Two numbers under the microscope

8059 /www/nrich/html/content/id/8059/ 1 2012-03-01T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Choose any two odd numbers, such as 5 and 9. Add them together. <mdo:image alt="" src="FivePlusNine.gif" style="width: 277px; height: 277px;"></mdo:image> Draw a picture or make a model to show how the numbers add together. Adam found some dominoes with 5 and 9 spots on them: <div class="toggle"><mdo:image src="5%2B9dominoes2.jpg"></mdo:image></div> Sarai made a model using Multilink cubes: <div class="toggle"><mdo:image src="5%2B9-multilink.jpg"></mdo:image></div> Abdul drew a picture of 5 add 9 like this: <div class="toggle"><mdo:image src="5%2B9drawing2.jpg"></mdo:image></div> What do you notice about the answer? Look closely at the models and pictures. Can you see anything in any of them that would work in exactly the same way if you used two different odd numbers? Can you use your one example to prove what will happen every time you add any two odd numbers? See if you can explain this to someone else. Are they convinced by your argument? Once you can convince someone else, see if you can find a way to show your argument. You might draw something or take a photo of things you have used to prove that your result is always true from your example. Tell us about it by <a href="http://nrich.maths.org/8059/submitsolution">submitting your solution</a>.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">More than $30$ solutions were sent in but the activity had asked for some kind of proof and many solutions unfortunately consisted of examples of numbers only, so here are the ones that said a bit more. Firstly Amrit from Newton Farm Nursery, Infant and Junior School in the United Kingdom who wrote; Every odd number can be written in the form $2a + 1$, so let the two odd numbers be $2a + 1$ and $2b + 1$. Thus their sum is $2(a + b + 1)$. As every even number can be written in the form $2n$, the sum of two odd numbers is always even. From James from St. Johns School in Northwood, England we had; They always add up to an even number because if you add together the even numbers that are one less than each number you'll have an even number. Then if you add together the two that are left you get an even number. If you do this with three odd numbers it would give you an odd number because you'll have three left to add together. If you add up an even number of odd numbers you get an even number but if you have an odd number of odd numbers you get an odd number. From Madison at Norwayne School in the USA we had: Any odd number always ends up in pairs of two when added together. Like if you take $3$ and $9$ they end up in pairs of two. Or if you take $5$ and $9$ they end up in pairs of two. The reason for that is if you add any odd number with a other odd number then the sum is even. So you can split the sum up into to even group. From Peter at the British International School in Istanbul in Turkey we were sent: An odd number actually is an "even number plus $1$" (or minus $1$). So if you add two odd numbers. Both "$1$"s from both numbers togeher make a "$2$", which is an even number; and the remaining part of the both numbers were even anyway, so the total is always even. This is best shown by the picture of Abdul with the orange and green coloured squares. Christopher from Lyngby/Lundtofte  in Denmark sent in some good material but unfortunately I could not open open the file. Thanks to all of you who submitted solutions. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Two Numbers Under the Microscope</h2> Choose any two odd numbers, such as 5 and 9. Add them together. <mdo:image alt="" src="FivePlusNine.gif" style="width: 277px; height: 277px;"></mdo:image> Draw a picture or make a model to show how the numbers add together. Adam found some dominoes with 5 and 9 spots on them: <div class="toggle"><mdo:image src="5%2B9dominoes2.jpg"></mdo:image></div> Sarai made a model using Multilink cubes: <div class="toggle"><mdo:image src="5%2B9-multilink.jpg"></mdo:image></div> Abdul drew a picture of 5 add 9 like this: <div class="toggle"><mdo:image src="5%2B9drawing2.jpg"></mdo:image></div> What do you notice about the answer? Look closely at the models and pictures. Can you see anything in any of them that would work in exactly the same way if you used two different odd numbers? Can you use your one example to prove what will happen every time you add any two odd numbers? See if you can explain this to someone else. Are they convinced by your argument? Once you can convince someone else, see if you can find a way to show your argument. You might draw something or take a photo of things you have used to prove that your result is always true from your example. Tell us about it by <a href="http://nrich.maths.org/8059/submitsolution">submitting your solution</a>. </div> <h3>Why do this problem?</h3> This problem supports the development of the idea of generic proof with the children. This is a tricky concept to grasp but it draws attention to mathematical structures that are not often addressed at primary school level. It is possible that only very few children in the class may grasp the idea but this is still a worthwhile activity which provides opportunities for children to explore odd and even numbers and the relationship between them. Proof is a fundamental idea in mathematics and in encouraging them to do this problem you will be helping them to behave like mathematicians. By addressing the case of adding two odd numbers, a generic proof that adding two odd numbers always gives an even answer is developed based on the structure of odd and even numbers. The article entitled <a href="http://nrich.maths.org/8081">Take One Example</a> will help you understand how this problem supports the development of the idea of generic proof with the children. Reading it will help you to see what is involved. <h3>Possible approach</h3> Ask the children to choose two odd numbers and add them together. It is probably easiest if they choose ones that are easy to model and numbers that they are secure with. Suggest that they make a model of their numbers using apparatus that is widely available in the classroom. Resist pointing them in specific directions unless they become stuck, but if they are then resources such as Multilink cubes, Numicon or squared paper will be helpful. After some time exploring they may need some prompting to move them towards looking at the pairing of their dots or cubes. These <a class="pdflink" href="/content/id/8059/Odd&Even-1-6.pdf">pictures</a> may help. The idea is that they take a particular example and then see if they can see the general structure within that one example. <h3>Key questions</h3> How would you like to show these numbers? What do you notice about the answer? Can you see anything in your example that would work in exactly the same way if you used two different odd numbers? Can you say what will happen every time you add any two odd numbers? Can you convince your friend that this is true? <h3>Possible extension</h3> <h5>Even plus even</h5> See what happens if you add two even numbers such as 4 and 12. Can you 'see' in this example what will happen every time you add two even numbers? You may find it helpful to work in a similar way to the way you worked for odd numbers. <h5>Adding an odd and an even.</h5> See what happens if you add an odd and an even or an even and an odd such as 6 and 9. Can you 'see' in this example what will happen every time you add an odd and an even or an even and an odd number? You may find it helpful to work in a similar way to the way you worked before. You could go on to look at <a href="http://nrich.maths.org/8063">Take Three Numbers</a>. <h3>Possible support</h3> It may be helpful to encourage children to use paper cut outs of the numbers such as <a class="pdflink" href="/content/id/8059/Odd&Even-1.pdf">these</a>. In these the oddness of the odd numbers is very clear. Laminated sets of these cards could be a very useful addition to your classroom resources. Numicon would be an alternative resource to use.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">If you are stuck, try looking at your model or picture in different ways to see if that helps. Do any of the bits of it match up in pairs? Think about what you know about the properties of odd and even numbers.</mdoxml> 2 4 1 0 0 0 0 Two numbers under the microscope This investigates one particular property of number by looking closely at an example of adding two odd numbers together. Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Numbers and the Number System Properties of numbers Numbers and the Number System Odd and even numbers Admin Lower primary mapping document