8280
/www/nrich/html/content/id/8280/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p><em>This problem follows on from <a href="/7405">What Numbers Can We Make?</a></em></p>
<p> </p>
<p>The interactivity below creates sets of bags similar to those in <a href="/7405">What Numbers Can We Make?</a></p>
<p>Investigate what numbers you can make when you choose three numbers from the bags. Once you think you know what is special about the numbers you can check your answer.</p>
<p>You can also find what is special when you choose four, five or six numbers.</p>
<p>When you have an efficient strategy, test it by choosing 99 or 100 numbers.<br></br>
<br></br>
<em>Always enter the biggest possible multiple. The "plus" number may include zero.</em></p>
<p> </p>
<div style="border-width:5px; border-style:groove; padding:20px;padding-bottom:20px;width:550px;height:350px;position:relative;">
<input id="newquestion" type="button" onclick="generate()" value="New Numbers" />
<p></p>
<div style="color:#000000;font-size:16px;font-family:helvetica,arial,sans-serif">
<form id="frm1">
Choose
<select id="howmany" style="color:#000000;font-size:16px;font-family:helvetica,arial,sans-serif" onchange="wipe()">
<option value="3">three</option>
<option value="4">four</option>
<option value="5">five</option>
<option value="6">six</option>
<option value="99">99</option>
<option value="100">100</option>
</select>
numbers from the bags and find their total.</form>
</div>
<p></p>
<div style="position:relative; text-align:center">
<img id="Bags" src="what numbers bag.png" />
<span id="bag1" style="position:absolute; top:65px;left:136px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag2" style="position:absolute; top:60px;left:239px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag3" style="position:absolute; top:55px;left:347px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag4" style="position:absolute; top:62px;left:438px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag1a" style="position:absolute; top:45px;left:108px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag2a" style="position:absolute; top:40px;left:217px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag3a" style="position:absolute; top:35px;left:317px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag4a" style="position:absolute; top:32px;left:420px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag1b" style="position:absolute; top:86px;left:108px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag2b" style="position:absolute; top:80px;left:215px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag3b" style="position:absolute; top:82px;left:302px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag4b" style="position:absolute; top:80px;left:405px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag1c" style="position:absolute; top:65px;left:98px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag2c" style="position:absolute; top:60px;left:197px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag3c" style="position:absolute; top:59px;left:302px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag4c" style="position:absolute; top:58px;left:405px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag1d" style="position:absolute; top:88px;left:140px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag2d" style="position:absolute; top:85px;left:242px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag3d" style="position:absolute; top:80px;left:340px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
<span id="bag4d" style="position:absolute; top:85px;left:430px; font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center"></span>
</div>
<p></p>
<div style="color:#000000;font-size:16px;font-family:helvetica,arial,sans-serif">
The answer is always <input id="remainder" type="text" size="3" style="font-size:16px; font-family:helvetica,arial,sans-serif; text-align:center" />
<select id="plusminus" style="color:#000000;font-size:16px;font-family:helvetica,arial,sans-serif" >
<option value="plus">more</option>
<option value="minus">less</option>
</select>
than a multiple of
<input id="multiple" type="text" size="4" style="font-size:16px;font-family:helvetica,arial,sans-serif; text-align:center" />
<p></p>
<input id="check" type="button" onclick="check()" value="Check" />
<p>
<span id="feedback" style="font-size:16px;font-family:helvetica,arial,sans-serif;"></span>
</p>
<p>
</p>
</div>
</div>
<p><br></br>
<span style="font-weight: bold;">Can you explain your strategies?</span><br></br>
<br></br>
<span style="font-weight: bold;">And finally...</span><br></br>
<br></br>
If the bags contained 3s, 7s, 11s and 15s, can you describe a quick way to check whether it is possible to choose 30 numbers that will add up to 412?<br></br>
<br></br>
<br></br>
There are a few related problems that you might like to try:</p>
<p><a href="/6713">Shifting Times Tables</a><br></br>
<a href="http://nrich.maths.org/7024&part=">Charlie's Delightful Machine</a><br></br>
<a href="/7016">A Little Light Thinking</a><br></br>
<a href="http://nrich.maths.org/1783&part=">Remainders</a><br></br>
<a href="http://nrich.maths.org/746&part=">Where Can We Visit?</a><br></br>
<a href="http://nrich.maths.org/1864&part=">Cinema Problem</a></p>
<script src="8280-script.js" type="text/javascript">
</script></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><span class="editorial">Henry, from St. Hugh's, answered our question at the end:<br></br>
<br></br>
<em>If the bags contained 3s, 7s, 11s and 15s, can you describe a quick way to check whether it is possible to choose 30 numbers that will add up to 412?</em><br></br>
<br></br>
He said the following:</span><br></br>
<br></br>
It is impossible to make 412.<br></br>
The starting number is 3 and the difference between the numbers is 4.<br></br>
If I choose 30 numbers and add them all up, I will get a number that is 30x3=90 more than a multiple of 4.<br></br>
But, 90÷4=22 remainder 2, so I will get a number that is 2 more than a multiple of 4.<br></br>
But since 412 is a multiple of 4 (not 2 more than a multiple of 4), it won't work.<br></br>
<br></br>
<span class="editorial">Well spotted! Luke, from Cottenham Village College, said this in a more algebraic way, using a tool called 'modular arithmetic':</span><br></br>
<br></br>
All of our numbers are of the form 4x-1, for x=1, 2, 3 or 4. Therefore the sum of 30 of them comes to $4(x_1+x_2+\dots +x_{30})-30$, which is 2 mod 4 <span class="editorial">(i.e. 2 more than a multiple of 4)</span>. But 412 is 0 mod 4 <span class="editorial">(i.e. 0 more than a multiple of 4)</span>, so this sum cannot be equal to 412.<br></br>
<br></br>
<span class="editorial">If you are unfamiliar with Modular Arithmetic, you might like to take a look at <u><a href="/4350">this introductory article</a></u>.<br></br>
<br></br>
Luke also gave his thoughts on the interactivity:</span><br></br>
<br></br>
The numbers in the bag always form part of a linear arithmetic sequence, and so the number in the x-th bag is mx+c. Consecutive numbers are always a fixed distance m apart. This means that we can read off the value of m easily, and then find the value of c. We can then conclude that, if you choose z numbers from the bags, their sum will be of the form $m(x_1+x_2+\dots +x_z) + cz$, as all the
numbers are of the form mx+c, for different values of x. This is obviously cz more than a multiple of m.<br></br>
<br></br>
To illustrate this method with an example, take the numbers in the bag to be 1, 4, 7 and 10, as in the original example, with z=4.<br></br>
Their differences are all a multiple of 3, so by analysing them mod 3 <span class="editorial">(i.e. by looking at their remainders when they are divided by 3)</span> we find that they are all of the form 3x+1.<br></br>
As z=4 and c=1, they must be cz=4 more than a multiple of 3.<br></br>
Since 4 is 1 mod 3 <span class="editorial">(i.e. dividing 4 by 3 gives remainder 1)</span>, 4 numbers selected from the bags must add to give 1 more than a multiple of 3.<br></br>
<br></br>
<span class="editorial">Nice!</span><br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><h3>Why do this problem?</h3>
<p>This problem offers students the opportunity to consider the underlying structure behind multiples and remainders, as well as leading to some very nice generalisations and justifications.<br></br>
</p>
<h3>Possible approach</h3>
<div> </div>
<div>This problem should be attempted after working on <a href="/7405">What Numbers Can We Make?</a></div>
<div> </div>
<div>Start by showing the interactivity from the <a href="/8280">problem</a>, and clicking on 'New Numbers' several times:</div>
<div>"This interactivity can generate lots of different sets of bags like the set we worked on last lesson. Later on I'm going to generate a set of bags and ask you what is special about the total when I choose three, four, five, six... 99 or 100 numbers. To prepare a strategy for answering these questions, here are some bags to get you started."</div>
<div> </div>
<div>Display this <a href="/content/id/8280/all%20bags.png">image</a> (available as a <a class="powerpoint" href="/content/id/8280/What%20Numbers2.ppt">PowerPoint</a>). Then arrange the class in pairs or small groups, and allocate one or two sets of bags to each.<br></br>
</div>
<div><strong>"In a while, you'll need to be able to explain to the rest of the class what happens when you add together three, four, five, six... 99, 100 numbers from your set of bags, and how you worked it out."</strong></div>
<div> </div>
<div>While groups are working, circulate and listen for any useful insights to bring out in the whole class discussion later. If anyone finishes their set of bags early, they can apply their strategy to someone else's set.</div>
<div> </div>
<div>Bring the class together, and invite groups to share what they found. Then allow groups a few minutes to discuss a general strategy for answering the questions generated by the interactivity.</div>
<div> </div>
<div>Finally, display the interactivity again. Generate new questions, and invite the groups to use their strategy to work out what happens for three, four, five, six, 99 or 100 numbers. Check their answers, and then repeat, giving each group a chance to have a go at answering a 99 or 100 question.</div>
<div> </div>
<div>You could finish off by asking the final question from the problem:</div>
<div>"If the bags contained 3s, 7s, 11s and 15s, can you describe a quick way to check whether it is possible to choose 30 numbers that will add up to 412?"</div>
<div> </div>
<h3>Key questions</h3>
<div> </div>
<div>If I choose 5 numbers that are each one more than a multiple of 5, what is special about their total? Why?<br></br>
</div>
<h3>Possible extension</h3>
<p>There are a few related problems that students could work on next:</p>
<p><a href="http://nrich.maths.org/1866&part=">Take Three from Five</a><br></br>
<a href="/6713">Shifting Times Tables</a><br></br>
<a href="http://nrich.maths.org/7024&part=">Charlie's Delightful Machine</a><br></br>
<a href="/7016">A Little Light Thinking</a><br></br>
<a href="http://nrich.maths.org/1783&part=">Remainders</a><br></br>
<a href="http://nrich.maths.org/746&part=">Where Can We Visit?</a><br></br>
<a href="http://nrich.maths.org/1864&part=">Cinema Problem</a></p>
<p> </p>
<h3>Possible support</h3>
<p>Begin by asking students to explore what happens when they add numbers chosen from a set of bags containing 2s, 4s, 6s and 8s.</p>
<p>They could then consider what happens when they add numbers chosen from a set of bags containing 1s, 11s, 21s and 31s.</p>
<p>Can they explain their findings?</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p>This problem follows on from <a href="/7405">What Numbers Can We Make?</a>, so perhaps you might explore that first.</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"> </mdoxml>
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What numbers can we make now?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Using, Applying and Reasoning about Mathematics
Mathematical reasoning & proof
Numbers and the Number System
Divisibility
Numbers and the Number System
Factors and multiples
Using, Applying and Reasoning about Mathematics
Generalising
Numbers and the Number System
Modulus arithmetic