1. Teaching : (a+b)2
1. Ask student about the answer regarding how students incorrectly square the sum of two numbers
(as in (a + b)2) to get the sum of the squares of the individual numbers (as in a2 + b2).
2. This lesson will try to make sense out of this inequality:
3. Before explaining why it isn't true, I think it's important to understand why so many people think
it is true. There are several reasons a student might think it's true, and each of those reasons has
its own logic.
(1) I know it's true for multiplication:
so why isn't it true if the symbol between them is addition instead?
4. Let the students to make understand that Squaring means multiplying an expression times itself:
(ab)2 means (ab)(ab). But crucially, that's a whole bunch of multiplications. And we all know
that WE can regroup and rearrange items being multiplied at will (formally, we say that
multiplication is associative and commutative). And when we do regroup and rearrange, you see
there are two a's and twob's all multiplied together; thus a2b2.
5. But look at what (a + b)2 means: (a + b)(a + b).
6. What does (a + b)2 equal, then? Or, put another way, is there an expression without parentheses
that always has the exact same value? The formal algebra is below. It uses the distributive
property of multiplication over addition that we saw above.
7. The popular term for the process that you might know is Following:

2. 8. Again, be sure you understand every step in that process.
9. Now, a2 + 2ab + b2 isn't exactly an obvious way to rewrite (a + b)2, but it does have the
advantage of being correct! If you try to interpret that in a different way, though (geometrically,
for example), it can make much more sense.
10. The natural way to understand the concept of squaring is through looking at the area of a
square—which is calculated by squaring. So below is a picture of a square whose sides are
each a + b long. To make that more clear, those sides are broken up into their
separate a and b parts.
11. Asking about (a + b)2, then, is just like asking about the area of that whole square. But the whole
square is broken up into smaller squares and rectangles, and we know enough information to
calculate each of those smaller parts separately. The areas of the two smaller squares are
calculated below.

3. 12. Notice that the areas of the two smaller squares together come nowhere close to totaling the area
of the large square.
13. In algebra terms, we'd have to say that (a + b)2 must simply be greater than a2 + b2. Of course
that means they can't be equal, which is exactly what we've been trying to understand!
14. This picture actually tells us even more, though. It tells us how much greater. Each of the blue
rectangles has a length of a and a width of b, so they each have an area of a times b. And there's
two of them. Which means precisely that (a + b)2 = a2 + 2ab + b2, just as we saw in the algebra.
15. That is, I hope, enough to convince you.