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/www/nrich/html/content/98/02/bbprob1/
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2011-02-01T00:00:01
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<mdoxml version="1.0"><p>Uncle Raj has three children. Next year, when they've had their birthdays, Naomi will be $5$, Alex will be $6$ and Chris will be $7$. The family has decided on something rather unusual for part of their presents.<br></br>
<br></br>
All three children have their birthday in the late spring and since they are keen on gardening they are going to buy some plants for the garden, one for each year they have been alive.<br></br>
<br></br>
Here is the plan of their house and garden:</p>
<p> </p>
<div style="text-align: center;"><mdo:image alt="Plan of house and garden" src="BBhome.gif"></mdo:image></div>
<br></br>
<p>You notice that there are three circular paths that cross over each other. Each child is to have a circle but there will be some bits that are shared, around the middle.</p>
<p>When the time comes, the four of them go off to the garden centre to choose the plants. They do not have a lot of money so they're looking for special offers. They find a very special offer which gives a good discount if you buy ten plants altogether. The three children say that that is no good because they need more than ten. But Uncle Raj realises they can manage with only ten.</p>
<p>They go to the cafe and have some cool drinks, and Uncle Raj draws a plan of the three paths and puts little marks to show the plants.</p>
<p>Here is his idea:</p>
<p style="text-align: center;"> <mdo:image alt="the garden" height="388" src="the%20garden.jpg" width="413"></mdo:image></p>
<p> </p>
<p>The children are fascinated to see that Naomi has $1$ and shares $4$, Alex has $2$ and shares $4$ and Chris has $3$ and shares $4$. They think that's rather cool and it saves them a lot of money. So they finish their drinks and off they go to buy their ten plants.</p>
<p>Well now it's your turn to have a go and find some different solutions.<br></br>
REMEMBER:- You must use exactly ten plants (no more, no less)<br></br>
REMEMBER:- The circles must contain $5$, $6$ and $7$ plants (no more, no less).</p>
<p>As you try, you may find that you are developing a system for getting the next one. If so, we'd love to hear about it. You might like to try to find them all, and write about all the things you notice about each solution.</p>
<p>You could print off <a href="/content/98/02/bbprob1/Plants%20Recording%20Sheet.pdf">Plants Recording Sheet.pdf</a> to help you record them, if you like. However, you might find a different way of recording them altogether. Some people find it's easiest to do it quite large and have ten objects to move around in different places.</p>
<p>As with most of these challenges you can and should ask "I wonder what would happen if ...?" Well you might try a different number of objects (plants). You could try different numbers for each circle, as if the children were different ages.</p>
<p>Good luck and don't forget about sending in any results you have.</p>
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<mdoxml version="1.0"><div><span class="editorial">The first solutions we rceived seemed to focus on getting $3, 4$ and $5$ plants in the different areas while sharing one. Perhaps we made it look as though there was some calulating to do.</span></div>
<div><span class="editorial">To achieve solutions for the three children getting $5, 6$ and $7$ using just $10$ plants, the most successful solutions were practical. Kagan at St. Michael's Sandhurst used buttons in circles to show a good example of a correct solution - well done, and here it is.</span></div>
<div> </div>
<div style="text-align: center;"><mdo:image src="plats%20solA.jpg"></mdo:image></div>
<div> </div>
<div><span class="editorial">We've also had a number of other responses which focused on working systematically to get all the solutions. The best results so far have been from Misha and Marjolaine of Highgate School, who both found 34 different arrangements!</span></div>
<div> </div>
<div><span class="editorial">Some years ago the same challenge generated these solutions from a group of children in Somerset:</span></div>
<div> </div>
<div>
<p style="margin: 0px; padding: 1ex 0px 0px;"><mdo:image alt="" src="plant.gif" style="margin: 0px; padding: 0px; border: 0px solid rgb(255, 255, 255); cursor: default;"></mdo:image></p>
<p style="margin: 0px; padding: 1ex 0px 0px;"> </p>
<p style="margin: 0px; padding: 1ex 0px 0px;"> </p>
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<h2>Plants</h2>
<p>Uncle Raj has three children. Next year, when they've had their birthdays, Naomi will be $5$, Alex will be $6$ and Chris will be $7$. The family has decided on something rather unusual for part of their presents.<br></br>
<br></br>
All three children have their birthday in the late spring and since they are keen on gardening they are going to buy some plants for the garden, one for each year they have been alive.<br></br>
<br></br>
Here is the plan of their house and garden:</p>
<p> </p>
<div style="text-align: center;"><mdo:image alt="Plan of house and garden" src="BBhome.gif"></mdo:image></div>
<br></br>
<p>You notice that there are three circular paths that cross over each other. Each child is to have a circle but there will be some bits that are shared, around the middle.</p>
<p>When the time comes, the four of them go off to the garden centre to choose the plants. They do not have a lot of money so they're looking for special offers. They find a very special offer which gives a good discount if you buy ten plants altogether. The three children say that that is no good because they need more than ten. But Uncle Raj realises they can manage with only ten.</p>
<p>They go to the cafe and have some cool drinks, and Uncle Raj draws a plan of the three paths and puts little marks to show the plants.</p>
<p>Here is his idea:</p>
<p style="text-align: center;"> <mdo:image alt="the garden" height="388" src="the%20garden.jpg" width="413"></mdo:image></p>
<p> </p>
<p>The children are fascinated to see that Naomi has $1$ and shares $4$, Alex has $2$ and shares $4$ and Chris has $3$ and shares $4$. They think that's rather cool and it saves them a lot of money. So they finish their drinks and off they go to buy their ten plants.</p>
<p>Well now it's your turn to have a go and find some different solutions.<br></br>
REMEMBER:- You must use exactly ten plants (no more, no less)<br></br>
REMEMBER:- The circles must contain $5$, $6$ and $7$ plants (no more, no less).</p>
<p>As you try, you may find that you are developing a system for getting the next one. If so, we'd love to hear about it. You might like to try to find them all, and write about all the things you notice about each solution.</p>
<p>You could print off <a href="/content/98/02/bbprob1/Plants%20Recording%20Sheet.pdf">Plants Recording Sheet.pdf</a> to help you record them, if you like. However, you might find a different way of recording them altogether. Some people find it's easiest to do it quite large and have ten objects to move around in different places.</p>
<p>As with most of these challenges you can and should ask "I wonder what would happen if ...?" Well you might try a different number of objects (plants). You could try different numbers for each circle, as if the children were different ages.</p>
<p>Good luck and don't forget about sending in any results you have.</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=36&part=">activity</a> is a good one to choose when wanting to encourage children to think creatively and 'have a go' at problem solving. It is also accessible to a wide range of attainment levels. This activity can also be a platform from which to give opportunities for children to talk freely about their thinking and to
encourage them <em>really</em> to listen to each other.</div>
<div> </div>
<h3>Possible approach</h3>
<div>Before you embark upon this activity with pupils, I do suggest that you play around with it a bit on your own or with colleagues. This is often a good idea, but in this case I think it is particularly valuable as you find that interesting strategies come into play that you probably would not have thought about had you not indulged yourself.</div>
<br></br>
<div>You could, with some youngsters, just introduce three hoops and a number of counters (to represent plants) and, having labelled the hoops with the quantities that need to be inside, ask the youngsters to see if they can find a solution. With the suggested $5$, $6$ and $7$ for each of the three circles, you could explore the activity using between $7$ and $18$ counters. It would be quite
something for some of your pupils to explore all the different possibilities with each of the numbers from $7$ through to $18$, particularly with the emphasis on talking, listening and discussion.</div>
<br></br>
<h3>Key questions</h3>
<div>How many are there now in this part?</div>
<div>How could we make this part have a correct number in it?</div>
<br></br>
<h3>Possible extension</h3>
<div>a. This idea actually came from a pupil doing this investigation. She labelled each area and then produced a table to show the number of plants in each.</div>
<div> </div>
<div style="text-align: center;"><mdo:image alt="3circles" height="208" src="3Circles.jpg" width="212"></mdo:image> </div>
<div style="text-align: left;">Pupils might consider the areas D, E, F and G as "worth" more than $1$, (D, E, F being $2$ and G $3$) and record results in a table. There's a lot to explore in such tables, and it's interesting at the start to find out <em>how</em> the pupils explorate to get the table. Some may be using a spreadsheet, or mental calculations, or just 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.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">b. Explore other groups of numbers instead of just $5$, $6$ and $7$ - what about numbers going up in $2$s, $4$s, $6$s and $8$s or any other number?</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">c. If pupils have happily constructed tables in which every possibility is discovered, you might explore the number of possibilities according to the difference between the total for the three circles if they were no overlaps, ($5+6+7=18$) and the number of items used. For example there were seven solutions for a difference of two:</div>
<div style="text-align: left;"> </div>
<div style="text-align: center;"><mdo:image alt="3circlesolut" height="127" src="3CirclesSolut.jpg" width="232"></mdo:image></div>
<div style="text-align: center;"> </div>
<div>Further exploration 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 ($18 - 13$) five items needed.</div>
<br></br>
<h3>For the exceptionally mathematically able</h3>
Obvious extension work can be looked at by considering four or more areas - so take yourselves to <a href="http://nrich.maths.org/7543&part=">More Children and Plants</a>.<br></br>
<h3><br></br>
Possible support</h3>
<div>I feel that the best support is to be alongside the children being the 'more mature thinker' in that you can more easily help them to keep track of where they are and prompt them to suggest the next move.</div>
<p><br></br>
<br></br>
</p>
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<mdoxml version="1.0">How many plants have you got in each circle?<br></br>
What could you do to make the correct totals?<br></br></mdoxml>
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<div style="text-align: center;"> </div>
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<p> </p>
<br></br>
<table style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; font-size: 0.95em; border-top-color: rgb(211, 211, 211); border-right-color: rgb(211, 211, 211); border-bottom-color: rgb(211, 211, 211); border-left-color: rgb(211, 211, 211); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted;">
<tbody style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<tr style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">
<td style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 5px; padding-right: 5px; padding-bottom: 5px; padding-left: 5px; border-top-color: rgb(211, 211, 211); border-right-color: rgb(211, 211, 211); border-bottom-color: rgb(211, 211, 211); border-left-color: rgb(211, 211, 211); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted;">
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"><span class="editorial" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; color: rgb(68, 102, 68); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted; border-top-color: rgb(128, 128, 128); border-right-color: rgb(128, 128, 128); border-bottom-color: rgb(128, 128, 128); border-left-color: rgb(128, 128, 128);">
The problem with three circles containing 5, 6, and 7 plants has many solutions. The following solutions were found by children in</span> <em class="editorial" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; color: rgb(68, 102, 68); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted; border-top-color: rgb(128, 128, 128); border-right-color: rgb(128, 128, 128); border-bottom-color: rgb(128, 128, 128); border-left-color: rgb(128, 128, 128);">Somerset</em> <span class="editorial" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; color: rgb(68, 102, 68); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted; border-top-color: rgb(128, 128, 128); border-right-color: rgb(128, 128, 128); border-bottom-color: rgb(128, 128, 128); border-left-color: rgb(128, 128, 128);">in
February.</span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"><span class="editorial" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; color: rgb(68, 102, 68); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted; border-top-color: rgb(128, 128, 128); border-right-color: rgb(128, 128, 128); border-bottom-color: rgb(128, 128, 128); border-left-color: rgb(128, 128, 128);">
Unfortunately the only name I have of one of the children is</span> <em class="editorial" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; color: rgb(68, 102, 68); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted; border-top-color: rgb(128, 128, 128); border-right-color: rgb(128, 128, 128); border-bottom-color: rgb(128, 128, 128); border-left-color: rgb(128, 128, 128); font-style: normal;">Peter
who preferred to be called Retep</em><span class="editorial" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; color: rgb(68, 102, 68); border-top-width: 1px; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-top-style: dotted; border-right-style: dotted; border-bottom-style: dotted; border-left-style: dotted; border-top-color: rgb(128, 128, 128); border-right-color: rgb(128, 128, 128); border-bottom-color: rgb(128, 128, 128); border-left-color: rgb(128, 128, 128);">,
that is Peter backwards. Well why not?!</span></p>
</td>
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</tbody>
</table>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"><mdo:image alt="" src="plant.gif" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-top-style: solid; border-right-style: solid; border-bottom-style: solid; border-left-style: solid; border-top-color: rgb(255, 255, 255); border-right-color: rgb(255, 255, 255); border-bottom-color: rgb(255, 255, 255); border-left-color: rgb(255, 255, 255); cursor: default;"></mdo:image></p>
<div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">Here are the number of solutions possible when considering how many extra items are needed - see C in Notes.</div>
<div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: center;"><mdo:image alt="Nos of Sols" height="232" src="Nos%20of%20Sols.jpg" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-top-style: solid; border-right-style: solid; border-bottom-style: solid; border-left-style: solid; border-top-color: rgb(255, 255, 255); border-right-color: rgb(255, 255, 255); border-bottom-color: rgb(255, 255, 255); border-left-color: rgb(255, 255, 255); cursor: default;" width="121"></mdo:image></div>
<div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: left;"> </div>
<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"></br>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 1ex; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;">These can be split in two - odds and evens for further exploration.</p>
<div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; text-align: center;"><mdo:image alt="CircleSols" height="260" src="CircleSolDiff.jpg" style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-top-style: solid; border-right-style: solid; border-bottom-style: solid; border-left-style: solid; border-top-color: rgb(255, 255, 255); border-right-color: rgb(255, 255, 255); border-bottom-color: rgb(255, 255, 255); border-left-color: rgb(255, 255, 255); cursor: default;" width="521"></mdo:image></div>
<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px;"></br>
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Plants
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Working systematically
Decision Mathematics and Combinatorics
Combinations
Calculations and Numerical Methods
Addition & subtraction
Handling, Processing and Representing Data
Venn diagrams
Using, Applying and Reasoning about Mathematics
Recording mathematics
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Lower primary mapping document
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Upper primary mapping document