difference of two squares

You may wish to look at the problem What's Possible? before trying this one. Choose a number in the 3 times table. Take the numbers on either side of your chosen number and find the difference between their squares. Take the numbers on either side of your chosen number and find the difference between their squares. Instead of taking the numbers on either side of your starting number, investigate what happens if you take the numbers two above and two below your starting number and then work out the difference between their squares... If you enjoyed this problem you may like to try Why 24? next. With thanks to Don Steward , whose ideas formed the basis of this problem. Darius from Victoria College in the Channel Islands and Maisy from Long Field Academy in England noticed that, when using the 3 times table, the difference is a multiple of 3. Maisy wrote: Sanika from India and Sarvitha from The Ellen Wilkinson School for Girls, James from High Storrs School and Alex from Long Field Academy, all in the UK, noticed that this always gives a multiple of 12. This is Sarvitha's work: Elissa from The Ellen Wilkinson School for Girls, James, Darius, Sanika and Alex all noticed and proved something slightly different. This is Alex's work: Alexis from Long Field Academy, James and Sarvitha noticed that they are in fact multiples of 20. In fact, you can see this from Darius' proof. James, Alex and Elissa all proved this. Alex's proof is the same as Darius' proof, but noting that 4αn =4×αn is four times the original number. James ignored the times table, which resulted in simpler notation: Let m = any multiple of a number. The differences of the squares of the numbers one digit above and one digit below this number can be represented, once again using algebra, as (m+1)2−(m−1)2 =(m+2m+1)−(m−2m+1) which equals m−−2m =4m What about other gaps? Jonty from Long Field Academy, Sanika and Alex tried using gaps of 2 instead of 1. This is Alex's work: I do believe there [are similar relationships for other numbers] as it works for all possibilities I've tried. Amy from Long Field Academy and Daruis both proved some facts about any gap. This is Amy's work: The formula to get the answer for this question is 4xn, where x is the number that you start with and n = the amount that you count either side of that number. When there are two subtraction signs next to each other, they would become an addition sign and so, you would get the answer 4xn. Freya from Long Field Academy made a different observation by subtracting squares of consecutive numbers, rather than squares of numbers two apart. Click to see Freya's work. That looks very convincing, but we have seen from the examples above that it isn't always true! For example, 62−42 =36−16 =20, which is not a multiple of three. Can you spot the trick in Victor's proof? Click to see an explanation. Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom. This problem is an excellent context for observing, conjecturing and thinking about proof. It offers an opportunity for purposeful practice of algebraic manipulation of quadratic expressions. "Choose any multiple of 3, take the numbers on either side of your chosen number, square them, and find the difference. Give students some time to think about explanations, and circulate to listen to what they come up with. If no-one thinks of using algebra, pose the questions: Once students have engaged with the algebra, bring the class together once more and invite a couple of students out to the board to show their proofs. Pair Products is a similar problem, but with a little more structure and support.