7344
/www/nrich/html/content/id/7344/
1
2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
<p><span style="font-style: italic;">The ideas in this problem lead on from</span> <a href="http://nrich.maths.org/7342&part=" style="font-style: italic;">Which Is Cheaper?</a><br></br>
<br></br>
<span style="font-weight: bold;">Which is bigger, $n+10$, or $2n+3$?</span><br></br>
How did you decide?<br></br>
<br></br>
<br></br>
Charlie said: </p>
<br></br>
<div style="text-align: center;">"I wonder what happens when $n=4$."</div>
<div style="text-align: center;">"$4+10=14$ but $2 \times 4 + 3$ is only $11$."</div>
<div style="text-align: center;">"So it looks like $n+10$ is bigger."</div>
<br></br>
<br></br>
<p>Alison said:</p>
<div style="text-align: center;">"I wonder what happens when $n=10$."</div>
<div style="text-align: center;">"$10+10=20$ but $2 \times 10 +3$ is $23$."</div>
<div style="text-align: center;">"So it looks like $2n+3$ is bigger."</div>
<br></br>
<p><span style="font-weight: bold;">Can you explain why they have come to different conclusions?</span><br></br>
<span style="font-weight: bold;">Is there a diagram you could draw that would help?</span><br></br>
<br></br>
<br></br>
<br></br>
For the following pairs of expressions, can you work out when each expression is bigger?<br></br>
</p>
<br></br>
<div style="text-align: center;">$2n+7$ and $4n+11$</div>
<div style="text-align: center;">$2(3n+4)$ and $3(2n+4)$</div>
<div style="text-align: center;">$2(3n+3)$ and $3(2n+2)$</div>
<div style="text-align: center;"> </div>
<div><span style="font-weight: bold;">Here are some challenges to try:</span></div>
<div> </div>
<div>Find two expressions so that one is bigger whenever $n< 5$ and the other is bigger whenever $n> 5$.</div>
<div>Find three expressions so that the first is biggest whenever $n< 0$, the second is biggest whenever $n$ is between 0 and 4, and the third is biggest whenever $n> 4$.</div>
<div>Find three expressions so that the first is biggest whenever $n< 3$, the second is biggest when $n> 3$, and the third is never the biggest.</div>
<p>Find three expressions so that one of them is the biggest regardless of the value of $n$.</p>
<p><br></br>
<a href="http://nrich.maths.org/8017">Click here for a poster of this problem</a>.</p></mdoxml>
<mdoxml version="1.0"><br></br>
<span class="editorial">Mike, James, Toby, Molly and Harry from
Highfields School, Owen, Estee and Emma from Montessori School,
Connor from Gladesmore School, and Karnan from Stag Lane Junior all
came up with a similar explanation for the first part of the
problem.</span><br></br>
<br></br>
<span class="editorial">Jasmine from Highfields
explained:</span><br></br>
<br></br>
I plotted both equations on a graph. From this I found out that
they both intercept when x=7 Therefore from the graph I have found
that:<br></br>
When n=7,both equations are the same value<br></br>
When n< 7, n+10 is the biggest<br></br>
When n> 7, 2n+3 is the biggest<br></br>
Charlie and Alison got different answers because one person chose a
value below 7 and one person chose a value above 7. This method
works with every set of equations where you have to find the
biggest.<br></br>
<br></br>
<span class="editorial">Zak explained how a graph
helped:</span><br></br>
<div style="text-align: center;"><mdo:image height="303" width="390" alt="" src="Zak.jpg"></mdo:image></div>
From looking at this graph we can see that the line with the
equation 2n+3 has a larger gradient and so overtakes the line with
the equation n+10 even though it starts off lower, so because 2n+3
has a larger gradient it will be larger after n=7.
<br></br>
<br></br>
<span class="editorial">We also received some solutions as
documents which you can download below:</span><br></br>
<br></br>
<a href="/content/id/7344/Rajeev.doc">Rajeev
from Haberdashers' Aske's Boys' School</a><br></br>
<a href="/content/id/7344/Niharika.pdf">Niharika
from Leicester High for Girls</a><br></br>
<a href="/content/id/7344/Shivin.doc">Shivin from the British
School, Manila</a><br></br>
<a href="/content/id/7344/siddhartha%20solution.pdf">Siddhartha
from Beijing City International School</a><br></br>
<a href="/content/id/7344/Amar.doc">Amar from Wilson's
School</a><br></br>
<br></br>
<span class="editorial">Well done to you all!</span><br></br>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
This problem requires students to appreciate the significance of
variables in an algebraic expression. Working on the challenges
will offer students the opportunity to apply their understanding of
equations of straight lines and simultaneous equations. The problem
could also be used in preparation for work on inequalities.<br></br>
<h3>Possible approach</h3>
<div style="font-weight: bold;">Start the lesson by posing the
question:</div>
<div>"Which is bigger, $n+10$, or $2n+3$?"</div>
<div> </div>
<div>Give students a short amount of time to decide, and then ask
them to discuss the justification for their answer in pairs. Look
out for any pairs using a graphical argument to support their
conclusions.</div>
<div> </div>
<div>Share these discussions as a class. (If everyone agrees that
one particular expression is bigger, use Charlie and Alison's
example in the problem to generate some controversy.) </div>
<div> </div>
<div>"Is there any way we could represent what's going on visually,
to convince ourselves that the first expression is bigger when
$n< 7$ and the second expression is bigger when $n>
7$?"</div>
<div> </div>
<div><span style="font-weight: bold;">Once there is an
understanding</span> that comparison of the expressions depends on
the variable $n$, and that the comparison can be done graphically,
set the next task:</div>
<div> </div>
<div>"For each pair, can you work out when each expression is
bigger?"</div>
<div style="text-align: center;">$2n+7$ and $4n+11$</div>
<div style="text-align: center;">$2(3n+4)$ and $3(2n+4)$</div>
<div style="text-align: center;">$2(3n+3)$ and $3(2n+2)$
</div>
<div> </div>
<div>Again, give students time to work on this in pairs, making
sure they are ready to justify their answers using the insights
they have gained.</div>
<div> </div>
<div><span style="font-weight: bold;">Finally</span>, set them to
work on the challenges offered in the problem. One nice way to
round off the task could be to set up a graph plotting program (<a href="http://www.geogebra.org/">Geogebra</a> is available to
download for free) and ask each pair of students to read out the
expressions they have found. As the expressions are plotted, the
class can quickly decide whether they satisfy the requirements.
This helps to capture the idea that there are infinitely many sets
of expressions that satisfy each condition.</div>
<h3>Key questions</h3>
<div>Is one expression always bigger?</div>
<div>How can you decide when each expression is bigger?</div>
<h3>Possible extension</h3>
<div>Introduce challenges that require quadratic expressions as
well as linear ones.</div>
<div>For example:</div>
<div>"Can you find two expressions so that the first is bigger for
$n< 0$ and $n> 3$, but the second is bigger when $n$ is
between $0$ and $3$?"</div>
<h3>Possible support</h3>
<div><a href="http://nrich.maths.org/5609&part=">Parallel
Lines</a> may be a suitable preliminary task for students who are
not yet confident at working with equations of straight
lines. </div>
<br></br></mdoxml>
<mdoxml version="1.0">Charlie and Alison have filled in their results in the table below.
<br></br>
Can you fill in the rest of the table?<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">
<div>$n$</div>
</td>
<td style="text-align: center;">$n+10$</td>
<td style="text-align: center;">$2n+3$</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">14</td>
<td style="text-align: center;">11</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;"> </td>
<td style="text-align: center;"> </td>
</tr>
<tr>
<td style="text-align: center;">6</td>
<td style="text-align: center;"> </td>
<td style="text-align: center;"> </td>
</tr>
<tr>
<td style="text-align: center;">7</td>
<td style="text-align: center;"> </td>
<td style="text-align: center;"> </td>
</tr>
<tr>
<td style="text-align: center;">8</td>
<td style="text-align: center;"> </td>
<td style="text-align: center;"> </td>
</tr>
<tr>
<td style="text-align: center;">9</td>
<td style="text-align: center;"> </td>
<td style="text-align: center;"> </td>
</tr>
<tr>
<td style="text-align: center;">10</td>
<td style="text-align: center;">20</td>
<td style="text-align: center;">23</td>
</tr>
</tbody>
</table>
<div> </div>
<div>What do you notice? </div>
<br></br></mdoxml>
2
4
0
0
0
1
0
Which is bigger?
Which is bigger, n+10 or 2n+3? Can you find a good method of
answering similar questions?
Sequences, Functions and Graphs
Gradients
Algebra
Solving equations graphically
Algebra
Simultaneous equations
Sequences, Functions and Graphs
Graphs