B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Rational Numbers: Positive and Negative Fractions and Decimals

B1.3About 40 minutes

Learning goals

I can explain what a rational number is.
I can read and write negative fractions and decimals.
I can place positive and negative fractions and decimals on a number line.
I can use rational numbers to describe real situations, like temperatures below zero.

Words to know

rational number
Any number that can be written as a fraction with an integer on top and a non-zero integer on the bottom. Examples: 3/4, −2.5, 7, −1 1/3.

integer
A whole number or its opposite: …, −3, −2, −1, 0, 1, 2, 3, …

opposites
Two numbers the same distance from 0 on opposite sides, like 3/4 and −3/4.

negative number
A number less than 0. It is written with a − sign, like −0.75.

Remember from Grade 6

In Grade 6 you learned about integers: whole numbers and their opposites, like −3 and 3. You placed them on horizontal and vertical number lines (temperatures, floors in a building). You also worked with positive fractions and decimals to thousandths.

But real life has numbers like −4.5°C and −3/4. These are negative numbers that are between the integers. Today you will put all of these together.

A chilly morning

On a January morning, the temperature in Thunder Bay is −4.5°C. That is four and a half degrees below zero. It is between −5°C and −4°C.

−4.5 is not an integer, but it is still a number we can show on a number line. It belongs to a big family called rational numbers.

Snow-covered rocks and ice along the shore of Lake Superior on a frosty winter morning, with steam rising from the water. — On a winter morning near Thunder Bay, the temperature can be a number like −4.5°C: a rational number between −5 and −4. AI-generated image

What is a rational number?

A rational number is any number that can be written as a fraction a/b, where a and b are integers and b is not 0.

Positive fractions: 3/4, 5/2
Negative fractions: −3/4, −1 1/2 (which is −3/2)
Decimals: 0.6 = 6/10, −2.25 = −9/4, and repeating decimals like 0.333… = 1/3
Integers: 7 = 7/1, −2 = −2/1, 0 = 0/1

So all the numbers you have used so far, whole numbers, integers, fractions and decimals, positive or negative, are rational numbers.

Every whole number is an integer, and every integer is a rational number. Fractions and decimals like 3/4 and −0.75 are rational but not integers.

Four rational numbers on a number line: −1 1/2, −0.25, 3/4 and 1.6. Each small tick is one quarter.

Reading and writing negative fractions and decimals

−0.75 is read "negative seventy-five hundredths" or "negative zero point seven five".
−3/4 is read "negative three-quarters". It equals −0.75.
−1 1/2 is read "negative one and one-half". It equals −1.5.

On a number line, a negative number is to the left of 0 (or below 0 on a vertical number line, like a thermometer). Opposites are the same distance from 0: −3/4 and 3/4 are each three-quarters of a unit from 0.

Situation	Rational number
4.5°C below zero	−4.5
A deposit of $12.75 in a bank account	+12.75 (or 12.75)
A withdrawal of $12.75	−12.75
2 1/4 m below the surface of a lake	−2 1/4
A rise of 3/4 of a degree	+3/4

Positive and negative rational numbers describe opposite directions.

Worked example

Example 1: Place numbers on a number line

Where do −1.2, −2/3 and 0.4 go on a number line?

−1.2 is between −2 and −1. It is 0.2 past −1, so it is just a little to the left of −1.
−2/3 is between −1 and 0. Split that space into thirds; −2/3 is two-thirds of the way from 0 to −1.
0.4 is between 0 and 1, a little less than halfway (0.5).

From left to right: −1.2, −2/3, 0.4.

Worked example

Example 2: Change the form

Write −3/4 as a decimal and −1.5 as a mixed number.

3/4 = 0.75, so −3/4 = −0.75. The sign stays the same.
1.5 = 1 5/10 = 1 1/2, so −1.5 = −1 1/2.

−3/4 = −0.75 and −1.5 = −1 1/2

Worked example

Example 3: Between which integers?

Between which two integers is −13/4?

13/4 = 3 1/4 (13 ÷ 4 = 3 R1).
So −13/4 = −3 1/4, which is 1/4 to the left of −3.

−13/4 is between −4 and −3, closer to −3.

The sign goes with the whole number

−1 1/2 means −(1 + 1/2), so it is −1.5, not −0.5. The negative sign applies to the whole mixed number.

Hands-on activity

A number line on the floor

You need: painter's tape or string, 7 sticky notes or small paper cards, and a marker.

Make a long number line on the floor with tape or string. Mark −3, −2, −1, 0, 1, 2, 3 about one big step apart.
Write these on cards: −2.5, −3/4, 1/2, −1 1/4, 0.8, 2.25, −0.1.
Place each card where it belongs. Say aloud which two integers it is between.
Pick a card and stand on it. Your grown-up stands on its opposite. Are you the same distance from 0?
In winter, check an outdoor thermometer or the weather forecast. Find a temperature with a decimal, like −7.4°C, and show where it goes.

What to look for: −2.5 is halfway between −3 and −2; −1 1/4 is just left of −1; −0.1 is very close to 0 on the left side.

Just for fun

What did 3/4 say to −3/4?

"We're total opposites... but we're exactly the same distance from zero, so let's be friends!"

Your turn

Write a rational number for 4.5°C below zero.

What is the opposite of −2/5?

Is −7 a rational number? Explain.

Between which two integers is −2.6?

Write −3/4 as a decimal.

Write −1.5 as a mixed number.

What number is halfway between −1 and 0?

More practice: warm-up

What is the opposite of 0.8?

Write a rational number for a drop of 2 1/4 m.

On a horizontal number line, is −0.5 to the left or right of 0?

Write −1/2 as a decimal.

Write 3 as a fraction.

More practice: on your own

Between which two integers is −13/5?

Which is farther from 0: −0.9 or 0.7? Explain.

Write rational numbers for: a $40 deposit and a $12.75 withdrawal.

On a thermometer from −5°C to 5°C, is −2.5°C above or below −3°C?

Write −0.125 as a fraction.

Challenge

Is −0.333… (the 3s go on forever) a rational number? Explain.

A point is 3/4 of the way from −2 to −1. What number is it?

A diver is 4.2 m below the surface of a lake. A loon is flying 3 1/2 m above the surface. Write each position as a rational number. How far apart are they?

Check your understanding

Quiz

Which rational number means 3.5 degrees below zero?

What is −3/4 as a decimal?

What is the opposite of −1 1/2?

−2.3 is between which two integers?

Which statement is false?

On a horizontal number line, where is −0.6?

For parents and teachers

Big idea: Grade 7 expectation B1.3 asks students to read, represent, compare, and order rational numbers, including positive and negative fractions and decimal numbers to thousandths, in various contexts. In Grade 6 your child learned integers and positive fractions and decimals. The new part is negative fractions and decimals, and the idea that all these numbers belong to one family.

Common difficulties:

Treating −1 1/2 as −1 + 1/2 (= −0.5). The sign belongs to the whole mixed number: −1 1/2 = −1.5.
Placing −1.2 between −1 and 0. It is past −1, between −2 and −1.
Thinking only fractions are rational. Integers and decimals are rational too.

How to help: Make the floor number line. Winter weather is full of negative decimals; read forecasts together. Comparing and ordering rational numbers is the next lesson.