Chapter 6 · Rational numbers
Reading and Choosing a Scale on a Number Line
Use the printed numbers on a number line to work out each step and name the number a point shows. Then show real-world amounts on a line you scale yourself.
You already know how to graph fractions and decimals on a number line. Now you will go the other way: a point is marked, and you name the number it shows. Then you will set up a line of your own for real-world amounts.
A number line has short marks called ticks. Each tick stands for one number. The move from one tick to the next is one step. On every line in this lesson the ticks are evenly spaced, so each step on a line is worth the same amount. That amount is the step size, also called the scale of the line. Here no step size is smaller than one hundredth ().
Key concept
Printed numbers give the scale; steps from zero give the value
A number describes its tick in two ways. Its sign tells you which side of the tick is on. The number without its sign tells you how far the tick is from . Zero has no sign: it is the point you count from. Any other number written with no sign is positive.
On a line whose ticks are evenly spaced, you can find both, as long as is a tick and the point sits exactly on a tick:
- Step size: pick two printed numbers that are next to each other. Divide how far apart they are by the number of equal steps between them.
- Distance from zero: count the equal steps from to the point. Multiply the number of steps by the step size.
- Sign: on a horizontal line where numbers grow to the right, right of is positive and left of is negative.
Worked example
Reading a point between two printed numbers
The number line above prints , , , and . Its ticks are evenly spaced, and Point A sits exactly on a tick between and . What number does Point A show?
Find the two printed numbers on either side of Point A. They are −1 and 0. The distance between them is 1, because −1 is exactly 1 to the left of 0.
Count the equal steps from −1 to 0. Count the spaces between ticks, not the ticks themselves: there are 4. Because the ticks are evenly spaced, the distance of 1 is shared equally among the 4 steps, so divide to get the step size. Multiplying back checks it: 4 steps of 0.25 make 1.
Count the equal steps from 0 to Point A. It is 3 steps away. Multiply the number of steps by the step size to get its distance from 0. Because 0.25 and one fourth are the same step size, 3 steps give the same distance either way: 0.75, or three fourths.
Decide the sign. Point A is to the left of 0, and on this line numbers to the left of 0 are negative. So Point A is the number that is 0.75 to the left of 0.
Check from the other printed number. The tick for −1 is 4 steps left of 0. Point A is 1 step to the right of −1, so it is 3 steps left of 0. Both ways of counting agree.
Answer
Point A shows , which is the same number as .
A step size is not always . It can be large or tiny, and you cannot tell which from how a line looks. Only the printed numbers decide.
The next line is zoomed in. It shows only the part of the number line from to . Its ticks are evenly spaced, is one of its ticks, and both marked points sit exactly on ticks. So the same three moves work: step size, distance from , sign.
Worked example
Reading two points on a zoomed-in line
The number line above shows only the part from to . It prints , , , and , and its ticks are evenly spaced. Point B is left of and Point C is right of . Each sits exactly on a tick. What numbers do Point B and Point C show?
Find the step size from two printed numbers that are next to each other: 0 and 0.1. They are 0.1 apart, with 5 equal steps between them. Think of 0.1 as 10 hundredths. Sharing 10 hundredths among 5 steps gives 2 hundredths each, which is 0.02. A step of 0.01 cannot be right: 5 such steps reach only 0.05, but the line prints 0.1 there.
Read Point B. It is 7 equal steps from 0. Multiply 7 by the step size to get its distance from 0: 7 groups of 2 hundredths make 14 hundredths. Point B is to the left of 0, so its number is negative.
Check Point B from the printed −0.1, which is 5 steps left of 0. Point B is 2 more steps to the left, so it is 5 + 2 = 7 steps from 0. Its distance from 0 is 0.1 plus 2 more steps of 0.02.
Read Point C the same way. It is 3 equal steps from 0, and it is to the right of 0, so its number is positive.
Answer
Point B shows and Point C shows .
Many real-world amounts come in two opposite directions, such as above and below zero. A number line with negative numbers can show both directions at once.
Before you draw one, decide three things:
- What zero means. It is the reference point: the place that both directions are measured from.
- Which direction is positive. Its opposite is then negative. On the horizontal lines in this lesson, positive amounts go to the right of and negative amounts go to the left.
- What the numbers count. Name the unit, such as degrees Celsius or dollars.
The table shows these choices for three situations.
| Situation and unit | What means | Positive, right of | Negative, left of |
|---|---|---|---|
Temperature, in degrees Celsius | degrees Celsius, the temperature at which water freezes | degrees above zero | degrees below zero |
Height compared with sea level, in feet | exactly at sea level | feet above sea level | feet below sea level |
Money in an account, in dollars | nothing in the account and nothing owed | dollars in the account | dollars owed |
Worked example
Showing two temperatures on one line
At 6 a.m. the temperature in a town is degrees Celsius below zero. At 2 p.m. it is degrees Celsius above zero. Represent both temperatures on one number line.
Say what 0 and each direction mean. Here 0 stands for 0 degrees Celsius, the temperature at which water freezes. It does not mean that there is no temperature. Above zero is positive, to the right of 0. Below zero is negative, to the left of 0. The numbers on the line count degrees Celsius.
Write each temperature as a positive or negative number. The number of degrees away from zero gives the distance from 0. The direction gives the sign: below zero is negative, and above zero is positive.
Choose a step size. Each point must sit exactly on a tick, and 0 must be a tick too. Reading a point multiplies: steps times step size gives the distance from 0. Placing a value undoes that with division. So each distance from 0, divided by the step size, must be a whole number of steps. A step of 1 fails, because 2.5 steps would end between two ticks.
Look for a step size that both distances are built from. The distance 7.5 is 3 groups of 2.5, and the distance 2.5 is 1 group of 2.5, so test a step of 2.5. Both divisions give whole numbers, so this step works. A step of 0.5 would also work, but the same line would need 5 times as many steps.
Choose the ends so both values fit, with room to spare. Start at −5, which is 2 steps left of 0, and end at 10, which is 4 steps right of 0. Print a number at every second tick: −5, 0, 5 and 10.
Place each point by counting equal steps from 0: 1 step left for −2.5 and 3 steps right for 7.5. Multiplying the steps by the step size gives back 2.5 and 7.5, so the divisions were right. On the line below, the point at −2.5 is halfway between the printed −5 and 0, and the point at 7.5 is halfway between the printed 5 and 10.
Answer
On a line from to with a step size of , the 6 a.m. temperature is the point at , one step left of . The 2 p.m. temperature is the point at , three steps right of .
Here is the idea to keep. On a number line with evenly spaced ticks, two printed numbers that are next to each other tell you the step size. When is a tick and a point sits exactly on a tick, count the equal steps from . The steps times the step size give the point's distance from , and its side of gives its sign.
To show a real-world amount, run the same idea backwards. Decide what means and which direction is positive. Write the amount as a positive or negative number. Then choose a step size that makes its distance from a whole number of steps, so the point lands exactly on a tick.