5576 /www/nrich/html/content/id/5576/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Four numbers in little boxes are put into a special big box that does a multiplication, then four new numbers come out at the end:<br></br> <br></br> <mdo:image width="528" height="534" alt="functionA" src="functionA.jpg"></mdo:image><br></br> We only used whole numbers to go in, so, what multiplication might have gone on in the big box to get the answers in the picture above?<br></br> <br></br> What was the largest number that could have been used to multiply by, in that big box?<br></br> <br></br> Imagine four new boxes now (with new numbers in) and the large box multiplying by a different number this time. The numbers that come out are these:<br></br> <br></br> <mdo:image width="375" height="151" alt="functionB" src="functionB.jpg"></mdo:image><br></br> What would be the number that the big box is multiplying by?<br></br> How are you working these out?<br></br> Discuss with others and see if there are different ways that you found the answers.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Rhiannon from St Mary Redcliffe Primary School thought hard about the first part of this problem:</p> I worked out all the numbers that would go into the smallest number (24) and tested each of them out by using a calculator to divide them into the other numbers (56, 112, and 216). <br></br> <div style="clear: both;">The largest was 8.</div> <div style="clear: both;"></div> <p class="editorial" style="clear: both;">That's a great method - well done, Rhiannon. (Luca from Devonshire Primary School told us that the possibilities for the first challenge were 1, 2, 4 and 8, so 8 is indeed the largest.)</p> <p class="editorial">In a great team effort, Class 7P at Loretto Junior School sent us a solution to the second part of the problem:</p> In the second problem the numbers coming out were all odd. So we thought the multiply number would be odd.<br></br> We knew it could not be 3 or 9 etc because the digit total of 143 was not a multiple of 3. <br></br> Graeme said maybe they were all prime numbers so the box number would be 1. <br></br> However Zabrina and Angus suddenly saw the common factor was 11!<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>What&#39;s in the Box?</h2> <br></br> Four numbers in little boxes are put into a special big box that does a multiplication, then four new numbers come out at the end:<br></br> <br></br> <mdo:image alt="functionA" height="534" src="functionA.jpg" width="528"></mdo:image><br></br> We only used whole numbers to go in, so, what multiplication might have gone on in the big box to get the answers in the picture above?<br></br> <br></br> What was the largest number that could have been used to multiply by, in that big box?<br></br> <br></br> Imagine four new boxes now (with new numbers in) and the large box multiplying by a different number this time. The numbers that come out are these:<br></br> <br></br> <mdo:image alt="functionB" height="151" src="functionB.jpg" width="375"></mdo:image><br></br> What would be the number that the big box is multiplying by?<br></br> How are you working these out?<br></br> Discuss with others and see if there are different ways that you found the answers.</div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>Of course <a href="http://nrich.maths.org/public/viewer.php?obj_id=5576&amp;part=">this problem</a> is rather like a function machine, but it can be more interesting for the pupils and easily extended to challenge a wide range of pupils. It could be used to introduce children to the idea of common factors.</div> <br></br> <h3>Possible approach</h3> <div>It may be necessary to introduce the class to just one number going in and to give them one outcome to start with so that they understand the process. Then, gradually increase the number of numbers going in until you reach four, as in the problem. Your own examples can be adjusted in complexity according to the level of your pupils.</div> <br></br> Once learners have had some time to work on the first part of the problem in pairs, ask them to share their ways of working with the whole group. Look out for those who give good reasons for choosing particular methods. At this stage, you could introduce the vocabulary of common factors if appropriate.<br></br> <br></br> <h3>Key questions</h3> <div>What might have gone on in the box to get this number answer?</div> <div>Could that have produced the other answers too?</div> <br></br> <h3>Possible extension</h3> <div>Outputs like $165, 45, 135$ and $315$ could obviously have "$ \times 5$" in the box, meaning the inputs were $33, 9, 27$ and $63$ but there is another possibility when fraction multiplication is allowed. If "$ \times 3.75$" was in the box then the input numbers would have been $44, 12, 36$ and $84$. In this example it would be appropriate to ask experienced pupils what was happening - in other words, encourage them to recognise that there are two solutions and ask them to explain how and why the numbers relate to each other. Challenging pupils in this way will almost certainly get them to consider number relationships very seriously, reinforcing what they have learnt and opening doors to further learning. Some pupils could go on to invent their own for others to do.</div> <br></br> <h3>Possible support</h3> <div>For just one number going in you can use counters and a cloth. Cover the counters with the cloth and then secretly add the required extra number of counters under the cover before revealing them to the pupil. Then a number of probing questions can be asked: How many counters now? What must have happened under the cover? As they tackle the main problem, some learners might find it useful to have a multiplication square or calculator available.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Remember, each of the four numbers that goes in is multiplied by the SAME number in the big box.<br></br> So, what is special about the four numbers that come out at the end ?<br></br> Why don't you try out some ideas to see whether they work?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> For the first example the multiplier could be <br></br> <div style="margin-left: 40px;">2 with 28,12,56,108 being the starting numbers</div> <div style="margin-left: 40px;">4 with 14,6,28,54</div> <div style="margin-left: 40px;">8 with 7,3,14,27</div> <div style="margin-left: 40px;"></div> <div>For the second example the multiplier would be</div> <div style="margin-left: 40px;">11 with 13,27,31,111 being the starting numbers.</div> <div style="margin-left: 40px;"></div> <br></br></mdoxml> 2 3 0 1 0 0 0 This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box? Calculations and Numerical Methods Multiplication & division Numbers and the Number System Factors and multiples Numbers and the Number System Common factors Admin Upper primary mapping document