Three neighbours

8108 /www/nrich/html/content/id/8108/ 1 2012-03-01T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Take three numbers that are 'next door neighbours' when you count. These are called consecutive numbers. Add them together. What do you notice? Take another three consecutive numbers and add them together. What do you notice? Can you prove that this is always true by looking carefully at one of your examples?</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There were many contributions to this activity. We were sent many from Monkfield Park Primary School, namely Kallum, Athur, Kiera and Oliver. Other good answers came from Maddie from Hurstpierpoint Prep School and Hannah, Sam and Gloria from Abington. Dina from the Beit Shvidler Primary School  in London sent in this good observation and a proof; In the series of consecutive numbers, for any $3$ consecutive numbers, if the first number is $a$, the next is $a+1$, and the next is $a+2$. The sum is $3a+3$. For the next group of $3$ numbers in the series, the $3$ consecutive numbers will be: $a+1, a+2, a+3$. This sum is $3a+6$. This proves that a sum of $3$ consecutive numbers will always be $3$ more than the sum of the previous $3$ consecutive numbers in the series. Amrit from Newton Farm Nursery, Infant and Junior School  wrote; Let the three consecutive numbers be $a - 1, a$, and $a + 1$. Thus their sum is $3a$. Thus the sum of any three consecutive numbers is a multiple of $3$. Aisling  from St Joseph's Catholic School, Dorking  showed a proof in another good way; You take your three consecutive numbers and add them together. Then you look at your answer. All the answers are multiples of $3$ eg $3+4+5=12$ Then you will notice that the middle number, if timesed by $3$ will equal a multiple of $3$ eg $3+4+5=12, 4$x$3=12$ Another way to work it out, is to look at the outer numbers. Then take $1$ away from the last number and add it to the first number. You will notice that all the numbers are now the same. Add them together and the answer will be the same as in the other ways eg $3+4+5$ gives $5-1=4$; $3+1=4$; $4+4+4=12$ Thank you for all your contributions. Keep them coming in!</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Three Neighbours</h2> Take three numbers that are 'next door neighbours' when you count. These are called consecutive numbers. Add them together. What do you notice? Take another three consecutive numbers and add them together. What do you notice? Can you prove that this is always true by looking carefully at one of your examples? </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 consecutive numbers and the relationship between them. Generic proof involves examining one example in detail to identify structures that will prove the general result. 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 three consecutive numbers, a generic proof that adding three consecutive always gives an answer that is a multiple of three is developed based on the structure of one example. The article entitled <a href="http://nrich.maths.org/8081">Take One Example</a> with its video clips 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 three consecutive numbers and 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 consecutive numbers? Can you say what will happen every time you add any three consecutive 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 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 starting number is odd or even? What would happen if we added four consecutive numbers? Or five? Or six? The possibilities are endless. <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/8108/Odd&Even-1.pdf">these</a> to support their thinking.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Try starting off with three easy numbers like 3, 4 and 5. Add them together and what do you get? Can you find three equal numbers with the same sum? Now try three more and see if there is a similar answer.</mdoxml> 2 4 0 1 0 0 0 Three neighbours Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice? Using, Applying and Reasoning about Mathematics Making and proving conjectures Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Numbers and the Number System Properties of numbers Numbers and the Number System Factors and multiples Admin Upper primary mapping document