6713
/www/nrich/html/content/id/6713/
1
2011-02-01T00:00:01
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<mdoxml version="1.0"><br></br>
The numbers in the four times table are<br></br>
$$4, 8, 12, 16... 36, 40, 44... 100, 104, 108...$$<br></br>
I could shift the four times table up by 3 and end up with<br></br>
$$7, 11, 15, 19... 39, 43, 47... 103, 107, 111...$$<br></br>
<strong>What do you notice about the differences between consecutive terms in each sequence?</strong><br></br>
<br></br>
The interactivity displays five numbers from a shifted times table.<br></br>
<em>On <strong>Levels 1 and 2</strong> it will always be the first five numbers.<br></br>
On <strong>Levels 3 and 4</strong> it could be <span style="font-weight: bold;">any</span> five numbers from the shifted times table.</em><br></br>
<br></br>
Use the interactivity to generate some sets of five numbers.<br></br>
<strong>Can you work out the times table and by how much it has been shifted?</strong><br></br>
<br></br>
<br></br>
<div style="background-color:#ffffff;border-width:5px; border-style:groove; padding:20px;padding-bottom:20px;width:550px;height:350px;position:relative;">
<div style="color:blue;font-size:32px;font-family:helvetica,arial,sans-serif; text-align:center">Shifting Times Tables</div>
<br></br>
<br></br>
<select id="gamechoice" style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif;">
<option value="level1">Level 1</option>
<option value="level2">Level 2</option>
<option value="level3">Level 3</option>
<option value="level4">Level 4</option>
</select> <input id="newquestion" onclick="generate()" type="button" value="New Numbers"></input><br></br>
<br></br>
<div> <span id="num1" style="color:blue; position:relative; left:15px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num2" style="color:blue; position:relative; left:105px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num3" style="color:blue; position:relative; left:195px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num4" style="color:blue; position:relative; left:285px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num5" style="color:blue; position:relative; left:375px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span></div>
<br></br>
<div style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">Always enter the biggest times table it could be.<br></br>
The shift is always less than the times table.</div>
<br></br>
<div style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">Table <input id="table" size="3" style="color:blue; font-size:20px;font-family:helvetica,arial,sans-serif;" type="text"></input> Shifted <select id="direction" style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif;">
<option value="up">up</option>
<option value="down">down</option>
</select> by <input id="shift" size="3" style="color:blue; font-size:20px; font-family:helvetica,arial,sans-serif" type="text"></input> <input id="checkbutton" onclick="check()" type="button" value="Check"></input>
<p> </p>
<div id="feedback" style="text-align:center"> </div>
<p> </p>
<p> </p>
</div>
</div>
<br></br>
<br></br>
<script src="shifting.js" type="text/javascript">
</script><br></br>
Once you are confident that you can work out the times table and the shift quite easily, <span style="font-weight: bold;">here are some questions to consider:</span><br></br>
<br></br>
What can you say if the numbers are all odd?<br></br>
What about if they are all even?<br></br>
Or a mixture of odd and even?<br></br>
<br></br>
What can you say if the units digits are all identical?<br></br>
What if there are only two different units digits?<br></br>
<br></br>
What can you say if the difference between two numbers is prime?<br></br>
What can you say if the difference between two numbers is composite (not prime)?<br></br>
<br></br>
Can you explain how you worked out the table and shift each time, and why your method will always work?<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0"><span class="editorial">Benita and Chloe, from Drumbowie, suggested the following method:</span><br></br>
<br></br>
We found the differences between the numbers and looked at that times table, and then we found out if the table we had was shifted up or down from the times table it was meant to be, and how much by.<br></br>
<br></br>
<span class="editorial">Nikita commented how much easier Levels 1 and 2 were because you simply need to find the nth term rule.</span><br></br>
<br></br>
<span class="editorial">Jamie gave an example:</span><br></br>
<br></br>
14, 24, 34, 44, 54<br></br>
If the unit digits are identical, the table will be a multiple of ten and the shifted up number will be the same as the unit digit - here, 4.<br></br>
<br></br>
<span class="editorial">He also gave some rules for helping to determine the times table:</span><br></br>
<ul>
<li>If the numbers are all odd, the original table will be even and the shift will be odd.</li>
<li>If the numbers are all even, both the table and the shift will be even.</li>
<li>If the numbers are a mixture of odd and even, the table will be odd and the shifted up number will be even.</li>
<li>If there are only two different unit digits then the table is probably a multiple of five.</li>
<li>If the difference between two numbers is prime, then the table will be that prime number.</li>
</ul>
<span class="editorial">Lucia and Matt gave similar rules. Matt went on to share a harder example:</span><br></br>
<br></br>
348, 92, 252, 284, 124<br></br>
<br></br>
The differences are even, so we can see that the times table is even. We can also see that this sequence has more than 2 different unit digits, so cannot be a multiple of either 5 or 10.<br></br>
<br></br>
To work out exactly what the times table is we need to start by subtracting the lowest number from the second lowest number: 124 - 92 = 32. From this we can tell that the times table must be a factor of 32. We then need to subtract the second lowest number from the third lowest number (252 - 124 = 128), and then the next pair (284 - 252 = 32), and then the final
pair (348 -284 = 64).<br></br>
<br></br>
This gives us four numbers: 32,128,32 and 64. The highest common factor of these numbers is 32. From this we can see that the times table is 32 as all the numbers are a multiple of 32 apart.<br></br>
<br></br>
To find the shift up or down we can divide one of the numbers in our times table and look at the remainder, e.g. 92/32 = 2 remainder 28. From this, we can see that if we were to move our number down 28 it would be a multiple of 32. This gives us our solution - the times table is 32 and you need to move it down 28.<br></br>
<br></br>
<span class="editorial">Well done to Sharanya who used the same method as Matt, and to everyone else who found a method.</span></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Shifting Times Tables</h2>
<br></br>
The interactivity displays five numbers from a shifted times table.<br></br>
<em>On <strong>Levels 1 and 2</strong> it will always be the first five numbers.<br></br>
On <strong>Levels 3 and 4</strong> it could be <span style="font-weight: bold;">any</span> five numbers from the shifted times table.</em><br></br>
<br></br>
Use the interactivity to generate some sets of five numbers.<br></br>
<strong>Can you work out the times table and by how much it has been shifted?</strong><br></br>
<br></br>
<br></br>
<div style="background-color:#ffffff;border-width:5px; border-style:groove; padding:20px;padding-bottom:20px;position:relative;">
<div style="color:blue;font-size:32px;font-family:helvetica,arial,sans-serif; text-align:center">Shifting Times Tables</div>
<br></br>
<br></br>
<select id="gamechoice" style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif;">
<option value="level1">Level 1</option>
<option value="level2">Level 2</option>
<option value="level3">Level 3</option>
<option value="level4">Level 4</option>
</select> <input id="newquestion" onclick="generate()" type="button" value="New Numbers"></input><br></br>
<br></br>
<div><span id="num1" style="color:blue; position:relative; left:5px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num2" style="color:blue; position:relative; left:45px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num3" style="color:blue; position:relative; left:85px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num4" style="color:blue; position:relative; left:125px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span> <span id="num5" style="color:blue; position:relative; left:165px; font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">?</span></div>
<br></br>
<div style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">Always enter the biggest times table it could be.<br></br>
The shift is always less than the times table.</div>
<br></br>
<div style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif; text-align:center">Table <input id="table" size="3" style="color:blue; font-size:20px;font-family:helvetica,arial,sans-serif;" type="text"></input><br></br>
Shifted <select id="direction" style="color:blue;font-size:20px;font-family:helvetica,arial,sans-serif;">
<option value="up">up</option>
<option value="down">down</option>
</select> by <input id="shift" size="3" style="color:blue; font-size:20px; font-family:helvetica,arial,sans-serif" type="text"></input> <input id="checkbutton" onclick="check()" type="button" value="Check"></input>
<p> </p>
<div id="feedback" style="text-align:center"> </div>
<p> </p>
<p> </p>
</div>
</div>
<br></br>
<br></br>
<script src="shifting.js" type="text/javascript">
</script><br></br>
<br></br>
</div>
<br></br>
<h3>Why do this problem?</h3>
<p><a href="http://nrich.maths.org/6713&part=">This problem</a> encourages students to think about the properties of numbers. It could be used as an introduction to work on linear sequences and straight line graphs.</p>
<br></br>
<h3>Possible approach</h3>
<div>"I'm thinking of a times table. I wonder if you can work out which it is? $6, 12, 18, 24$" (writing the numbers on the board as you say them.)</div>
<div>"What about $33, 44, 55, 66?$"</div>
<div>"$48, 54, 60, 66?$"</div>
<div>"$135, 150, 165, 180?$"</div>
<div>Keep going until the class are confident and fluent at working out the times tables - to avoid shouting out, students could write their answers on mini whiteboards.</div>
<br></br>
<div>"What if I give you some random numbers from a times table instead? $55, 40, 105, 60$"</div>
<div>"What about $90, 60, 105, 45?$" Discuss that these are all in the $3$, $5$ and $15$ times tables, but we're only interested in finding the <span style="font-weight: bold;">largest</span> possible times table, so we'll say these are numbers in the $15$ times table.</div>
<br></br>
<div>"What about $280, 160, 560, 720?$" We hope this will catch some students out, as they'll be tempted to suggest the $10$ or possibly the $20$ times table. In fact, these are all in the $40$ times table! This is another chance to highlight that we're interested in the <span style="font-weight: bold;">largest</span> possible times table.</div>
<br></br>
<p>Now show the interactivity from the problem and alert the students that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 or 2, and give the class a short time to discuss with their partner what they think the computer has done. Do the same a couple more times, without any whole-class sharing, but giving pairs a little time to refine
their ideas. Then bring the class together and discuss what they think is going on. Link what they say to the terminology of "Table" and "Shift" that the computer uses.<br></br>
<br></br>
Emphasise that the table should always be the <span style="font-weight: bold;">largest</span> possible, and the shift should always be less than the table. This example could be used to bring these ideas out:</p>
<div style="margin-left: 40px;">$82, 202, 122, 442$</div>
<br></br>
<p>Possible suggestions that might emerge:</p>
<div style="margin-left: 40px;">Table: 10, Shift: 2, or 12, or 22...</div>
<div style="margin-left: 40px;">Table: 5, Shift: 2, or 7, or 12...</div>
<div style="margin-left: 40px;">Table: 20, Shift: 2, or 22, or 42...</div>
<br></br>
<p>But we are interested in</p>
<div style="margin-left: 40px;">Table: 40, Shift: 2.</div>
<br></br>
<p>Ideally, each pair would now work at a computer to develop a method of finding the table and shift with ease. If that isn't possible, generate a dozen or so examples at appropriate levels, and write them on the board for the class to work on. Students could also work in pairs and create examples for their partners to work out.<br></br>
<br></br>
Once students are finding the table and shift easily, bring the class together. Generate a new example and ask a pair to talk through their thinking as they work towards the solution, but ask them to stop short of actually giving the answer. The rest of the class could write the answer on mini whiteboards once they've heard enough to work it out. Repeat, giving other pairs the opportunity to
share their thinking.<br></br>
<br></br>
Finally, allow the class some time to work in pairs on the questions at the bottom of the problem, and then discuss their ideas, emphasising the need to justify any conclusions they reach.<br></br>
<br></br>
<a href="/content/id/6713/Shifting%20times%20tables%20-%20final.pdf" style="font-style: italic;">Here</a> <span style="font-style: italic;">is an account of one teacher's approach to using this problem.</span></p>
<h3>Key questions</h3>
<div>What is the same between numbers in a times table and numbers in the shifted times table?</div>
<div>What can you learn from the difference between any two numbers in a shifted times table?</div>
<div>How do you find the shift once you've worked out the table?</div>
<h3>Possible extension</h3>
<div>Here are some follow-up resources that may build on students' thinking about this problem:</div>
<div><a href="/7405">What Numbers Can We Make?</a><br></br>
<a href="/7024">Charlie's Delightful Machine</a></div>
<div><a href="http://nrich.maths.org/6402&part=">The Remainders Game</a></div>
<div><a href="http://nrich.maths.org/582&part=">Expenses</a></div>
<div><a href="http://nrich.maths.org/4350&part=">Modular Arithmetic</a> (article)</div>
<br></br>
<h3>Possible support</h3>
<div>Perhaps start with the <a href="http://nrich.maths.org/5468&part=">Factors and Multiples Game</a> to practise working with multiples and factors. This could then be followed up by looking at the problem <a href="http://nrich.maths.org/1783&part=">Remainders</a>.</div>
<p><br></br>
<br></br>
</p></mdoxml>
<mdoxml version="1.0"><br></br>
<p>For the Level 3 and 4 problems, start by rearranging the numbers so that they are in order. Then look at the pairs of numbers that are closest together.</p></mdoxml>
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<mdoxml version="1.0"><p class="editorial">This problem encourages you to think about the characteristics of numbers and the times tables that they are part of. Noticing these patterns is not only fun, but will also help you with calculations in other work that you do.</p>
<p class="editorial">Many great solutions were submitted, with clear explanations, so thank you very much to everyone for this!</p>
<p class="editorial">Let us first consider levels one and two. In these levels, the numbers displayed are always the first five numbers from a shifted times table. A student from Mearns Castle High School explained:</p>
In Levels $1$ and $2$, if the first number is x and the second is $x+d$, then the d times table has been altered.<br></br>
<p class="editorial">Using this notation, we can also find an expression for the third number: $x+2d$. In fact, we can write a formula for any number. For example, to find the $n^{th}$ number, we can write a general formual: $x+(n-1)d$. Note that in levels one and two, the times table number is the same as "d". However, this may not be the case in levels three and four, as the numbers may not be
consecutive.</p>
<p><span class="editorial">The question also asks you to work out the number that the times table has been shifted by. Holly, Nick, and Lucy from Old Earth Primary School first found the times table. Then they subtracted this number (e.g. "$5$" for the five times table) from the first number of the sequence to find the shift. Sharumilan from Wilson's School, and James from Woodstock CE Primary
School used a similar method.</span></p>
<p><span class="editorial">When solving the problem, several students found it helpful to look at the properties of the numbers in the sequence. For example, they noted whether the units digits were all odd, or all even, or a mixture, and whether they were all identical or whether there were only two different digits. This can eliminate some possibilities for the times table and shift, whilst
pointing you in the right direction to find the answer. Other characteristics students examined included the type of difference between the numbers in the sequence (prime or composite).</span></p>
<p><span class="editorial">Emma from Tadcaster Grammar School answered the questions posed regarding the</span> <span class="editorial" style="text-decoration: underline;">Level 1 and 2 problems</span><span class="editorial">:</span></p>
<br></br>
1. WHAT CAN YOU SAY IF THE NUMBERS ARE ALL ODD?<br></br>
The times table is always even, and it has been shifted by an odd number.<br></br>
<br></br>
2. WHAT ABOUT IF THEY ARE ALL EVEN?<br></br>
The times table is always even, and it has been shifted by an even number.<br></br>
<br></br>
3. OR A MIXTURE OF ODD AND EVEN?<br></br>
The times table is always odd; the shift could be odd or even.<br></br>
<br></br>
4. WHAT CAN YOU SAY IF THE UNITS DIGITS ARE ALL IDENTICAL?<br></br>
The times table has a zero as its unit digit i.e. it is a multiple of ten.<br></br>
<br></br>
5. WHAT IF THERE ARE ONLY TWO DIFFERENT UNITS DIGITS?<br></br>
The times table has a five as its unit digit.<br></br>
<br></br>
6. WHAT CAN YOU SAY IF THE DIFFERENCE BETWEEN TWO NUMBERS IS PRIME?<br></br>
The times table is prime.<br></br>
<br></br>
7. WHAT CAN YOU SAY IF THE DIFFERENCE BETWEEN TWO NUMBERS IS COMPOSITE (NOT PRIME)?<br></br>
The times table is composite.<br></br>
<br></br>
<p class="editorial">Many others submitted correct answers to these questions. These include: Sharumilan from Wilson's School, Harry from Maidstone Grammar School, Stephen and Alexander from Tudhoe Grange, Andrew from Charters Secondary School, and Jack from Sir Harry Smith Community College. Several mentioned the importance of considering factors and multiples, in order to come to the correct
conclusions about the times tables. Emma mentioned this in her solution (see above).<br></br>
<br></br>
Harry from Maidstone Grammar School explained an approach that would work at all levels:</p>
<div style="font-style: italic;">Can you explain how you worked out the table and shift each time, and why your method will always work?</div>
<br></br>
<div>1. Put the numbers given in ascending order.</div>
<div>2. Find the difference between each number and the number next to it.</div>
<div>3. Find the highest common factor of all the differences (this is the table)</div>
<div>4. Divide one of the numbers by the highest common factor and find the remainder (this is the shift).</div>
<br></br>
<div>My method will always work because the differences between each number must be a multiple of the table, or the table itself, as otherwise you would not be able to increase the original number to the next.</div>
<br></br>
<div>e.g. 43 --- 55 --- 61 The difference between 43 and 55 is 12, the difference between 55 and 61 is 6, therefore the highest common factor is 6. 43 divided by 6 = 7 remainder 1 so the shift is 1!</div>
<br></br>
<p class="editorial">This is a lovely explanation, and a very good systematic way to approach the problem. Many students, like Harry, mentioned that the first step is to order the numbers. This is a useful start to very many problems, not just in mathematics, but in other subjects too. It is helpful to begin by logically organising the information that you are given so that patterns can be seen
more clearly and the information can be manipulated more easily.</p>
<p class="editorial"><span style="text-decoration: underline;">For levels 3 and 4</span>, the numbers given can be any five numbers; they are not necessarily in order. This means that the difference between two of the numbers may not be the actual number of the times table; there may be other numbers in between these, which have not been provided. This is why we need to find the highest common
factor of all of the differences. Some students just used the smallest difference between the numbers in the sequence and concluded that this is the times table number. However, this difference may in fact be a multiple of the "true" difference.</p>
<p class="editorial">Here is a solution that illustrates these points nicely. It was submitted by Luke, from Sawston Village College who participated in a Maths Masterclass:</p>
<p>For Levels three and four, where the numbers are in a random order, it is advisable to look at all of the differences, as sometimes they are different.</p>
<p>For example: $239, 459, 519, 579, 619$.</p>
<p>In this, the smallest difference is $40$, but there are also differences of $60$ and $220$.<br></br>
Because you must enter the largest times table that works for all of the numbers, you cannot enter $40$ (the smallest difference), as that doesn't work for all the differences.<br></br>
$20$ goes into all of these, so it is the largest times table that works for all of the differences.</p>
<p>However, if one of the differences you find is prime, there is no point in going through the process of looking at all of the differences, as it isn't divisible by anything apart from $1$ or itself, and therefore will not share a factor with any other differences you find. The shift is not altered by different differences.</p>
<p><span class="editorial">Thank you also to the following students, who submitted great explanations for working out the times table and the shift: the two "Toms" from Bassingbourn Village College, Stephen and Alexander from Tudhoe Grange School, and Holly, Nick and Lucy from Old Earth Primary School.</span></p>
<p><span class="editorial">If you enjoyed this problem and would like to follow it up, have a look at the article on</span> <a class="editorial" href="http://nrich.maths.org/4350&part=Modular%20Arithmetic">Modular Arithmetic</a><span class="editorial">.</span></p>
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<br></br></mdoxml>
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Shifting Times Tables
Can you find a way to identify times tables after they have been shifted up?
Sequences, Functions and Graphs
Sequences
Sequences, Functions and Graphs
Arithmetic sequence
Numbers and the Number System
Factors and multiples
Secondary Mapping Document
Patterns and sequences LS
Secondary Mapping Document
DisplayCabinet
Secondary processes
PM - Exploring and noticing structure
Secondary processes
PM - Working Systematically
ajk44
solution needs editing