Techniques for Subtracting with Ease

mental math series, part 3

By the end of this lesson, it will be easy to do that in your head. Really, with a couple tricks and a little practice you can do it! Have faith!

In the last lesson, we talked about using benchmarks as resting points in addition problems involving a single-digit value. We can apply benchmarks to subtraction too.

Example 1

The benchmark value is 10 because it is the closest multiple of 10 less than 15. From 15 to 10 is 5 units. Therefore, we have subtracted 5 of the 9 needed.

We need to subtract 4 more to have taken away a total of 9.

Mentally, you can take the subtraction one step at a time: subtract 5 from 15, then 4 from the result.

Counting Up

Another popular technique for mental subtraction is counting up. In this method you begin with the smaller number and count up to the larger number, using benchmarks as needed.

Begin with 9. It takes 1 unit to get from 9 to 10.

From 10 to 15 is 5 units.

We’ve added a total of 6 units, therefore 15 – 9 = 6.

Example 2

Step 1: Begin with the smaller number and count up to the first benchmark. In this case, the smaller number is 18 and the first benchmark is 20.

Step 2: There are 10 units between 20 and 30, so add 10 to get to 30.

Step 3: Lastly, count up from 30 to 33.

Step 4: Sum all the increments.

Example 3

Step 1: Begin with 59, count up to 60.

Step 2: Now count from 60 up to 120.

Ask yourself, “what is the difference between 6 tens and 12 tens?”

Step 3: Count up from 120 to 127.

Step 4: Add the increments.

Example 4

Begin with 328. Count up to 330.

Now count up to 400.

From 400 skip to 500.

Finally, count up to 572 from 500.

I have two 70's and two 4's, so group them together as 140 and 4.

Therefore, our final answer is:

Subtraction as Distance on a Number Line

We can also think of subtraction as the distance between two numbers. Begin by sectioning out the middle hundred.

Then count from the middle-out to determine the increments. For example, count from 500 to 570 and from 570 to 572, for a total of 72 units. Next count from 400 down to 33o, and 330 to 328, for a total of 72 as well. So far we have accounted for 172 units between the two values.

(I swear I didn’t plan this! Must be a mathematician’s intuition for symmetry or something…idk.)

Visually, we have:

I often imagine the number line to aid this process

We can now add another 72 to the 172 we’ve counted so far.

We could also have added the double numbers first, then the hundreds.

Next Lesson: 100 Prisoners and a Light Bulb

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