7819 /www/nrich/html/content/id/7819/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>A number in a little box is put into a wonderful big box that adds something to the number and then a new number comes out at the end:</p> <p style="text-align: center;"><br></br> <br></br> <mdo:image src="What%20was%20in%20the%20Box%20pic1.jpg"></mdo:image></p> <p><br></br> The first time this happens, 10 is put into the little box, so what happened in the big box to get the answer in the picture above?<br></br> <br></br> Now three more boxes with new numbers in, go into the wonderful box one at a time. It still does the same as before.</p> <p style="text-align: center;"><mdo:image src="What%20was%20in%20the%20Box%20pic2.jpg"></mdo:image></p> <p style="text-align: center;"> </p> <p> </p> <p>So, what were the three new numbers that went in? Remember that the wonderful big box did the same for all four numbers that went in.</p> <p> </p> <p style="text-align: center;">-----------</p> <p> </p> <p>Imagine four new boxes now (with new numbers in) and the wonderful box does a new and different add or take away this time. For one of these boxes the number 10 was put in. The numbers that come out are these:</p> <p> </p> <p> </p> <p style="text-align: center;"><mdo:image src="what%20else%20in%20the%20box%20pic%203.jpg"></mdo:image></p> <p><br></br> <br></br> What could have happened? How did you work these out?<br></br> Discuss with others and see if there were different ways that you found the answers.</p> <p> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">We had the following solution from pupils in the Philippines:</span></p> <p> </p> <p>Students of Olongapo City National High School, Philippines. Leader: Jumer Palacio<br></br> Members:Joachim, Lee Anthony, Meynard, Kimberly, Erwin, Arlan, Von Nairo<br></br> <br></br> The first time, 10 was put in a box and the result when it came out is 18.?? 10 ? 18 = -8<br></br> -8 is the difference of the two number before and after it was put in the box.</p> <p>We wonder if where we able use this number to our next line of numbers that have been put in the box.<br></br> The next line-up is 12, 8, 15. We minus -8 from the line of numbers written before this sentence.<br></br> 12 - 8 = 4       8 - 8 = 0      15 - 8 = 7<br></br> We thought of similarities between the given answers and we come up of this solution:</p> <p><mdo:image src="Whats%20in.jpg"></mdo:image>4 when you plus 8 the answer will 12, 12 when you minus 8 the answer will 4<br></br> 7 when you plus 8 the answer will 15, 15 when you minus 8 the answer will 7<br></br> Just like a triangle when you minus or plus something the answer will be inside of the triangle.</p> <p>The next line up of numbers are:  0, 19, 1, 11. First, we need to find out on which number 10 will be added on one of the number on the line.We put the 10 on 0 so it becomes 10.<br></br> The new line-up is 10,19, 1, 11. There is a new add and subtract of number on this line-up, the number 8.</p> <p><mdo:image src="What%27s%20in%20the.jpg"></mdo:image>We thought of what is the similarities to each other and we come up to this solution: 10+19+1+11=41      2+27-7+3+19=41<br></br> For the answer -8 for plusing and minusing for the numbers inside the box.THAT?S WHAT HAPPENED IN THE BOX!<br></br> <br></br> <span class="editorial">We also had the following from Carly in Ashington:</span></p> <p> </p> <p>The first numbers were adding 8,6,10,3</p> <p>The second numbers were subtracting and adding 10,9,9,1</p> <p> </p> <p><span class="editorial">Thank you for these solutions.</span></p> <p><span class="editorial">Alfie from Hunters Bar Infant School sent in a good very late solution.</span></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>What Was in the Box?</h2> <br></br> <p>A number in a little box is put into a wonderful big box that adds something to the number and then a new number comes out at the end:</p> <p style="text-align: center;"><br></br> <br></br> <mdo:image src="What%20was%20in%20the%20Box%20pic1.jpg"></mdo:image></p> <p><br></br> The first time this happens, 10 is put into the little box, so what happened in the big box to get the answer in the picture above?<br></br> <br></br> Now three more boxes with new numbers in, go into the wonderful box one at a time. It still does the same as before.</p> <p style="text-align: center;"><mdo:image src="What%20was%20in%20the%20Box%20pic2.jpg"></mdo:image></p> <p style="text-align: center;"> </p> <p> </p> <p>So, what were these three new numbers that went in and do you have a better idea of what  has been going on - remember it was the same for all four numbers that went in.</p> <p> </p> <p style="text-align: center;">-----------</p> <p> </p> <p>Imagine four new boxes now (with new numbers in) and the wonderful box does a new and different add or take away this time. For one of these boxes the number 10 was put in. The numbers that come out are these:</p> <p> </p> <p> </p> <p style="text-align: center;"><mdo:image src="what%20else%20in%20the%20box%20pic%203.jpg"></mdo:image></p> <p><br></br> <br></br> What could have happened? How did you work these out?<br></br> Discuss with others and see if there were different ways that you found the answers.</p> <p> </p> <p> </p> <p> </p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>Of course <a href="http://nrich.maths.org/7819">this problem</a> is rather like a function machine, but it can be more interesting and is easily extended to challenge a wide range of pupils. It could even be used to introduce children to the idea of addition and subtraction.</div> <h3>Possible approach</h3> <div>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> <p>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.</p> <p> </p> <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> <p>Pupils who have gone onto multiplication and division and are doing well may be moved onto further thinking by going to <a href="http://nrich.maths.org/5576">What&#39;s in the Box</a>.</p> <p> </p> <h3>Possible support</h3> <div>For just one number going in you can use counters and a cloth. Show the counters and then cover them with the cloth. 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?</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Remember, each of the four numbers that goes in has the SAME number added by the big box.</p> <p> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>For the first example the multiplier could be</p> <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;">Previous answers:-</div> <div style="margin-left: 40px;"> </div> <div style="margin-left: 40px;">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> The largest was 8.</p> That&#39;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>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!</div> <div style="margin-left: 40px;"> </div> <br></br></mdoxml> 2 3 1 0 0 0 0 This big box adds something to any number that goes into it. If you know the numbers that come out, what addition might be going on in the box? Calculations and Numerical Methods Addition & subtraction Applications Codes and cryptography Using, Applying and Reasoning about Mathematics Making and proving conjectures Admin Lower primary mapping document