The other day my son observed that since I was a certain age when he was born, and he was 9, then my age would be ... at which point he correctly determined my age. His distressingly accurate computation centered around a key concept: The difference between two quantities (in this case, our ages) is unchanged if we add the same amount to both. 

This is the basis for the compensation method of arithmetic, and the source of questions like the following:

Draw and label a tape diagram to show how to simplify the problem 370 – 190. Write the new number sentence, and then subtract.

In a tape diagram, the length of a tape corresponds to a number: thus the numbers 370 and 190 can be represented as tapes — 370 units and 190 units in length. To add 370 + 190, we would glue the two tapes together:

Meanwhile, to subtract 370 – 190, we might set the tapes down, one above the other, and find the difference:

What if we add the same amount to both tapes?  For example, if we add 10 to both tapes:

The difference in tape lengths is still the same, but the actual subtraction has been changed from 370 – 190 to 380 – 200.

The crucial observation is that the number we've added makes the number we're subtracting an benchmark number: a number that is easy to work with.  What makes a number “easy” varies from person to person, but a good guideline is that the more times 0 appears as a digit of a number, the easier the number is to work with.

One of the virtues of the common core is that the material is interconnected and designed to work together, with each part reinforcing the rest. In this case, counting by benchmarks, introduced early in the 2nd grade, prepares students for subtraction via compensation, introduced later in the 2nd grade. 

Counting by benchmarks involves counting up to the next ten; then to the next hundred; and so on.  For example, we can count from 368 onward as:

Note that we counted by ones to 370, our first benchmark number; and then by tens to 400, our next benchmark number.  We can read the numbers above the arrows as indicating that if we start with 368 and add 1 + 1 + 10 + 10 + 10 (which is to say, 32), we will get 400:  thus 368 + 32 = 400.

Now consider the difference 526 – 368.  We can represent this using a tape diagram as:

But if we count by benchmarks from 368 to 400, we can find a different subtraction that gives us the same difference:

Common core material is available at EngageNY under a Creative Commons non-commercial license: in effect, school districts can use the materials free of charge, as long as they attribute the material correctly. You can read more about counting by benchmarks in Grade 2, Module 3 and subtraction by compensation in Grade 2, Module 5.

Jeff Suzuki teaches mathematics at Brooklyn College, and is one of the founders of the Mid-Hudson Valley Math Teachers Circle, a group of teachers, professors, and mathematics aficionados working to promote mathematics education in the Hudson Valley.

Read more answers to Common Core questions