6151
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1
2011-02-01T00:00:01
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<ul id="stemLinks">
<li><a href="http://nrich.maths.org/7502">Warm-up</a></li>
<li><a href="http://nrich.maths.org/6493">Try this next</a></li>
<li><a href="http://nrich.maths.org/6160">Think higher</a></li>
<li><a href="http://plus.maths.org/content/graphical-methods-i-slug-wars">Read: mathematics</a></li>
<li><a href="http://plus.maths.org/content/light-attenuation-and-exponential-laws">Read: science</a></li>
<li><a href="http://en.wikipedia.org/wiki/Wikipedia:Modelling_Wikipedia's_growth">Explore further</a></li>
</ul>
<div> </div>
<p>These four graphs can be used to represent growth rates of certain organisms. How many possibilities can you suggest for each one?</p>
<p align="center"><mdo:image alt="growthRates" height="245" src="growthRates.JPG" width="432"></mdo:image></p>
<p align="center" style="text-align: left;">Can you suggest organisms which grow according to a qualitatively different type of graph?</p>
<p align="center" style="text-align: left;">The graphs are all without scale on the axes.</p>
<p align="center" style="text-align: left;">For any possibility you suggested for the graphs, what would be a sensible scale for the axes in each case to make the graphs make sense?</p>
<p align="center" style="text-align: left;"> </p>
<p align="center" style="text-align: left;"> </p>
<h6 align="center" style="text-align: left;"> </h6>
<h6 align="center" style="text-align: left;"> </h6>
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From this task you should appreciate that the growth rates of
certain organisms over variable time scales can be strikingly
different!<br></br>
<br></br>
The blue curve represents a situation of exponential growth. For
example the multiplication of a bacterium such as <span style="font-style: italic;">E. Coli</span> over time would follow
such a trend provided that adequate nutrition is available over the
timescale of the experiment. The trend could be represented by the
form 2$^{\frac {t}{T}}$, where t is time and T is the period of
division. Bacterial division does not occur entirely in synchrony
and so a smooth curve is produced, rather than the stepped graph
that would be observed during synchronised division.<br></br>
<br></br>
The red curve is seen to represent the seasonal pattern of plant
growth which is rapid during the spring and relatively stagnant
during autumn and winter. It is also seen that year upon year, the
rate of growth in the fast growth period (seen as 'steps')
increases.<br></br>
<span style="font-style: italic;">What do you think would happen to
the shape of the growth curve of an oak tree once it has reached
maturity?</span><br></br>
<br></br>
The brown curve depicts the growth of a crustaceans or arthropods.
There are many examples of organisms that fit this curve. A few are
tarantulas, lobsters and crabs. Crustaceans have a cuticle that is
often biomineralised with substances such as calcium carbonate to
produce a rigid exoskeleton. However this inhibits growth; the
exoskeleton must be shed through a process known as moulting.
Intake of water facilitates the rapid expansion of the softer new
cuticle before it hardens after the old cuticle is detached. This
process is depicted by the vertical 'step' portions within the
growth curve. Internal tissue growth occurs constantly.<br></br>
<span style="font-style: italic;">Why might the moulting period be
a dangerous time for an organism such as a shore crab?</span><br></br>
<br></br>
The black curve represents classical mammalian growth. The initial
portion of the curve depicts the rapid growth of a new-born infant.
This growth rate falls and then progressively
increases during adolescence. Eventually a stage of
maturity is reached where mitotic processes operate on the whole to
replenish cells within the individual.<br></br>
<br></br>
As an extension, look up the profile of a biphasic bacterial growth
curve and understand the conditions that produced such a curve.
Wikipedia is a useful place to start. Two clear phases of growth
are seen due to:<br></br>
<br></br>
1) The depletion of glucose from the nutrient medium<br></br>
2) Transcription of $\beta$-galactosidase and associated enzymes to
allow lactose metabolism<br></br>
<br></br>
<span style="font-style: italic;">Is there any similarity to some
of the curves given to you in the question?</span><br></br></mdoxml>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6151&part=">
This problem</a> encourages students to get into the real meaning
of graphical representation without getting bogged down in
algebraic calculations or falling back into blind computation. It
will also encourage the students to think about the various
differences and similarities between growth processes in the
sciences.
<h3>Possible approach</h3>
<div>This problem works well in group discussion. For each idea,
try to encourage students to explain their reasoning as precisely
and clearly as possible. You could split the class into different
groups and see who can produce the most valid examples for each
graph.</div>
<h3>Key questions</h3>
<ul>
<li>How many 'growth processes' in science can you think of. Would
any of these graphs match those processes?</li>
<li>How might you label the scales for each example?</li>
</ul>
<h3>Possible extension</h3>
<div>This type of problem is rich with extension possibilities. We
suggest two:</div>
<br></br>
<div><span style="font-weight: bold;">Extension 1:</span> Are there
other shapes of graph which could be used to model other natural
growth processes?</div>
<br></br>
<div>How might you describe these curves algebraically? Can you
write down equations, the graphs of which match the shape of the
curves in this question?</div>
<br></br>
<div><span style="font-weight: bold;">Extension 2:</span> Look up
the profile of a biphasic bacterial growth curve and understand the
conditions that produced such a curve. Wikipedia is a useful place
to start. Two clear phases of growth are seen due to:</div>
<br></br>
<div>1) The depletion of glucose from the nutrient medium</div>
<div>2) Transcription of $\beta$-galactosidase and associated
enzymes to allow lactose metabolism</div>
<br></br>
<div style="font-style: italic;">Is there any similarity to some of
the curves given to you in the question?</div>
<br></br>
<div>You might naturally try <a href="http://nrich.maths.org/public/viewer.php?obj_id=6160&part=">
Real-life equations</a> next.</div>
<h3>Possible support</h3>
<div>Let students leaf through a science textbook searching for
graphs and charts. Do they notice that the same shapes of charts
appear frequently? Can they match any to the graphs in this
question?</div>
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<mdoxml version="1.0">What parts of the graphs would correspond to growth spurts or slow
growth?<br></br>
<br></br>
Think of some common organisms and sketch their growth curves.
Could you fit these onto the graph shapes shown here?<br></br>
<br></br>
The horizontal axis is a time scale but the vertical axis could be
any measure of growth (height, mass, length, population size
...)<br></br></mdoxml>
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bio graphs
What biological growth processes can you fit to these graphs?
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