7502
/www/nrich/html/content/id/7502/
1
2011-03-14T16:50:38
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<ul id="stemLinks">
<li><a href="http://nrich.maths.org/7419">Warm-up</a></li>
<li><a href="http://nrich.maths.org/7415">Try this next</a></li>
<li><a href="http://nrich.maths.org/6500">Think higher</a></li>
<li><a href="http://plus.maths.org/content/understanding-uncertainty-how-long-will-you-live">Read: mathematics</a></li>
<li><a href="http://en.wikipedia.org/wiki/Physical_change">Read: science</a></li>
<li><a href="http://plus.maths.org/content/philosophy-applied-mathematics">Explore further</a></li>
</ul>
<div> </div>
<br></br>
<p>Here are some graphs of physical processes.<br></br>
</p>
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<td> <mdo:image alt="" height="206" src="Chart%201.png" width="309"></mdo:image></td>
<td> <mdo:image alt="" height="206" src="Chart%202.png" width="309"></mdo:image></td>
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<td> <mdo:image alt="" height="206" src="Chart%203.png" width="309"></mdo:image></td>
<td> <mdo:image alt="" height="206" src="Chart%204.png" width="309"></mdo:image></td>
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<td> <mdo:image alt="" height="206" src="Chart%205.png" width="309"></mdo:image></td>
<td> <mdo:image alt="" height="206" src="Chart%206.png" width="309"></mdo:image></td>
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<td> <mdo:image alt="" height="206" src="Chart%208.png" width="309"></mdo:image></td>
<td> <mdo:image alt="" height="206" src="Chart%209.png" width="309"></mdo:image></td>
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<br></br>
<br></br>
<p><b>Can you suggest plausible processes that could have given rise to each graph?</b><br></br>
<br></br>
Now click on the link below to see eight processes that give rise to the graphs.<br></br>
Can you match each process to a graph?<br></br>
<br></br>
<a href="#" id="hideShow1">PROCESSES</a></p>
<br></br>
<div id="hidden1" style="display: none;">
<ol>
<li>A cup of tea is made and the temperature measured in degrees Celsius every second. What would the temperature - time graph look like?</li>
<li>The height of the valve on a bicycle tyre above the ground is measured after each centimetre that the bicycle travels forwards. What would the height - distance graph look like?</li>
<li>I throw a tennis ball straight up into the air and catch it. The height of the ball from the ground is measured over the time of the journey using freeze-frame photography. What would the height - time graph look like?</li>
<li>I measure several objects using inches and then using metres, plot them on a scatter graph, and join the points. What would the metres - inches graph look like? </li>
<li>I jump out of a plane and the distance fallen from the plane is measured every $0.1$ second until I open my parachute. What would the distance fallen - time graph look like whilst in freefall?</li>
<li>I drive at 70 miles an hour along the motorway and note the reading on my odometer (mile counter) every 5 minutes. What would the odometer reading - time graph look like? </li>
<li>I blow up a roughly spherical balloon using a balloon pump. After each pump I measure the radius of the balloon in centimetres. What would the radius - number of pumps graph look like?</li>
<li>I suck water through a straw out of a large beaker at a constant rate and measure the volume of liquid remaining at various times. What would the volume-time graph look like?</li>
</ol>
</div>
<br></br>
<p><b>Can you suggest equations that could model each of these processes?</b><br></br>
<br></br>
Now click on the link below to see eight equations.<br></br>
Can you match each equation to the graphs and related processes?<br></br>
<br></br>
<a href="#" id="hideShow2">EQUATIONS</a></p>
<br></br>
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<td style="" width="200">$$y=A \sqrt[3]{x}$$</td>
<td style="" width="200">$$y = A \sin(Bx)$$ </td>
<td style="" width="200">$$y = Ax + B$$ </td>
<td>$$y=\frac{A}{e^x}+B$$ </td>
</tr>
<tr>
<td>$$y=Ax^2$$ </td>
<td>$$y = Ax^2 + Bx + C$$ </td>
<td>$$y = Ax$$ </td>
<td>$$y = Ax+B$$ </td>
</tr>
</tbody>
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</div>
<br></br>
<p><b>Extension</b><br></br>
Can you determine values for the constants A, B and C for each equation? For some, you will need to make some assumptions.<br></br>
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<span class="editorial">Elliot from Wilson's School sent the
following explanations:</span><br></br>
<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%201.png"></mdo:image><br></br>
The radius-number of pumps graph of the balloon links to the 1st
graph, as the radius would gradually increase by less as the volume
increases. <br></br>
<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%202.png"></mdo:image><br></br>
The temperature-time graph for the tea links to the 2nd graph, as
the temperature would decrease less as it cools down.<br></br>
<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%203.png"></mdo:image><br></br>
The odometer reading-time graph links to the 3rd graph, as the car
is moving at a constant speed, so the distance would also increase
constantly.<br></br>
<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%204.png"></mdo:image><br></br>
The height-distance graph for the bicycle valve links with the 4th
graph, as the valve's height would increase and decrease as the
wheel rotates around.<br></br>
<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%205.png"></mdo:image><br></br>
The distance fallen-time graph links to graph 5, as you would
accelerate, meaning your distance fallen would gradually increase
by more. <br></br>
<mdo:image height="206" width="309" alt="" src="Chart%206.png"></mdo:image><br></br>
The volume-time graph for the water links to graph 6, as the water
is sucked out at a constant rate, so the volume would decrease at a
constant rate.<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%208.png"></mdo:image><br></br>
The height-time graph for the tennis ball would be graph 7, as the
height would increase when you throw it, then gradually stop
increasing and decrease again as it falls.<br></br>
<mdo:image height="206" width="309" alt="" src="Chart%209.png"></mdo:image><br></br>
The metres-inches graph links to graph 8, as the number of inches
would be much greater than the number of metres, creating a very
shallow straight line.<br></br>
<br></br>
<a class="editorial" href="/content/id/7502/Niharika.pdf">Niharika</a> <span class="editorial">came up with some processes of her own that each
graph could represent.</span><br></br>
<a class="editorial" href="/content/id/7502/Linden.pdf">Linden</a><span class="editorial"> explained the shape of each graph and found
the correct equations.</span><br></br>
<br></br>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/7502">This problem</a> offers students the chance to explore functions and graphs in real-life contexts. There is an opportunity for students to use their scientific understanding as they are invited to suggest plausible processes for each graph.<br></br>
<h3>Possible approach</h3>
<div>Hand out <a href="/content/id/7502/What%27s%20That%20Graph%201.pdf">this worksheet</a> with copies of the eight graphs, and ask students to discuss in pairs what physical processes they could represent. Make it clear that there are a lot of possible answers:</div>
<div>"I want you to be prepared to justify that the processes you suggest correspond to the graphs you have been given".</div>
<br></br>
After students have had time to come up with processes for each graph, invite them to share their suggested processes and justifications. Encourage the class to be critical of the suggestions.<br></br>
<br></br>
Next, reveal that the graphs were actually generated from the processes listed on <a href="/content/id/7502/What%27s%20That%20Graph%202.pdf">this worksheet</a>. Hand out the worksheet, and ask students to work in pairs to match the processes (and equations if appropriate) to the graphs.<br></br>
<br></br>
To ensure that students think critically about matching the processes to the graphs, students could be required to present their solutions on a poster with each graph and process (and equation if appropriate) accompanied by a short sentence explaining WHY they match. <br></br>
<h3>Key questions</h3>
<div>What are the key features of each graph? </div>
<div> </div>
<div>For each process, roughly what shape graph would you expect? Which graphs or equations might be consistent with this?</div>
<br></br>
Where might the axes be placed on the graphs? What scale might you put on the axes?<br></br>
<h3>Possible extension</h3>
<div>Ask students to suggest realistic values for the constants A, B and C in the equations of the graphs.</div>
<div> </div>
<div><a href="http://nrich.maths.org/6500&part=">Whose Line Graph Is it Anyway?</a> is a similar but more challenging problem using functions met at Stage 5.</div>
<br></br>
<h3>Possible support</h3>
Start by giving students the processes and ask them to discuss and sketch what the graphs might look like first. Clarify the appropriate labels for each of the axes.<br></br>
<br></br>
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<mdoxml version="1.0">A 'temperature-time' graph has temperature on the vertical (y) axis
and time on the horizontal (x) axis. <br></br>
What would the temperature be after a short time has passed? <br></br>
What would happen to the temperature as time passes?<br></br>
<br></br>
A 'height-distance' graph has height on the vertical axis and
distance on the horizontal axis...<br></br>
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What's that graph?
Can you work out which processes are represented by the graphs?
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