Solved Exercises on Adding Integers in Grade 7

Master adding integers: positive, negative, zero pairs, absolute value, number line representation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Positive + Positive
Exercise 1
Calculate: \( (+7) + (+3) \)
Definition:

Positive integer: A whole number greater than zero (no sign or + sign)

Addition method:
  1. Add the absolute values of both numbers
  2. Keep the positive sign
  3. Result is always positive
Original Expression
\((+7) + (+3)\)
Add Absolute Values
\(7 + 3 = 10\)
Result
\(10\)
Step 1: Identify the signs

Both numbers are positive: \((+7)\) and \((+3)\)

Step 2: Add the absolute values

\(|7| + |3| = 7 + 3 = 10\)

Step 3: Keep the positive sign

Since both were positive, the result is positive: \(+10\)

Step 4: Simplify notation

Positive numbers can be written without the + sign: \(10\)

\( (+7) + (+3) = 10 \)
Final answer:

\( (+7) + (+3) = 10 \)

Applied rules:

• Same signs: Add absolute values, keep the common sign

• Positive + Positive: Always results in a larger positive number

• Simplification: Omit the + sign for positive results

2 Negative + Negative
Exercise 2
Calculate: \( (-5) + (-8) \)
Definition:

Negative integer: A whole number less than zero with a - sign

Original Expression
\((-5) + (-8)\)
Add Absolute Values
\(5 + 8 = 13\)
Keep Negative Sign
\(-13\)
Step 1: Identify the signs

Both numbers are negative: \((-5)\) and \((-8)\)

Step 2: Add the absolute values

\(|-5| + |-8| = 5 + 8 = 13\)

Step 3: Keep the negative sign

Since both were negative, the result is negative: \(-13\)

Step 4: Verify the concept

Think of it as moving left on the number line: 5 units left, then 8 more units left

\( (-5) + (-8) = -13 \)
Final answer:

\( (-5) + (-8) = -13 \)

Applied rules:

• Same signs: Add absolute values, keep the common sign

• Negative + Negative: Always results in a more negative number

• Number line: Adding negatives moves further left

3 Positive + Negative
Exercise 3
Calculate: \( (+9) + (-4) \)
Definition:

Absolute value: Distance from zero, always positive: \(|a|\)

Original Expression
\((+9) + (-4)\)
Compare Absolute Values
\(9 > 4\)
Subtract Smaller from Larger
\(9 - 4 = 5\)
Take Sign of Larger
\(5\)
Step 1: Identify the signs

One positive \((+9)\) and one negative \((-4)\)

Step 2: Compare absolute values

\(|+9| = 9\) and \(|-4| = 4\), so \(9 > 4\)

Step 3: Subtract smaller from larger

\(|9| - |4| = 9 - 4 = 5\)

Step 4: Take the sign of the number with larger absolute value

Since \(9 > 4\), take the sign of \(+9\), which is positive

Step 5: Write the final answer

\(+5\) or simply \(5\)

\( (+9) + (-4) = 5 \)
Final answer:

\( (+9) + (-4) = 5 \)

Applied rules:

• Different signs: Subtract absolute values, take sign of larger

• Positive + Negative: Result depends on which has larger absolute value

• Concept: Think of it as combining gains and losses

Rules and methods, laws,...
\( (+a) + (+b) = +(a + b) \)
Same Signs Addition
Positive + Positive
\( (+a) + (+b) = +(a + b) \)
Add absolute values, keep positive sign
Negative + Negative
\( (-a) + (-b) = -(a + b) \)
Add absolute values, keep negative sign
Different Signs
\( (+a) + (-b) = ±(a - b) \)
Subtract, take sign of larger absolute value
Key definitions:

Integer: Whole number including positive, negative, and zero

Absolute value: Distance from zero on number line, always non-negative

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

Zero pair: A positive number and its negative counterpart that sum to zero

Integer addition methods:
  1. Same signs: Add absolute values, keep the common sign
  2. Different signs: Subtract absolute values, take sign of number with larger absolute value
  3. Number line: Start at first number, move in direction of second number
  4. Zero pairs: Cancel out equal positive and negative amounts
Tip 1: Remember: "Same signs add, different signs subtract"
Tip 2: Always take the sign of the number with the larger absolute value
Tip 3: Use a number line to visualize when confused
Tip 4: Positive numbers can be written without the + sign
Common errors: Forgetting to consider signs, mixing up addition/subtraction rules, not taking the correct sign.
Exam preparation: Practice all three scenarios, memorize the rules, use number lines for verification.
Formulas to know by heart:

• Same signs: \( (+a) + (+b) = +(a + b) \) and \( (-a) + (-b) = -(a + b) \)

• Different signs: \( (+a) + (-b) = ±(a - b) \) (sign depends on larger absolute value)

• Opposite numbers: \( (+a) + (-a) = 0 \)

• Identity: \( a + 0 = a \)

Solution: Exercises 4 to 5
4 Negative + Positive
Exercise 4
Calculate: \( (-7) + (+12) \)
Definition:

Commutative property: Order doesn't matter in addition: \(a + b = b + a\)

Original Expression
\((-7) + (+12)\)
Compare Absolute Values
\(12 > 7\)
Subtract Smaller from Larger
\(12 - 7 = 5\)
Take Sign of Larger
\(5\)
Step 1: Identify the signs

One negative \((-7)\) and one positive \((+12)\)

Step 2: Compare absolute values

\(|-7| = 7\) and \(|+12| = 12\), so \(12 > 7\)

Step 3: Subtract smaller from larger

\(|12| - |7| = 12 - 7 = 5\)

Step 4: Take the sign of the number with larger absolute value

Since \(12 > 7\), take the sign of \(+12\), which is positive

Step 5: Write the final answer

\(+5\) or simply \(5\)

\( (-7) + (+12) = 5 \)
Final answer:

\( (-7) + (+12) = 5 \)

Applied rules:

• Different signs: Subtract absolute values, take sign of larger

• Commutative property: This equals \((+12) + (-7)\)

• Concept: Think of losing 7 then gaining 12, net gain of 5

5 Opposite Numbers
Exercise 5
Calculate: \( (+8) + (-8) \)
Definition:

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

Original Expression
\((+8) + (-8)\)
Same Absolute Value
\(8 = 8\)
Subtract
\(8 - 8 = 0\)
Result
\(0\)
Step 1: Identify the signs

One positive \((+8)\) and one negative \((-8)\)

Step 2: Notice they are opposites

\(|+8| = 8\) and \(|-8| = 8\), so they have the same absolute value

Step 3: Apply the rule for opposites

When adding opposites, subtract their absolute values: \(8 - 8 = 0\)

Step 4: Understand the concept

This creates a "zero pair" - equal amounts in opposite directions cancel out

Step 5: Write the final answer

The result is always zero when adding opposite numbers

\( (+8) + (-8) = 0 \)
Final answer:

\( (+8) + (-8) = 0 \)

Applied rules:

• Opposite numbers: \( (+a) + (-a) = 0 \)

• Zero pair: Equal positive and negative amounts cancel out

• Identity element: Zero is the additive identity

Key Concepts: Laws, Methods, Rules, Definitions
\( (+a) + (-b) = ±(a - b) \)
Different Signs Addition
Key definitions:

Integer: A whole number that can be positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...)

Absolute value: The distance of a number from zero on the number line, denoted as \(|a|\), always non-negative

Positive integer: A number greater than zero, often written without a sign

Negative integer: A number less than zero, written with a minus sign

Opposite numbers: Two numbers that are the same distance from zero but in opposite directions

Zero pair: A positive number and its negative counterpart that sum to zero

Complete addition methodology:
  1. Identify signs: Determine if numbers have the same or different signs
  2. Same signs: Add absolute values and keep the common sign
  3. Different signs: Subtract absolute values and take the sign of the number with larger absolute value
  4. Special cases: Opposite numbers always sum to zero
  5. Verify: Check with number line or mental math
Tip 1: Same signs → Add and keep the sign; Different signs → Subtract and take the larger sign
Tip 2: Use a number line to visualize: start at first number, move in direction of second number
Tip 3: Remember that adding a negative is the same as subtracting a positive
Tip 4: Opposite numbers (like +5 and -5) always sum to zero
Common errors: Forgetting to consider signs, adding when should subtract, taking wrong sign.
Exam preparation: Master the three addition scenarios, practice with number lines, verify answers.
Formulas to know by heart:

• Same signs: \( (+a) + (+b) = +(a + b) \) and \( (-a) + (-b) = -(a + b) \)

• Different signs: \( (+a) + (-b) = ±(a - b) \) (sign depends on larger absolute value)

• Opposite numbers: \( (+a) + (-a) = 0 \)

• Identity: \( a + 0 = a \)

• Commutative: \( a + b = b + a \)

• Associative: \( (a + b) + c = a + (b + c) \)

Exercise with Visualization: Integer Addition Patterns
Exercise 6: Integer Addition on Number Line
Visualize these additions on a number line:
\( (+3) + (-5) \)
\( (-2) + (+7) \)
\( (-4) + (-3) \)

Analysis: The chart shows how integer addition works on the number line.

  • \( (+3) + (-5) = -2 \): Start at +3, move 5 left to -2
  • \( (-2) + (+7) = 5 \): Start at -2, move 7 right to 5
  • \( (-4) + (-3) = -7 \): Start at -4, move 3 more left to -7

Questions & Answers

Question: I get confused when adding integers with different signs. How do I remember which sign to take for the answer?

Answer: This is a very common question! Here's the key rule:

  • When adding integers with different signs, subtract their absolute values
  • Then take the sign of the number with the larger absolute value

Memory trick: "Find the difference, take the sign of the bigger guy!"

Example: For \((+9) + (-4)\)

  • Absolute values: \(|+9| = 9\) and \(|-4| = 4\)
  • Subtract: \(9 - 4 = 5\)
  • Which number had the larger absolute value? \(+9\) did
  • So take the sign of \(+9\), which is positive
  • Answer: \(+5\) or just \(5\)

The number with the larger absolute value "wins" in determining the sign of the result!

Question: Why does adding a negative number make the result smaller? It seems backwards.

Answer: Great observation! Adding a negative number is actually equivalent to subtracting a positive number:

  • \(5 + (-3)\) is the same as \(5 - 3 = 2\)
  • You're essentially removing 3 from 5, which makes the result smaller

Real-world analogy:

  • If you have $5 and you add a debt of $3, you now have $2
  • If the temperature is 5°C and it drops by 3°, the new temperature is 2°C

Think of adding a negative as "taking away" rather than "gaining," which explains why the result is smaller.

On a number line, adding a negative number moves you to the left (decreasing values).

Question: I hear about "zero pairs" in integer operations. What are they and why are they important?

Answer: A zero pair is a positive number combined with its negative counterpart that sums to zero:

  • \((+1) + (-1) = 0\)
  • \((+5) + (-5) = 0\)
  • \((+100) + (-100) = 0\)

Why they're important:

  • They help simplify complex addition problems
  • They provide a visual way to understand integer operations
  • They explain why opposite numbers cancel each other out

Practical use: When solving problems like \((+7) + (-3) + (-4)\), you might notice that \((-3) + (-4) = -7\), and \((+7) + (-7) = 0\), so the total is 0.

Zero pairs represent the fundamental concept that equal amounts in opposite directions neutralize each other.