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/www/nrich/html/content/01/04/penta5/
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2011-02-01T00:00:01
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<h4><em><em>If you are a teacher, click <a href="http://nrich.maths.org/1053&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...</em></em></h4>
<p> </p>
<p>Each symbol has a numerical value. The total for the symbols is written at the end of each row and column.</p>
<p><strong>Can you find the missing total that should go where the question mark has been put?</strong></p>
<p><mdo:image alt="Grid containing symbols." src="board3.gif"></mdo:image></p>
<div> </div>
<div>Once you've had a go at the problem, click below to see how Charlie, Alison and their friends began working on the task.</div>
<div>
<div> </div>
<div><strong>Can you take each of their starting points and develop them into a solution?</strong></div>
</div>
<div> </div>
<div>Method 1</div>
<div class="toggle"> <mdo:image src="small%20pic.png"></mdo:image> <mdo:image src="A1.png"></mdo:image></div>
<div> </div>
<div>Method 2</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image> <mdo:image src="A2.png"></mdo:image></div>
<div> </div>
<div>Method 3</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image><mdo:image src="A3.png"></mdo:image></div>
<div> </div>
<div>Method 4</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image><mdo:image src="A4.png"></mdo:image></div>
<div> </div>
<div>Method 5</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image> <mdo:image src="B1.png"></mdo:image></div>
<div> </div>
<div>Method 6</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image> <mdo:image src="B2.png"></mdo:image></div>
<div> </div>
<div>Method 7</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image> <mdo:image src="B3.png"></mdo:image></div>
<div> </div>
<div>Method 8</div>
<div class="toggle"><mdo:image src="small%20pic.png"></mdo:image> <mdo:image src="B4.png"></mdo:image></div>
<div> </div>
<div> </div>
<div><strong>Can you find any other methods to solve the problem?</strong></div>
<div> </div>
<div> </div>
<div>You can apply these methods to other versions of the problem. <a href="/content/01/04/penta5/Whats%20it%20worth%20extra%20questions2.xls">This spreadsheet</a> can be used to generate lots of different examples.</div>
<div> </div>
<div> </div>
<div>Here are three other problems that can be solved using similar strategies:</div>
<div><a href="http://nrich.maths.org/2030&part=">Sweet Shop</a></div>
<div><a href="http://nrich.maths.org/850&part=">Letterland</a></div>
<div><a href="http://nrich.maths.org/2031&part=">Children at Large</a></div>
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<mdoxml version="1.0"><p><span class="editorial">Wow! We received loads of great solutions to this - thanks to everyone who took part, tried our problem and submitted a solution.</span></p>
<p><span class="editorial">The answer, as given correctly by almost everyone who responded, is 21. Congratulations to everyone who got the right answer! We didn't ask for them, but in case you were wondering, the individual symbols have values as follows: the circle is 4; the triangle is 6; the hexagon is 7; the square is 8. Below are just a few of the solutions we were given.</span></p>
<p><span class="editorial">Henry, Jack and Tobi from Junior King's Canterbury used a trial-and-improvement method, but spotted the following, which helped them find the right answer more quickly:</span></p>
<p>From the information in the third row down, we see that triangle has to be in the three times table. We tried 3, and that didn't work, but then 6 did work.</p>
<p><span class="editorial">Simran, from SS Peter and Paul, also used trial-and-improvement, but spotted the following way of estimating the value of certain shapes quickly:</span></p>
<p>First I looked for a row (or column) that had most shapes the same. When I found a row (or column) like this, I divided the total of the row (or column) by the number of shapes which were the same in that row (or column).</p>
<p><span class="editorial">Emma, from Walton High School, used that reasoning when she looked at the third row which contained three circles:</span></p>
<p>To begin with I looked at the third row and saw that the circle had to be 5 or under because if it was 6 or more then the triangle wouldn't be worth anything.</p>
<p><span class="editorial">Leo and Jenni, from the Russell School, noticed the following way of speeding things up:</span></p>
<p>First of all we looked at the 2nd row: two squares and two hexagons. The total was 30 so we divided that by 2, which is 15. The whole numbers that are closest to half of 15 are 7 and 8. So we started by approximating the hexagon as 7 and the square as 8.</p>
<p><span class="editorial">Many students followed some of the other methods we gave, and worked out exactly the values of the shapes without trial-and-improvement. For example, Laura from Beaconsfield High School used simultaneous equations:</span></p>
<p>Let T stand for Triangle, S for Square, etc.</p>
<p>Using row 1: 2T + 2S = 28, so T + S = 14.</p>
<p>Using column 4: T + S + 2C = 22, so 2C = 8, and so C = 4...</p>
<p><span class="editorial">Huy, from Vietnam, wrote:</span></p>
<p>Comparing the two rows at the bottom of the table, I find out that the square is worth 2 more than the triangle.</p>
<p>Then looking at the first row, I can figure out that the value of the triangle is 6:</p>
<p>2 squares + 2 triangles = 4 triangles + 4 = 28,<br></br>
so triangle = 6</p>
<p>Then I can figure out the value of the square: 6+2 = 8.</p>
<p><span class="editorial">Eddie from Wilson's School gave a comparison of the various methods described:</span></p>
<p>Methods 1 and 5 just use trial and error.</p>
<p>Methods 2, 3, 4, 6 and 7 compare various rows and columns and use some more advanced logic.</p>
<p>Method 8 is the most straightforward, and yet probably the most overlooked.</p></mdoxml>
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<h3>Why do this problem?</h3>
<p><a href="http://nrich.maths.org/public/viewer.php?obj_id=1053">This problem</a> provides a good challenge in reasoning working with multiple unknowns. There are issues of redundancy to think about in the provision of too much information.</p>
<p>This is a pre-algebra task that can introduce students to the sort of manipulations that can be used to solve simultaneous equations.</p>
<h3> </h3>
<h3>Possible approach</h3>
<p>Show the grid below, or hand out <a href="/content/01/04/penta5/whatsitworthlarge.pdf">this worksheet</a>, or display <a href="/content/01/04/penta5/whatsitworthslide.ppt">this PowerPoint slide</a>.</p>
<p>"Each symbol has a numerical value. The total for the symbols is written at the end of each row and column."</p>
<p>"This is a really interesting problem because it can be solved in lots of different ways. Can you find the missing total that should go where the question mark has been put?"</p>
<p>"Can you find more than one way to do it?"</p>
<p> </p>
<p><mdo:image alt="Grid containing symbols." src="http://nrich.maths.org/content/01/04/penta5/board3.gif"></mdo:image></p>
<div> </div>
<p>After everyone has spent a few minutes on the problem, hand out <a href="/content/01/04/penta5/whatsitworthmethods.pdf">this worksheet</a>, containing eight different ways of starting to solve the problem.</p>
<p>"Here are the first few lines of the solutions found by some friends working on this problem. Can you complete their solutions?"</p>
<p><em>You may choose to give all students all eight solutions, or share them out so different groups work on different solutions.</em></p>
<p>"In a while, I'm going to invite you up to the board to present your favourite method as a completed solution, so be ready to explain every part of your thinking clearly, and to justify why it is your preferred approach."</p>
<p><strong>Here are some prompts</strong> that could be offered to students working on the different approaches if they get stuck:</p>
<p>"If you could work out the value of the square/hexagon/circle/triangle, how could you then work out the value of the triangle/circle/hexagon/square?"</p>
<p>"What can you deduce by comparing the bottom two rows?"</p>
<p>For the trial and improvement methods: "If this one didn't work, what value does it make sense to try next?"</p>
<p>Finish off by inviting students to the board to share the completed methods, together with any other methods they thought of for themselves.</p>
<p>The same techniques can be applied to some follow-up tasks. <a href="/content/01/04/penta5/Whats%20it%20worth%20extra%20questions2.xls">This spreadsheet</a> can be used to generate other versions of the same problem. <a href="/content/01/04/penta5/whatsitworthgrids.pdf">Here</a> are some grids that could be printed off for students to annotate with the new values.</p>
<p>Here are another three problems that can be solved using similar strategies:</p>
<div><a href="http://nrich.maths.org/2030&part=">Sweet Shop</a></div>
<div><a href="http://nrich.maths.org/850&part=">Letter Land</a></div>
<div><a href="http://nrich.maths.org/2031&part=">Children at Large</a></div>
<h3> </h3>
<h3>Possible extension</h3>
<p>Using the spreadsheet, can students work out how much information is necessary to solve the problems uniquely, and identify redundant information?</p>
<p> </p>
<h3>Possible support</h3>
<p>Students could work in pairs to make sense of the different methods. The trial and improvement methods (1 and 5) are perhaps the most accessible starting point.<br></br>
<br></br>
</p>
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<p>What can you deduce by comparing the bottom two rows?</p>
<p>If you could work out the value of the square, how could you then work out the value of the triangle?</p>
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<p>This was the problem that attracted most solutions this month.
You obviously like to play with the ideas of finding out the values
shapes that stand in for numbers. Do you realise that you were
doing algebra type work here? Well done to all of you!! Especially
to our regular contributors from <strong style="font-weight: 400;">Yarm Primary School, Moorfield Primary
School</strong> and <strong style="font-weight: 400;">Tattingstone
Primary School</strong> . Thank you and welcome to <strong style="font-weight: 400;">Seacrest Country Day School</strong>, in
Naples, Florida, everybody from <strong style="font-weight: 400;">Dalwood</strong> Primary School, the pupils of
Worth School and those from <strong style="font-weight: 400;">Mef
School</strong> in Turkey also to our e-mailer <strong style="font-weight: 400;">p4239</strong>.</p>
<p>The favourite strategy seemed to be 'trial and error'; or
guessing at numbers, seeing if they work and then adjusting your
guess. <strong style="font-weight: 400;">Moorfield</strong> Primary
School describe this as "trial and improvement", an excellent name.
They calculated the values of most shapes and were able to work as
number detectives: "... we worked out that circle had to be low
(value) because it was with two big numbers". Good thinking! They
arrived at the same answers as <strong style="font-weight: 400;">Bijan, John</strong> and <strong style="font-weight: 400;">Paul</strong> from <strong style="font-weight: 400;">Yarm</strong> Primary School, who said: "We
worked the answer out by knowing one value and working out wards to
find the values of the other shapes".</p>
<p><strong style="font-weight: 400;">Carla</strong> and
<strong style="font-weight: 400;">Luke</strong> , pupils from
<strong style="font-weight: 400;">Tattingstone</strong> Primary
School, drew a table to help show his thinking, while
<strong style="font-weight: 400;">James</strong> and <strong style="font-weight: 400;">Matthew</strong> from <strong style="font-weight: 400;">Worth</strong> School, wrote equations to
clearly and fully explain their solutions. They used letters to
represent the various shapes:</p>
<p>C = Circle<br></br>
S = Square<br></br>
H = Hexagon<br></br>
T = Triangle</p>
<p>To begin with we looked at the bottom two rows which were:<br></br>
C+T+C+C=18<br></br>
and<br></br>
C+S+C+C=20</p>
<p>We realised that the square must be 2 more than the
triangle.<br></br>
Knowing this we looked down the second column from the left which
was:<br></br>
S+S+T+S=30</p>
<p>So, we realised that it must be 4S-2=30.<br></br>
Then we did 30+2 divided by 4 which gave S=8, so we took away two
to equal T=6.</p>
<p>We then looked at the second row down which was:<br></br>
H+S+H+S=30<br></br>
We converted this into:<br></br>
2S+2H=30</p>
<p>We halved each side which gave:<br></br>
S+H=15<br></br>
Then subtracted 8 (S) and so we got<br></br>
H=7</p>
<p>We then looked at the first column:<br></br>
from the left which was:<br></br>
T+H+C+C=?<br></br>
this, in numbers, is:<br></br>
6+7+4+4</p>
<p><strong style="font-weight: 400;">Sibel</strong> and
<strong style="font-weight: 400;">Yeseren</strong> of Mef School in
Turkey gave their solution as: <strong style="font-weight: 400;">?
= 21</strong> .<br></br>
Because as, <strong style="font-weight: 400;">James</strong> and
<strong style="font-weight: 400;">Matthew</strong> of
<strong style="font-weight: 400;">Worth School</strong> tell us,
and the students from <strong style="font-weight: 400;">Seacrest</strong> explained:</p>
<p>A triangle is worth 6<br></br>
A square is worth 8<br></br>
A circle is worth 4<br></br>
A hexagon is worth 7</p>
<p>So, does <strong style="font-weight: 400;">?</strong> equal
<strong style="font-weight: 400;">21</strong> ?</p>
<p><strong style="font-weight: 400;">Let's check:</strong></p>
<table>
<tbody>
<tr>
<td><mdo:image alt="green triangle." src="triangle.gif"></mdo:image></td>
<td>+</td>
<td></td>
<td><mdo:image alt="yellow hexagon." src="hexagon.gif"></mdo:image></td>
<td>+</td>
<td></td>
<td><mdo:image alt="red circle." src="circle.gif"></mdo:image></td>
<td>+</td>
<td></td>
<td><mdo:image alt="red circle." src="circle.gif"></mdo:image></td>
<td>= 21</td>
</tr>
</tbody>
</table>
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What's it worth?
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