Solved Exercises on Multiplying Integers in Grade 7

Master multiplying integers: positive, negative, zero, absolute value through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Positive × Positive
Exercise 1
Calculate: \( (+4) \times (+3) \)
Definition:

Integer multiplication: Repeated addition of the same number

Multiplication method:
  1. Multiply the absolute values of both numbers
  2. Apply the sign rule: positive × positive = positive
  3. Result is always positive
Original Expression
\((+4) \times (+3)\)
Multiply Absolute Values
\(4 \times 3 = 12\)
Apply Sign Rule
Positive
Result
\(12\)
Step 1: Identify the signs

Both numbers are positive: \((+4)\) and \((+3)\)

Step 2: Multiply the absolute values

\(|4| \times |3| = 4 \times 3 = 12\)

Step 3: Apply the sign rule

Positive × Positive = Positive

Step 4: Write the final answer

\((+4) \times (+3) = +12\) or simply \(12\)

\( (+4) \times (+3) = 12 \)
Final answer:

\( (+4) \times (+3) = 12 \)

Applied rules:

• Sign rule: Positive × Positive = Positive

• Multiplication: Multiply absolute values first

• Positive result: Always results in a positive number

2 Negative × Positive
Exercise 2
Calculate: \( (-5) \times (+2) \)
Definition:

Sign rule: When multiplying integers with different signs, the result is negative

Original Expression
\((-5) \times (+2)\)
Multiply Absolute Values
\(5 \times 2 = 10\)
Apply Sign Rule
Negative
Result
\(-10\)
Step 1: Identify the signs

One negative \((-5)\) and one positive \((+2)\)

Step 2: Multiply the absolute values

\(|-5| \times |+2| = 5 \times 2 = 10\)

Step 3: Apply the sign rule

Negative × Positive = Negative

Step 4: Write the final answer

\((-5) \times (+2) = -10\)

\( (-5) \times (+2) = -10 \)
Final answer:

\( (-5) \times (+2) = -10 \)

Applied rules:

• Sign rule: Negative × Positive = Negative

• Multiplication: Multiply absolute values first

• Different signs: Always results in a negative number

3 Positive × Negative
Exercise 3
Calculate: \( (+6) \times (-3) \)
Definition:

Commutative property: Order doesn't matter in multiplication: \(a \times b = b \times a\)

Original Expression
\((+6) \times (-3)\)
Multiply Absolute Values
\(6 \times 3 = 18\)
Apply Sign Rule
Negative
Result
\(-18\)
Step 1: Identify the signs

One positive \((+6)\) and one negative \((-3)\)

Step 2: Multiply the absolute values

\(|+6| \times |-3| = 6 \times 3 = 18\)

Step 3: Apply the sign rule

Positive × Negative = Negative

Step 4: Write the final answer

\((+6) \times (-3) = -18\)

\( (+6) \times (-3) = -18 \)
Final answer:

\( (+6) \times (-3) = -18 \)

Applied rules:

• Sign rule: Positive × Negative = Negative

• Multiplication: Multiply absolute values first

• Different signs: Always results in a negative number

Rules and methods, laws,...
\( (+a) \times (+b) = +(a \times b) \)
Same Signs Multiplication
Positive × Positive
\( (+a) \times (+b) = +(a \times b) \)
Multiply absolute values, keep positive sign
Negative × Negative
\( (-a) \times (-b) = +(a \times b) \)
Multiply absolute values, result is positive
Different Signs
\( (+a) \times (-b) = -(a \times b) \)
Multiply absolute values, result is negative
Key definitions:

Integer: A whole number including positive, negative, and zero

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

Sign rule: Rules for determining the sign of the product based on signs of factors

Commutative property: Order doesn't matter in multiplication: \(a \times b = b \times a\)

Integer multiplication methods:
  1. Multiply absolute values: Ignore signs initially
  2. Apply sign rule: Determine sign of result
  3. Same signs: Result is positive
  4. Different signs: Result is negative
Tip 1: Remember: "Same signs give positive, different signs give negative"
Tip 2: Always multiply absolute values first, then determine the sign
Tip 3: Negative × Negative = Positive (two negatives make a positive)
Tip 4: Use the commutative property to rearrange factors if needed
Common errors: Forgetting sign rules, multiplying signs instead of absolute values, mixing up multiplication and addition rules.
Exam preparation: Practice all sign combinations, memorize the sign rules, use absolute values first.
Formulas to know by heart:

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

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

• Identity: \( a \times 1 = a \)

• Zero: \( a \times 0 = 0 \)

Solution: Exercises 4 to 5
4 Negative × Negative
Exercise 4
Calculate: \( (-4) \times (-7) \)
Definition:

Negative × Negative: This always results in a positive number

Original Expression
\((-4) \times (-7)\)
Multiply Absolute Values
\(4 \times 7 = 28\)
Apply Sign Rule
Positive
Result
\(28\)
Step 1: Identify the signs

Both numbers are negative: \((-4)\) and \((-7)\)

Step 2: Multiply the absolute values

\(|-4| \times |-7| = 4 \times 7 = 28\)

Step 3: Apply the sign rule

Negative × Negative = Positive

Step 4: Write the final answer

\((-4) \times (-7) = +28\) or simply \(28\)

\( (-4) \times (-7) = 28 \)
Final answer:

\( (-4) \times (-7) = 28 \)

Applied rules:

• Sign rule: Negative × Negative = Positive

• Multiplication: Multiply absolute values first

• Same signs: Always results in a positive number

5 Multiplying by Zero
Exercise 5
Calculate: \( (-9) \times 0 \)
Definition:

Multiplication by zero: Any number multiplied by zero equals zero

Original Expression
\((-9) \times 0\)
Apply Zero Property
Any number × 0 = 0
Result
\(0\)
Step 1: Identify the factors

One negative \((-9)\) and zero \((0)\)

Step 2: Apply the zero property

Any number multiplied by zero equals zero

Step 3: Write the final answer

\((-9) \times 0 = 0\)

Step 4: Understand the concept

Multiplying by zero means no groups of the number, so the result is always zero

\( (-9) \times 0 = 0 \)
Final answer:

\( (-9) \times 0 = 0 \)

Applied rules:

• Zero property: \(a \times 0 = 0\) for any number \(a\)

• Identity: Zero is the multiplicative absorption element

• Independence: Sign of other factor doesn't matter when multiplying by zero

Key Concepts: Laws, Methods, Rules, Definitions
\( (-a) \times (-b) = +(a \times b) \)
Negative × Negative Rule
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

Sign rule: Rules for determining the sign of the product based on the signs of the factors

Commutative property: \(a \times b = b \times a\) - order doesn't matter in multiplication

Complete multiplication methodology:
  1. Identify signs: Determine the signs of both factors
  2. Multiply absolute values: Multiply the absolute values of both numbers
  3. Apply sign rule: Determine the sign of the product based on the signs of factors
  4. Same signs: Result is positive
  5. Different signs: Result is negative
  6. Special cases: Any number × 0 = 0
  7. Verify: Check with repeated addition or mental math
Tip 1: Same signs → Positive result; Different signs → Negative result
Tip 2: Remember: "Two wrongs make a right" - Negative × Negative = Positive
Tip 3: Always multiply absolute values first, then apply the sign rule
Tip 4: Multiplying by zero always results in zero, regardless of the other factor's sign
Common errors: Forgetting sign rules, applying addition rules to multiplication, not multiplying absolute values first.
Exam preparation: Master the sign rules, practice all four scenarios, memorize the zero property.
Formulas to know by heart:

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

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

• Zero property: \( a \times 0 = 0 \)

• Identity: \( a \times 1 = a \)

• Commutative: \( a \times b = b \times a \)

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

Exercise with Visualization: Integer Multiplication Patterns
Exercise 6: Multiplication Sign Patterns
Observe these multiplication patterns:
\( (+2) \times (+3) = +6 \)
\( (-2) \times (+3) = -6 \)
\( (+2) \times (-3) = -6 \)
\( (-2) \times (-3) = +6 \)

Analysis: The chart shows how signs affect multiplication results.

  • \( (+2) \times (+3) = +6 \): Same positive signs → Positive result
  • \( (-2) \times (+3) = -6 \): Different signs → Negative result
  • \( (+2) \times (-3) = -6 \): Different signs → Negative result
  • \( (-2) \times (-3) = +6 \): Same negative signs → Positive result

Questions & Answers

Question: Why does multiplying two negative numbers give a positive result? It doesn't make sense!

Answer: This is a common source of confusion! Here are a few ways to understand it:

  1. Pattern recognition: Look at this sequence:
    \(3 \times 2 = 6\)
    \(3 \times 1 = 3\)
    \(3 \times 0 = 0\)
    \(3 \times (-1) = -3\)
    \(3 \times (-2) = -6\)
    Now continue the pattern: \((-3) \times (-2) = 6\)
  2. Real-world analogy: If you lose $3 per day, and someone says "cancel" (negative) that loss for 2 days (negative), you gain $6.
  3. Mathematical consistency: This rule maintains the distributive property: \((-a) \times (-b) = ab\) keeps all our mathematical rules working together.

Remember the phrase: "Two wrongs make a right" - two negative signs create a positive result!

Question: What's the easiest way to remember the sign rules for multiplying integers?

Answer: Here are some memorable ways to remember the sign rules:

  • Same signs → Positive: "Friends of friends are friends" and "Enemies of enemies are friends"
  • Different signs → Negative: "Friends of enemies are enemies" and "Enemies of friends are enemies"
  • Simple rhyme: "Same signs positive, different signs negative"
  • Visual: Think of a number line where same directions lead to positive, opposite directions lead to negative

Sign Rules Summary:

  • Positive × Positive = Positive
  • Negative × Negative = Positive
  • Positive × Negative = Negative
  • Negative × Positive = Negative

Always multiply the absolute values first, then apply the sign rule!

Question: How do I know if my multiplication of integers is correct?

Answer: Here are several ways to verify your integer multiplication:

  1. Sign check: Verify you applied the correct sign rule
  2. Estimation: Round numbers to check if answer is reasonable
  3. Repeated addition: For small numbers, think of multiplication as repeated addition
  4. Reverse operation: Divide your answer by one factor to get the other

Example verification: For \((-4) \times (-7) = 28\)

  • Sign check: Negative × Negative = Positive ✓
  • Estimation: \(4 \times 7 = 28\), so 28 is reasonable ✓
  • Reverse: \(28 \div (-4) = -7\) or \(28 \div (-7) = -4\) ✓

The most important check is ensuring you applied the correct sign rule!