5958
/www/nrich/html/content/id/5958/
1
2011-02-01T00:00:01
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<mdo:image alt="Weights" height="127" src="weights_img.gif" width="350"></mdo:image><br></br>
<br></br>
Imagine you have <span style="font-weight: bold;">two of each</span> of the 'weights' above.<br></br>
Different combinations of the weights available allow you to make different totals.<br></br>
<br></br>
Here are some examples:<br></br>
<div style="margin-left: 40px;">$B + C = 6$<br></br>
$B + 2C = 15$<br></br>
$A + 2B + C = 4$<br></br>
$2A + B + 2C + D =$ $-10$</div>
<br></br>
The largest total you can make is $20$ (check you agree).<br></br>
The smallest total you can make is $-60$ (again, check you agree).<br></br>
<br></br>
Can you make all the numbers in between?<br></br>
Can you show us how?<br></br>
<br></br>
Is there always a unique way of producing a total, or can different combinations produce the same total?<br></br>
<br></br>
<span style="font-weight: bold;">Extension:</span><br></br>
<br></br>
Imagine you are allowed just three different weights this time ($E$, $F$ and $G$), and at least one must be a negative weight, but you are allowed to have up to <span style="font-weight: bold;">three of each</span> .<br></br>
<br></br>
For example, if you choose:<br></br>
<div style="margin-left: 40px;">$E = 1$<br></br>
$F =$ $-4$<br></br>
$G = 5$</div>
<br></br>
you can make $7$ and $-10$:<br></br>
<br></br>
<div style="margin-left: 40px;">$E + F + 2G = 7$<br></br>
$2E + 3F =$ $-10$</div>
<br></br>
Choose your three weights and test out which totals you can make.<br></br>
<br></br>
<span style="font-weight: bold;">Which set of three weights ($E$, $F$ and $G$) allows you to make the largest range of totals with no gaps in between?</span><br></br>
<br></br>
<br></br>
<span class="editorial">This problem is based on one of Don Steward's problems published by MEDIAN.</span><br></br>
<span class="editorial">Don can be contacted at William Brookes School in Shropshire where he is now based.</span><br></br></mdoxml>
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<p><span class="editorial">We received a complete set of
solutions,showing that all the numbers between -60 and 20 could be
done, from Fred, Chester and Tom from Hotwells Primary School and
from Josh B, Matthew H, Seb W, Ben S, Max G and Jamie W, all in 8P1
at Hove Park School.</span> <a class="editorial" href="/content/id/5958/YEAR%208%20WEIGHTS%20SOLUTION.xls">Here</a>
<span class="editorial">is the solution from the Hove Park students
(addition signs have been omitted so it is easier to see the
pattern that forms).</span></p>
<p class="editorial">The Maths Challenge Group at Colyton Grammar
School explained how they approached the problem:</p>
<div>First of all we tried to find the solution by dividing all of
the numbers between us and finding ways to reach these numbers.
e.g.</div>
<div>-20 = D+C+B+A</div>
<div>-21 = D+C+B</div>
<div>-22 = D+C+2B+2A</div>
<div>-23 = D+C+2B+A</div>
<br></br>
<div>Using this method we couldn't find any solutions that didn't
occur. However, we noticed a pattern in the weights, that each
weight was -3 times the previous weight. In effect this means that
the solutions are the numbers written in base -3.</div>
<div>This means that the first weight (A) are 1s, (B) are -3s,
etc.</div>
<div>Since for each weight we can have 3 possible values (0, 1 or 2
weights), the number required for a -3 based system, we can make
any of the numbers in the range.</div>
<br></br>
<div>In this base system any number can be written in only one way
- just as in base 10.</div>
<br></br>
<div>Using this theory we quickly worked out the best values for
the extension.</div>
<div>Since for each weight there are 4 possible values (None, 1, 2
or 3 weights) this will be a base -4 system, so the weights must be
(since the most basic unit 1 is required) 1, -4 and 16.</div>
<div>This could also be done with a positive 4, giving the same
range but no negative values.</div>
<br></br>
<div><span class="editorial">Thank you and well done to you
all.</span></div>
<br></br></mdoxml>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=5958">This
problem</a> offers students an opportunity to add positive and
negative numbers while challenging them to work
systematically.<br></br>
<h3>Possible approach</h3>
Introduce the scenario as it appears in the problem. Negative
weights could be seen as strong helium balloons lifting up the pan
of the weighing scales. <br></br>
<br></br>
Students could suggest a few examples for the class to work through
and clarify the procedure. Ask about the largest total and the
smallest total, giving time for students to explain to their
partners why $20$ and $^-60$ are correct.<br></br>
Set the class working on the main task:<br></br>
Can you make all the numbers in between? Can you show us how?
<br></br>
Is there always a unique way of producing a total, or can different
combinations produce the same total?<br></br>
<br></br>
There are different strategies for tackling this problem so if you
observed that students approached the task in different ways, ask
(carefully selected) students to describe how they did it. This
could lead to a discussion of the merits of the different
approaches.<br></br>
<h3>Key questions</h3>
Can you make all the numbers in between? Can you show us how?
<br></br>
Is there always a unique way of producing a total, or can different
combinations produce the same total?<br></br>
<h3>Possible extension</h3>
<div>The extension task in the main problem is suitable here.</div>
<div>Some students may learn from seeing how other students have
thought about this, and how they have explained their ideas. Direct
them to the solutions page for this problem.</div>
<h3>Possible support</h3>
Organise students into small teams with a big central sheet of
paper. Teams can write the numbers from $20$ to $^-60$, and then
students can fill in any sum that they find. Ask students to pass
their suggested sum to a team mate for checking before it is
written on the team sheet. If appropriate, the teams could be
racing each other to get as many totals as possible, but organise a
penalty system for errors (thus really encouraging the checking
process!).<br></br>
<br></br>
Teachers may like to take a look at the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=5947&part=">Adding
and Subtracting Negative Numbers</a> <br></br></mdoxml>
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The largest total you can make is $20$ ($2A + 2C$). <br></br>
<br></br>
The smallest total you can make is $-60$ ($2B + 2D$). <br></br>
<br></br>
There are some totals that are easier to produce than others
(perhaps those that require just one or two weights), so it may be
a good idea to start with those and then build on them with a few
extra weights.<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0">I can make all the totals except for zero.<br></br>
I can make each totoal in just one way.<br></br></mdoxml>
2
4
0
0
1
0
0
Weights
Different combinations of the weights available allow you to make
different totals. Which totals can you make?
Numbers and the Number System
Positive-negative numbers
Using, Applying and Reasoning about Mathematics
Working systematically
Calculations and Numerical Methods
Addition & subtraction
Information and Communications Technology
smartphone
Secondary Mapping Document
Place value, ordering and rounding LS