Math For Elementary Teachers

Part 2vi: Area Model of Multiplication

Further Explanation for this Clip:

Mentally Computing Two-Digit Squares
In the previous clip I discussed how to use the identity (a+b)^2=a^2 +2ab+b^2 to mentally compute two-digit squares such as 93^2, but I never got around to demonstrating how to do so. So here is a step-by-step explanation:

In order to use the identity, we need to think of 93^2 as (90 +3)^2. Now we apply the above identity, with 90 playing the role of a, and 3 playing the role of b:

(90 +3)^2=90^2 +2 \cdot 90 \cdot 3 +3^2

Notice that although there are several computations on the right hand side of the equation, they are all relatively simple mental math problems:
90^2=90\cdot 90=9\cdot 10\cdot 9\cdot 10=9\cdot 9\cdot 10\cdot 10=81\cdot 100=8100
2\cdot 90 \cdot 3 = 2 \cdot 9 \cdot 10 \cdot 3 = 2 \cdot 9 \cdot 3 \cdot 10 = 54 \cdot 10 = 540
3^2=9

Therefore, 93^2 = 8100 + 540 + 9 = 8649.

Notice that there was nothing special about the number 93; if you want to square any two-digit number, the ‘a^2+2ab+b^2’ part of the computation will always be similar to our above computations.  Since we are using expanded form, our ‘a’ will always be a multiple of 10, and our ‘b’ will always be a one-digit number; thus a^2 will always be the square of a multiple of 10 (a relatively easy mental computation), 2ab will always be the product of two one-digit numbers (2 and b) and a multiple of 10 (again, a relatively easy mental computation), and b^2 will always be a one-digit square.

Try a few on your own, and once you have the hang of it you will be able to wow and amaze your friends and colleagues with your mental math prowess!

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