1103
/www/nrich/html/content/02/03/penta5/
1
2011-02-01T00:00:01
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<p>We have a game which has a number of discs in
seven different colours. These are kept in a flat square
box with a square hole for each disc. There are $10$ holes in each
row and $10$ in each column. So, there would be $100$ discs
altogether, except that there is a square booklet which is kept in
a corner of the box in place of some of the holes.</p>
<p>We haven't drawn a grid to show all the holes because that would
give the answer away!</p>
<br></br>
<div style="text-align: center;"><mdo:image height="291" width="291" alt="box" src="box.gif"></mdo:image></div>
<br></br>
There is a different number of discs of each of the seven colours.
<p>Half ($\frac{1}{2}$ ) of the discs are red, $\frac{1}{4}$ are
black and $\frac{1}{12}$ are blue.</p>
<mdo:image alt="" src="pieces1.gif"></mdo:image><br></br>
<p>One complete row (of $10$ holes) of the box is filled with all
the blue and green discs.</p>
<mdo:image alt="" src="pieces2.gif"></mdo:image><br></br>
<p>One of the shortened rows (that is where the booklet is) is
exactly filled with all the orange discs.</p>
<mdo:image alt="" src="pieces3.gif"></mdo:image><br></br>
<p>Two of the shortened rows are filled with some of the red discs
and the rest of the red discs exactly fill a number of complete
rows (of $10$) in the box.</p>
<p>There is just one white disc and all the rest are yellow.</p>
<mdo:image alt="" src="pieces4.gif"></mdo:image><br></br>
<p>How many discs are there altogether?<br></br>
What fraction of them are orange?<br></br>
What fraction are green? Yellow? White?</p>
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<p class="editorial">This was a tricky problem. Well done to those
of you who had a go. We had some very clearly explained answers.
The key was to work out the size of the booklet first.</p>
<p class="editorial">Rachel, Ol, Jack and Alex from
Moretonhampstead Primary said:</p>
<p class="editorial"></p>
<div>First we worked out how many squares the booklet is. The
number has to be a square number and has to be even and
$\frac{1}{4}$ of that number has to be even again. The only number
possible for that is $1$6 (four squared).</div>
<div>Then we took $16$ (that was how big the booklet was) from a
$100$ which is $84$ (there are $84$ squares to play the game with).
With $84$ we can answer the first question - how many discs are
there altogether? ($84$).</div>
<div>After that we worked out how many discs there would be for the
colours. We worked out there would be $42$ discs ($\frac{1}{2}$ of
$84$), $21$ black discs ($\frac{1}{4}$ of $84$) and $7$ blue
($\frac{1}{12}$ of $84$).</div>
<div>We put that on the grid as it says on the sheet. Next we used
the last full column for blue and green. We know that there are $7$
blues (because of what we worked out earlier) which means there are
$3$ green discs ($7 +3 =10$ (which is how many in a column)).</div>
<div>There were five squares left. It says that there is $1$ white
square then the leftovers are yellow so we had $4$ yellows and $1$
white disc.</div>
<div>Now we hade completed the grid we could answer question $2$
and $3$. For question $2$ we counted up all the orange squares
($6$) and the fraction is $\frac{6}{84}$ but we had to simplify it
to $\frac{1}{14}$ (the answer).</div>
<div>Lastly we did question $3$. The fraction of green is
$\frac{3}{84}$ and simplified for the actual answer is
$\frac{1}{28}$.</div>
<div>The fraction of yellow is $\frac{4}{84}$ and simplified for
the answer is $\frac{1}{21}$.</div>
<div>There is only one white square so the answer is
$\frac{1}{84}$, for white.</div>
<p class="editorial"></p>
<p class="editorial">Hamish, Rory, Sarah, Jesse and Samuel from
Rutherglen Primary also reasoned very clearly and they sent us a
picture of the full box which they modelled using cubes:</p>
<p class="editorial"></p>
<p class="editorial"><mdo:image height="534" width="400" alt="" src="HamishSol.jpg"></mdo:image></p>
<p class="editorial"></p>
<p class="editorial">Sophie and Claire from The Downes School
wrote:</p>
<blockquote>$1\times 1$ didn't work because it said that
<strong>two</strong> shortened rows have red discs.<br></br>
$2\times 2$ didn't work because you need <strong>two</strong>
shortened rows of red and <strong>one</strong> of orange.<br></br>
$3\times 3$ didn't work because the total number of discs would be
odd and you couldn't halve it. This means all odd numbers didn't
work.<br></br>
$4\times 4$ did work because you had the right amount of shortened
rows.<br></br>
$6 \times 6$ didn't work because you can't divide $64$ by
$12$.<br></br>
$8 \times 8$ didn't work because you need <strong>six</strong>
whole rows.</blockquote>
<p class="editorial">Emma, Abi, Matthew B and Yuji from Moorfield
Junior School; Keshinie and Sharon at Kilvington GGS Victoria,
Australia; Gideon from Newberries Primary School and Hannah,
Georgia, Patrick; Hana from Bali International School and Matthew
from Brighton College Prep School realised that the number left
after taking away the booklet must be a multiple of $12$. Keshinie
and Sharon describe how they continued from there:</p>
<blockquote>So that made it $84$.<br></br>
Half of the disks are red so that made the amount of red
$42$.<br></br>
Then it said that a quarter is black so that made it $21$.<br></br>
Then it said that one twelfth is blue so that made it $7$.<br></br>
Then it said that one complete row was filled with all of blue and
green and the remainder of $10$ if you take away $7$ made it $3$
green.<br></br>
Then it said that one of the shortened rows is exactly filled with
all the orange disks so that makes it $6$.<br></br>
Then it said that there was only one white disk.<br></br>
Then we added all the numbers together making $80$ disks so there
was a remainder of $4$ which had to be yellow.
<p>We divided the $84$ disks by the $6$ orange ones that made it
$14$. So the fraction of orange had to be $1$ out of $14$
($\frac{1}{14}$).<br></br>
We divided the $84$ disks by the $3$ green disks making the answer
$28$. So the fraction of green had to be $\frac{1}{28}$.<br></br>
We already knew that the fraction of white disk was
$\frac{1}{84}$.<br></br>
We divided the $84$ disks by the $4$ yellow ones making it $21$ so
the fraction of yellow had to be $\frac{1}{21}$.</p>
</blockquote>
<p class="editorial">James from the Charter School explained very
well how he went about the problem:</p>
<br></br>
<div>Each of the sides is $10$ units and I called each of the sides
of the booklet $x$. this means that the equation for finding the
number of discs was $N=100-x^2$. ($N$ being the number of remaining
discs).</div>
<div>The amount of Blue discs was $\frac{N}{12}$ meaning that $N$
was a multiple of $12$. So I then collected all of the multiples of
$12$: $12$, $24$, $36$, $48$, $60$, $72$, $84$, $96$. I then
eliminated those that did not fit the earlier equation because
there was not a square number that fitted. This left: $36$, $84$,
$96$.</div>
<div>I eliminated $96$ because $x$ had to be more that three for
there was one complete row of orange discs and two of red
discs.</div>
<div>This left: $36$, $84$. From this I deduced that $x$ had to be
$4$ or $8$. This means that the amount of red disks had to end in a
$2$ or a $4$, because there are two incomplete rows of red these
either have to be a length of $6$ or $2$. Since the amount of red
discs is half of $N$ I halved both my possible $N$'s which came up
with $18$ and $42$. This means that the amount of reds was $42$ and
$N$ was $84$.</div>
<div>This means that $x$ is $4$ and that the amount of orange discs
was $6$ meaning the fraction is $\frac{1}{14}$.</div>
<div>The amount of blue was $\frac{N}{12}$ which was $7$.</div>
<div>This means that the amount of green discs was $3$ because
blues + greens = $10$. That means the fraction was
$\frac{1}{28}$.</div>
<div>The amount of white was $1$ meaning the fraction was
$\frac{1}{84}$.</div>
<div>Finally the rest were yellow.</div>
<div>Red was $42$. Blacks was $\frac{N}{4}$ which was $21$. Blue
was $7$. Orange was $6$. Green was $3$. White was $1$.</div>
<div>If you take all those away from $84$ you end up with $4$. That
is the amount of yellows. This means the fraction is
$\frac{1}{21}$.</div>
<br></br>
<p class="editorial">Well done too to Harriet and Harah from
Greenacre School for Girls, Ruairidh from St Mary's High School,
Anne-Marie, Emma, Katherine, Laura from Gorseland Primary and
Eulalie and Holly who go to Lympstone Primary School.</p>
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<h2>Fractions in a Box</h2>
<br></br>
<p>We have a game which has a number of discs in seven different colours. These are kept in a flat square box with a square hole for each disc. There are $10$ holes in each row and $10$ in each column. So, there would be $100$ discs altogether, except that there is a square booklet which is kept in a corner of the box in place of some of the holes.</p>
<p>We haven't drawn a grid to show all the holes because that would give the answer away!</p>
<br></br>
<div style="text-align: center;"><mdo:image alt="box" height="291" src="box.gif" width="291"></mdo:image></div>
<br></br>
There is a different number of discs of each of the seven colours.
<p>Half ($\frac{1}{2}$ ) of the discs are red, $\frac{1}{4}$ are black and $\frac{1}{12}$ are blue.</p>
<mdo:image alt="" src="pieces1.gif"></mdo:image><br></br>
<p>One complete row (of $10$ holes) of the box is filled with all the blue and green discs.</p>
<mdo:image alt="" src="pieces2.gif"></mdo:image><br></br>
<p>One of the shortened rows (that is where the booklet is) is exactly filled with all the orange discs.</p>
<mdo:image alt="" src="pieces3.gif"></mdo:image><br></br>
<p>Two of the shortened rows are filled with some of the red discs and the rest of the red discs exactly fill a number of complete rows (of $10$) in the box.</p>
<p>There is just one white disc and all the rest are yellow.</p>
<mdo:image alt="" src="pieces4.gif"></mdo:image><br></br>
<p>How many discs are there altogether?<br></br>
What fraction of them are orange?<br></br>
What fraction are green? Yellow? White?</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=1103&part=index">This problem</a> gives practice in calculating with fractions in a challenging setting. It also requires the use of factors and multiples. While doing the problem learners will need to express a smaller whole number as a fraction of a larger one and find equivalent fractions. This activity will require some estimating
and trial and improvement, combined with working systematically.<br></br>
<h3>Possible approach</h3>
<div>You could start by showing the picture in the problem and explaining the task orally to the group. Give them a chance to think on their own for a minute then ask them to talk to a partner about how they might start the problem. Sharing some ideas will help you ascertain whether learners understand the task and it will give you the chance to talk through any misconceptions they may have. It
may be helpful to collate some known facts or suggestions on the board. At this stage, welcome all ideas for how to begin as long as they are backed up by logical reasons. It is likely that many will involve testing a size for the booklet, but different children may have different starting points for the size. Others might want to test total numbers of counters.</div>
<br></br>
<div>After that encourage them to work in pairs on the problem from a printed sheet (<a href="/content/02/03/penta5/1103.pdf">this sheet</a> is photocopiable) so that they are able to talk through their ideas with a partner. Make sure that learners have access to any resources that they require, such as squared paper, coloured pens/pencils, mini-whiteboards, plain paper, counters ... Warn them
that you would like them to be able to explain how they approached the problem at the end of the lesson. You could ask each pair or group to produce a poster of their working.</div>
<br></br>
In the plenary, you could invite the class to look at each other's posters and ask each other questions about the methods used.<br></br>
<h3>Key questions</h3>
<div>What shape is the booklet? How are you going to work out its size?</div>
<div>How many holes <span style="font-style: italic;">could</span> the booklet could take up?</div>
<div>What do you know about the factors of the total number of discs?</div>
<br></br>
<h3>Possible extension</h3>
Learners could find out possible fractions for the differently coloured discs if the square booklet was larger, say $6 \times 6$.<br></br>
<h3>Possible support</h3>
Some children my find it helpful for you to structure the approach a little. You could suggest making a list of the possible sizes for the square booklet then working out the number of remaining small square holes for the coloured discs.<br></br>
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Remember the booklet in the corner of the box is square, so how
many holes <span style="font-style: italic;">could</span> it take
up? <br></br>
The whole box, without the space for the booklet, would fit $100$
discs, so the number of discs must be $100$, less the square number
taken up by the booklet.<br></br>
What do you know about the factors of the total number of
discs?<br></br></mdoxml>
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Fractions in a Box
The discs for this game are kept in a flat square box with a square
hole for each disc. Use the information to find out how many discs
of each colour there are in the box.
Numbers and the Number System
Square numbers
Fractions, Decimals, Percentages, Ratio and Proportion
Calculating with fractions
Numbers and the Number System
Factors and multiples
Using, Applying and Reasoning about Mathematics
Trial and improvement
Using, Applying and Reasoning about Mathematics
Working systematically
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