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Teacher Guide: Adding and Subtracting Integers

Learn how to add and subtract positive and negative numbers using number line strategies and rules.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Negative Numbers. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Add two integers with the same sign
  • Add two integers with different signs
  • Subtract integers by adding the opposite
  • Apply integer operations to real-world contexts
  • Explain why subtracting a negative results in addition
Prerequisites
  • • Understanding of positive and negative numbers
  • • Ability to locate integers on a number line
  • • Knowledge of absolute value
  • • Comparing integers using inequality symbols
Discussion Starters
  • 1. Why does 'subtracting a negative' feel counterintuitive at first? How can we make it make sense?
  • 2. Can you think of a situation where you'd need to add two negative numbers?
  • 3. If someone says 'two negatives make a positive,' is that always true? When does it apply?
  • 4. How would you explain 5−8=−35 - 8 = -3 to a younger student using a story?
Common Misconceptions

Thinking 'two negatives always make a positive'

Remediation: Clarify that this rule only applies when multiplying or dividing, or when subtracting a negative. (−3)+(−5)=−8(-3) + (-5) = -8, not positive!

Confusing the operation sign with the number's sign

Remediation: Use parentheses consistently. Write 5+(−3)5 + (-3) not 5+−35 + -3. Show that the ++ is the operation and −3-3 is a negative number.

Forgetting to apply the sign rule when subtracting

Remediation: Always convert to addition first. Drill: 'Keep, Change, Change' - keep the first number, change subtraction to addition, change the sign of the second number.

Differentiation Ideas

For Struggling Students:

  • • Use physical number lines and counters/chips
  • • Start with single-digit integers only
  • • Focus on one rule at a time (same signs first, then different signs)
  • • Use the 'KCC' method: Keep-Change-Change for subtraction

For On-Level Students:

  • • Work with two-digit integers
  • • Solve multi-step problems with mixed operations
  • • Create their own real-world scenarios
  • • Practice without number line scaffolding

For Advanced Students:

  • • Work with three or more integers in one expression
  • • Explore integer operations with decimals
  • • Solve equations involving integer addition/subtraction
  • • Investigate patterns in integer sums
Standards Alignment
  • 7.NS.A.1 (CCSS.MATH.CONTENT.7.NS.A.1)

    Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers

  • 7.NS.A.1.B (CCSS.MATH.CONTENT.7.NS.A.1.B)

    Understand p + q as the number located a distance |q| from p, in the positive or negative direction

  • 7.NS.A.1.C (CCSS.MATH.CONTENT.7.NS.A.1.C)

    Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q)

Lesson Resources
  • visualInteractive Number Line

    Students practice moving left (subtract) and right (add)

  • activityInteger Chip Model

    Use two-color chips to model adding and subtracting

  • gameInteger War Card Game

    Students draw cards and compute sums/differences

  • worksheetReal-World Integer Problems

    Temperature, elevation, and money scenarios

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Adding integers means combining positive and negative numbers. Subtracting integers means finding the difference, which we can rewrite as adding the opposite.
The absolute value ∣a∣|a| is the distance of aa from zero: ∣−5∣=5|-5| = 5 and ∣5∣=5|5| = 5.

Key Rules

Adding Integers

  • Same signs: Add the absolute values, keep the sign
  • 5+3=85 + 3 = 8 (both positive)
  • (−5)+(−3)=−8(-5) + (-3) = -8 (both negative)
  • Different signs: Subtract the smaller absolute value from the larger, keep the sign of the larger
  • 5+(−3)=25 + (-3) = 2 (positive wins)
  • (−5)+3=−2(-5) + 3 = -2 (negative wins)

Subtracting Integers

  • Subtracting is adding the opposite: a−b=a+(−b)a - b = a + (-b)
  • 5−3=5+(−3)=25 - 3 = 5 + (-3) = 2
  • 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8
  • (−5)−3=(−5)+(−3)=−8(-5) - 3 = (-5) + (-3) = -8

Worked Examples

Calculate (−4)+(−7)(-4) + (-7)
1

Identify the signs

Both numbers are negative → Same signs
2

Add the absolute values

∣−4∣+∣−7∣=4+7=11|-4| + |-7| = 4 + 7 = 11 → Sum is 11
3

Apply the common sign

Both negative, so result is negative → −11-11

Common Mistakes

Thinking 5−(−3)=25 - (-3) = 2 instead of 88

Why it's wrong: Forgetting that subtracting a negative number means adding its opposite: the subtraction sign and the negative sign together become a plus.

Correct: 5−(−3)=5+3=85 - (-3) = 5 + 3 = 8. Subtracting −3-3 is the same as adding 33.

Writing (−4)+(−6)=−2(-4) + (-6) = -2 instead of −10-10

Why it's wrong: Mistakenly subtracting instead of adding when both numbers are negative.

Correct: Same signs means ADD absolute values: 4+6=104 + 6 = 10, then keep the negative: −10-10.

Getting the wrong sign when subtracting like 3−8=53 - 8 = 5

Why it's wrong: Forgetting to check which number has the larger absolute value.

Correct: 3−8=3+(−8)=−53 - 8 = 3 + (-8) = -5 because 8>38 > 3 and we keep the negative sign.

Confusing −5+3-5 + 3 with −5−3-5 - 3

Why it's wrong: Not carefully reading the operation sign.

Correct: −5+3=−2-5 + 3 = -2 (moving right on number line), but −5−3=−8-5 - 3 = -8 (moving left).

Why It Matters

Integer operations are essential in everyday life:
  • Temperature changes: If it's −3°C-3°C and drops 5 more degrees: −3+(−5)=−8°C-3 + (-5) = -8°C
  • Money and debt: You owe 50 dollars and borrow 30 more: (−50)+(−30)=−80(-50) + (-30) = -80, a debt of 80 dollars
  • Elevators: From floor 2, going down 5 floors: 2+(−5)=−32 + (-5) = -3 (basement level 3)
  • Football: A team gains 8 yards then loses 12: 8+(−12)=−48 + (-12) = -4 yards total
Understanding these operations is the foundation for algebra!

Real World Applications

Banking Transactions

Banks use integer operations to track deposits (positive) and withdrawals (negative).

Example:

If your account has 120 dollars and you withdraw 85 dollars, then deposit 40 dollars: 120+(−85)+40=75120 + (-85) + 40 = 75 dollars.

1Try It Yourself

Your bank account shows −45-45 dollars (overdraft). You deposit 80 dollars.

What is your new balance?

Step 1: Write the mathematical expression

Calculate: −45+80-45 + 80

Altitude and Depth

Pilots and divers use integers to track altitude above sea level (positive) or depth below (negative).

Example:

A submarine at −80-80 meters dives 25 meters deeper: −80+(−25)=−105-80 + (-25) = -105 meters.

2Try It Yourself

A submarine is at −120-120 meters. It rises 75 meters toward the surface.

What is the new depth?

Step 1: Write the mathematical expression

Calculate: −120+75-120 + 75

Game Scores

Many games track gains and losses as positive and negative points.

Example:

In a trivia game, you earn 50 points, lose 30, then earn 45: 50+(−30)+45=6550 + (-30) + 45 = 65 points.

3Try It Yourself

Your game score is −15-15 points (in the negative). You answer correctly for 40 points, then incorrectly and lose 10.

What is your final score?

Step 1: Write the mathematical expression

Calculate: −15+40+(−10)-15 + 40 + (-10)

Key Takeaways

  • 1Same signs: Add absolute values and keep the common sign (−4+(−3)=−7-4 + (-3) = -7)
  • 2Different signs: Subtract absolute values and keep the sign of the larger (−4+7=3-4 + 7 = 3)
  • 3Subtracting = adding the opposite: a−b=a+(−b)a - b = a + (-b)
  • 4Double negative: a−(−b)=a+ba - (-b) = a + b
  • 5Use a number line: right is positive, left is negative

Frequently Asked Questions

Why does subtracting a negative give a positive?

Think of it as removing a debt. If you owe 5 dollars (−5-5) and someone cancels that debt (subtracts −5-5), you gain 5 dollars. 0−(−5)=0+5=50 - (-5) = 0 + 5 = 5. Subtracting negative = adding positive!

How do I know which sign the answer will have?

For addition with different signs, the answer has the sign of the number with the larger absolute value. −8+5=−3-8 + 5 = -3 because ∣−8∣=8>5=∣5∣|-8| = 8 > 5 = |5|, so the answer is negative.

Is there a quick trick for subtracting integers?

Yes! Change subtraction to addition and flip the sign of the second number. Then follow the addition rules. Example: 7−(−3)7 - (-3) becomes 7+3=107 + 3 = 10.

Glossary

Integer
A whole number that can be positive, negative, or zero (e.g., −5-5, 00, 1212)
Absolute value
The distance of a number from zero, never negative (e.g., ∣−7∣=7|-7| = 7, ∣0∣=0|0| = 0)
Opposite
A number that is the same distance from zero but on the other side (e.g., opposite of −3-3 is 33)
Additive inverse
Another name for the opposite; a number plus its additive inverse equals zero (5+(−5)=05 + (-5) = 0)

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