Take three numbers

8063 /www/nrich/html/content/id/8063/ 1 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Choose any two odd numbers and one even number, such as 3, 5 and 2. How would you like to represent these numbers? Try adding them together and draw/make the representation of their sum. What do you notice about the answer? Look closely at your model. Would it work in exactly the same way if you used different numbers but still two odds and one even? Can you use your example to prove what will happen every time you add two odd numbers and one even number? 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 the argument on paper. 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/8063/submitsolution">submitting your solution</a>.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">We had nearly $100$ solutions from across the world sent in to us. From Hymers College Junior School we had solutions sent in from all of these pupils, Lucy, Max, Abdul, Rikki, Charlie, Helena, Imogen, Louisa, Helena, Eva, Harley, Rory, Thomas, Matthew, Khushi, Adi, Amaan, Mariana, Andrew, Sunnie , Sophie, Amy, Rashi, Eilza, Signe, Claire, Adnan, Amar, James, Adam, Amelia, Ayoun and Amy . Elliott from Solihull School in England sent in a good idea linked with Pascals Triangle;  <mdo:image src="pascal.jpg"></mdo:image>I've coloured in the even numbers in Pascal's Triangle In this simple calculation I will explain ........ how adding $2$ odd Numbers and an Even number together always adds up to an even number at the end. First of all I will demonstrate the one on the website as you see it is even More equations $5+9+4=18$ and $9+20101=20110+2=20112$ Amrit from Farm Nursery, Infant, and Junior School  wrote; All even numbers can be represented as $2a$, and all odd numbers as $2b + 1$, where a and b are integers. Thus the sum is $2b + 1 + 2c + 1 + 2a = 2(a + b + c + 1)$. Thus the answer is always even. Isabel and Caner  from St Theresa's, Finchley in the UK gave this very thorough explanation Odd+Odd always Equals even because the next number is always even. For example, if you add $7+3$ you are adding $3$ which is $2$(even)+$1$ and $7$ which is $6$(even)+$1$ even + even = even and $1+1=2$(even) Therefore odd=Odd must always = Even When you add an even number to an even number it always equals an even number this is because if you add $6$ and $2$ the next number from $6$ is $7$(odd) and the next number from $2$ is $3$(odd) We already proved in our explanation above that odd+odd = even, therefore if you add the 1 more from the even number in both parts of the equation, you will make two, which will combine with the even total of the odd numbers to form another even. An odd number + an odd number + an even number always equals an even number this is because odd is 1 away from even so it's an even number but when you add an even number + an even number the answer is an even number,when you add an odd number + an odd number the answer is even and when you add even number's together your answer has to be even because your adding an even number to an even number. Shivek from Monkfield Park Primary School sent in this interesting account;  Example 1. $5+9 = 14$ ; $14+ 22=36.$ Example 2. $1+1 = 2.$; $2+4=6$ Example 3. $99+99=198$; $198+198=396$. How I did it? Even numbers end in $0,2,4,6,8$ and odd numbers end in $1,3,5,7,9$. Lets add $2$ odd numbers like: $1+3$. You will get $4$ Add $3+5$. You will get $8$ and Add $5+7$. You will get $12$ So, you see: When you add any odd number to another odd number you get an Even number. Now when you add $2$ even numbers like $2+2, 4+2 or, 6+4 or 8+4$ you will always get an even number answer. Like ($2+2=4, 4+2=6 or, 6+4=10 or 8+4=12$.) As you can see above all addition of even numbers ends in $0, 2, 4, 6 or 8$ which are even numbers. So, Odd Number + Odd Number = Even Number and Even number + even number = Another even number. QED (quite easily done) by Shivek. Finally from Joanne I think she was rather young and this was written by a helper (?), it says; Jo used a selection of coloured elastic bands to test out what happens: $4$ Green bands, $5$ Blue bands, $3$ Yellow bands GGBBYY GGBBBY total = $12$ = even $6$ Green bands, $7$ Blue bands, $7$ Yellow bands GGGBBBBYYY GGGBBBYYYY total = $20$ = even She said: 'The number of bands is always even. I put them in $2$ rows and the two rows are the same length, so it is an even number.' Jo tried a bigger number in her head to test her theory: $10 + 21 + 35 = 66$ = even Jo tried to work out why her number would be even by working out a rule: even + odd + odd = even; (even + odd = odd, then odd + odd = even) She made up these rules for adding two numbers together: even + odd = odd; Even + even = even; odd + odd = even Jo had a go at trying out what would happen if she changed the odd and even numbers using the rules she had worked out: even + even + odd = odd; even + odd + even + odd = even odd + odd + odd + even = odd (odd+odd=even, then +odd=odd, then +even=odd) Well done to these and all the others who sent in so very many solutions, most with very good thoughts and understanding.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><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. Generic proof involves examining one example in detail to identify structures that will prove the general result. 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 helping them to do this problem you will be encouraging them to behave like mathematicians. By addressing the case of adding two odd numbers and an even number, a generic proof that adding two odd numbers and an even number 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 an even number 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. If they are stuck then resources such as Multilink cubes, Numicon or squared paper will be helpful. 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 represent 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 three different numbers but still one even and two odds? Can you say what will happen every time you add any two odd numbers and one even numbers? Can you convince your friend that this is true? <h3>Possible extension</h3> When adding three numbers there are a number of different combinations of odds and evens that are possible. Ask the children to explore what they are. Get them to identify the possible combinations and the features of those combinations that matter. Does it matter whether the numbers are odd or even? How many different cases can you find? To work on the generic proofs for each case the children will need to consider them separately. Can you create a proof for each case using one example? A possible extension would be to look at <a href="http://nrich.maths.org/8108">Three Neighbours.</a> Some other problems at higher stages that may also be worth exploring are <a href="http://nrich.maths.org/7208">Always a Multiple</a>, <a href="http://nrich.maths.org/6401">Power Mad!</a> or <a href="http://nrich.maths.org/8111">Seven Squares</a>. <h3>Possible support</h3> It may be helpful to return to <a href="http://nrich.maths.org/8059">Two Numbers Under the Microscope</a> if the children are struggling with adding three numbers. This might help them to feel more comfortable with the rules they have proved in that problem and so build the foundations for this one. The children may find it helpful to use representations of numbers such as <a class="pdflink" href="/content/id/8063/Odd&Even-1.pdf">these</a> to support their thinking.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">What kind of number do you get when you add two odd numbers together? How about adding an odd number to an even number? Can you use these results to help you to see what is happening in your addition of two odd numbers and one even number?</mdoxml> 2 3 0 1 0 0 0 Take three numbers What happens when you add three numbers together? Will your answer be odd or even? How do you know? Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Using, Applying and Reasoning about Mathematics Making and proving conjectures Using, Applying and Reasoning about Mathematics Working systematically Calculations and Numerical Methods Addition & subtraction Numbers and the Number System Odd and even numbers