5714 /www/nrich/html/content/id/5714/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> The coloured shapes stand for eleven of the numbers from $0$ to $12$. Each shape is a different number.<br></br> <br></br> Can you work out what they are from the multiplications below?<br></br> <br></br> <mdo:image alt="multiplications using shapes" height="435" src="shapes.gif" width="550"></mdo:image><br></br> <br></br> <p style="font-style: italic;">This activity is featured in the hands-on Brain Buster Maths Boxes, developed by members of the NRICH Team and produced by BEAM. For more details see <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&amp;part=index">here</a> .</p> <p style="font-style: italic;"> </p> <p style="font-style: normal;"><a href="http://nrich.maths.org/public/viewer.php?obj_id=6653&amp;part=">Click here for a poster of this problem</a>.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">We had some very well-reasoned solutions to this problem - thank you to those of you who wrote a lot of detail. Unfortunately there are too many of you to mention everyone by name. Suryasnato from Limes Farm Junior School wrote:</p> The upside down isosceles triangle is 0 because every time it is in a question it is the answer. The only number that rule applies to is 0.<br></br> The diamond is 1 because every time it is in a question the other number in the question is the answer. This can only happen if the number is one. <br></br> The square is 2 because something cubed is equal to something. All the numbers must be under twelve so the squares must be 2 or 1 because when those numbers are cubed, the answer is under twelve. It can't be 1 because 1 cubed is one and the answer was a different shape from the shape in the question. The semi-circle was 8 because 2 cubed is 8. <br></br> The oval was 4 because the second question on the left hand row was 2 times the oval is 8 and 2 times 4 is 8.<br></br> The circle is double the triangle because the rectangle times 4 is equal to the circle while the rectangle times 2 is equal to the triangle. Now we need to find a number and its double. It can't be 7 and 14 because all the numbers must be twelve or below. It can't be five and ten because the triangle and the circle are in sums which involve the 2 times tables and the 4 times tables and 5 isn't in any of them. It isn't 4 and 8 because the semi-circle and oval are those numbers and you can't have two different symbols which are the same number. It can't be 3 and 6 because they aren't in the 2 or 4 times table. It can't be 2 and 4 or 1 and 2 because all those numbers are other symbols. It also can't be 0 and 0 because they are different symbols so they must be different numbers. The only possible pair is 6 and 12 so the triangle is 6 and the circle is 12. <br></br> The circle is 12 and the oval is 4 so 12 divided by 4 is the value of the rectangle. 12 divided by 4 is 3 so the rectangle is 3. <br></br> The second sum on the top is rectangle times rectangle is equal to something. Since the rectangle is 3, it is 3 times 3 which is equal to 9. So the star with more sides is 9. <br></br> That leaves the hexagon and the other star. It says 2 times star equals hexagon (see the second question on the right hand side) so we have to search for a pair of doubles. The only remaining pair of doubles under 12 is 5 and 10.<br></br> <br></br> <p><span class="editorial">Alex and Bradley from Losley Fields Primary did it in a very slightly different order again. They sent in</span> <a href="/content/id/5714/Alex_Bradley.doc" class="editorial">this Word document</a> <span class="editorial">which explains their thought-processes very well.</span></p> <p><span class="editorial">Rhiannon from St Mary Redcliffe Primary also had a different way. She sent in</span> <a href="/content/id/5714/Rhiannon.pdf" class="editorial">this pdf</a> <span class="editorial">to tell us how she did it.</span></p> <p class="editorial">Some of you said that you worked out the solution by trial and improvement - trying one number for a particular shape to see whether it worked. That can be a useful strategy although it might take a little longer than some of the ways already described.</p> <p class="editorial">Here is a summary picture of the value of each shape:</p> <br></br> <br></br> <mdo:image width="295" height="479" alt="solution" src="sol.gif"></mdo:image><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Shape Times Shape</h2> <br></br> The coloured shapes stand for eleven of the numbers from $0$ to $12$. Each shape is a different number.<br></br> <br></br> Can you work out what they are from the multiplications below?<br></br> <br></br> <mdo:image alt="multiplications using shapes" height="435" src="shapes.gif" width="550"></mdo:image><br></br> <br></br> <p style="font-style: italic;">This activity is featured in the hands-on Brain Buster Maths Boxes, developed by members of the NRICH Team and produced by BEAM. For more details see <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&amp;part=index">here</a> .</p> <p style="font-style: italic;"> </p> <p style="font-style: normal;"><a href="http://nrich.maths.org/public/viewer.php?obj_id=6653&amp;part=">Click here for a poster of this problem</a>.</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5714&amp;part=index">This problem</a> helps children become familiar with the idea of a symbol (in this case a shape) representing a number. It is also an opportunity for pupils to practise using multiplication and division in a challenging context.</div> <br></br> <h3>Possible approach</h3> <div>In order to introduce the idea of a shape representing a number, you could start the lesson by having some shapes representing numbers in addition sums. For example:</div> <div><mdo:image alt="square + rectangle=5, circle + square = 11, rectangle + rectangle = 8" height="178" src="intro.gif" width="251"></mdo:image></div> <div>Ask partners to talk to each other about how they would work out what each shape stands for in the calculations above and share their ideas amongst the whole group. In this case the last sum is actually the most helpful to start with.</div> <br></br> <div>You could then introduce the Shape Times Shape problem by displaying it on the whiteboard and explaining the task. (It might also be useful for pairs of children to have a paper copy.) Ask the children to think on their own about where they might start. Invite them to share their ideas with a partner, then discuss the options amongst the whole group. Look out for good reasoning, based on their knowledge of number properties, and encourage everyone to strive for this level of explanation.</div> <br></br> <div>Set them off in pairs to tackle the problem indicating that the plenary will focus on <span style="font-style: italic;">how</span> they went about solving it. You may want to stop them after a few minutes to find out how they are recording their work and to share some efficient ways.</div> <br></br> <h3>Key questions</h3> <div>Which sum is useful to start with? Why?</div> <div>Now that we know that shape, which sum could we look at next? What does that tell us? How do we know?</div> <div>How are you keeping track of what you have done?</div> <br></br> <h3>Possible extension</h3> <div>Learners could make up their own problem using shapes as symbols and test it on a friend. The problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=1053&amp;part=index">What&#39;s it Worth?</a> requires similar thinking processes to this problem and would be a good one for pupils to try next.</div> <br></br> <h3>Possible support</h3> <div>Children might find it easiest to have numbered counters or cards available so that they can physically form the sums to check their reasoning. You might want to support their recording by giving out a sheet with each of the shapes drawn on, like <a href="/content/id/5714/recording%20sheet.doc">this</a> . Alternatively, <a href="/content/id/5714/ShapeTimesShapeRecording.pdf">this sheet</a> has a copy of the question and space for recording (thank you to Rose Prentice for sharing it with us).</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Where could you start?<br></br> What can you tell from the first multiplication with three purple squares and a yellow semi-circle?<br></br> Perhaps you can use what you now know to help you with another sum?<br></br> How will you remember what you have tried and what you have found out? You could use <a href="/content/id/5714/recording%20sheet.doc">this sheet</a> to keep a note of what you think the shapes stand for.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <br></br></mdoxml> 2 3 0 1 0 0 0 These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are? Algebra Introducing algebra Calculations and Numerical Methods Multiplication & division Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document