Spot Thirteen

7513 /www/nrich/html/content/id/7513/ 1 2011-06-29T13:19:32 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">On the grid below, choose $13$ spots by clicking on them. You will see that as you click more spots, your score changes. Can you work out how the scoring system works? We would love to hear what you did in order to test out your ideas. Which spots should you choose to produce the maximum possible score? How do you know that this is the maximum? <mdo:flash height="500" id="/content/id/7513/PegPad3.swf" width="650"><param name="allowfullscreen" value="true"></param><param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/7513/PegPad3.swf"></param> <param name="allowfullscreen" value="true"></param> <param name="flashplayerversion" value="10"></param> </mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This problem was perhaps more challenging than it looks! Sophie from King's Junior School in Chester told us: If you have three in a row you get three points, if you get four in a row you get four points etc. I think the highest possible score is $40$. The Super Six at Coldean agreed with Sophie and clarified that the line could be horizontal, vertical or diagonal. Angus from Australia also articulated the rules very clearly. Will from Holmfirth High School suggests a way of making a score of $40$: $5$ in a row first, which gives you $5$ points. Make two rows of $4$ which gives $4$ points each. This will give you $13$ points altogether so far. There are nine rows and diagonals of three which give $3$ points each so altogether you get $40$!! Here is a picture of Will's description: <mdo:image src="solution.png"></mdo:image> Phoebe from Walton High School described a different way: If you place $3$ dots in a row you will score $3$ points, if you place $4$ dots in a row you will score $4$ points and so on. The shape you need to make to score the highest result consists of a $3 \times 4$ rectangle on the grid, with the final dot above the top right dot on the rectangle. This should leave you with the highest score of $40$. Here is a picture of Phoebe's arrangement: <mdo:image src="PhoebeSol.png"></mdo:image> Are Phoebe's and Will's ways different from each other, do you think? Millie and Charlotte from Princes Risborough School sent a very comprehensive solution. They sent this series of pictures to illustrate the scoring system: <mdo:image src="MillieCharlotte1.png"></mdo:image> <mdo:image src="MillieCharlotte2.png"></mdo:image> <mdo:image src="MillieCharlotte3.png"></mdo:image> <mdo:image src="MillieCharlotte4.png"></mdo:image> <mdo:image src="MillieCharlotte5.png"></mdo:image> What do you think about their way of getting $40$ compared with Pheobe's and Will's? Millie and Charlotte, Hannah, Pete and Will, and Krystof from Prague all noticed a mistake in the Poly Plug Rectangle interactivity. They sent in this screenshot: <mdo:image src="Mistake.png"></mdo:image> With the scoring system you have identified, we would expect this arrangement to score $3$ points but it doesn't. This was completely our fault - sorry if this caused you confusion. It should score $3$ points. Thank you for pointing out our mistake - we will put it right ASAP! Hopefully by the time you read this, it will be correct. Lucy from King Athelstan Primary School found two other ways of getting $40$ points: <mdo:image src="Lucy.png"></mdo:image> I wonder whether there are other ways of getting $40$ points? How do we know that it is impossible to get more points? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Spot Thirteen</h2> On the grid below, choose $13$ spots by clicking on them. You will see that as you click more spots, your score changes. Can you work out how the scoring system works? We would love to hear what you did in order to test out your ideas. Which spots should you choose to produce the maximum possible score? How do you know that this is the maximum? <mdo:flash height="500" id="/content/id/7513/PegPad3.swf" width="650"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="fullscreen" value="true"></param> <param name="movie" value="/content/id/7513/PegPad3.swf"></param> <param name="flashplayerversion" value="10"></param> </mdo:flash> </div> This activity has been inspired by Doug Williams' Poly Plug resource. You can find out more details, including how to order sets of Poly Plug, on the <a href="http://www.blackdouglas.com.au/taskcentre/polyplug.htm">Mathematics Task Centre website</a>. However, you do not need sets of Poly Plug to have a go at this activity - please see below and take a look at the Teachers' Notes. Why do this problem? <a href="/7513">This challenge</a> encourages learners to make hypotheses, devise ways to test them and then refine their thinking based on new evidence. <h3>Possible approach</h3> Display the interactivity on the board and invite a pupil to select $13$ spots. The simple fact of the score going up will hook children into the activity. Explain that their task is to work out how the scoring system works. Give them a few minutes to chat to someone else about any initial thoughts. Draw the whole group together and take a few suggestions. Then, give pairs time to discuss what they want to do in order to find out more and test their ideas. Explain that you will want to hear (a) what they'd like to do and (b) what information this will give them. At this point, if you have a access to a computer suite, you may wish the class to work in pairs at a computer, trying out their own ideas. If this is not possible, then bring everyone together again and invite some pairs to offer their thoughts. As a whole class, come to an agreement about what you are going to try next, and give it a go. Some pupils may wish to record different arrangements and their different scores to refer back to later. <a href="/content/id/7513/SpotThirteen.pdf">This sheet of blank $5$ by $5$ grids</a> may be useful for that purpose but children will have other methods of their own too. Continue in this way with pairs making suggestions for arrangements to test until the whole group is sure that the scoring system is understood. You could choose $13$ spots of your own and show the group your arrangement on paper or by drawing it on the board, asking them to predict the score before trying it out using the interactivity. The challenge can then focus on obtaining the maximum possible score. Pairs could use the <a href="/content/id/7513/SpotThirteen.pdf">blank $5$ by $5$ grids</a> to record different arrangements and make a note of their calculated scores. After a suitable length of time, you could find out who has the very high scores so far and test them on the computer. This could be a 'simmering' activity so that children are encouraged to work on the maximum score over a period of a few days or weeks. You could create some wall space for them to post up their current top scores and pictures of the arrangements for the whole class to return to at a specified time. <h3>Key questions</h3> What could you try next? Why is that a good arrangement to try? What do you think the score will be? <h3>Possible extension</h3> Some children may enjoy investigating the maximum score on a differently sized grid and possibly creating their own scoring system for someone else to decipher. <h3>Possible support</h3> Working at a computer with a partner will give instant feedback which may help some children who might not persevere otherwise. The image in the <a href="/7513&part=clue">Hint</a> might also be useful. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Have you watched the way the score changes as you add more and more spots? This might give you a clue as to what is going on. Trying to make a very low score might help. Here is a picture showing the first $9$ spots I tried (in blue): <mdo:image alt="" src="Hint1.png" style="width: 200px; height: 201px;"></mdo:image> What do you think the score is so far? Check using the interactivity. How does this help? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Get 5 points for 5 in a line with no gap, 4 points for 4 in a line, 3 points for 3 in a line, no points for 1/2 in a line </mdoxml> 5 3 0 1 0 0 0 Spot Thirteen Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score? Using, Applying and Reasoning about Mathematics Making and testing hypotheses Information and Communications Technology Interactivities Admin Upper primary mapping document