## Notes on Representation – The Number Line

One representation of quantity that is central in math class at many different grade levels is the number line.

Depending on the grade level, different aspects of the number line are emphasized or played down.  In the lower grades, when a number line is used, negative numbers aren’t usually shown, zero may or may not be shown, and the line itself – as opposed to the markings for each counting number – may not be shown.  Somewhere along the way, negative numbers take their place on the number line, and somewhere along the way, fractions and decimal numbers take their place.  Later still, room is found on the number line for non-repeating decimal numbers such as the ones that correspond to the square root of 5.

We could say, in the hindsight of somebody completing high school that the way the number line represent quantity is by a distance, as well as a direction, of the marker to the marker showing zero.  At this point, the notion of distance has become rich enough to allow for the idea that the distance between two particular points might not be expressible as a finite decimal number, nor as a repeating decimal number.  In particular, students have seen that a distance (or equivalently, a length) of $\sqrt{5}$ can easily be forged by drawing a right triangle where one side is 1 and the other side is 2.  The length of the diagonal is exactly $\sqrt{5}$ no matter if you ever try to measure its length using a ruler.

In earlier grades, the way the number line represent quantity isn’t necessarily as a distance.  For first or second graders, the way the number line represents quantity may well be through its position, specifically its relative position .  The marker for “4” is to the right of the marker for “3”, and to the left of the marker for “5”.  That matches what we know or are learning about the numbers 3, 4 and 5, and what we know or are learning about the quantities 3, 4 and 5.  That the markers for the numbers are also carefully evenly spaced may not be important to the student nor particularly relevant to what they are doing.  For example, if the number line is used to show the addition of 4 to 3, the student may start at 4 and move to the right: one, two, three, ending up at the marker for 7.

In later grades, the similarity of a number line and a ruler becomes more important.  A child’s familiarity with a ruler marked in inches (with typical markings for half inches, quarter inches, perhaps even eighth inches), and the availability of rulers marked in centimeters (with typical markings for millimeters) becomes important as a representation for certain fractions and certain decimals.  Though the inches ruler is not directly showing fractions like $\frac{1}{3}$ and the metric ruler is not directly showing decimals like .001, the number line can still be thought of containing all of those, in a spacing that is carefully proportional.  Kids learn how to put a marker on the number line where one isn’t already shown, for 1/7 or for .234, and this becomes important for drawing graphs in two dimensions.  Conversely, having kids draw a graph for something like $y = \frac{12}{x}$ is usually both good practice on the one hand and good evidence for kids’ understanding of number lines, rationals, graphs, and smoothly decreasing functions, on the other hand.  Depending on the curriculum, this may happen in seventh or eighth grade.  It is very interesting to see kids self-correct when the graph doesn’t appear sufficiently smooth to them, and go back to check both numbers and spacings.

Number lines, from the lowest grades on, represent which number is bigger, and continue to do so through higher grades.  After kids are introduced to negative numbers, the comparison of -10 and 7 challenges kids to come to grips with two different notions of bigger and to learn to keep them straight.  On the number line, one of these notions has to do with the distance of the marker to the zero marker (regardless of direction) and solidifies the relationship of quantity with distance; the second of these notions has to do with the direction of one marker relative to the other, and solidifies the notion that the bigger number is to the right of the smaller number regardless of where each number is located with respect to zero.

Number lines are a fairly natural way to introduce the notion of embedding without ever needing to emphasize it or make it explicit.  The way the number line is used in the lower grades allows for the new numbers (fractions, decimals, negative numbers, irrational numbers, non-repeating decimals) to be “penciled in” later, without needing to displace the counting numbers that were on the number line all along.