B. Number · Number Sense: Numbers to One Million, Integers, Decimals and Fractions

Integers: Numbers Below Zero

B1.2 A1About 40 minutes

Learning goals

I can explain what positive and negative integers are.
I can read and write integers like −7 and +3.
I can show integers on horizontal and vertical number lines.
I can find the opposite of an integer.
I can describe where integers are used in real life, like temperature and elevation.

Words to know

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

positive integer
An integer greater than zero, like 5 or +5.

negative integer
An integer less than zero, like −5. Read it "negative five".

zero
The integer that is neither positive nor negative.

opposites
Two integers the same distance from zero on opposite sides, like −4 and 4.

sea level
The height of the surface of the ocean. Heights are measured up or down from sea level, which counts as 0 m.

A cold January morning

It's a January morning in Thunder Bay. Before school, the thermometer outside the kitchen window shows 6 degrees below zero. By the afternoon it has warmed up to 4 degrees above zero.

How do we write a temperature that is below zero? We use a negative sign: −6 °C. We read it as "negative six degrees Celsius". The afternoon temperature is 4 °C (we could also write +4 °C).

A snowy street on a frosty, sunny winter morning, with chimney smoke rising and a lake in the distance. — On a cold January morning in Northern Ontario, the temperature is often below zero. We write those temperatures with a negative sign, like −6 °C. AI-generated image

A vertical number line, like a thermometer. −6 °C is 6 steps below zero. 4 °C is 4 steps above zero.

What are integers?

Integers are the counting numbers, their opposites, and zero:

…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …

Positive integers are greater than zero: 1, 2, 3, … We can write them with or without a plus sign: 3 or +3.
Negative integers are less than zero: −1, −2, −3, … They always need the negative sign.
Zero is neither positive nor negative.

Integers never have fractions or decimals. So −6 is an integer, but 2.5 and −1/2 are not.

A horizontal number line. Negative integers are to the left of zero, positive integers to the right. −4 and 4 are opposites: each is 4 steps from zero.

Opposites

Two integers are opposites if they are the same distance from zero on opposite sides. The opposite of 4 is −4. The opposite of −9 is 9. The opposite of 0 is 0.

Opposite words help you choose the sign: above/below, gain/lose, up/down, deposit/withdraw. Above, gain, up are usually positive. Below, lose, down are usually negative.

Situation	Integer
5 degrees below zero	−5 °C
A plane flying 3000 m above sea level	+3000 m (or 3000 m)
A diver 12 m below the surface of the ocean	−12 m
Parking level 2 below the ground floor	−2
A hockey team scores 3 more goals than it allows	+3
Sea level	0 m

Integers in everyday life.

Worked example

Example 1: Read and write integers

Write each as an integer: (a) 8 degrees below zero (b) 150 m above sea level (c) losing 6 points in a game.

"Below" means less than zero: (a) is −8 °C.
"Above" means greater than zero: (b) is +150 m (or 150 m).
"Losing" points goes down: (c) is −6.

(a) −8 °C (b) +150 m (c) −6

Worked example

Example 2: Use a vertical number line

In an office building, the ground floor is 0. Floors above ground are positive and parking levels below ground are negative. Lin parks on level −3. Her office is on floor 5. Show both on a vertical number line.

Draw a vertical line with 0 at the ground floor.
Count 5 floors up for her office: mark 5.
Count 3 levels down for the parking: mark −3.

The office is 5 above zero, and the parking is 3 below zero. Up is positive; down is negative.

A vertical number line for a building. Higher is greater; lower is less.

Worked example

Example 3: Integer tiles

Show −3 and +2 with two kinds of objects.

Choose one kind for positive (for example, buttons) and one for negative (for example, dry beans). Keep small objects like these away from little brothers and sisters.
−3 is 3 beans. +2 is 2 buttons.
One button and one bean together make a zero pair: +1 and −1 cancel out to make 0.

−3 = 3 negative objects. +2 = 2 positive objects. A positive and a negative object together make zero.

Stuck? Picture it

If an integer question feels confusing, sketch a quick number line or thermometer. Seeing where the number sits makes most questions easier. Feeling stuck is normal; try a picture, then try again.

Just for fun

Why did the car on parking level −3 look so gloomy?

It was feeling a little negative. Then it drove up to floor 5 and cheered right up!

Hands-on activity

Weather watch on a vertical number line

You need: a long strip of paper (or tape stuck on a door frame), a pencil, a ruler and a weather forecast (radio, TV, newspaper or a website with a grown-up).

Draw a vertical number line from −20 at the bottom to +20 at the top. Make tick marks 1 cm apart and label every 5: −20, −15, −10, −5, 0, 5, 10, 15, 20.
Find the forecast low and high temperatures for today or for the next 5 days in your area.
Mark each one on your line and label it with the day.
Answer: Which day is the coldest? Which is the warmest? Are any temperatures opposites?

If it's a warm month, use these made-up numbers: −14, −9, −2, 0, 3. (In late fall, winter or early spring, you could also check the forecast for a Northern community such as Iqaluit or Inuvik.)

Math talk: Why is −14 colder than −9, even though 14 is bigger than 9?

Your turn

Write as an integer: 12 degrees below zero.

Write as an integer: a gain of 7 points.

What is the opposite of −15?

Is 0 positive, negative or neither?

A hiker is at 400 m above sea level. A submarine is at 400 m below sea level. Write both as integers. What do you notice?

Which of these are integers: −8, 3.5, 0, 17, −1/2?

On a vertical number line, is −2 above or below 0? How many steps?

More practice: warm-up

Read −9 out loud.

Write as an integer: 3 floors up from the ground floor.

What is the opposite of 6?

Is −20 positive or negative?

What integer stands for sea level?

More practice: on your own

Write as an integer: a temperature drop to 25 degrees below zero.

A whale dives to 200 m below the surface. An eagle flies 200 m above the surface. Write both integers and explain how they are related.

On a horizontal number line, −7 is how many steps from zero? Which side?

Describe a real situation for −4.

Use objects: buttons = positive, beans = negative. You have 2 buttons and 2 beans. What integer is that? Why?

Challenge

A thermometer reads −3 °C. It warms up by 5 degrees. Then it cools down by 8 degrees. Use a vertical number line. What does it read now?

Two integers are opposites. They are 18 steps apart on a number line. What are they?

Write the integer shown by 5 beans (negative) and 3 buttons (positive). Explain with zero pairs.

Check your understanding

Quiz

How do you write 10 degrees below zero?

What is the opposite of −8?

Which number is not an integer?

On a vertical number line, where is −4?

A diver is 6 m below sea level. Which integer shows this?

Zero is...

For parents and teachers

Big idea: B1.2 asks Grade 6 students to read and represent integers using tools such as horizontal and vertical number lines. In Grade 6, integers are about understanding and representing; adding and subtracting integers comes in later grades.

Common difficulties:

Reading the sign. Encourage "negative six" rather than "minus six" when talking about the number −6. ("Minus" means subtract.)
Thinking −8 is bigger than −3. This comes up in the next lesson; a thermometer helps: −8 °C is colder.
Forgetting zero. Zero is an integer, but it is neither positive nor negative.

How to help: Canadian winters are perfect for integers! Read the outdoor temperature together, talk about how many degrees below zero it is, and check the forecast. Elevators and parking levels (P1, P2) are also great examples of vertical number lines.