Factor track

7468 /www/nrich/html/content/id/7468/ 1 2011-06-01T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Factor Track is not a race but a game of skill.<br></br> Below is a simple training version. The idea is to go round the track in as few moves as possible, keeping to the rules.<br></br> <br></br> <div style="text-align: center;"><mdo:image width="315" height="257" src="Ftrack.png" alt="track"></mdo:image> </div> <br></br> <br></br> Rules:<br></br> You start on the (yellow) $60$ and must make your way round to the (red) 'end' square.<br></br> You can move any factor of the number you are on, except $1$.<br></br> You must land exactly on each green square, so you can't go round corners in one move.<br></br> <br></br> Have a go at moving round this 'training' track following the rules. <br></br> Can you do it in fewer moves?<br></br> <br></br> When you feel ready, try this more complicated track where there are possible short cuts. You will have to work out whether they are worthwhile.<br></br> <br></br> <div style="text-align: center;"> <mdo:image width="292" height="325" src="Ftrack2.png" alt="bigger track"></mdo:image></div> <br></br> You might find it easier to print out <a href="/content/id/7468/Ftrack4.pdf">this larger copy</a> of the track which also has the rules on it.<br></br> <br></br> You can do this on your own or with a friend. Who can get round in the least number of moves?<br></br> <br></br> What is the best route to take to do it in the least number of moves?<br></br> Which squares do you need to land on?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">J Darcy from Pulner Junior School and Otto completed the first Factor Track in the following 9 moves:</span><br></br> <br></br> Divide 60 by 2 and move on 2 spaces, then divide 14 by 7 and move on 7 spaces. 7 is a factor of 28 so move on 7 spaces. 18 divides into 3 so move on 3 spaces. 14 divides into 2 so move on 2 spaces. 24 divides into 3 so move to 36. 36 divides into 3 to move to 32. 32 divides by 4 so move to 14. 14 divides by 7 to reach the end!<br></br> <br></br> <span class="editorial">Lots of you went straight to the more challenging one and managed it in 10 moves including students from Wilson's School, St Helen's C of E Primary School, Baston C of E Primary School, MacDiarmid Primary School and Highfield School.</span><br></br> <br></br> <span class="editorial">Ultra Violet Class from Unicorn explain their solution here:</span><br></br> <br></br> Issy, Laura and Anna worked as a team to solve this problem. They found that doing the training track gave them a very good understanding of the problem. On the more complicated track they found the shortest route involved 10 steps. They used a counter and recorded each step by writing the sequence of numbers linked by arrows with the number of moves written above:<br></br> <br></br> 60 (3 moves) 48 (6 moves) 28 (7 moves) 51 (3 moves) 96 (4 moves) 81 (3 moves) 54 (2 moves) 87 (3 moves) 72 (4 moves) 49 (7 moves) END<br></br> <br></br> They noted that the only point they could not go from corner to corner was on the first row where they had to use 2 moves between the numbers but there were lots of options:<br></br> <br></br> 60 to 14 to 28<br></br> 60 to 48 to 28<br></br> 60 to 25 to 28<br></br> 60 to 45 to 28<br></br> <br></br> They found using divisibility rules to check for factors of the larger numbers a very useful approach. The most helpful one being the divisibility rule for the number 3.<br></br> <br></br> Alex, Jack and Jamie also came up with the shortest route of 10 steps. Peter, Ben and Jack all tried going the longer route and found they could do it in 18, 17 and 16 steps. The shortest route of 16 steps was achieved by moving from corner to corner in all but two cases. This was the first row as described above and the very last column where they had to go 12 to 49 to END.<br></br> <br></br> <span class="editorial">Mr Bouchard's Class from Richmon Elementary School, USA, sent us a picture of their solution:</span><br></br> <br></br> <mdo:image height="471" width="479" src="Factor%20Track%20-%20Richmond.jpg" alt="rich"></mdo:image><br></br> <br></br> <span class="editorial">And Krystov's explanation can be found</span> <a class="editorial" href="/content/id/7468/factor%20-%20krystov.pdf">here</a><span class="editorial">.</span><br></br> <br></br> <span class="editorial">Students from Greenacre School for Girls decided to make their own Factor Tracks. See if you can complete them.</span><br></br> <span class="editorial"> </span> <br></br> <a href="/content/id/7468/factor%20-%20grace%20and%20katie.pdf" class="editorial">1 by Erica and Helena</a><br></br> <a class="editorial" href="/content/id/7468/factor%20-%20grace%20and%20katie.pdf">2 by Grace and Katie</a><br></br> <a class="editorial" href="/content/id/7468/factor%20-%20rebecca%20and%20elizabeth.pdf">3 by Rebecca and Elizabeth</a><br></br> <a class="editorial" href="/content/id/7468/factor%20-%20Rosie.pdf">4 by Rosie</a><br></br> <a class="editorial" href="/content/id/7468/factor%20-%20sophie%20and%20esme%20and%20josie.pdf"> 5 by Sophie, Esme and Josie</a><br></br> <a class="editorial" href="/content/id/7468/factor%20-%20zoe%20and%20rachel.pdf">6 by Zoe and Rachel</a><br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Factor Track</h2> <br></br> Factor Track is not a race but a game of skill.<br></br> Below is a simple training version. The idea is to go round the track in as few moves as possible, keeping to the rules.<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="track" height="257" src="Ftrack.png" width="315"></mdo:image></div> <br></br> <br></br> Rules:<br></br> You start on the (yellow) $60$ and must make your way round to the (red) 'end' square.<br></br> You can move any factor of the number you are on, except $1$.<br></br> You must land exactly on each green square, so you can't go round corners in one move.<br></br> <br></br> Have a go at moving round this 'training' track following the rules. <br></br> Can you do it in fewer moves?<br></br> <br></br> When you feel ready, try this more complicated track where there are possible short cuts. You will have to work out whether they are worthwhile.<br></br> <br></br> <div style="text-align: center;"> <mdo:image alt="bigger track" height="325" src="Ftrack2.png" width="292"></mdo:image></div> <br></br> You might find it easier to print out <a href="/content/id/7468/Ftrack4.pdf">this larger copy</a> of the track which also has the rules on it.<br></br> <br></br> You can do this on your own or with a friend. Who can get round in the least number of moves?<br></br> <br></br> What is the best route to take to do it in the least number of moves?<br></br> Which squares do you need to land on?</div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/7468&part=">This problem</a> will help to improve learners' knowledge of factors, especially those in the usual multiplication tables, and encourages them to use trial and improvement. The competitive element can bring out the best in some pupils.<br></br> <h3>Possible approach</h3> <div>You could start by showing the group the 'training track' given in the problem, working on this so that they are able to see the rules in action. <a href="/content/id/7468/Ftrack3.pdf">This sheet</a> has the 'training track' on, if you want it. Once they have had a go, you could spend a short time discussing the reasons for their choices and ultimately the minimum number of moves that will take you round the track. </div> <div> </div> <div>Next, introduce the trickier version of the track, giving pairs a copy of <a href="/content/id/7468/Ftrack4.pdf">this sheet</a>. (The sheet gives the rules as well as the full track). This could be printed out in an enlarged version and could also be laminated for a longer life. Allow children to choose any other tools that they feel would be helpful. Some may want to mark their chosen squares in pencil on the track, some may want to record the numbers or calculations on a separate sheet or whiteboard, some will have other ideas!</div> <div> </div> <div>At the end of the lesson all the learners could come together to discuss the best and shortest route. Is there a consensus about the best moves to take?</div> <br></br> <h3>Key questions</h3> <div>Which tables will you find this number in?</div> <div>What are the factors of that number?</div> <div>Can you think of any more? How do you know you've thought of all the factors?</div> <div>How are you keeping track of the route you're taking?</div> <div>How do you know that you have found the best route?</div> <br></br> <h3>Possible extension</h3> <div>Learners could make their own 'factor track' for others to try. Alternatively, the problem <a href="http://nrich.maths.org/5578&part=">Factor-multiple Chains</a> offers another interesting way to explore factors.</div> <br></br> <h3>Possible support</h3> You could suggest that learners record all factors of the number in green squares so they are able to keep track of the ones they have tried more easily. The problem <a href="http://nrich.maths.org/7471&part=">Jumping Squares</a> has the same idea of getting round a track in as few moves as possible but focuses on counting, rather than factors.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You might try writing down all factors of the numbers in the green squares.<br></br> How will you know which routes you have tried?<br></br></mdoxml> 5 4 0 1 0 0 0 Factor track Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules. Using, Applying and Reasoning about Mathematics Trial and improvement Numbers and the Number System Factors and multiples Admin Upper primary mapping document