1150 /www/nrich/html/content/03/01/penta2/ 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 height="167" width="371" alt="" src="SpBub3.gif"></mdo:image></div> <div style="text-align: center;"></div> <p>Try with the squares of the numbers between $4$ and $20$.</p> <p>Did you find any square numbers which cannot be made by adding two prime numbers together?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>This question has been very well answered - thank you to everyone who sent us solutions. David, Jack and Matthew, and Beth, Harry and Henry from St. Nicolas CE Junior School, Newbury explain how they tackled it:</p> <blockquote>We listed the prime numbers up to 100 and then we listed the squares of the numbers between 4 and 20. Then we started working out the answers by adding two prime numbers together to make the squares until we reached 400. We found out that two squares were impossible to make. When we had an odd square number, we had to have a number that adds on to 2 to make the square number and sometimes this number wouldn't be prime. For example, to make 121 with two primes, one of the primes has to be 2, and 119 is not prime.</blockquote> <blockquote> <p>Our results were:<br></br> 25 = 2 + 23<br></br> 36 = 13 + 23<br></br> 49 = 2 + 47<br></br> 64 = 17 + 47<br></br> 81 = 2 + 79<br></br> 100 = 3 + 97<br></br> 121 impossible<br></br> 144 = 47 + 97<br></br> 169 = 2 + 167<br></br> 196 = 29 + 167<br></br> 225 = 2 + 223<br></br> 256 = 89 + 167<br></br> 289 impossible<br></br> 324 = 101 + 223<br></br> 361 = 2 + 359<br></br> 400 = 41 + 359</p> </blockquote> <p>Martha from Tattingstone School went about it in a slightly different way, but came to the same conclusions:</p> <blockquote> <p>Firstly I looked to see what the square of 20 was, 400. Then I found the nearest prime number that was below 400 and found it was 397. I then took 397 away from 400 and found it was 3 which is also a prime number so I added them together to get 20². I did that on all of them and got:</p> <p><mdo:image width="200" height="374" src="Tatt2solcropped.gif" alt="121 and 289 not possible, all others written as sum of two prime numbers"></mdo:image> </p> <p>I noticed that all the odd numbers that needed to be squared e.g. 19, 17, 15, 13 etc. needed to be added by two because it is the only even number that is a prime number as well.</p> </blockquote> <p>Thanks to Alice and James from Tattingstone, also William from Sydney Grammar School, St Ives who sent in correct answers as well.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Two Primes Make One Square</h2> <br></br> <div style="text-align: center;"><mdo:image alt="" height="167" src="SpBub3.gif" width="371"></mdo:image></div> <div style="text-align: center;"> </div> <p>Try with the squares of the numbers between $4$ and $20$.</p> <p>Did you find any square numbers which cannot be made by adding two prime numbers together?</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=1150&amp;part=index">This problem</a> brings together two important classes of numbers: primes and squares. By working with them in an investigative way, learners will become more familiar with their properties and also discover some interesting number facts. This investigation will encourage children to be persistent as they work at mathematics and it gives you an opportunity to highlight different methods of approach.<br></br> <h3>Possible approach</h3> <div>As a starter, ask the class to find all the different $3$-digit numbers they can make from the digits $1$, $3$ and $5$, then to talk in pairs about which are prime. This should bring up some divisibility rules and remind pupils of the properties of prime numbers, so that in fact they will conclude that none of them are prime. You could also use the <a href="http://www.standards.dfes.gov.uk/primary/teachingresources/mathematics/nns_itps/number_grid/">interactive teaching programme number grid</a> from the Standards site to discuss prime numbers further by, for example, shading all the numbers that are multiples of $2$, $3$, $5$ etc.</div> <br></br> <div>Pose the question and if necessary discuss square numbers. You may like to ask pupils to try a few on mini-whiteboards as a start to check that they understand the task, and then leave them to explore the challenge. It would be worth saying that you are not expecting learners to find solutions straight away - they may need to persevere!</div> <br></br> <div>A mini-plenary after $10$ minutes or so could focus on different ways of approaching this task. Some children might make lists of the square numbers first and work that way, others might start with pairs of prime numbers. It would be worth sharing these different strategies and drawing attention to any useful systems that the pupils have developed, for example starting with the smallest square number and working upwards.</div> <br></br> <div>As they work, you could invite children to write up their sums on the board or on a wall for everyone to see. This would give a focus for a plenary as you can talk about the square numbers that haven&#39;t been made. Are they impossible to make or is it just that no-one has found a way yet? Why?</div> <br></br> <h3>Key questions</h3> <div>What are the squares of the numbers from $4$ to $20$?</div> <div>What happens when you add two prime numbers together?</div> <div>What happens if you add two different prime numbers together?</div> <div>Why can&#39;t you make that square number?</div> <br></br> <h3>Possible extension</h3> Some children might like to explore all the different ways that each square number can be made. How do they know they have them all?<br></br> <h3>Possible support</h3> Many pupils will find that calculators, number squares and multiplication squares will be useful tools. You could encourage children to make a list of the first six square numbers and the primes up to $36$.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> What are the squares of the numbers from $4$ to $20$? <br></br> What happens when you add two prime numbers together? <br></br> What happens if you add two different prime numbers together? <br></br></mdoxml> 2 4 0 1 0 0 0 Can you make square numbers by adding two prime numbers together? Using, Applying and Reasoning about Mathematics Working systematically Calculations and Numerical Methods Addition & subtraction Numbers and the Number System Factors and multiples Numbers and the Number System Prime numbers Numbers and the Number System Square numbers Admin Upper primary mapping document