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B2.4 · Model and solve integer addition and subtraction problems

Learn to model and solve integer addition and subtraction problems through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Number

Model integer operations with number lines and everyday situations

Integers are whole numbers and their opposites. They include numbers below zero, zero, and numbers above zero. You may see them when describing a temperature, a change in height, or a game score. In this lesson, you will use a number line and simple situations to model adding and subtracting integers. A number line has zero at the centre, positive numbers to the right, and negative numbers to the left.

What you will learn

  • Identify positive integers, negative integers, and zero in a situation.
  • Model integer addition and subtraction with a number line.
  • Solve integer problems and explain what the answer means.

1. Understand integers in a situation

A positive integer can represent a gain or a position above a chosen starting point. A negative integer can represent a loss or a position below that point. Zero means no change from the starting point. The situation tells you what the signs mean.
For example, let sea level be the starting point. A location 4 metres above sea level is represented by +4+4. A location 3 metres below sea level is represented by −3-3. The numbers 4 and 3 tell how far from sea level; the signs tell which direction.
On a number line, moving right means the value increases. Moving left means the value decreases. Each equal step changes the value by 1. You can use this line to show both the starting value and each move.
…,−2,−1,0,1,2,…\ldots,-2,-1,0,1,2,\ldots
  • Positive integers are to the right of zero.
  • Negative integers are to the left of zero.
  • Zero is neither positive nor negative.

2. Model addition

Think of addition as joining changes. On a number line, start at the first integer. For a positive integer, move right by that many steps. For a negative integer, move left by that many steps. The place where you stop is the sum.
For example, start at −2-2 and add +5+5. Move five steps to the right. You pass zero and stop at 3. So the sum is 3. The direction and number of steps come from the second integer.
When both integers have the same sign, their moves go in the same direction. When they have different signs, their moves go in opposite directions. The number line shows why one direction can partly undo the other. You do not need a special trick: follow the moves and read the stopping point.
a+ba+b
  • For addition, start at the first integer.
  • A positive addend moves right; a negative addend moves left.
  • The final position is the sum.

3. Model subtraction

Subtraction asks for the change from the number being subtracted to the starting number, or it can describe taking away a signed amount. A useful number-line model is to ask: what move takes me from the second number to the first? The signed size of that move is the difference.
For example, to find 3−53-5, begin at 5 and ask what move reaches 3. Moving 2 places left does that, so the difference is −2-2. This gives the same result as starting at 3 and taking away 5 units.
There is another helpful way to model subtraction: subtracting a number has the same result as adding its opposite. The opposite of a number is the same distance from zero on the other side. For example, the opposite of −4-4 is +4+4. So subtracting −4-4 means adding +4+4. Use this idea to rewrite a subtraction as addition, then follow the addition moves.
Keep the sign attached to the number. In 7−(−3)7-(-3), the first minus sign tells you to subtract. The sign inside the brackets belongs to the number being subtracted. Rewriting gives 7+37+3, which equals 10.
a−b=a+(−b)a-b=a+(-b)
  • The opposite of a number is the same distance from zero in the other direction.
  • Rewrite subtraction as addition of the opposite when it helps.
  • Check that the final sign matches the direction of the net move.

4. Choose a model and check the meaning

A number line is useful when direction and movement matter. A story can also help you decide which integer to use. First name the starting amount. Then describe each change with a sign. Finally, combine the changes and explain what the answer means in the story.
For a temperature, a rise of 6 degrees is a positive change and a drop of 8 degrees is a negative change. If the starting temperature is −2-2 degrees, the new temperature is found by adding both changes to the starting value. The answer is a temperature, not just a calculation.
Before accepting an answer, ask whether its size and sign make sense. If several leftward moves outweigh the rightward moves, the result should be negative. If the moves balance, the result should be zero. You can also check subtraction by adding the difference to the number you subtracted: it should return the starting number.
b+(a−b)=ab+(a-b)=a
  • Translate each change into a signed integer.
  • Use the situation to interpret the answer.
  • Check the direction and size of the result.

Worked example

Add integers on a number line

Find −3+7-3+7. Use a number line to show the moves.
Start at −3 and move 7 steps right
Start at −3 and move 7 steps rightThe arrow ends at 4, the sum.-4-3-2-1012345+7

The arrow ends at 4, the sum.

  1. Start at the first integer
    The first integer is −3-3, so begin three steps left of zero.
    −3-3
  2. Follow the second integer
    The second integer is positive, so move 7 steps right. You cross zero and stop at 4.
    −3+7=4-3+7=4
  3. State the sum
    The number line ends at 4, so the sum is positive 4.
    −3+7=4-3+7=4
Answer: The sum is 4.
Check: There are 7 rightward steps and 3 leftward steps. The rightward movement exceeds the leftward movement by 4.

Worked example

Subtract a negative integer

Find 5−(−4)5-(-4). Show why the answer is greater than 5.
Start at 5 and move 4 steps right
Start at 5 and move 4 steps rightSubtracting −4 is represented by adding its opposite, +4.45678910+4

Subtracting −4 is represented by adding its opposite, +4.

  1. Find the opposite
    The opposite of −4-4 is +4+4. Subtracting −4-4 is the same as adding its opposite.
    5−(−4)=5+45-(-4)=5+4
  2. Model the addition
    Start at 5. Adding positive 4 means moving four steps right, so you stop at 9.
    5+4=95+4=9
  3. Interpret the result
    The result is greater than 5 because the rewritten operation adds a positive amount.
    5−(−4)=95-(-4)=9
Answer: The answer is 9.
Check: Adding the opposite of the answer to 5 is not the check needed here. Instead, use the rewrite: subtracting negative 4 becomes adding positive 4, which gives 9.

Worked example

Combine temperature changes

At dawn, the temperature is −2-2 °C. It rises by 6 °C, then drops by 9 °C. What is the temperature after both changes?
Temperature changes from −2 °C
Temperature changes from −2 °CMove 6 right to 4, then 9 left to −5.-6-5-4-3-2-1012345-3 °C

Move 6 right to 4, then 9 left to −5.

  1. Write the starting value and changes
    The starting temperature is −2-2 °C. A rise is a positive change, and a drop is a negative change.
    −2+6+(−9)-2+6+(-9)
  2. Combine the first two values
    Starting at −2-2 and moving 6 places right ends at 4.
    −2+6=4-2+6=4
  3. Apply the final change
    A drop of 9 means move 9 places left from 4. You end at −5-5.
    4+(−9)=−54+(-9)=-5
Answer: The temperature is −5-5 °C.
Check: The 9-degree drop is larger than the earlier 6-degree rise, leaving a net change of 3 degrees downward from −2-2 °C.

Common mistakes and how to avoid them

Moving left when adding a positive integer.
Correction: A positive addend moves right on the number line. A negative addend moves left.
Treating 5−(−4)5-(-4) as 5−45-4.
Correction: The number being subtracted is negative. Rewrite the subtraction as adding its opposite: 5+45+4.
Dropping the sign when writing a change in a story.
Correction: Decide whether the change is a rise or a drop, gain or loss, before calculating.

Lesson summary

  • Integers include positive numbers, negative numbers, and zero.
  • For addition, begin at the first integer and move according to the sign of the second.
  • For subtraction, add the opposite of the number being subtracted.
  • Use the story and a number-line model to check what the answer means.

Check your understanding

Question 1

What is −6+2-6+2?
  1. −8-8
  2. −4-4
  3. 44
  4. 88
Show answer and explanation
−4-4
Start at −6-6 and move 2 places right. You stop at −4-4.

Question 2

What is 3−(−5)3-(-5)?
  1. −2-2
  2. 22
  3. 88
  4. −8-8
Show answer and explanation
88
Subtracting −5-5 is the same as adding +5+5, so the result is 8.

Question 3

A diver is at −7-7 metres. The diver rises 3 metres. What is the new position?
  1. −10-10 metres
  2. −4-4 metres
  3. 44 metres
  4. 1010 metres
Show answer and explanation
−4-4 metres
Rising is a positive change: −7+3=−4-7+3=-4. The diver is still below the starting level.

Key terms

Integer
A whole number, its negative, or zero.
Opposite
A number the same distance from zero as another number, but on the other side.
Sum
The result of adding numbers.
Difference
The result of subtracting one number from another.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B2.4. It is a study resource, not an official curriculum publication.

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