6345
/www/nrich/html/content/id/6345/
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2011-02-01T00:00:01
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Imagine you have a large supply of 3kg and 8kg weights.<br></br>
<br></br>
Four 3kg weights and one 8kg weight have an average weight of 4kg.<br></br>
<mdo:image alt="4 3kg qnd 1 8kg weight balancing 5 4kg weights" height="223" src="weights400.jpg" width="400"></mdo:image><br></br>
<br></br>
How many of each weight would you need for the average (mean) of the weights to be 6kg?<br></br>
<br></br>
If you had other combinations of the 3kg and 8kg weights, what other whole number averages could you make?<br></br>
What's the smallest?<br></br>
What's the largest?<br></br>
Can you make all the whole number values in between?<br></br>
<br></br>
What if you have a different pair of weights (for example 2kg and 7kg)?<br></br>
What averages can you now make?<br></br>
<br></br>
Try other different pairs of weights.<br></br>
<br></br>
Do you notice anything about your results?<br></br>
Do they have anything in common?<br></br>
Can you use what you notice to find, for example, the combination of 17kg and 57kg weights that have an average of 44kg......of 52kg.......of 21kg.....?<br></br>
<br></br>
Explain an efficient way of doing this.<br></br>
Can you explain why your method works?<br></br>
<br></br>
<a href="http://nrich.maths.org/7204&part=">Click here for a poster of this problem</a>.<br></br></mdoxml>
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<p class="editorial">Most of you sent us the right answer to the
first question of the problem ('How many of 3kg and 8kg weights
would you need for the average (mean) of the weights to be 6kg?').
Some found it by trial and error, for example, Ellen from
Shincliffe Primary:</p>
<br></br>
It is all about trial and error. I thought I had the answer when I
had fourof 3kg weights and threeof 8kg weights. But then I noticed
that I had seven weights. Then I kept my three of 8kg which is 24kg
and all I had to do was add 2 of 3kg weights. And I had my
answer.<br></br>
3 x 8kg + 2 x 3kg =30, and 30/5 = 6kg.<br></br>
<br></br>
<p class="editorial">Andy (Garden International School) pointed
out:</p>
<br></br>
The weight averages are from 3 to 8<br></br>
<p class="editorial"></p>
<p class="editorial">Rosie from St Bartholomew's Cof E Primary
School gave us the answers to some whole-number averages in between
3 and 8:</p>
<br></br>
For an average of 7, you would need 4 of 8kg weights and 1 of 3kg
weight.<br></br>
For an average of 6, you would need 9 of 8kg weights and 6 of 3kg
weights.<br></br>
For an average of 5, you would need 2 of 8kg weights and 3 of 3kg
weights.<br></br>
<br></br>
<p class="editorial">A general solution was provided by Hyeon
(British School Muscat):</p>
<br></br>
Imagine that the lighter weight is a and the heavier weight is
b.<br></br>
As long as a < b, the smallest average you can get is a and the
biggest average you can get is b.<br></br>
It is possible to get every single number in between. There are
[(b-a)+1] averages.<br></br>
<br></br>
<p class="editorial">Hyeon also noticed something important from
the results of her trial with different weights:</p>
<br></br>
If the total number of averages is odd, then 1 of a and 1 of b
would give the middle average weight.<br></br>
<br></br>
The amount of a and the amount of b used should add up to a factor
or a multiple of the difference between a and b for the average to
be a whole number.<br></br>
<br></br>
<p class="editorial">Here are some of her trials:</p>
<div>For the 1kg and 5 kg weights</div>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">1 kg</td>
<td style="text-align: center;">5 kg</td>
<td style="text-align: center;">Total</td>
<td style="text-align: center;">Average</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">0</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">1</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">8</td>
<td style="text-align: center;">2</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">6</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">3</td>
<td style="text-align: center;">16</td>
<td style="text-align: center;">4</td>
</tr>
<tr>
<td style="text-align: center;">0</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">5</td>
<td style="text-align: center;">
<div>5</div>
</td>
</tr>
</tbody>
</table>
<a href="" class="control" onclick="tablealter(2)"></a><br></br>
<br></br>
<div>For the 17 kg and 57 kg weights</div>
<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">17 kg</td>
<td style="text-align: center;">57 kg</td>
<td style="text-align: center;">Total</td>
<td style="text-align: center;">Average</td>
</tr>
<tr>
<td style="text-align: center;">9</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">210</td>
<td style="text-align: center;">21</td>
</tr>
<tr>
<td style="text-align: center;">13</td>
<td style="text-align: center;">27</td>
<td style="text-align: center;">1760</td>
<td style="text-align: center;">44</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">7</td>
<td style="text-align: center;">1020</td>
<td style="text-align: center;">52</td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
<div><span class="editorial">Although we received many answers to
this problem, most of the answers simply stated the results rather
than providing a general strategy for finding them. However,
Anurag, from Queen Elizabeth's Grammar School in Horncastle, did
draw some general conclusions. You can see his working</span>
<a href="/content/id/6345/Searching%20for%20Mean%28ing%29%20solution.doc">here</a></div>
<br></br>
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<h3>Why do this problem?</h3>
<div>Students are often asked to calculate the average (mean) of
sets of whole numbers. But what happens when the numbers vary?
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6345&part=index">
This problem</a> offers students a chance to consolidate their
understanding of average as a central measure, representative of
the set.</div>
<br></br>
<div>It also offers a chance to rehearse the key mathematical
processes of exploring, conjecturing, generalising and
justifying.</div>
<h3>Possible approach</h3>
Introduce the problem by asking the students to imagine they have
an infinite supply of 3kg and 8kg weights. Can they find a
combination of these weights that has an average of 4kg?<br></br>
<br></br>
Allow some time for students to work individually or in pairs and
then collect solutions. Confirm that there are many correct
possibilities but that you would like to focus on the one that
involves the least number of weights. <br></br>
<br></br>
"If you had other combinations of the 3kg and 8kg weights, what
other whole number averages could you make? <br></br>
What's the smallest? What's the largest? <br></br>
Can you make all the whole number values in between?" <br></br>
<br></br>
Allow some time for the students to work in pairs. <br></br>
Collect the results on the board for future reference. <br></br>
Some students may wish to comment on patterns that they notice
(e.g. for all possible whole number averages, the number of 3kg and
8kg weights adds up to 5).<br></br>
<br></br>
"What if you have a different pair of weights? What averages can
you now make?"<br></br>
<br></br>
Encourage students to work in small groups and each choose a
different pair of weights (perhaps suggesting that they restrict
themselves to weights less than 15kg).<br></br>
<br></br>
"Share your results with your group. What do you notice? Do your
results have anything in common?" <br></br>
<br></br>
Draw the groups together and share ideas and conjectures. (e.g.
students may notice a connection between the number of weights used
and the values of those weights)<br></br>
<br></br>
Encourage students to explain their findings.<br></br>
Offering a <a href="/content/id/6345/Average4.doc">visual image</a>
may be helpful. Can they adapt this image to explain how to work
out the other averages?<br></br>
<br></br>
Students may suggest an image like <a href="/content/id/6345/Average5.doc">this</a> .<br></br>
<br></br>
"Could you make any predictions about what combinations you need to
make all possible whole number averages for any pair of weights?
Can you use what you notice to find, for example, the combination
of 17kg and 57kg weights that have an average of 44kg......of
52kg.......of 21kg.....?"<br></br>
<br></br>
Encourage students to test and explain their predictions. <br></br>
<h3>Key questions</h3>
<div>What's the smallest average you can make? What's the largest?
How do you know?</div>
<div>Can you explain how to make all the whole number averages in
between?</div>
<h3>Possible extension</h3>
<div>Given the original 3kg and 8kg weights, can you find
combinations that produce averages of 4.5kg ... of 7.5kg ... of
4.2kg ...of 6.9kg ...? Convince yourself that all averages between
3kg and 8kg are possible.</div>
<br></br>
<div>What averages are possible if you are allowed a negative
number of 3kg and 8kg weights?</div>
<br></br>
<div>Students could be directed to <a href="http://nrich.maths.org/public/viewer.php?obj_id=4838&part=">Litov's
Mean Value Theorem</a> for a suitable follow-up problem.</div>
<h3>Possible support</h3>
<div>You may initially wish to restrict the weights used to those
which have a difference of 2kg, then 3kg, then 4kg, etc. in order
to model working systematically, and to make the pattern of results
more obvious.</div>
<br></br>
<div>Some students may find multilink cubes useful to support their
visual images.</div>
<br></br>
<div><a href="http://www.shodor.org/interactivate/activities/PlopIt/">This
interactivity</a> might be useful for whole class or independent
exploration.</div>
<br></br></mdoxml>
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Start with weights that differ by 2kg. What whole number averages
can you make?<br></br>
<br></br>
Then try weights that differ by 3kg, then by 4kg...<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0">We can make all the whole number averages in between 3 kg and 8 kg,
using combinations of those two weights:<br></br>
<table border="1">
<tr>
<td style="text-align: center;">3 kg weights</td>
<td style="text-align: center;">8 kg weights</td>
<td style="text-align: center;">Average</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">4</td>
<td style="text-align: center;">7</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">3</td>
<td style="text-align: center;">6</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">2</td>
<td style="text-align: center;">5</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">4</td>
</tr>
</table>
<br></br>
<div>We can make all the whole number averages in between 2 kg and
7 kg, using combinations of those two weights:</div>
<br></br>
<table border="1">
<tr>
<td style="text-align: center;">2 kg weights</td>
<td style="text-align: center;">7 kg weights</td>
<td style="text-align: center;">Average</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">4</td>
<td style="text-align: center;">6</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">3</td>
<td style="text-align: center;">5</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">2</td>
<td style="text-align: center;">4</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">3</td>
</tr>
</table>
<div>We can make all the whole number averages in between 3 kg and
10 kg, using combinations of those two weights:</div>
<br></br>
<table border="1">
<tr>
<td>3 kg weights</td>
<td>10 kg weights</td>
<td>Average</td>
</tr>
<tr>
<td>1</td>
<td>6</td>
<td>9</td>
</tr>
<tr>
<td>2</td>
<td>5</td>
<td>8</td>
</tr>
<tr>
<td>3</td>
<td>4</td>
<td>7</td>
</tr>
<tr>
<td>4</td>
<td>3</td>
<td>6</td>
</tr>
<tr>
<td>5</td>
<td>2</td>
<td>5</td>
</tr>
<tr>
<td>6</td>
<td>1</td>
<td>4</td>
</tr>
</table>
<br></br>
<div>In each case, the total number of weights required is equal to
the difference between the values of the two
weights.</div>
<br></br>
<div>So, if we have 17 kg and 57 kg weights, we can make whole
number averages with 40 weights:</div>
<br></br>
<table border="1">
<tr>
<td style="text-align: center;">17 kg weights</td>
<td style="text-align: center;">57 kg weights</td>
<td style="text-align: center;">Average</td>
</tr>
<tr>
<td style="text-align: center;">36</td>
<td style="text-align: center;">4</td>
<td style="text-align: center;">21</td>
</tr>
<tr>
<td style="text-align: center;">13</td>
<td style="text-align: center;">27</td>
<td style="text-align: center;">44</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">35</td>
<td style="text-align: center;">52</td>
</tr>
</table>
<br></br>
<div>We can also make non-whole number averages:</div>
<div style="text-align: center;"></div>
<table border="1">
<tr>
<td style="text-align: center;">3 kg weights</td>
<td style="text-align: center;">8 kg weights</td>
<td style="text-align: center;">Averages</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">9</td>
<td style="text-align: center;">7.5</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">7</td>
<td style="text-align: center;">6.5</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">5</td>
<td style="text-align: center;">5.5</td>
</tr>
<tr>
<td style="text-align: center;">7</td>
<td style="text-align: center;">3</td>
<td style="text-align: center;">4.5</td>
</tr>
<tr>
<td style="text-align: center;">9</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">3.5</td>
</tr>
</table>
<br></br>
<div>and other decimal averages:</div>
<br></br>
<table border="1">
<tr>
<td style="text-align: center;">3 kg weights</td>
<td style="text-align: center;">8 kg weights</td>
<td style="text-align: center;">Averages</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">49</td>
<td style="text-align: center;">7.9</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">48</td>
<td style="text-align: center;">7.8</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">47</td>
<td style="text-align: center;">7.7</td>
</tr>
</table>
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Searching for mean(ing)
Imagine you have a large supply of 3kg and 8kg weights. How many of
each weight would you need for the average (mean) of the weights to
be 6kg? What other averages could you have?
Handling, Processing and Representing Data
Mean
Handling, Processing and Representing Data
Averages
Using, Applying and Reasoning about Mathematics
Generalising
Using, Applying and Reasoning about Mathematics
Making and testing hypotheses
Information and Communications Technology
smartphone
Secondary Mapping Document
Statistics – processing and representing