7543 /www/nrich/html/content/id/7543/ 1 2011-07-25T08:24:42 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p><span style="font-style: italic;">This activity has been particularly created for the most able. (The pupils that you come across in many classrooms just once every few years</span>.) <span style="font-style: italic;">It is seen as a possible follow-on from</span> <a href="http://nrich.maths.org/36" style="font-style: italic;">Plants</a><span style="font-style: italic;">.</span></p> <p> </p> <p>In "<a href="http://nrich.maths.org/36">Plants</a>" we had three children sharing $10$ plants in the three overlapping circles.</p> <p>This particular challenge is about extending that, so that we consider four and then maybe five children in a similar way.</p> <p>In considering these larger numbers we have to examine a different arrangement of the circles (possibly changed into slightly different shapes).</p> <p>You will need to draw these four (first of all) areas in such a way that there is a section for each of the sharing situations. In the case of plants there were seven sections - allowing for each child to have an overlapping part with each and all of the other children.</p> <p>Once you have drawn an arrangement for four areas I suggest that you start with allocating $4, 5, 6, 7$ to the areas.</p> <p>As before, where can a certain number of plants go? I suggest you start with a number like $19$ for the total number plants.</p> <p>Find all the answers that satisfy the requirements of having $4, 5, 6, 7$ shared in the different regions using $19$ plants.<br></br> <br></br> Want to go still further? Then go to <a href="http://nrich.maths.org.uk/7676">More Plant Spaces.</a></p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="/7543">This activity</a> provides a good extension to the <a href="/36">Plants</a> activity when you have an exceptional pupil who you think could go so much further. It offers opportunities for both graphical and spreadsheet work. As with the more accessible version of this activity, it provides an opportunity for pupils to think creatively about those aspects of mathematics in which they may not have had any instruction.</div> <div> </div> <h3>Possible approach</h3> <div>Introduce the <a href="/36">Plants</a> activity and discuss with the pupil how this could be extended using this challenge. </div> <br></br> <h3>Key questions</h3> <div>What ways are you finding for drawing the appropriate diagrams?</div> <div>Tell me about the areas you have obtained.</div> <div>What have you set up on your spreadsheet/in your table?</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <br></br> <p>In order to have four overlapping areas we need something like this:</p> <p><mdo:image src="4%20sets.jpg"></mdo:image></p> <p>showing $15$ areas.</p> <p>Or, when moving on to five we need something like this:</p> <p style="text-align: center;"><mdo:image src="5%20sets.jpg"></mdo:image></p> <p>Showing $31$ areas.</p> <p>These are just suggestions and there are other ways of constructing a suitable arrangement.</p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> From notes of Plants<br></br>  <br></br> <br></br> So if pupils consider the areas D, E, F and G as &quot;worth&quot; more than 1, (D, E, F being 2 and G 3), then tables like this can sometimes result. There's a lot to explore in these tables, and it's interesting at the start to find out how the pupils do the explorations to get the table. Some may be using a spreadsheet, mental calculations, looking at the picture of three overlapping circles whilst others may use something practical to check that all is well with their ideas. Some interesting discussions may arise from some pupils who work very arithmetically and come up with a system but unfortunately ignore the maximum number allowed in each circle. B/ Explore other groups of numbers instead of just 5, 6 and 7 - what about numbers going up in 2s, 4s, 6s and 8s or just random numbers 3s, 6s and 7s? C/ If pupils have happily constucted tables like those above in which every possibility is discovered you might explore the number of possibilities according to the difference between the total for the three circles, (5+6+7) and the number of items used. For example there were 7 solutions for a difference of two. <br></br>  <br></br> <div style="text-align: center;"> <mdo:image width="232" height="127" src="3CirclesSolut.jpg" alt=""></mdo:image></div> <div>Further exporation will reveal the number of solutions when 3, 4, 5 etc extra ones are needed (e.g. when 13 items are used with 5, 6 and 7 circles then there an extra (18 - 13) 5 items are needed.</div> <div>For the highest-attaining Obvious extension work can be looked at by considering four areas - though not all are circles in this diagram - and asking similar questions.</div> <br></br>  <br></br></mdoxml> 5 5 0 1 1 0 0 This challenge extends the Plants investigation so now four or more children are involved. Using, Applying and Reasoning about Mathematics Recording mathematics Using, Applying and Reasoning about Mathematics Working systematically Using, Applying and Reasoning about Mathematics Investigations Decision Mathematics and Combinatorics Combinations Handling, Processing and Representing Data Venn diagrams Calculations and Numerical Methods Addition & subtraction