While tutoring students who are in the process of multiplying two numbers, I sometimes challenge myself by trying to do the multiplication in my head instead of using a calculator. I don’t know where I learned it, but I usually take a divide-and-conquer approach to multiplying large numbers, a large number for me being anything greater than 12, save for a few exceptions.* In general, my strategy has been to break the original multiplication problem into simpler parts that I can then solve easily and combine for the answer. I’ve noticed that I usually either “round down” a multiplying number to the nearest ten, break it apart and add up the resulting products or I “round up” a multiplying number, break it apart and subtract the resulting products.
That last sentence is probably impossible to parse without an example, so let’s look at 42 x 8, first using the round-down strategy.
Method 1: Round 42 down to nearest ten and add †
Here, the implied first step is to round 42 to 40, but that doesn’t mean we forget about the 2. Instead, we recognize that 40 + 2 = 42 and rewrite the equation with 42 broken up. Then it’s a question of solving 40 x 8 and 2 x 8, both of which should be easy enough to do mentally.‡ The final step is to add the products, 320 and 16 together.
Now let’s try using the second method by rounding up a multiplying number.
Method 2: Round 8 up to nearest ten and subtract
In this method, we round 8 up to 10, but recognize that 42 x 10 won’t give us the right answer. 42 x (10-2) will, though. Thus, we distribute the 42 and get two simpler products that we can solve one after the other: 42 x 10 and 42 x 2. We can then subtract the results to get the answer.
For problems involving multiplying two two-digit numbers, I find I unconsciously start with one method, but then often mix in the other in subsequent steps. Take 42 x 88:
|Start by rounding 42 down||Start by rounding 88 up|
No matter how I started, I rounded up/down/subtracted/added where I saw fit. In retrospect, I may have taken another approach completely by rounding 88 to 100.
*Exceptions including certain perfect squares you just see over and over again while tutoring (e.g. 152, 252) and multiples of “nice” numbers such as 10, 25, 100, etc.
†The rounding-down method is effectively separating the tens from the ones and adding the products together. This is equivalent to the long multiplication method a lot of us, I assume, learned in school but were never explained how it worked (at least I wasn’t, or I just can’t remember). The virtue of the above presentation over the school method is that it makes sense and simpler to do mentally—it’s not so easy to line up columns, carry ones and shift numbers in your head as it is on paper.
‡To use these methods, you should be okay adding and subtracting whole numbers 1 to 100, as well as knowing your times tables by heart to at least until 9 x 9. You should also know tricks when multiplying by multiples of 10, e.g 9 x 60 is just 9 x 6, 54, with a zero added at the end, 540.