# How many multiplications are needed?

In year 7, we do a maths project called Diagonal Differences. It used to be a GCSE coursework piece before they got scrapped and involves picking a 2 by 2 box out of a 10 by 10 number grid. You multiply the opposite corners of your chosen box and find the difference between the two answers.

The main purposes behind the investigation are to get students to think mathematically, investigating an unknown situation. They are encouraged to make conjectures and generalise their findings. We want them to find patterns and consider describing them algebraically. We make use of an algebraic proof to show that a 2 by 2 box will always have a difference of 10. We want them to extend the project. It’s full of mathematical opportunities.

We also want them to show us they can multiply two digit numbers. And, here’s the problem.

This website (http://www.subtangent.com/maths/ig-diagdiff.php) can do all the calculations for you. You can tell it what size grid and box you want, drag the grid to the right place and it’ll do the sums for you.

If someone can multiply a two digit number by another one confidently, should they just use this site (or a calculator) to help them discover patterns more quickly?

The website won’t do the algebra for them. It does give a hint if you click the ‘show algebra’ button:

I think I feel like they should be allowed to use the site once they’ve convinced me they can do the multiplying. If they really can multiply efficiently, I don’t need them to keep doing it and I do want them to be able to look for patterns and describe them algebraically.

So, the question is, how many multiplications are needed to convince me? I don’t know the answer to that. I have some ideas but I’d be interested in your thoughts.