot things down to help? Are any of them able to use appropriate mathematical symbols to represent what they are doing? It
is important that all their ideas are valued so you give them confidence to 'have a go' rather than worry about the method that they think you want to see.</div>
<div> </div>
<div>When it comes to sharing eight biscuits between three children, allow learners to come to their own decisions regarding the possibility of cutting biscuits. Again, this is a good opportunity to assess their understanding of sharing 'equally' or 'fairly' and possibly fractions.</div>
<br></br>
<h3>Key questions</h3>
<div>How do we know how many biscuits there are?</div>
<div>What could we do to share out the biscuits fairly?</div>
<div>Tell me about what you're doing.</div>
<br></br>
<h3>Possible extension</h3>
Encourage learners to find more than one answer to the final part of the problem and then even to generalise: some children might be able to articulate that if they bake more biscuits in sixes, then they will always be able to share them out fairly.<br></br>
<h3>Possible support</h3>
Some children might find <a href="/content/id/6883/Bix.pdf">this sheet</a> useful which has the pictures of the two trays of biscuits on it.<br></br>
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Could you find some 'pretend' biscuits to help you with this
problem? <br></br>
Or could you draw a picture? <br></br></mdoxml>
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Lots of Biscuits!
Help share out the biscuits the children have made.
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Counting
Numbers and the Number System
Odd and even numbers
Numbers and the Number System
Factors and multiples
Using, Applying and Reasoning about Mathematics
Recording mathematics
Admin
Lower primary mapping document