Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 25.1 Carry the One (or Maybe Not)

The idea of "carrying the one" is one of the many phrases that are taught when it comes to arithmetic. We are going to take an unusual approach in that we're going to talk about what it is and how it works, but then we're going to talk about why it's not that important in practice (at least these days).

Before we can talk about "carrying the one," we first need to think about what addition is. One way to think about addition is that you're starting with a certain quantity, and then we're going to combine it with another quantity and look at how much we have in total. When working with young children, the picture looks something like this:

But as the numbers get larger, our methods need to get more sophisticated. First, instead of just scattered blocks as we have above, we're going to use base-10 blocks. And then we're going to have to think logically about how we organize that information. Here is a diagram for \(37 + 46\text{:}\)

The most natural thing for us to do here is to reorganize the information so that the unit cubes are together and the tens rods are together.

This diagram highlights the reason that "carrying the one" is part of the process. We have run into the situation that we have "too many" unit cubes. They spill over to a number larger than what we can account for with the place value system. And so that's where we trade in 10 units for 1 rod, which is the concept that is attached to "carrying the one."

Activity 25.1.1. Addition with Base-10 Blocks.

Try it!

Draw a base-10 blocks diagram to represent \(35 + 17\) and compute the result.

Solution.

The idea of "carrying the one" was developed for the purpose of pencil-and-paper arithmetic. If you needed to add a long column of numbers (and there were once good jobs out there for people who can do this quickly and accurately), then you needed a notation for the number of groups of 10 that needed to be accounted for in the next larger place value. But at this time, there is not a lot of value in this particular calculation because we can have computers do it many times faster than we can and they do it with perfect accuracy.

But this does not mean that there is zero value in humans performing arithmetic. There are times when it's handy to be able to do a 2-digit or 3-digit addition problem mentally instead of having to reach for a calculator. Some people are able to add in columns in their heads, but many find it difficult to keep track of all the different digits floating around. So we are going to work with addition on the number line as our model for thinking about mental arithmetic.

Addition on the number line is about movement. The first number represents your starting position and the second number represents how far to the right you move from that position. Here is what \(2+3\) looks like:

For small numbers, it is easy enough to just count out the steps. But for a larger calculation, counting is simply too slow. And in order to simplify the diagram, we're going to use just the part of the number line that's relevant. Here is the setup to calculate \(37 + 46\text{:}\)

Of course, the challenge is to figure out what the value of is going to be.

Remember that our goal is to set ourselves up for mental arithmetic. So we are going to set up this picture in a way that can be done with simple mental calculations, rather than working with base-10 blocks or digit manipulations. The key trick is to break the motion of 46 steps to the right into two separate motions: 40 steps to the right followed by 6 more steps to the right. In the diagram below, see if you can work out the values of both ? and ?? by thinking through the picture.

With a little bit of mental effort, you should be able to determine that ? is 77 and ?? is 83.

What's interesting about this is that you probably didn't have to think about "carrying the one" at all when going from to When looking at the number line, you would never make the mistake of thinking that \(77 + 6\) is 73 (forgetting to "carry the one") or 713 (incorrectly placing the between the ones digit and tens digit). But when we teach children to add in columns, they make these mistakes with regularity. And this highlights the difference between working with adults and working with children. Many children are still developing their basic number sense, but most adults already have it, and so we can leverage that number sense into methods that feel far more intuitive for adults than children.

Activity 25.1.2. Two-Step Addition on a Number Line.

Here is the full diagram for the calculation \(37 + 46\) using a number line.

Try it!

Calculate \(39 + 27\) using a number line.

Solution.

Activity 25.1.3. Visualizing Larger Addition Calculations.

The same idea can work for larger numbers, and it's not significantly more mentally taxing.

Try it!

Calculate \(271 + 119\) using a number line.

Solution.