1079
/www/nrich/html/content/01/11/penta1/
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2011-02-01T00:00:01
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<p>Tom's Dad sowed some tomato seed in February. He gave Tom one of
the tomato plants in a pot.</p>
<p>At the beginning of May Tom put his tomato plant outside. On the
same day he sowed a bean in another pot.</p>
<p>Ten days later the bean plant was just $1$ cm (centimetre) above
the soil surface. Tom measured his tomato plant which was already
$38$ cm tall.</p>
<p>Each evening Tom measured his two plants.</p>
<p>On the evening of the next day the little bean plant had grown
another $2$ cm so it was $3$ cm high. Each day it continued to grow
double the amount it had grown the day before.</p>
<p>The tomato plant grew at a steady $5$ cm a day.</p>
<p style="text-align: center;"><mdo:image height="271" width="226" src="plants.jpg" alt="plants"></mdo:image></p>
<br></br>
After how many days were the two plants the same height when Tom
measured them in the evening? How high were they?<br></br></mdoxml>
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<p class="editorial">The solutions we had in, varied in the ways
they were expressed.</p>
<p class="editorial">We had many contributions from Crudgington
Primary School in Shropshire, U.K.</p>
<div class="editorial">Hannah</div>
Firstly, I drew two pictures one of a tomato plant and one of the
bean plant. Then I made myself a little chart like this ...<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="86" width="348" src="Han%27Pic.jpg" alt="Han's"></mdo:image></div>
<br></br>
... and so on. As you can see that on the section where I worked
out the bean plant's results the first number I wrote was how much
the plant would need to increase in size. I found that this helped
me a lot. Then I wrote the height of the plant and finally how many
days it took for the plant to reach that height. I did this same
method for the tomato plant until the height was the same. So the
answer is: the height is $63$cm after $15$ days of growing.<br></br>
<br></br>
<br></br>
<p><span class="editorial">Tom</span></p>
<p>First I looked at the information and saw that the bean plant
grew only $1$cm after $10$ days! So I started from $10$ days. And
after the bean plant grew $1$cm the tomato plant had already grown
$38$cm. And then after each day the bean plant doubled its current
size and the tomato plant grew $5$cm. So after $15$ days both
plants are $63$cm.</p>
<br></br>
<br></br>
<p class="editorial">From Curl Curl North School in New South Wales
we had two answers that at first looked different but were just
being expressed from a different point of view. Christopher,
Ashton, Grace, Charlotte, Kaden, Ben, Kate, Charlie, Dylan, Josh,
Matthew and Mitchell's team, said;</p>
$6$ days from the time the bean was first measured.<br></br>
<p><span class="editorial">And from Sean, Emma, Emily, Amelia, Ed
and Stacey's team we had:</span></p>
$5$ days because you count from when the bean was $3$ cm
tall.<br></br>
<br></br>
<p class="editorial">I wonder if they thought the other team had
the wrong answer?</p>
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<h2>The Tomato and the Bean</h2>
<br></br>
<p>Tom's Dad sowed some tomato seed in February. He gave Tom one of the tomato plants in a pot.</p>
<p>At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot.</p>
<p>Ten days later the bean plant was just $1$ cm (centimetre) above the soil surface. Tom measured his tomato plant which was already $38$ cm tall.</p>
<p>Each evening Tom measured his two plants.</p>
<p>On the evening of the next day the little bean plant had grown another $2$ cm so it was $3$ cm high. Each day it continued to grow double the amount it had grown the day before.</p>
<p>The tomato plant grew at a steady $5$ cm a day.</p>
<p style="text-align: center;"><mdo:image alt="plants" height="271" src="plants.jpg" width="226"></mdo:image></p>
<br></br>
After how many days were the two plants the same height when Tom measured them in the evening? How high were they?</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=1079&part=">This problem</a> can help to give learners an understanding of the difference between a steady and an increasing rate of growth. It could be used when looking at number patterns especially doubling, putting them in a real context, and can be used to demonstrate the use of tables in problem solving. It will also be
fascinating for you as the teacher to see how children represent the problem as they work on it.<br></br>
<h3>Possible approach</h3>
<div>You could start this problem as a whole class activity by introducing the growth rate of other plants as examples. (These plants could be real or imagined!) For example:</div>
<div>"There is a plant that is $3$ centimetres tall. If it grows $4$ centimetres a day how tall will it be the next day?" "And the day after that?" And so on ...</div>
<br></br>
<div>Then you can introduce a different type of example:</div>
<div>"There is a plant that is $5$ centimetres tall. If it doubles in height each day how tall will it be the next day?""And the day after that?" And so on ...</div>
<br></br>
<div>Then a new question needs to be asked to help in approaching the problem as written. For example:</div>
<div>"It starts at $5$ centimetres and is $10$ centimetres the next day. How much has it grown?"</div>
<div>"If it grows double that the next day, how tall will it be?"</div>
<br></br>
<div>You could then ask learners to do the given problem in pairs and encourage them to use anything they like to help them with the solution. Some may draw pictures and use numbers, some may make a list, some may use multilink to physically represent the plants, some will be able to make a table of the results, others may be happier if given <a href="/content/01/11/penta1/The%20Tomato%20and%20the%20Bean%20Table.pdf">this sheet</a> of a table to be completed.</div>
<br></br>
<h3>Key questions</h3>
<div>At what height did the bean start?</div>
<div>How tall was it the next day?</div>
<div>How much did it grow by the next day? So how tall would it be that day?</div>
<div>And the day after that?</div>
<div>At what height did the tomato start?</div>
<div>How tall would it be on the second day? And the day after that?</div>
<br></br>
<h3>Possible extensions</h3>
Learners can be encouraged to open out this activity further by considering plants that increase their rate of growing by trebling/quadrupling the growth each year. Each year's height could be calculated and learners can see what happens.
<div>Doubling goes:- $1$, $3$, $7$, $15$, $31$, $63$ ...</div>
<div>Trebling goes:- $1$, $4$, $13$, $40$ ...</div>
<div>Quadrupling goes, $1$, $5$, $21$, $85$ ...</div>
<br></br>
<div>To open this problem out further, children may like to explore the digital roots of the numbers in their results. <a href="/content/01/11/penta1/The%20Tomato%20and%20the%20Bean%20Extension.pdf">This sheet</a> gives some suggestions as to how this might be done.</div>
<br></br>
<h3>Possible support</h3>
Suggest using <a href="/content/01/11/penta1/The%20Tomato%20and%20the%20Bean%20Table.pdf">this sheet</a> and filling in first the days ($1$-$8$) and the heights on the first day. The bean's growth is easier to work out than the tomato's so this can be tackled next.<br></br>
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How tall will the bean plant be the next day? <br></br>
How tall will the tomato plant be on that day?<br></br>
How about the day after that?<br></br></mdoxml>
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<p class="editorial">We received many correct answers to this
problem. Most of you had done it a similar way, by drawing a
table.</p>
<p class="editorial">Joanne and Liam from Moorfield Junior School
explained their approach very clearly:</p>
<p>We each worked on a different plant and each time we worked out
the height for that day we checked to see if we had the same answer
until we did. This is what it looked like.......</p>
<table cellspacing="0" cellpadding="5">
<tbody>
<tr>
<th>Tomato plant</th>
<th>Bean plant</th>
</tr>
<tr>
<td>1 cm</td>
<td>38 cm</td>
</tr>
<tr>
<td>3 cm</td>
<td>43 cm</td>
</tr>
<tr>
<td>7 cm</td>
<td>48 cm</td>
</tr>
<tr>
<td>15 cm</td>
<td>53 cm</td>
</tr>
<tr>
<td>31 cm</td>
<td>58 cm</td>
</tr>
<tr>
<td>63 cm</td>
<td>63 cm</td>
</tr>
</tbody>
</table>
<p>The answer is six days and the plant would be 63 cm tall</p>
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The tomato and the bean
At the beginning of May Tom put his tomato plant outside. On the
same day he sowed a bean in another pot. When will the two be the
same height?
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