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 isnt true, I think its important to understand why so many people think it is true. There are several reasons a student might think its true, and each of those reasons has its own logic. (1) I know its true for multiplication: so why isnt 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, thats 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 as and twobs 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:
8. Again, be sure you understand every step in that process.9. Now, a2 + 2ab + b2 isnt 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.
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, wed have to say that (a + b)2 must simply be greater than a2 + b2. Of course that means they cant be equal, which is exactly what weve 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 theres 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.