6500
/www/nrich/html/content/id/6500/
1
2011-02-01T00:00:01
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<ul id="stemLinks">
<li><a href="http://nrich.maths.org/6493">Warm-up</a></li>
<li><a href="http://nrich.maths.org/6427">Try this next</a></li>
<li><a href="http://nrich.maths.org/6583">Think higher</a></li>
<li><a href="http://nrich.maths.org/7352">Read: mathematics</a></li>
<li><a href="http://nrich.maths.org/7352">Read: science</a></li>
<li><a href="http://nrich.maths.org/7352">Explore further</a></li>
</ul>
<div> </div>
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<p>Scientific processes involving two variables can often be represented using equations and line graphs.<br></br>
<br></br>
In this problem, $9$ processes, their equations and graphs have been mixed up and shown below. In each case, the two variables are represented by the letters $x$ and $y$ and the labels from the axes of the graphs have been removed.<br></br>
<br></br>
Which can you match up? What is the interpretation of the variables $x$ and $y$ in each case?<br></br>
<br></br>
Can you identify the physical interpretation of three key points on each of the graphs?<br></br>
<br></br>
<span style="font-weight: bold;">Processes</span></p>
<ol>
<li>Number of rapidly dividing bacteria present in a food-limited environment, starting from a small initial sample.</li>
<li>Concentration in the blood of a drug following an injection.</li>
<li>Angle of oscillation of a real pendulum of length $1$m in air.</li>
<li>Volume (litres) against pressure (atmospheres) for $1$ mole of an ideal gas at $0^\circ$ C.</li>
<li>Vertical distance travelled by a small, heavy ball dropped from a plane.</li>
<li>Rate of reaction of a catalysed reaction in terms of the concentration of reagent.</li>
<li>Number of rapidly dividing bacteria present in a food-rich environment, starting from a small initial sample.</li>
<li>Hours of daylight per day in a town in the far northern hemisphere.</li>
<li>Model of the distance of the Earth from the sun in astronomical units.</li>
</ol>
<p><span style="font-weight: bold;">Line Graphs</span> <mdo:image alt="" height="783" src="charts.jpg" width="629"></mdo:image><br></br>
<span style="font-weight: bold;">Equations</span></p>
<br></br>
<div>A: $y(x) = 4.9 x^2$</div>
<br></br>
<div>B: $y(x) =500 \times 2^{-0.6667x}$</div>
<br></br>
<div>C: $y(x) =1- 0.01671\cos(0.0172 x)$</div>
<br></br>
<div>D: $y(x) = 12+10\sin(0.121 x)$</div>
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<div>E: $y(x) = 5\cos(3.13 x)e^{-0.05x}$</div>
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<div>F: $y(x) = (11.3 x)/(2.1+x)$</div>
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<div>G: $y(x) =10 \times 2^{4x}$</div>
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<div>H: $y(x) = \frac{1000000}{10+(100000-10)2^{-4x}}$</div>
<br></br>
<div>I: $y(x)x = 22.4133$</div>
<br></br>
<br></br>
<p>The numbers have been carefully chosen to represent certain time/length/unit scales for particular physical phenomena. Can you deduce the reason for the choice of any of the numbers?<br></br>
<br></br>
<br></br>
</p>
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<mdo:image height="809" width="650" alt="Numbered charts" src="charts_numbered_2.jpg"></mdo:image><br></br>
<table border="1">
<tbody>
<tr>
<td style="font-weight: bold;">Process</td>
<td style="font-weight: bold;">Graph</td>
<td style="font-weight: bold;">Equation</td>
<td style="font-weight: bold;">Explanation</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">6</td>
<td style="text-align: center;">H</td>
<td>In a food-limited environment bacteria tend to maximum number
after initial exponential growth (following lag phase)</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">5</td>
<td style="text-align: center;">B</td>
<td>Concentration will exponentially decay as drug is
metabolised</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">9</td>
<td style="text-align: center;">E</td>
<td>Pendulum will have an sinusodial motion with a decaying
amplitude due to damping from air resistance</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">7</td>
<td style="text-align: center;">I</td>
<td>Ideal gas equation $pV = nRT$ therefore pressure and volume are
in a reciprocal relationship</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">2</td>
<td style="text-align: center;">A</td>
<td>Ball is small and heavy, therefore assume air resistance is
negligible at first and so acceleration is constant. $s = ut +
\frac{1}{2}at^2$ therefore for $u=0;\ a=g \Rightarrow
s=\frac{1}{2}gt^2$ (qualitatively vertical speed represented by
gradient of graph is increasing)</td>
</tr>
<tr>
<td style="text-align: center;">6</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">F</td>
<td>As concentration of reagent increases so does reaction rate,
however as increase continues concentration of the catalyst becomes
the limiting factor (saturation)</td>
</tr>
<tr>
<td style="text-align: center;">7</td>
<td style="text-align: center;">4</td>
<td style="text-align: center;">G</td>
<td>When not food limited, bacteria follow exponential growth after
initial lag phase</td>
</tr>
<tr>
<td style="text-align: center;">8</td>
<td style="text-align: center;">3</td>
<td style="text-align: center;">D</td>
<td>Hours of daylight varies from a maximum at mid-summer to
minimum at mid-winter, with a mean value of 12</td>
</tr>
<tr>
<td style="text-align: center;">9</td>
<td style="text-align: center;">8</td>
<td style="text-align: center;">C</td>
<td>Earth's orbit is not perfectly circular therefore small
oscillations about mean distance (1 AU) with period of one year
(note non-zero origin on vertical axis)</td>
</tr>
</tbody>
</table>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6500&part=">This problem</a> requires students to engage with the relationship between algebra, graphs and the physical processes from which they are derived. There is a strong modelling aspect to the problem which requires students to relate the decimal numbers in the expressions to key physical concepts.<br></br>
<br></br>
<h3>Possible approach</h3>
<div>This is well-suited to a card sorting approach. You can use these printouts of the <a href="/content/id/6500/Whose%20line%20graph%20-%20equations%20NEW.pdf">equations</a>, the <a href="/content/id/6500/Whose%20line%20graph%20charts%20-%20New.jpg">graphs without axes</a> , the <a href="/content/id/6500/Whose%20line%20graph%20-%20charts.jpg">graphs with axes</a> and the <a href="/content/id/6500/Whose%20line%20graph%20-%20processes%20NEW.pdf">processes</a>. There is quite a lot of information for students to grapple with at the start of the problem and initially they might try to get a feel for the structure of the problem before attempting to pair cards together.</div>
<br></br>
<div>The simplest way into the problem is first to match the graphs and equations and then to match the processes onto these. Matching equations and graphs by a process of elimination is a good idea and students will need to realise that substituting $x=0$ or $x$ equal to the largest value on the horizontal axis is the best way to do this.</div>
<br></br>
<div>Students should be encouraged to try to get into the equations by looking at key points on the graphs (such as turning points, roots, asymptotes) and then deciding whether this might link well with a physical process.</div>
<br></br>
<div>Of course, some parts can be matched by process of elimination. However, the problem could be taken one step further: once students are confident that they have matched a process, graph and equation they should think about the match more carefully. Can they provide several different numerical/visual/physical common sense checks that the match is correct?</div>
<br></br>
<div>The most challenging aspect of the problem is deciding on the origin of the unusual-looking numbers, such as $3.13$ or $0.01671$. Reinforce to students that most of the numbers do result naturally from the physical process: all of the long decimals have an 'obvious' scientific origin. However, uncovering this origin will require students to think carefully about the physical process; there
will doubtless be gaps in their scientific knowledge which make it likely that they will only be able to understand some of these fully.</div>
<br></br>
<h3>Key questions</h3>
<div>How might you rule out certain pairs of graphs and equations?</div>
<div>Where are the key points on each graph?</div>
<div>For each process, roughly what shape graph would you expect? Which graphs or equations might be consistent with this?</div>
<div>Once you have matched a process, equation and graph how would you construct very convincing evidence that the match is correct?<br></br>
</div>
<h3>Possible extension</h3>
<div>At a higher level, students might question the modelling assumptions giving rise to the graphs and equations. First they will have to work out what modelling assumptions were made; then they can challenge any or all of these.<br></br>
</div>
<h3>Possible support</h3>
<div>The introductory task <a href="/7502">What's That Graph</a> has the same structure but uses processes with a simpler functional form.<br></br>
<br></br>
Suggest that students try to substitute the values $x=0$ and the largest value on the horizontal axes of the graphs into the equations. Which match up?</div>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">How might you rule out certain pairs of graphs and equations?<br></br>
<br></br>
Where are the key points on each graph?<br></br>
<br></br>
For each process, roughly what shape graph would you expect?<br></br>
<br></br>
Which graphs or equations might be consistent with this?<br></br>
<br></br>
Once you have matched a process, equation and graph how would you
construct very convincing evidence that the match is correct?
<br></br></mdoxml>
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Whose line graph is it anyway?
Which line graph, equations and physical processes go together?
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