Once again, the title of my post seems to suggest a much more difficult concept than it is. To make things a little clearer, imagine you're asked to do 19 x 21. The number that 19 & 21 are surrounding is 20. Here we will illustrate DiffSquares1. This concept showed up 9 times this year on middle school tests, always between questions 22 & 37 (inclusive), with the median at question 23.
Number Dojo Level: 167
This process works best (and is the easiest) when the number directly in the middle of (and surrounded by) the two factors is a multiple of 10. In other words, it is best when the average of the two factors is a multiple of 10. I will show the steps of why this works first, but don't get bogged down in the details. The actual steps performed on a number sense test will be much quicker.
Why it Works:
19 x 21 can be rewritten as (20 - 1) x (20 + 1). Using the FOIL method of multiplication:
- F refers to First (20 x 20), which is 400.
- O refers to Outside (20 x 1), which is 20.
- I refers to Inside (-1 x 20), which is -20.
- L refers to Last (-1 x 1), which is -1.
So why is this process called the difference of squares? Because what you were really left with during the FOIL process was:
- Determine the average of the two factors, and square this number.
- Determine the difference between either factor and this average, and square this difference.
- Subtract the result from step 2 from the result from step 1.
Example 1: 38 x 42
Example 2: 67 x 73
Example 3: 42 x 58
Example 4: 39 x 21
Example 5: 15 x 19
Here's a free worksheet to help you practice DiffSquares1: