Placing and comparing integers on a number line

When you reach for it

Any time numbers that run both ways from zero have to be pictured, read or put in order. A drawing of a volcanic island shows its summit above sea level and its foot on the sea floor below. The marks on a ship’s hull climb up from the waterline and run down beneath it. Each is a number line, laid flat or stood on its end, and the questions are always the same two: where does this number go, and which of these two numbers is the greater?

Negative numbers brought in the numbers below zero and the idea that a number to the left is less. This lesson turns both into a dependable skill for the Integers, the whole numbers 0, 1, 2, 3, \ldots together with their negatives -1, -2, -3, \ldots: placing them on lines whose ticks count in ones, twos or hundreds, flat or upright, and comparing any two of them, including two negatives, with < and >.

The procedure

The skill is taught over two days, one part each.

Part 1: placing and reading integers

Every gap is worth the same. The ticks on a number line are evenly spaced, and the amount one gap stands for, the scale, is the same everywhere on the line, on both sides of zero. That is what lets you count. With a scale of 1, the point three ticks left of 0 is -3, because “three units to the left of zero” is what -3 means. With a scale of 5, three ticks left of 0 is three steps of 5, so 15 units to the left, which is -15.

So to place an integer, find 0, find the scale, and count steps: to the right for a positive integer, to the left for a negative one. Reading a point is the same count run backwards: count the gaps from 0 to the point, multiply by the scale to get the number of units, and let the side of zero give the sign.

Finding the scale. Often only two ticks are labeled. The scale is the number of units between those two ticks shared equally among the gaps between them. When both labels lie on the same side of zero, take the smaller distance from zero away from the larger: -50 is 50 units left of 0 and -20 is 20 units left, so 30 units lie between them. When the labels lie on opposite sides of zero, the two distances add: from -6 to 9 is 6 units to reach zero and 9 more, 15 in all.

Between the ticks. A point halfway between two neighboring ticks is half a step from each. On a line with a scale of 2, there is exactly one integer between neighboring ticks, at the halfway point. If half a step is not a whole number, as with a scale of 5, the halfway point is not an integer at all; those numbers belong to negative fractions and decimals.

Choosing a scale. When you draw the line yourself, the scale is your choice. The line has to reach from the least number to the greatest. A scale that is too small needs a great many ticks; one that is too large squeezes the numbers into a few gaps. A scale that every number is a multiple of puts each number on a tick, and a scale twice that size puts some numbers halfway between ticks, which is still easy to read.

Standing the line on end. A vertical number line works the same way, with up as the positive direction: positive integers above 0, negative integers below. It suits anything measured up and down from a level, such as heights from sea level or marks above and below a waterline. Because the scale is the same on both sides of zero, a number and its opposite sit the same number of ticks from 0, one above and one below.

Part 2: comparing integers

Further right is greater. As negative numbers showed, numbers grow to the right, so of two integers the one further right is the greater; on a vertical line, higher is greater. A negative integer lies left of 0, so it is less than 0 and less than every positive integer. Of two negative integers, the one with more units between it and zero is further left, so the one further from zero is the less.

Here -8 is 8 units left of 0 and -3 is only 3 units left, so -8 is further left, and -8 < -3.

Two ways to write one fact. The statements “-8 is less than -3” and “-3 is greater than -8” describe the same two positions, read from opposite ends. In both symbols the narrow point faces the smaller number.

Neighbors, and the integers in between. Integers sit one unit apart, so each has exactly one neighbor on each side: the one just to its right is the next greater integer, and the one just to its left is the next smaller. The next integer greater than -15 is -14, one step to the right. Between two integers there are only finitely many integers, and you can list them by stepping from one end to the other. Whether the ends themselves count depends on the words: “greater than -5” leaves -5 out.

Worked examples

Examples 1 to 4 go with Part 1. Examples 5 to 7 go with Part 2.

Boundaries

Equal steps on both sides of zero. Counting ticks gives the right number only if every gap is worth the same, on both sides of zero. A line marked 0, 1, 2, 3 to the right and -10, -20, -30 to the left is not one number line, and counting across it gives wrong answers.

Integers sit on whole steps. On a line with a scale of 1, the integers are exactly the ticks, and a point between two ticks is not an integer. Numbers such as -2.5 are placed in negative fractions and decimals.

Which way is positive. “Further right is greater” and “higher is greater” rest on the usual choice: numbers grow to the right, and upward on a vertical line. A scale can be set the other way, for instance depth below the surface counted downward as positive, and then deeper means greater. Check the direction of a scale before comparing positions on it.

Greater is about position, not distance from zero. A comparison says which integer is further right. It does not say which is further from zero: -100 < -1, yet -100 is far further from zero. How far a number is from zero is a different question, answered by absolute value.

The symbols are strict. -4 < -4 is false, since no integer lies to the left of itself.

Where this goes next

The distance of an integer from zero, whichever side it is on, is its absolute value, read straight off the same line. Fractions and decimals take their places between the integers in negative fractions and decimals, and then a whole mixed list can be put in order in comparing and ordering rational numbers. A horizontal and a vertical number line crossing at their zeros make the coordinate plane, where every point is placed with two integers at once, one read along and one read up. Later, adding and subtracting integers reads each sum as a move along this line.

Exercises

Part 1: reading a number line

1. On a line with ticks one unit apart, name the integer (a) fourteen ticks left of 0, (b) nine ticks right of 0, (c) two ticks right of -5.
2. On a line with a scale of 3, name the integer on (a) the fourth tick left of 0, (b) the seventh tick left of 0.

3. Find the integers at the points P, Q, R and S on the line above.
4. Name the integer halfway between the ticks (a) -24 and -22, (b) -70 and -50.
5. Name the two integers that are neighbors of -99 on the number line.

Part 1: scales, flat and upright

6. Five equally spaced ticks are labeled -12 at the first and 12 at the last. Label the other three.
7. Two ticks are labeled -40 and -10, with two unlabeled ticks evenly spaced between them. Label those two, and say where 0 is.
8. Xia draws a number line for -35, 20, -5 and 45, with a tick every 5 from -35 to 45. How many ticks does she need, and which tick, counting from the left, is 0?
9. On a made-up volcanic island, heights are measured from sea level: a seabird nest 210 meters up, a steaming crack in the rock 120 meters up, a sunken lava flow 60 meters down and a telephone cable 180 meters down. Write each as an integer, and say how many ticks above or below 0 each sits on a vertical line with a scale of 30.
10. Calla’s boat has marks every 50 centimeters, numbered from the waterline with up as positive. A patch of barnacles is on the fourth mark below the waterline, and the boat’s name starts at the third mark above it. Write each height in centimeters.

Part 2: comparing pairs

Write < or > between each pair.

11. -19 \;\square\; {-6}
12. -40 \;\square\; {-48}
13. 7 \;\square\; {-70}
14. -24 \;\square\; 0
15. Compare -23 and -32, and write the fact both ways, once with < and once with >.
16. True or false: (a) -17 > -10, (b) -2 < -20, (c) -60 < 6.

Part 2: ordering and the integers between

17. Write from least to greatest: -16, 9, -31, 0, -5, 14.
18. Write from greatest to least: -50, -7, 12, -18, -1.
19. List every integer that is greater than -4 and less than 3.
20. Find the greatest integer less than -20, and the least integer greater than -20.
21. How many integers are greater than -14 and less than -4?
22. On the island of Exercise 9, a diving cave is at -35 meters and an old anchor rests at -52 meters. Which is lower? Write the comparison with <.
23. On a ferry, heights are marked in decimeters (tenths of a meter) from the waterline, up positive: a vent at 15, a rust spot at -2, a dent at -11 and the tip of the rudder at -27. Write the four heights in one chain from highest to lowest, using >.

Reasoning

24. Yeva says that of two negative integers, the one closer to zero is always the greater. Explain why she is right, using the number line.
25. Is there a greatest negative integer? Is there a least one? Explain.