4 ms·

I "see" that m^2 + 2m is almost a square. Just one unit square missing: ---------------- |< m >| | ---------------- | |^| |

• ColinWright 12y ago
  Indeed. As taught, children get shown that a^2-b^2=(a+b)(a-b). This is extremely useful in all sorts of factorizations and calculations. That means that when you see two numbers that are close together you can think of them as (a+b) and (a-b) where b is small, and a is their average. The challenge is to see this in all its variations. We can see that 53^2 - 3^2 = (53-3)(53+3), hence 53^2 = 50x56 + 9 which in turn is 100x28 + 9 which is 2809. So 53^2 is 2809, something I needed a short time ago. It would've taken longer to get a phone, calculator, or computer, than it did to just do it in my head. So (m)(m+2) is the product of two numbers close to each other, hence it's (a+b)(a-b) where a is the average, and b is small. In this case a=m+1 and b=1, so (m)(m+2)=(m+1)^2-1. This is also related to the usual quadratic formula, which is done by completing the square. Same ideas, different disguise.
  • jonsen 12y ago
    The danish minister of finance once impressed the audience at a tv show by multiplying two big numbers in his head very fast. He then explained that this was the method he used.