943 /www/nrich/html/content/99/04/penta5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div>Put operations signs ($+$ or $-$ or $\times$ or $\div$) between the numbers 3, 4, 5, 6 to make the highest possible number and lowest possible number.<br></br> <br></br> How about trying with numbers 1, 2, 3, 4, 5 and 6?</div> <div style="text-align: center;"><br></br> <br></br> <mdo:image src="numbers%201-6.png"></mdo:image></div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">We had nearly $100$ solutions submitted to us with acceptable ideas for the highest and lowest. Five schools sent in many submissions and quite a few students went on to explore what they could do with $1, 2, 3, 4, 5$ and $6$.<br></br> <br></br> Banstead Junior School pupils George, Marcus, Ellanah, Ellie, Grace and Lauren sent in individual solutions that showed $360$ as the highest number and $0$ as the lowest. They went on to show that with more numbers they could get as high as $720$.<br></br> <br></br> Norwayne Elementary School in the USA pupils, Kyler, Tim, Wesley and Grace also sent in individual solutions and extended the lowest down to $0.025$.  When using the $6$, numbers got as low as $1-65432 = -65431$.</span><br></br> <br></br> <span class="editorial">Hamilton Academy pupils Saihaan, Sean, Theo and Huda individually took the lowest number down to $-36$.<br></br> <br></br> Bowes Primary School pupils Yasara, Chloe, Shiv, Rhea, Lotte, and Aan individually took  it</span> <span class="editorial">down to $-19$</span> <span class="editorial">using the six numbers.<br></br> <br></br> Heage Primary School pupils Fraser, Brogan, Miles and Phoebe individually took using six numbers up to $1080$.<br></br> <br></br> <span class="editorial">Matthew from  Beechwood Park School
 sent in this which is typical of many of the solutions sent in;</span></span><br></br> <br></br> Highest: $(2$x$3)$x$(4$x$5)$x$6=720$   Lowest:$1-2-3-4-5-6=-21$<br></br> <br></br> <span class="editorial">Annabelle from Toongabbie Christian School
 in Australia
 sent in this for the first part of the problem which represents many students&#39; workings:</span><br></br> <br></br> $3$x$4$x$5$x$6$ highest             $3-4-5-6$ lowest<br></br> <br></br> <span class="editorial">Tritham from St.Christopher&#39;s 
School Penang, Malyasia sent in the following for the smallest answer:</span><br></br> <br></br> The smallest answer is $6-5-(4-3)=0$<br></br> <br></br> <span class="editorial">Finally at the end of the month we had this submission from Year $5$ at St. Ambrose School:</span><br></br> <br></br> Highest:  $63$x$54= 3402$  and Lowest: $3-654= -651$.<br></br> <br></br> <span class="editorial">Well you certainly worked hard on this activity. Well done all of you. We hope to hear from you again soon.</span><br></br>  </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Highest and Lowest</h2> <div style="text-align: center;"><br></br> <mdo:image src="numbers%201-6.png"></mdo:image></div> <br></br> Put operations signs ($+$ or $-$ or $\times$ or $\div$) between the numbers 3, 4, 5, 6 to make the highest possible number and lowest possible number.</div> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This activity gives a good opportunity to explore using the knowledge and skills the pupils already have in a "safe" environment.<br></br>  </div> <h3>Possible approach</h3> <div>Start off by writing the four numbers down in order and putting the same sign between each pair. Which operation gives the highest total and which the lowest?<br></br> The children can vary the order themselves either working in pairs or individually. After a short period of independent work ask some of the children to explain their thinking to the others before continuing to see what the highest and lowest possible solutions are.<br></br> Having tried this challenge, many children will be able to explore further some of the attributes associated with the four rules of number and place value.</div> <br></br> <h3>Key questions</h3> <div>Could you make this answer bigger somehow?</div> <div>Could you make this answer smaller somehow?</div> <div>How have you got your ideas?<br></br> How do you know that this is the biggest possible answer?<br></br> How do you know that this is the smallest possible answer?</div> <br></br> <h3>Possible extension</h3> <div>Other forms of number manipulation may be applied e.g. using powers. The children could also try to make a range of target numbers using the same numbers or alternatively choose a different set of starting numbers and see what results they can make.</div> <br></br> <h3>Possible support</h3> <div>Many pupils will benefit from using a calculator so that their energies can be applied to exploring ideas and reasoning rather than just be taken up with calculating. However if your focus is on gaining fluency in calculating skills then it would be better to restrict the set to just three of the numbers such as $3$, $4$ and $5$. A further reduction in the challenge would be to take two numbers and consider all the different solutions that could be made by using any of the operations on them.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">As is often the case with solving problems, there are several ways of looking at the problem. For example, Mark from St Johns, Walsall Wood kept the digits separate and did:<br></br> <br></br> "Highest: $3 \times4 \times5 \times6 = 360$<br></br> <br></br> Lowest: $3 - 4 \times5 \times6 = - 117$"<br></br> <br></br> But Jason from Dunstable put some digits together:<br></br> <br></br> "I made the highest number $3258$ by simply multiplying $543$ by $6$.<br></br> <br></br> I made the lowest number $-651$ by simply subtracting $654$ from $3$".</mdoxml> 2 3 0 1 0 0 0 Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number. Numbers and the Number System Comparing and Ordering numbers Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Trial and improvement Calculations and Numerical Methods Multiplication & division Admin Upper primary mapping document