6349
/www/nrich/html/content/id/6349/
1
2011-02-01T00:00:01
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<ul id="stemLinks">
<li><a href="http://nrich.maths.org/7500">Warm-up</a></li>
<li><a href="http://nrich.maths.org/6151">Try this next</a></li>
<li><a href="http://nrich.maths.org/6505">Think higher</a></li>
<li><a href="http://nrich.maths.org/7605">Read: mathematics</a></li>
<li><a href="http://en.wikipedia.org/wiki/Logarithmic_scale">Read: science</a></li>
<li><a href="http://scaleofuniverse.com/">Explore further</a></li>
</ul>
<div> </div>
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<p>This activity will test your knowledge of powers of 10 in scientific contexts. See the <a href="http://nrich.maths.org/public/viewer.php?time=1256645007&obj_id=6349&part=clue">hints</a> if you need some help understanding the notation.<br></br>
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Will you be able to judge the length of the physical objects, with sizes ranging from 1 Angstrom to 1 million km, with no wrong attempts?<br></br>
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<h3>You will possibly need to view this resource in <a href="/content/id/6349/scale.swf">full screen</a> mode.</h3>
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<h6>Send <a href="mailto:post16.nrich@maths.org?subject=Feedback%20on%20NRICH%20problem%20id%206349">feedback</a> on this problem</h6>
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<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6349&part=">
This activity</a> gives practice in working with powers of 10 and
tests awareness of relative sizes of physical objects encountered
in scientific contexts. The numbers are given in scientific notion,
such as 1.2e3 for 1200 or 26e-3 for 0.026, the familiarity with
which is essential in the sciences.</div>
<br></br>
<div>You can print out this activity for <a href="/content/id/6349/MatchingTables.pdf">use on cards</a> if you
wish.</div>
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<h3>Possible approach</h3>
<div>The activity could work well as a starter activity with
students working individually on computers or working
collaboratively to try to place the objects on the scale with as
few wrong answers as possible. Alternatively, it could be played
competitively with pairs of students taking it in turns to place
the objects and checking to see if they are right. There is the
opportunity for lots of discussion to justify where students wish
to place things on the scale, referring to sizes they know and
deciding whether the named objects are smaller or larger.</div>
<br></br>
<div>The activity generates a variety of questions so could be used
over a series of lessons with students trying to beat their best
scores, helping them to build up a fluency with powers of 10 and a
better awareness of the size of things.</div>
<br></br>
<div>There are various levels at which the activity could be used.
First students could look at the list of scientific quantities and
disentangle which of them they first of all recognise and,
secondly, which of them they could place on the scale. Once the
easier ones are placed students could take it in turns to take
sensible guesses with the harder ones, adjusting the answers in
turn until the cards are correctly placed. Finally, the numbers
could be revealed to allow the placement of the most difficults
cards. This is still non-trivial because quantities are given in
standard forms such as 260e-9 (260 nanometres) which would
correspond to 2.6e-7.</div>
<h3>Key questions</h3>
<div>Are there any objects in the list whose size I am fairly sure
about?</div>
<div>Is this object bigger or smaller than a metre? How much bigger
or smaller?</div>
<h3>Possible extension</h3>
<div></div>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6140&part=">
Big and Small Numbers in Biology</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=6504&part=">
Big and Small Numbers in Physics</a> are two problems which give
practice in estimation and calculations using very large and very
small numbers.</div>
<h3>Possible support</h3>
Start by arranging the objects whose size is well known, and use
the checking facility to see how many are right so far. Then
gradually add in the others.<br></br>
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'Order of magnitude' in science is a very useful concept: we are
often not necessarily interested in the exact measurement of a
quantity but rather whether it is 'about a metre' or
'about a kilometre' or 'about a nanometre' etc.<br></br>
<br></br>
Orders of magnitude makes the use of scientific notation. For any
two numbers $X$ and $Y$ we use the notation $X$e$Y$ to mean
$X\times 10^Y$. In case you are wondering, the letter $e$ stands
for 'exponent' and is sometimes written $E$ instead. In standard
notation, the number $X$ must be between $1.0$ and $9.99...$ and
the exponent a whole number.<br></br>
<br></br>
For example, $1.2$e$3$ is $1.2\times 10^{3}$, which is the same as
$1200$.<br></br>
<br></br>
The power of $10$ can also be negative, so that
$6.8$e$-2$ means $6.8\times 10^{-2}$, which is the same as
$0.068$.<br></br>
<br></br>
In science, certain exponents are more frequently used. Powers of
$\pm 3,\pm 6, \pm 9, \pm 12$ are standards, which is why you will
see (non-standard) measurements such as $375$e$-9$m, which
scientists would refer to 'Three hundred and Seventy Five
Nanometres'. You will need to convert such numbers to standard form
before placing them on the scale.<br></br>
<br></br>
Are there any objects whose size you are confident you know? Are
the other objects larger or smaller than these?<br></br></mdoxml>
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In standard scientific notation, numerical quantities are written
as<br></br>
$$<br></br>
a\times 10^b<br></br>
$$<br></br>
where $b$ is an integer and $a$ is a decimal number less than $2$
and not less than $1$.<br></br>
<br></br>
Construct your best estimates of the lengths of the quantities on
the slips of paper and rank them according to their exponent $b$
when written in scientific notation.<br></br>
<br></br>
Use units of metres, so that, for example, $1.0 \times 10^3$ would
correspond to $1$km.<br></br>
<br></br>
<span style="font-weight: bold;">The challenge is to correctly rank
all of the quantities.</span> <br></br>
<br></br>
<span style="font-style: italic;">Note that there is one slip of
paper per exponent, so you might be able to use detective work to
rank unfamiliar quantities.</span><br></br>
<br></br>
<span style="font-style: italic;">Extension task: Construct your
own set of questions for this activity.</span><br></br>
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A question of scale
Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?
Measures and Mensuration
Length/distance
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