2727 /www/nrich/html/content/id/2727/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <div style="text-align: center;"><mdo:image width="466" height="150" src="FuncMach.gif" alt="Function machine: add 3, multiply by 5, take away 2. Numbers in are 5, 7 and 4"></mdo:image></div> <br></br> <br></br> This one is a bit more of a puzzle.<br></br> <div style="text-align: center;"></div><mdo:image width="550" height="165" src="functfinal2.gif" alt="Function machine. Numbers in: 2, 3, 4, 5. Numbers out in same order: 22, 28, 34, 40"></mdo:image><br></br> <br></br> Can you work out what happens in the three circles?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">Well done to all of you who gave a correct solution to this problem. The correct answer to the first part of the question is $33, 48, 38$.</span><br></br> <p><span class="editorial">There are several correct answers to the second part of the problem including:</span></p> <p class="editorial"><mdo:image width="259" height="94" src="solution%20stage1%201star.JPG" alt="example solutions"></mdo:image></p> <p class="editorial">Richard from Burlingame School used a systematic approach to help him find the answer to the second part of the problem:</p> <p>What I did was multiply each number by $2$, then $3$, then $4$, etc till I found a pattern. If you multiply the number by $6$, you come out with a number $10$ less than the answer. I simply added ten, then subtracted $0$ to give me the answer.</p> <p class="editorial">Quite a few of you used a really useful technique to help find the solution more efficiently including Marley and Jake from Swarcliffe Primary and Shiv from Mayplace Primary School. Ester sent in a nice solution:</p> <p>The numbers are CONSECUTIVE going in and HAVE A DIFFERENCE OF $6$ coming out. This tells us that there must be a multiplication by $6$ somewhere.</p> <p class="editorial">She also spotted that there was more than one solution:</p> <p>If $\times 6$ is in the middle circle some solutions could be</p> <p>$+2, \times 6, -2$</p> <p>$+1, \times 6, +4$</p> <p>$-1, \times 6, +16$</p> <p>If you multiply by six first you have to add or subtract in the last two circles so that the result is $+10$</p> <p>$\times ,+6,+4$</p> <p>$\times 6,+11,-1$</p> <p>You cannot make the $\times 6$ the last thing you do because the numbers coming out are not in the six times table.</p> <p class="editorial">As you can see, there are lots of possible solutions using multiply by $6$ as one of the functions, but you can also split multiplying by $6$ into multiplying by $3$ and multiplying by $2$, or as several pupils from the Mount School in York observed, multiplying by $12$ and dividing by $2$.</p> <br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><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=2727&amp;part=index">This problem</a> gives a good deal of practice in addition, subtraction, multiplication and division in an interesting and unusual format. The second part of the problem involves much reasoning about numbers.<br></br> <h3>Key questions</h3> <div>Would it be a good idea to start with $5$?</div> <div>What do you do next?</div> <div>What comes out?</div> <div>How could you get from $2$ to $22$ ? Will those same operations turn $3$ into $28$?</div> <div>Would it be a good idea to multiply each number by $2$, then $3$, then $4$, etc until find a pattern?</div> <div>What do you notice about the numbers going in? What do you notice about the numbers coming out?</div> <br></br> <h3>Possible extension</h3> Learners could create their own function machines for friends to complete. Asking them to make examples like the second part of the problem will involve a great deal of mental calculation and reasoning about numbers.<br></br> <h3>Possible support</h3> Suggest trying just the first part of the problem which is relatively easy. They could then go on to making their own &#39;function machine&#39; [as in the second part of the problem] with two, or even just one, circle.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You might like to start with $5$. Follow the operations in order - add $3$, multiply by $5$, take away $2$. What comes out?<br></br> How <span style="font-style: italic;">could</span> you get from $2$ to $22$ ? Will those same operations turn $3$ into $28$?<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> 38, 48, 33<br></br>x3 +5 +2<br></br></mdoxml> 2 4 0 1 0 0 0 If the numbers 5, 7 and 4 go into this function machine, what numbers will come out? Calculations and Numerical Methods Multiplication & division Calculations and Numerical Methods Addition & subtraction Sequences, Functions and Graphs Sequences