Use a Vertical Number Line

Integers Made Easy: Use a Vertical Number Line.

Many students have first learned integers using a number line like the one shown below. It's actually not the kind I prefer, but it's probably familiar.

This is the most common kind of number line.

It is useful to notice that negative numbers are to the right of zero, and positive numbers are to the left. However, my students seem to better understand the relationship between positive and negative numbers when the number line is turned vertically (up and down).

A vertical number line is easier for many students.

This is a bit easier to understand because it looks more like an important application of integers in real life: a thermometer. If the temperature is -3, we say it is 3 degrees below 0. On this line, the negative numbers are literally below zero, and the positive numbers are above 0. Positive numbers can go up forever and negative numbers can go down forever.

Thinking about positive numbers as up also helps us think about elevation (height) of a place. Elevation is a very common application for integers. Positive elevations are above sea level, and negative elevations is below sea level. Sea level is 0. So the surface of the ocean is at sea level (zero). A mountain might 200 meters above sea level (positive 200). In some places, like New Orleans, the elevation might be negative (like -6), because they are actually below sea level.

It is very important to realize that zero is neither positive nor negative: it is neutral. This is easy to remember by visualizing the number line. Zero isn't negative because negative numbers are below 0. (Zero isn't below itself.) Zero isn't above 0, so it isn't positive either. It's just plain 0.

Comparing Integers

This is easy if you remember your vertical number line. Numbers that are greater are higher on the number line (or warmer on the thermometer). Positive 4 is greater than -4 because +4 is higher. Zero is greater than all of the negative numbers because 0 is higher than all of them. Positive 3 is greater than 2, but -2 is actually greater than -3. If you look at the vertical number line, -2 will actually be higher up than -3.

Numbers that are less are lower on the number line (or colder on the thermometer). So one is less than four because +1 is lower than +4. On the other hand, -4 is less than -1 because -4 is lower. All of the negative numbers are less than zero.

In math, we write greater and less with < and > signs.

4 > -4

0 > -3

3 > 2

-2 > -3

1 < 4

-4 < -1

-3 < 0