Course 1 Chapter 3 Operations with Integers Answer Key - Addition, Subtraction, and Estimation

Complete answer key for operations with integers including detailed solutions for addition, subtraction, and estimation problems.

Answer Key: Exercises 1 to 3
1 Integer Addition
Exercise 1
Solve: 5 + (-3) = ?
Definition:

Integer addition: Adding positive and negative numbers using absolute values and sign rules.

Integer addition method:
  1. Identify the signs of both numbers
  2. Find the absolute values
  3. Subtract the smaller absolute value from the larger
  4. Give the result the sign of the number with the larger absolute value
Problem
5 + (-3)
Absolute Values
|5| = 5, |-3| = 3
Answer
2
Step 1: Identify signs

5 is positive, -3 is negative

Step 2: Find absolute values

|5| = 5, |-3| = 3

Step 3: Subtract absolute values

5 - 3 = 2

Step 4: Assign sign

Since 5 > 3, the result takes the sign of 5 (positive)

5 + (-3) = 2
Final answer:

5 + (-3) = 2

Applied rules:

• Sign rule: Different signs → subtract absolute values

• Sign determination: Result gets sign of number with larger absolute value

2 Integer Subtraction
Exercise 2
Solve: 8 - (-4) = ?
Definition:

Integer subtraction: Subtracting integers by changing the operation to addition and changing the sign of the second number.

Problem
8 - (-4)
Convert to Addition
8 + 4
Answer
12
Step 1: Change subtraction to addition

8 - (-4) becomes 8 + 4

Step 2: Change sign of second number

-(-4) becomes +4

Step 3: Perform addition

8 + 4 = 12

Step 4: Read final answer

8 - (-4) = 12

8 - (-4) = 12
Final answer:

8 - (-4) = 12

Applied rules:

• Subtraction rule: a - b = a + (-b)

• Double negative: -(-b) = +b

3 Integer Estimation
Exercise 3
Estimate: (-23) + 18. Then find the exact answer.
Definition:

Integer estimation: Approximating the result of integer operations by rounding numbers to make calculations easier.

Problem
(-23) + 18
Estimate
(-20) + 20 = 0
Exact
-5
Step 1: Round to nearest ten

-23 rounds to -20
18 rounds to 20

Step 2: Add rounded numbers

(-20) + 20 = 0

Step 3: Find exact answer

(-23) + 18 = -5

Step 4: Compare estimates

Estimated: 0, Exact: -5 → Reasonable approximation

Estimated: 0, Exact: -5
Final answer:

Estimate: 0, Exact: (-23) + 18 = -5

Applied rules:

• Estimation: Round to nearest convenient value

• Reasonableness: Check if estimate is close to exact answer

Integer Operations Answer Key
Complete Answer Key:
Exercise 1: 5 + (-3) = 2
Exercise 2: 8 - (-4) = 12
Exercise 3: (-23) + 18 = -5
Exercise 4: (-15) - 7 = -22
Exercise 5: (-9) + (-4) = -13
a + (-b) = a - b
Addition Rule
a - (-b) = a + b
Subtraction Rule
Rule 1
Same Signs → Add
KEEP SIGN
Rule 2
Different Signs → Subtract
KEEP LARGER SIGN
Rule 3
Double Negative → Positive
-(-a) = +a
Key definitions:

Integers: Whole numbers including positive numbers, negative numbers, and zero

Absolute Value: Distance of a number from zero, always positive

Opposites: Numbers that are the same distance from zero but in opposite directions

Sign Rules: Rules governing how signs interact in operations

Integer operation methods:
  1. Addition with same signs: Add absolute values, keep common sign
  2. Addition with different signs: Subtract smaller absolute value from larger, keep sign of larger
  3. Subtraction: Change to addition and change sign of second number
  4. Estimation: Round numbers to convenient values before operating
Tip 1: Think of integers as money: positive = earned, negative = owed.
Tip 2: Remember: subtracting a negative is the same as adding a positive.
Tip 3: Always check if your answer makes sense in the context of the problem.
Tip 4: Use a number line to visualize integer operations.
Common errors: Forgetting sign changes, misapplying rules, not considering absolute values.
Practice strategies: Use number lines, practice with real-world contexts, verify each step.
Rules to know by heart:

• Addition rule: Same signs → add and keep sign; Different signs → subtract and keep larger sign

• Subtraction rule: Change to addition and change sign of subtrahend

• Double negative: -(-a) = +a

• Absolute value: Always positive, measures distance from zero

Detailed Summary:

Key Concepts:

  • Integers are whole numbers including positive, negative, and zero
  • Absolute value is the distance from zero, always positive
  • Sign rules determine the outcome of operations with integers
  • Number line visualization helps understand integer relationships

Core Rules:

  • Addition with same signs: Add absolute values, keep common sign
  • Addition with different signs: Subtract absolute values, keep sign of larger
  • Subtraction: Change to addition and change sign of second number
  • Double negative: -(-a) = +a

Step-by-Step Method:

  1. Identify the operation and signs of numbers involved
  2. Apply the appropriate rule based on signs
  3. Perform the calculation with absolute values
  4. Determine the correct sign for the result
  5. Verify the answer makes sense

Examples Progression:

  • Simple: 5 + 3 = 8 (both positive)
  • With negatives: 5 + (-3) = 2 (different signs)
  • Subtraction: 8 - (-4) = 12 (double negative)
  • Both negative: (-9) + (-4) = -13 (same signs)

Memory Tips:

  • "Same signs, add and keep; different signs, subtract and keep larger"
  • "Subtracting a negative is adding a positive"
  • Think of positive as "have" and negative as "owe"
Answer Key: Exercises 4 to 5
4 Mixed Operations
Exercise 4
Solve: (-15) - 7 = ?
Definition:

Mixed integer operations: Operations involving both positive and negative integers requiring careful attention to signs.

Problem
(-15) - 7
Convert to Addition
(-15) + (-7)
Answer
-22
Step 1: Change subtraction to addition

(-15) - 7 becomes (-15) + (-7)

Step 2: Apply addition rule for same signs

Both numbers are negative, so add absolute values: 15 + 7 = 22

Step 3: Keep the common sign

Since both were negative, result is -22

Step 4: Verify the result

Check: (-22) + 7 = (-15) ✓

(-15) - 7 = -22
Final answer:

(-15) - 7 = -22

Applied rules:

• Subtraction conversion: a - b = a + (-b)

• Same signs addition: Add absolute values and keep common sign

5 Double Negative
Exercise 5
Solve: (-9) + (-4) = ?
Definition:

Double negative addition: Adding two negative integers results in a more negative number.

Problem
(-9) + (-4)
Add Absolute Values
9 + 4 = 13
Apply Sign
-13
Step 1: Identify signs

Both numbers are negative: (-9) and (-4)

Step 2: Add absolute values

|(-9)| + |(-4)| = 9 + 4 = 13

Step 3: Apply common sign

Since both were negative, result is -13

Step 4: State the answer

(-9) + (-4) = -13

(-9) + (-4) = -13
Final answer:

(-9) + (-4) = -13

Applied rules:

• Same signs addition: Add absolute values and keep common sign

• Negative plus negative: Results in a more negative number

Integer Operations Guide & Practice
Answer Verification:
5 + (-3) = 2 ✓
8 - (-4) = 12 ✓
(-23) + 18 = -5 ✓
(-15) - 7 = -22 ✓
(-9) + (-4) = -13 ✓
|a| = distance from zero
Absolute Value
Key definitions:

Integers: The set of whole numbers {..., -3, -2, -1, 0, 1, 2, 3, ...}

Absolute value: The non-negative value of a number without regard to its sign

Opposite numbers: Numbers that are equidistant from zero on the number line

Complete methodology:
  1. Identify operation: Determine if addition, subtraction, or mixed operations
  2. Recognize signs: Note the signs of all numbers involved
  3. Apply rules: Use appropriate integer operation rules
  4. Calculate: Perform the arithmetic with absolute values
  5. Determine sign: Apply the correct sign to the result
  6. Verify: Check the answer using inverse operations
Tip 1: Visualize on a number line: move right for positive, left for negative.
Tip 2: Remember: subtracting a negative is equivalent to adding a positive.
Tip 3: Always double-check your sign before writing the final answer.
Tip 4: Use real-world analogies like temperature changes or bank balances.
Common errors: Misapplying sign rules, forgetting absolute values, incorrect subtraction conversions.
Verification methods: Use inverse operations, number line checks, estimation.
Formulas to know by heart:

• Addition with same signs: |a| + |b| with common sign

• Addition with different signs: |larger| - |smaller| with sign of larger

• Subtraction: a - b = a + (-b)

• Double negative: -(-a) = a

Integer Operations Workflow
🔄
Step-by-Step Process
1
IDENTIFY
2
SIGN CHECK
3
APPLY RULE
4
CALCULATE
5
VERIFY
Sign Rules

Positive + Positive = Positive

Negative + Negative = Negative

Positive + Negative = Sign of Larger Absolute Value

Subtracting Negative = Adding Positive

✅ ANSWERS VERIFIED!

Questions & Answers

Question: I don't understand why subtracting a negative number makes the answer bigger. It seems backwards!

Answer: This is a common confusion! Think of it this way:

  • Subtracting means "taking away"
  • When you subtract a negative, you're "taking away" something that's already negative
  • This is like removing a debt or undoing a loss
  • Example: If you owe $5 (-5) and someone forgives your debt (-(-5)), you gain $5

So: 8 - (-4) = 8 + 4 = 12. You're adding because you're removing a negative situation.

Question: When I add a positive and negative number, how do I know which sign to use for the answer?

Answer: Here's the rule for adding numbers with different signs:

  • Subtract the smaller absolute value from the larger absolute value
  • Give the answer the sign of the number with the larger absolute value
  • Example: 7 + (-3): 7 > 3, so answer is positive: 7 - 3 = 4
  • Example: (-7) + 3: 7 > 3, so answer is negative: -(7 - 3) = -4

The number with the larger absolute value "wins" the sign battle!

Question: My child is struggling with integer operations. How can I help them understand this concept?

Answer: Here are effective ways to help with integer operations:

  • Use a number line: Show movement right (positive) and left (negative)
  • Real-world examples: Temperature changes, bank balances, elevator floors
  • Manipulatives: Use colored chips (red for negative, blue for positive)
  • Memory aids: "Same signs, add and keep; different signs, subtract and keep larger"
  • Practice gradually: Start with simple problems and build complexity

Patience and consistent practice with visual aids are key to developing integer operation fluency.