B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Adding Integers

B2.4 A1About 45 minutes

Learning goals

I can use integer chips (zero pairs) to add integers.
I can use a number line to add integers.
I can write and solve addition equations with integers.
I can solve problems about temperature, elevation and money with integers.

Words to know

integer
A positive or negative whole number, or zero: ..., −3, −2, −1, 0, +1, +2, +3, ...

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

zero pair
A positive chip and a negative chip together. They cancel out: (+1) + (−1) = 0.

integer chips
Two-sided counters or two kinds of small objects used to show positive and negative numbers.

Remember from Grade 6

In Grade 6 you learned that integers are the whole numbers, their opposites, and zero. You placed them on a number line and compared them: −8 is less than −2 because it is farther left. You saw them in temperatures below zero and in elevations below sea level. In Grade 7 you will add and subtract integers.

Let's start with a cold morning

On a January morning in northern Ontario, the temperature is −12°C. By the afternoon it has gone up 9 degrees. What is the afternoon temperature?

Start at −12 on a thermometer and go up 9: −11, −10, ... , −3. The afternoon temperature is −3°C. As an equation: (−12) + (+9) = −3.

Model 1: Integer chips and zero pairs

Use one kind of chip for +1 and another kind for −1. A positive chip and a negative chip together make a zero pair: they add to 0, so you can remove them.

To add, put all the chips together, remove the zero pairs, and count what's left.

Example: (+5) + (−3). Put out 5 positive and 3 negative chips. Make 3 zero pairs. 2 positive chips are left, so the sum is +2.

Five + chips and three − chips. Each dashed loop is a zero pair. Removing the 3 zero pairs leaves two + chips, so (+5) + (−3) = +2.

Model 2: The number line

Start at the first number. Adding a positive integer moves you right (up). Adding a negative integer moves you left (down).

Start at +3. Adding −7 means moving 7 to the left. You land on −4, so (+3) + (−7) = −4.

Signs	What happens	Example
Both positive	Add the distances from zero. The answer is positive.	(+4) + (+3) = +7
Both negative	Add the distances from zero. The answer is negative.	(−4) + (−3) = −7
One of each	Make zero pairs. Subtract the distances; the answer takes the sign of the number farther from zero.	(+6) + (−2) = +4 and (−6) + (+2) = −4
Opposites	They make all zero pairs.	(−5) + (+5) = 0

Patterns you will notice when you add integers with chips or a number line.

Worked example

Example 1: Two negatives

Find (−4) + (−3).

Chips: put out 4 negative chips and 3 negative chips. There are no zero pairs.
Count: 7 negative chips.
Number line: start at −4 and move 3 to the left. You land on −7.

(−4) + (−3) = −7

Worked example

Example 2: More than two integers

A scuba diver is at −20 m (20 m below the surface of a lake). She rises 8 m, then sinks 5 m. Where is she now?

Write an equation: (−20) + (+8) + (−5).
Add from left to right: (−20) + (+8) = −12. (8 zero pairs, 12 negatives left.)
(−12) + (−5) = −17.

She is at −17 m, 17 m below the surface.

Worked example

Example 3: Group the positives and the negatives

Find (+7) + (−9) + (+6) + (−3).

Adding can be done in any order, so group the positives and the negatives.
Positives: (+7) + (+6) = +13. Negatives: (−9) + (−3) = −12.
(+13) + (−12) = +1.

Think like a mathematician

Picture it before you calculate. Ask: "Are there more positives or more negatives?" That tells you the sign of the answer. Then decide how far from zero it is. Writing the brackets, like (−4) + (−3), keeps the signs from getting mixed up with the operation.

Hands-on activity

Red and black card game

You need: a deck of playing cards with the face cards (jacks, queens and kings) removed. Aces count as 1. Black cards (clubs and spades) are positive. Red cards (hearts and diamonds) are negative. Paper and a pencil.

Each player draws 2 cards and writes an addition equation. For example, a black 7 and a red 4: (+7) + (−4) = +3.
The player with the greater sum wins the round. (Remember: −2 is greater than −9.)
After a few rounds, play again drawing 3 cards each.
If you have small objects instead (like two kinds of buttons or dried beans and pasta), model each sum with zero pairs to check.

What to look for: the correct sign on every answer, and using zero pairs or a number line to explain.

Math talk: "How can you tell the sign of the sum before you work it out?"

Just for fun

A positive chip and a negative chip walk into a party. They pair up and... poof! Zero. Zero pairs are the masters of the disappearing act.

Your turn

(+4) + (+3) = ?

(−4) + (−3) = ?

(+6) + (−2) = ?

(−6) + (+2) = ?

(−5) + (+5) = ?

(−8) + (+3) + (+7) = ?

The temperature is −12°C and rises 9 degrees. Write an equation and find the new temperature.

More practice: warm-up

(+2) + (−2) = ?

(−1) + (−4) = ?

(+7) + (−3) = ?

(−3) + (+1) = ?

(+5) + (+5) = ?

More practice: on your own

(−15) + (+9) = ?

(−7) + (−11) = ?

(+12) + (−20) = ?

(−4) + (+9) + (−6) = ?

You owe your sister $3, which you can write as −3. Then you earn $7 shovelling a neighbour's driveway. Write an equation. How much money do you have after you pay her back?

Challenge

Find the missing integer: (−9) + ? = +4.

A research submarine is at −250 m. It rises 80 m, then dives 45 m. Write one equation and find its new depth.

Jordan says, "When you add a positive and a negative integer, the answer is always negative." Is he right? Give an example.

Check your understanding

Quiz

What is (+6) + (−9)?

What is (−5) + (−8)?

What does a zero pair show?

On a number line, you start at −2 and add +5. Where do you land?

The temperature at midnight is −7°C. By noon it rises 10 degrees. What is the temperature at noon?

What is (+4) + (−6) + (+2)?

For parents and teachers

Big idea: The Ontario curriculum (B2.4) asks Grade 7 students to use objects, diagrams and equations to represent, describe and solve situations involving addition and subtraction of integers. Models (integer chips and number lines) come first, so that the patterns make sense before any rules are memorized. Multiplying and dividing integers is not part of Grade 7.

Common difficulties:

Adding the distances when the signs are different ((+6) + (−9) = 15).
Getting the sign wrong. Ask: "Are there more positives or more negatives?"
Thinking −9 is greater than −2 because 9 is greater than 2.

How to help: Play the red-and-black card game, and talk about real temperatures in winter ("It's −6°C now; it will warm up 4 degrees"). This lesson builds A1 skills: representing with models, explaining with math language, and noticing patterns.