Back in the summer (80 lessons ago), we first combined the concepts of multiplication and estimation (in EstMultWhole). I can't remember seeing a number sense test that did NOT contain this concept. Today we will look at a specific subset of these problems--where we are asked to multiply 3 numbers that are close (or somewhat close) together. This concept appeared 8 times last year, mostly at either question # 30 or question # 60.
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Number Dojo Level: 241
In many cases when these questions appear, the 3 numbers will be consecutive integers (or consecutive even or odd integers). It may be sufficient to simply cube the middle number. Or it may be easier to multiply 2 of the numbers together, and then round and multiply the result by the third number.
* NOTE: The greater the spacing between the numbers, the less likely the answer will be within range if you simply cube the middle number. Let's take a look at some examples:
Example 1: * 11 x 12 x 13 = ___
- Cube the middle number. (You should have this one memorized). Your answer is 1728.
- The exact answer is 1716, and the acceptable (+/- 5%) range is from 1631 to 1801.
Example 2: * 13 x 14 x 15 = ___
- Cube the middle number. If you don't have this one memorized, try this: Square 14 to get 196. Round this to 200, and then multiply by 14 again to get 2800.
- The exact answer is 2730, and the acceptable (+/- 5%) range is from 2594 to 2866.
Example 3: * 11 x 14 x 17 = ___
- The spacing here is dangerous. Let's try multiplying 11 x 14 to get 154. Round this down to 150, and then multiply by 17 to get 2550.
- The exact answer is 2618, and the acceptable (+/+ 5%) range is from 2488 to 2748.
- * Notice that our answer from Example 2 would've been out of range here.
Example 4: * 20 x 25 x 30 = ___
- Personally, I wouldn't estimate this because the multiplication is easy enough. Multiply 20 by 25 to get 500. Then multiply this by 30 to get 15000.
- The acceptable range is from 14250 to 15750. If we had cubed 25, we would've gotten 15625, which would be within range, but barely.
Example 5: * 9 x 12 x 15 = ___
- I wouldn't estimate this one either, but we can. Let's multiply 9 x 12 to get 108. Round this to 110.
- Write down our 0, and then multiply 11 x 15 to get 165; write this in front. Our answer is 1650.
- The exact answer is 1620, and the acceptable (+/- 5%) range is from 1539 to 1701.