Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 26.1 Borrowing is Overrated

The first concept of subtraction that students learn is the idea of "taking away" objects from a collection. In fact, many children learn to speak the subtraction calculation \(5 - 3\) as "5 take away 3". Here is one way to represent this graphically:

Subtraction is interesting because the "taking away" step isn't an object, but an action. We have used an X to symbolize that idea because it's visually easy to think about the final picture with parts crossed off. But that is not to say that the X is itself an object. It doesn't "count" towards the final tally. It is an action that is performed on the collection.

The extension of this idea is to do it with base-10 blocks. This mixes the concept of "taking away" with our method of organizing numbers.

Activity 26.1.1. Subtraction with Base-10 Blocks.

Here's another example, but this time we will encounter something different.

We can see that we still need to remove some unit cubes, but we don't have any unit cubes left. So we will convert one of the rods into 10 unit cubes so that we can finish removing the pieces.

The step of converting a rod into individual unit cubes is an example of the "borrowing" step for subtraction. We reduced the number of rods by 1 and increased the number of unit cubes by 10.

Try it!

Draw a base-10 blocks diagram to represent \(42 - 28\) and compute the result.

Solution.

Just like "carrying the one," the step of "borrowing" is a bookkeeping step. When subtracting in columns, if you don't cross out that number, you might forget that you borrowed and then make a mistake down the line. This is why the process was so heavily emphasized. But you don't actually have to do it that way at all. There are natural ways to keep track of that information. We're going to once again look at the number line for insight.

One way to look at subtraction on the number line is to think about movement. But while addition moved to the right, subtraction moves to the left. We will start with a diagram of \(5 - 3\text{:}\)

For small numbers, counting out the individual steps is reasonable. But for larger values, we're going to employ an organization technique similar to what we did with addition. Here is the setup for \(53 - 28\text{:}\)

Just as before, we're going to break this out into two steps.

Once again, by simply applying a bit of mental thought, you can determine the unknown values. The value of ? is 33 and the value of ?? is 25.

Just as "carrying the one" represented an intuition of increasing numbers, "borrowing" is a representation of an intuition of decreasing numbers. Starting from the number 33 and moving to the left, you will basically never make the mistake of somehow increasing the value to 35. Your brain can do this on its own because you have enough experience with numbers.

Activity 26.1.2. Two-Step Subtraction on a Number Line.

Here is the full diagram for the calculation \(53 - 28\) using the number line.

Try it!

Calculate \(62 - 17\) using a number line.

Solution.

Activity 26.1.3. Visualizing Larger Subtraction Calculations.

Larger numbers can be done in the same way.

Try it!

Calculate \(623 - 376\) using a number line.

Solution.