1070 /www/nrich/html/content/01/09/penta2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Have a look at this table square or matrix:<br></br> <br></br> <br></br> <div style="text-align: center;"><mdo:image alt="" height="143" src="1070B.png" width="142"></mdo:image></div> <br></br> Can you see how it has been constructed?<br></br> Why are some numbers in black and some in red?<br></br> Can you explain why the red $6$ is in that particular square?<br></br> Why is there a $45$ in the bottom right-hand corner?<br></br> <br></br> You will notice that the numbers $2$ - $9$ are used to generate the matrix and only one of these numbers is used twice (the $2$).<br></br> <p>Can you fill in the matrix (table square) below?</p> <p>The numbers $2$ -$12$ were used to generate it with, again, just one number used twice.</p> <p style="text-align: center;"><mdo:image alt="" height="399" src="1070C.png" width="399"></mdo:image></p> <p style="text-align: center;"> </p> <p style="text-align: center;"> </p> <p style="text-align: left;"><a href="http://nrich.maths.org/public/viewer.php?obj_id=6823&amp;part=">Click here for a poster of this problem</a>.</p> <p style="text-align: center;"> </p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">I have to say that I am very impressed by the number and quality of the solutions you have sent in to this Mystery Matrix problem. I am afraid I can't mention everyone individually but special mention must go to pupils at Wharncliffe Side Primary School, Stourport Primary School, How Wood Primary and Mount Pleasant Juniors, all of whom sent in well-explained solutions.</p> <p class="editorial">Rukmini from Roberts Elementary School wrote:</p> This is my solution to the Mystery Matrix:<br></br> <br></br> I first realized that if you put your finger on a square and track its row and column, the number in the square would be the product of the two numbers - the one at the head of the row and the one at the head of the column.<br></br> <br></br> I knew that each column was part of a multiplication table. I realized which numbers were needed to multiply to get that product;<br></br> <br></br> e.g. $32$ was a given number and I knew that $4 \times 8 = 32$, so I put $4$ and $8$ as the heads of the row and the column. $8$ needed to go at the head of the column and $4$ needed to go at the head of the row. $40$ was a given number and the head of its column was $10$. That helped me decide where to put the $4$ and $8$. If I switched $4$ and $8$ then I wouldn't be able to track $4 \times 10$ to $40$. I knew where to put the $4$ and $10$ because if I switched them I wouldn't be able to track the $4$ and the $8$ to $32$.<br></br> <br></br> $49$ was the number I tackled first. I knew that $7 \times 7$ was $49$. So it didn't matter which $7$ I put at the head of the row or the head of the column.<br></br> <br></br> Then I did $32$ as I already explained.<br></br> <br></br> Then $7 \times 8 = 56$, so I put $56$ in the blank box tracked from heads $7$ and $8$.<br></br> <br></br> Using the rest of the given numbers I figured out what the rest of the heads were. I used the heads to help me figure out what the rest of the numbers were.<br></br> <br></br> <p class="editorial">Mrs Beacham's class from Round Top Elementary School, Blythewood, USA also sent a very clear account of the way they solved the problem with a picture of the finished matrix:</p> <div><mdo:image height="360" width="432" src="MrsBeachamClass.gif" alt=""></mdo:image></div> <br></br> <div>First, we came up with the factors for each number in the matrix.</div> <div>We looked at the hint and figured out that $7$ had to be the repeated digit, because there is only one way to get $49$.</div> <div>Then we compared the numbers in each row and column where there were two numbers. For the numbers, $22$ and $24$, $15$ and $27$, $49$ and $42$ we realized they only had one factor in common. When we compared $22$ and $24$ we determined that they were both multiples of two. So $2$ had to go in this column, which meant that $11$ had to be the missing factor for $22$, and $12$ had to be the missing factor for $24$. $15$ and $27$ share the factor $3$, so we placed a $3$ in that row, and above the $15$ we placed a $5$, above the $27$ we placed a $9$.</div> <div>For $49$ and $42$, we already knew they shared the factor $7$, so we thought what times $7$ will give us $42$, and that was $6$.</div> <div>The only digits left to use in the matrix were $4$, $8$, and $10$. When we started, $32$ and $40$ were the hardest to figure out because these numbers shared more than one factor. If we put the factor $8$ in the row, that would have meant using the factor $5$ again, so we realized that the factor $4$ had to be placed in that row. $8$ had to go above the $32$, and we placed $10$ above the $40$.</div> <div>Finally, we multiplied the factors in each row and column to come up with the missing products.</div> <br></br> <p class="editorial">Anhar from Holy Cross Primary School, Oldham, also sent a very clear explanation of the steps he followed to get to a solution which are in a slightly different order compared with those of Mrs Beacham's class:</p> <br></br> <div>The first thing I did was work out that $22$ and $24$ are both in the $2$ times table. I put $2$ at the top of the third column.</div> <div>The second thing I did was notice that $49$ and $42$ are both in the $7$ times table. So I put $7$ at the top of the fifth column.</div> <div>The next thing I noticed was that $15$ and $27$ were both in the $3$ times table. I put $3$ in the fourth row down.</div> <div>Now I have worked out that $7$ is the fifth column, I know it must be $7 \times 6$ that equals $42$. So $6$ goes into the bottom row.</div> <div>Using the same method we knew $7 \times 7$ equals $49$, so we put $7$ in the second row down. This also meant that $7$ had to be the number we used twice.</div> <div>Now that we know the third column is $2$, we can work out that $2 \times 11$ equals $22$ and $2 \times 12$ equals $24$. So $11$ goes into the third row and $12$ goes into the fifth row.</div> <div>Knowing $3$ is the fourth row down, we can work out that $3 \times 9$ equals $27$. So I put $9$ in the last column.</div> <div>We also worked out that $3 \times 5$ equals $15$, and placed $5$ in the second column.</div> <div>We then worked out that $12 \times 2 = 24$, so I placed $2$ in the third column.</div> <div>I knew that $8 \times 4 = 32$. However, I did not know which to put in the column and which in to the row. I then tried putting $8$ in the column and $4$ in the row. But that would have meant putting a $5$ in the remaining column, meaning I would have used $5$ twice as well as $7$ twice. I switched the numbers around, $4$ to the column and $8$ to the row.</div> <div>I placed $10$ in the remaining column and completed the number matrix.</div> <br></br> <p class="editorial">Fantastic! Thank you, Anhar. Finally, Juliet and Lauren from Whipton Barton Junior School sent in a solution in which they made it clear that they were using a 'trial and improvement' approach, which can be a very powerful way of going about solving a problem like this one. Here is what they wrote:</p> <div>We solved this problem by working out what could go down the sides.</div> <div>First of all we worked with $32$. We did this by thinking of factors of $32$ and we found $8$ and $4$. We then thought about other factors of the numbers on the grid. It was pot luck whether we got the numbers the right way round along the sides at first, but if we noticed a problem we could change it.</div> <div>We noticed that $49$ was a square number and we knew that the two factors timesed together was $7$ times $7$. This meant $7$ was the only number that appeared twice.</div> <div>Also, we could fill the middle of the grid in using our times tables facts.</div> <br></br> <p class="editorial">Once again, thank you to everyone who sent in a solution which explained your method.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Mystery Matrix</h2> <br></br> Have a look at this table square or matrix:<br></br> <br></br> <br></br> <div style="text-align: center;"><mdo:image alt="" height="143" src="1070B.png" width="142"></mdo:image></div> <br></br> Can you see how it has been constructed?<br></br> Why are some numbers in black and some in red?<br></br> Can you explain why the red $6$ is in that particular square?<br></br> Why is there a $45$ in the bottom right-hand corner?<br></br> <br></br> You will notice that the numbers $2$ - $9$ are used to generate the matrix and only one of these numbers is used twice (the $2$).<br></br> <p>Can you fill in the matrix (table square) below?</p> <p>The numbers $2$ -$12$ were used to generate it with, again, just one number used twice.</p> <p style="text-align: center;"><mdo:image alt="" height="399" src="1070C.png" width="399"></mdo:image></p> <p style="text-align: center;"> </p> <p style="text-align: center;"> </p> <p style="text-align: left;"><a href="http://nrich.maths.org/public/viewer.php?obj_id=6823&amp;part=">Click here for a poster of this problem</a>.</p> <p style="text-align: center;"> </p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1070&amp;part=index">This problem</a> requires learners to apply knowledge of multiplication facts and stresses the importance of knowing the squares of numbers, particularly the squares of primes. It will increase their familarity with this way of representing multiplication facts, as a grid or matrix.<br></br> <h3>Possible approach</h3> <div>You could start by giving the whole group some numbers such as $18$, $33$, $35$, $48$ and $56$ and asking for the multiplication facts that generate them. Include some squares such as $25$, and even $49$, if you feel the group might have a problem identifying it.</div> <br></br> <div>You could then introduce the first matrix given in the top example and get the group to help fill it in. Point out the $4$ which is generated by $2 \times 2$ and therefore a square and the only number which has been used twice. As you go round, ask the learners how they know which number should go in the various square spaces. You could make a list of all the numbers from $2$ - $9$ and cross them off as they are used. As an alternative, you could give them the completed example and ask them to talk about it in pairs so that in a whole-group discussion, you come to an agreement about how the matrix represents these multiplication facts. All this should help them to be able to tackle the main part of the problem with confidence.</div> <br></br> <div>After this learners could work in pairs on the actual problem so that they are able to talk through their ideas with a partner. They could use <a href="/content/01/09/penta2/1070.pdf">this sheet</a> which has two copies of the main puzzle matrix on it.</div> <br></br> <div>At the end of the lesson the whole group could come together again to discuss, not only the answers, but how they worked them out and which multiplication tables they needed to know well in order to do a problem such as this.</div> <br></br> <h3>Key questions</h3> <div>Where might you start? Why?</div> <div>How can you decide which number is used twice?</div> <div>Which multiplication table does that number come into? Is that the only table it is in?</div> <div>How can you keep a check on which numbers you have used?</div> <br></br> <h3>Possible extension</h3> <div>Learners could make a similar puzzle with the least amount of information given so that it can still be done. They could then swap puzzles with someone else.</div> <br></br> <h3>Possible support</h3> Some learners might find it helpful to start by looking for the number which is used twice. You could, perhaps, suggest they find all the numbers given on a multiplication square and write down the factors that are less than $13$. Get them to a make a list of all the numbers from $2 - 12$ and cross the numbers off as they use them.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Can you work out which number is used twice? This will get you going.<br></br> How can you keep a check on which numbers you have used?<br></br> You could print off <a href="/content/01/09/penta2/1070.pdf">this sheet</a> which has two copies of the puzzle on it.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>This was a very popular problem and lots of you were successful in solving it. How did you keep a check on which numbers you had used? How did you decide which number was used twice?</p> <p>Katherine and Carl, pupils at St Aldhelm's School, made a list of all the numbers and crossed the numbers off as they used them.</p> <p>Abbi from The Mount School, York sent this solution:</p> <p><mdo:image alt="Abbi's solution." src="sol1.gif"></mdo:image></p> <br></br></mdoxml> 2 4 0 1 0 0 0 Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice. Calculations and Numerical Methods Multiplication & division Numbers and the Number System Factors and multiples Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document