Number Tracks

2285 /www/nrich/html/content/id/2285/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Ben's class were making number tracks and cutting them up. <mdo:image width="404" height="32" align="bottom" bgcolor="" src="0_12.gif" alt="number track of 0 to 12"></mdo:image> ... ... ... First they cut them into twos. Ben started with zero <mdo:image width="449" height="32" align="bottom" bgcolor="" src="pairs1.gif" alt="Pairs of numbers: 0,1 2,3 4,5 etc"></mdo:image> ... ... ... but Miles started with one: <mdo:image width="449" height="32" align="bottom" bgcolor="" src="pairs2.gif" alt="numbers in pairs: 1,2 3,4 5,6 etc"></mdo:image> ... ... ... Then they both added up the numbers on each piece. Ben wrote: $0 + 1 = 1$, $2 + 3 = 5$, $4 + 5 = $... ... ... Miles wrote: $1 + 2 = 3$, $3 + 4 = 7$, $5 + 6 =$ ... ... ... What patterns could they see? Alice cut her number track into $3$s and added up the numbers on each one. <mdo:image width="421" height="32" align="bottom" bgcolor="" src="threes1.gif" alt="numbers in threes: 0,1,2 3,4,5 etc"></mdo:image> ... ... ... Winston made a longer number track which he cut into $5$s and he added up the numbers on each one. <mdo:image width="504" height="32" align="bottom" bgcolor="" src="fives.gif" alt="numbers in fives: 0,1,2,3,4 5,6,7,8,9 etc"></mdo:image> ... ... ... What could they discover about the sum of the numbers on their strips of number track? What other patterns can you find? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Kaan from from FMV Erenkoy Isik School in Turkey looked at this problem step by step. He wrote out the numbers on each piece of cut strip and then wrote their totals. Here is what he sent for the first strip which Ben cut into twos starting with zero: <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 0 1 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 2 3 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 4 5 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 6 7 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 8 9 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 10 11 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 12 13 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 1 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 5 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 9 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 13 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 17 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 21 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 25 </td> </tr> </tbody> </table> This is for Miles' strip which started at 1 and was cut into twos also: <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 1 2 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 3 4 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 5 6 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 7 8 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 9 10 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 11 12 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 13 14 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 3 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 7 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 11 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 19 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 23 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 27 </td> </tr> </tbody> </table> Here are the totals for Alice's strip, starting at zero and cut into threes: <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 0 1 2 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 3 4 5 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 6 7 8 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 9 10 11 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 12 13 14 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 15 16 17 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 18 19 20 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 3 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 12 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 21 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 30 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 39 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 48 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 12%;"> 57 </td> </tr> </tbody> </table> Then Kaan worked out what would happen if he had a strip cut into threes but started at one: <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 1 2 3 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 4 5 6 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 7 8 9 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 10 11 12 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 13 14 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 16 17 18 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 6 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 24 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 33 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 42 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 14%;"> 51 </td> </tr> </tbody> </table> Kaan carried on like this, cutting the strips into fours, fives, sixes and sevens! Here are the totals: <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 0 1 2 3 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 4 5 6 7 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 8 9 10 11 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 12 13 14 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 16 17 18 19 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 6 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 22 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 38 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 54 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 66 </td> </tr> </tbody> </table> <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 1 2 3 4 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 5 6 7 8 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 9 10 11 12 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 13 14 15 16 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 17 18 19 20 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 10 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 26 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 42 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 58 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 16%;"> 74 </td> </tr> </tbody> </table> <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 0 1 2 3 4 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 5 6 7 8 9 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 10 11 12 13 14 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 15 16 17 18 19 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 10 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 35 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 60 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 85 </td> </tr> </tbody> </table> <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 1 2 3 4 5 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 6 7 8 9 10 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 11 12 13 14 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 16 17 18 19 20 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 40 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 65 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 20%;"> 90 </td> </tr> </tbody> </table> <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 0 1 2 3 4 5 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 6 7 8 9 10 11 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 12 13 14 15 16 17 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 15 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 51 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 87 </td> </tr> </tbody> </table> <table border="0" align="center" style="border-width: 0pt; border-style: solid; width: 80%;"> <tbody> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> Numbers </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 0 1 2 3 4 5 6 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 7 8 9 10 11 12 13 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 14 15 16 17 18 19 20 </td> </tr> <tr> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> Totals </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 21 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 70 </td> <td valign="middle" align="center" style="border-width: 0pt; border-style: solid; width: 25%;"> 119 </td> </tr> </tbody> </table> Kaan then wrote: I found that if I grouped the numbers as pairs, I found every sum increases by 4. If I made groups of 3, I found every sum increases by 9 (3 x 3). If I made groups of 4, I found every sum increases by 16 (4 x 4). If I made groups of 5, I found every sum increases by 25 (5 x 5). If I made groups of 6, I found every sum increases by 36 (6 x 6). If I made groups of 7, I found every sum increases by 49 (7 x 7). If I made groups of 8, I found every sum increases by 64 (8 x 8). You have been very systematic in your working, Kaan, well done. Ian from Cleveland High School was able to explain why this pattern occurs. He says: If the track contains two numbers, then the difference between the nth number of the previous track and the nth number of the next track is 2. So the difference in the sums of tracks increase by 2 x 2 = 4. Well reasoned, Ian. We can then see why for groups of three numbers, the sum increases by 3 x 3 = 9 and for fours by 4 x 4 = 16 etc. A group from Bali International School noticed that for Ben's strip and Miles' strip, you can find the sum of each cut piece by multiplying the first number on the piece by $2$ and then adding $1$. Well spotted! Can you explain why this is always the case? They also noticed that for Alice's strip, the sum of each cut piece can be found by multiplying the first number on the piece by $3$ and then adding $3$. Again, I wonder why this always works? And for Winston's strip, the sum of each cut piece can be found by multiplying the first number on the piece by $5$ and then adding $10$. What fantastic pattern identification. I'd be interested to hear from anyone who could explain why these patterns occur. I wonder whether anybody has noticed any other kind of patterns? Do the totals for the strips cut into threes have anything else in common? <a href="mailto:nrich@damtp.cam.ac.uk">Email</a> us if you do spot anything else. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Number Tracks</h2> Ben's class were making number tracks and cutting them up. <mdo:image align="bottom" alt="number track of 0 to 12" bgcolor="" height="32" src="0_12.gif" width="404"></mdo:image> ... ... ... First they cut them into twos. Ben started with zero <mdo:image align="bottom" alt="Pairs of numbers: 0,1 2,3 4,5 etc" bgcolor="" height="32" src="pairs1.gif" width="449"></mdo:image> ... ... ... but Miles started with one: <mdo:image align="bottom" alt="numbers in pairs: 1,2 3,4 5,6 etc" bgcolor="" height="32" src="pairs2.gif" width="449"></mdo:image> ... ... ... Then they both added up the numbers on each piece. Ben wrote: $0 + 1 = 1$, $2 + 3 = 5$, $4 + 5 = $... ... ... Miles wrote: $1 + 2 = 3$, $3 + 4 = 7$, $5 + 6 =$ ... ... ... What patterns could they see? Alice cut her number track into $3$s and added up the numbers on each one. <mdo:image align="bottom" alt="numbers in threes: 0,1,2 3,4,5 etc" bgcolor="" height="32" src="threes1.gif" width="421"></mdo:image> ... ... ... Winston made a longer number track which he cut into $5$s and he added up the numbers on each one. <mdo:image align="bottom" alt="numbers in fives: 0,1,2,3,4 5,6,7,8,9 etc" bgcolor="" height="32" src="fives.gif" width="504"></mdo:image> ... ... ... What could they discover about the sum of the numbers on their strips of number track? What other patterns can you find? </div> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=2285&part=index">This problem</a> is a great opportunity for children to explore numbers in a familiar context in a way that is likely to be unfamiliar. This problem will also help to reinforce important number concepts and vocabulary such as odd and even numbers, factors and multiples, sequences and square numbers. It provides a way in which learners can visualise number patterns and sequences. <h3>Possible approach</h3> <div>You could start with the whole group by cutting up a large size number strip and adding the two resulting numbers together and writing these on the board. Alternatively, you could use the illustrations in the problem itself.</div> <div>NOTE: These strips are not number lines which are counted from the division between the spaces, but number tracks in which the spaces themselves are numbered.</div> <div style="text-align: center;"><mdo:image alt="lines/strips" height="36" src="TN2285.png" width="271"></mdo:image></div> <div>Before starting on the actual problem it might be useful to count together in twos, both even and odd numbers, and perhaps in threes and fives as well.</div> <div>After this learners should work in pairs on the problem with scissors and plenty of number strips. <a href="/content/id/2285/numberstrips.pdf">These sheets</a>, which can be printed off and enlarged if necessary, have strips which go from both $0 - 19$ and $1 - 20$. If learners work in pairs they are able to talk through their ideas with their partner. Listening to the pairs talking should help you to assess their understanding of multiples and other sequences.</div> <div>At the end of the lesson there should be a good discussion on the various number patterns that have been found. How do they think strips cut into sevens would go? What about fours and other even numbers?</div> <div>The cut strips and resulting added numbers could make an interesting display.</div> <h3>Key questions</h3> <div>Are you starting with zero or one this time?</div> <div>Why don't you write down the numbers you have found in a list?</div> <div>How does your sequence of numbers go?</div> <div>How do the numbers increase each time?</div> <div>In what multiplication tables are the numbers you get from adding the $3$ strips and the $5$ strips?</div> <div>In what way do your list of numbers increase each time?</div> <h3>Possible extension</h3> Learners could try higher numbers without the support of actually cutting the number tracks. Can they find a pattern for cutting numbers into fours and sixes? <h3>Possible support</h3> This activity is so accessible that the beginning should not prove too difficult for anyone. Suggest writing down the result of adding the two numbers in order and saying them out loud. Where have they heard these numbers before? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> A good look should give you at least two different patterns for the sums of the pairs of numbers. In what multiplication tables are the numbers you get from adding the $3$ strips and the $5$ strips? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> They are all odd numbers. They go up in 4s because each number on each strip is 2 more than the one on the previous strip and there are two numbers 2 x 2 = 4 All the 3 bits are divisible by 3 and the 5 bits divisible by 5. The sums increase by 3 x 3 and 5 x 5 on each strip because each number on each strip is 3 (5) more than the corresponding one on the previous strip. With strips of length 4n + 1 and 4n + 3 the sums are always in the multiplication tables. If the length of the strip is of the order 4n + 2 you always get an odd total. If the length of the strip is 4n you always get an even total and the totals are not in the corresponding multiplication tables. </mdoxml> 2 3 0 1 0 0 0 Number Tracks Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see? Numbers and the Number System Odd and even numbers Calculations and Numerical Methods Multiplication & division Numbers and the Number System Factors and multiples Calculations and Numerical Methods Addition & subtraction Sequences, Functions and Graphs Sequences Using, Applying and Reasoning about Mathematics Generalising Admin Upper primary mapping document