# The Number Line - subtraction, and measurement

## The number line in mathematics

The number line is not just a school object. It is as much a mathematical idea as functions. Unlike the Number Line Hotel, hundreds charts, Cuisenaire rods, and base ten blocks, the number line is not just a pedagogical aid used only to help children learn; mathematicians refer to it, too.

The number line is a geometric “model” of all real numbers -- including 0 1, 2, 25, 374 trillion, and -5, but also 1/2, –17.359, 0.0000000000000001, and pi/6. Unlike counters, which model only counting, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measure using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.

Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number line model as well as counters.

## The number line in teaching mathematics

One reason to use this mathematical object with students is that they need to see arithmetic in both contexts: counting and measuring. At the beginning, children may sometimes use the number line to find answers to arithmetic problems (e.g., figuring out what 3 + 10 is, before that becomes automatic to them) but that is never its purpose. We don’t rely on the number line for getting answers -- for that, we want the children to know basic facts and methods and use their heads -- but we do use the number line to understand things about the operations (addition and subtraction) and to understand what the answers mean.

For example, the answer to the subtraction problem 92 – 49 is the distance between those two numbers on the number line. That image can greatly help mental computation: 49 to 50 is one step, 50 to 90 is another forty steps, and 90 to 92 is another two steps, so altogether 43 steps. (See "Number line in addition and subtraction" below.) It also makes arithmetic with negative numbers a snap when students later learn about those numbers. And the number line is essential for full understanding of fractions and decimals. In fact, a ruler (a number line!) is one of the important places students encounter and need to use fractions.

### In counting

Number lines are first used just to show sequence—numbers standing on line in order! At this stage, neither the straightness of the line nor the distance between numbers is mathematically important, though our images are always standard anyway. Some of the strategy we used was drawn from the work of Frédérique Papy, whose pre-K and K students would even draw these in spirals and zig-zags. Children will look at chunks of the line, not always starting at 1, and will work forwards or backwards from some number that is placed on the line. They are learning about sequence and order, and that develops somewhat independently from counting. As you know (and I mentioned over the phone to assure you that I knew, too) kids are often able to recite the numbers—“counting,” as their parents might say—before they can actually count objects correctly, one-to-one.

### In measurement

K students also learn about intervals on the number line, but just begin that process. (I have only grades 1-5 with me, so I can’t say where in K.) That takes time, and is harder to develop than it was forty (or so) years ago. Kids used to gain the interval idea (in a slightly different form) from their experience playing board games. They knew that when they rolled a 5, they had to count their five spaces beginning with the next space. That is, they were counting moves, not positions. Today’s children—at all SE levels, to only slightly different degrees—have had far less board-game experience, so they need to get that idea in school. Measurement depends on it. Addition and subtraction with counters does not depend on it, but those operations on the number line do depend on it.

If we can add and subtract with counters, why use the number line? To connect these operations with measurement, and also because the counters no longer suffice when we get to fractions, decimals, and negative numbers. Over time, kids will connect number line images with thermometers, clocks, rulers (with fractional inches)… Coordinate graphs are based on perpendicular number lines; even bar graphs require the measurement idea more than the count idea, although they can begin with count.

Addition and subtraction, or comparison, of distance is also why we use Cuisenaire rods instead of Unifix cubes. (Unifix cubes emphasize counting—one can’t know relationships among Unifix rods without counting because there are no fixed lengths.) Part-part-whole in red rod + green rod = yellow rod is easy to see without caring what numbers they represent. And if R + G = Y then Y – R = G. (See, for example, Gr 2, Ch 8, Lesson 1, but kids see this in Gr 1 as well.) Adding distance is further developed on open number lines. Students develop many ways to subtract (in second grade, they learn to subtract 8 from anything by subtracting 10 and then compensating, and then they extend that idea to other additions and subtractions). In Grade 2, Ch 2, Lesson 8, they develop the idea on the number line

and in third grade, they develop it further and use it to subtract much larger numbers.

Its value as a model is that it continues to work for negative numbers as well.

### And in decimals and fractions

Locating fractions on the number line has been noted by several mathematicians and researchers as essential for understanding them as numbers and not just parts of pizza.

Students need to learn to locate a fraction on a number line.

Many misconceptions about the meaning of "one half" become apparent when students are asked to locate it on a number line. As a number it is less than 1 and greater than 0, in fact "half" way between them, but it is quite common for students, the first time they are asked to place it on the number line, to locate it between 1 and 2, possibly because those are the two digits used in writing the symbol '1/2' or possibly because of the commonly-taught strategy of thinking of 1/2 as meaning "one out of two." Students who know about negative numbers before first encountering the question "where would 1/2 fit on the number line" sometimes place it exactly on top of 0, because they see the zero as the mark that divides the number line "in half." These are, respectively, strategies based on how the number is written, and how a fraction represents a "part of" a geometric object, both of which tend to be taught earlier and more intensively than messages about fractions being numbers, in their own right, and having magnitude and order (like all other real numbers).

Addition and subtraction with fractions can also benefit from a number line image. The calculation $10 - 2\frac 1 2$ can be represented, as can any other subtraction problem, as distance on a number line.

Having a clear sense of what the computation means makes it easier to interpret, and solve mentally, problems like the following, without the baggage of converting mixed numbers to improper fractions and converting back.

The object is not merely to make the computation easier, nor is it to render obsolete (which it does not) the converting-mixed-numbers-to-improper-fractions method, but to make clear what the computation means so that one can better understand the algorithm in those cases where it is necessary because the numbers are, without conversion, too hard to manage.

### Developmental issues

In Grade 1 (Chapter 2, lesson 1), students “record jumps” on a number line. The image

presents considerable cognitive challenge to kids. What stands out visually is the dots, not the spaces between them. Some children see the arrows as skipping “4 dots” because they encompass four, or skipping “2 dots” because the skip over two. Seeing the intervals is harder, and takes time, but is essential for measurement. Many children will not be secure experts at the end of this lesson, but observation shows that even 6th graders who have not had this experience before will take a while to get it. It is not the only reason children typically do poorly on measurement tasks on tests—a major reason is that they don’t get time doing real things that require measuring—but this appears to be one important reason.

In Grade 3 of Think Math!, students “zoom out” on the number line (Chap 4 Lesson 7), looking at large numbers, and setting the stage for “zooming in” on the number line (Gr4, Ch8 L3).