Integer Multiplication Slides Middle School Mathematics
Unit: The Number System
Multiplying Integers
Mastering the sign rules, uncovering patterns, and evaluating expressions with confidence.
Conceptual Understanding Clear Sign Rules
Standard: 7.NS.A.2
Essential Foundations
The Golden Rules of Multiplying Integers
RULE 1: SAME SIGNS
Always POSITIVE (+)
When multiplying two numbers with the same sign, the product is always positive.
\( (+a) \times (+b) \) = + Product
\( (-a) \times (-b) \) = + Product
RULE 2: DIFFERENT SIGNS
Always NEGATIVE (−)
When multiplying two numbers with different signs, the product is always negative.
\( (+a) \times (-b) \) = − Product
\( (-a) \times (+b) \) = − Product
Memory Trick: Same signs agree to be Positive . Opposites clash to be Negative .
Conceptual Proof
Why Does a Negative × Negative = Positive?
Observing the Pattern:
\( 3 \times (-4) \) = −12 +4 each step up
\( 2 \times (-4) \) = −8 +4
\( 1 \times (-4) \) = −4 +4
\( 0 \times (-4) \) = 0 +4
\( (-1) \times (-4) \) = +4 Pattern continues!
\( (-2) \times (-4) \) = +8 +4
Real-World Direction
Think of math as film played forwards or backwards:
• Walking backwards (negative) while running a tape in reverse (negative) looks like walking forwards (positive)!
• Removing (negative) a debt (negative) adds to your wealth (positive).
Key Conclusion: Math rules are consistent. Mathematics preserves continuity so patterns never break!
Pattern logic verifies: negative × negative = positive Keep moving forward →
Modeling Instruction
The 2-Step Solving Method
Example A Different Signs
\( (-7) \times 6 \)
1 Multiply absolute values: \( 7 \times 6 = 42 \)
2 Determine the sign: Negative × Positive are different → Negative
Final Answer: −42
Example B Same Signs
\( (-8) \times (-5) \)
1 Multiply absolute values: \( 8 \times 5 = 40 \)
2 Determine the sign: Negative × Negative are same → Positive
Final Answer: +40 (or 40)
Common Trap: Do not confuse multiplication rules with addition rules! (-8) + (-5) = -13, but (-8) × (-5) = +40.
Advanced Rule
Multiplying 3 or More Integers
The "Count the Negatives" Strategy
Multiply the absolute values normally, then simply count how many negative factors appear in the expression:
EVEN Number of Negatives (0, 2, 4, 6...)
Result is POSITIVE (+)
\( (-2) \times (-3) \times (-4) \times (-1) = \mathbf{+24} \)
4 negative signs (even) → Positive
ODD Number of Negatives (1, 3, 5, 7...)
Result is NEGATIVE (−)
\( (-2) \times (-3) \times (-4) = \mathbf{-24} \)
3 negative signs (odd) → Negative
Why? Negative pairs cancel each other into positives: (-) × (-) = (+). An odd sign leaves one unpaired negative!
Active Check
Quick-Fire Whiteboard Check
Solve each expression silently, then hold up your answer on signal.
1
\( 9 \times (-4) \)
Target: Sign first, then number
2
\( (-6) \times (-7) \)
Target: Same signs rule
3
\( (-3) \times (-2) \times (-5) \)
Target: Odd count rule
Ready to practice independently? Grab your Integer Multiplication Practice Worksheet !
Solutions: -36 | +42 | -30
Integer Multiplication Worksheet Multiplying Integers
Practice & Mastery Worksheet
Name:
Date:
Period:
Core Multiplication Rules: Read Before Solving
Memorize This
SAME SIGNS → ALWAYS POSITIVE (+)
If both factors have the exact same sign, the product is positive.
\( (+a) \times (+b) = \mathbf{+} \) | \( (-a) \times (-b) = \mathbf{+} \)
DIFFERENT SIGNS → ALWAYS NEGATIVE (−)
If the factors have opposite signs, the product is negative.
\( (+a) \times (-b) = \mathbf{-} \) | \( (-a) \times (+b) = \mathbf{-} \)
Part 1: Sign Diagnostic — Determine Sign Only (+ or −) Do not compute the full product yet.
\( (-5) \times 8 \)
Sign: _____
\( (-9) \times (-3) \)
Sign: _____
\( 12 \times (-4) \)
Sign: _____
\( (-7) \times (-7) \)
Sign: _____
\( (-11) \times 6 \)
Sign: _____
\( 15 \times 4 \)
Sign: _____
Part 2: Two-Factor Multiplication Fluency Determine the sign first, then multiply.
\( 6 \times (-8) \) = ______
\( (-5) \times (-9) \) = ______
\( (-12) \times 4 \) = ______
\( (-7) \times (-8) \) = ______
\( (-10) \times 11 \) = ______
\( (-4) \times (-15) \) = ______
\( 13 \times (-3) \) = ______
\( (-25) \times (-4) \) = ______
\( 0 \times (-18) \) = ______
\( (-14) \times (-2) \) = ______
\( (-9) \times 12 \) = ______
\( (-1) \times (-56) \) = ______
Turn page over for multi-factor problems, word problems, and error analysis. Page 1 of 2
Part 2 of 2
Extended Applications & Problem Solving
Name:
Multi-Factor Count Rule:
Even # of negatives → Positive (+) Odd # of negatives → Negative (−)
Part 3: Multiple Factors & Powers Count negatives first.
\( (-2) \times (-3) \times (-4) \) = ______
\( (-5) \times 2 \times (-3) \) = ______
\( (-1) \times (-1) \times (-1) \times (-1) \) = ______
\( (-6)^2 = (-6) \times (-6) \) = ______
\( (-2)^3 = (-2) \times (-2) \times (-2) \) = ______
\( 4 \times (-3) \times (-2) \times (-1) \) = ______
Part 4: Real-World Applications & Modeling
19. Deep Sea Exploration: A research submarine starts at sea level and descends at a rate of 18 meters per minute (\(-18\) m/min). Where is the submarine located relative to sea level after 6 minutes ?
Equation:
20. Budget Deficit: A community club has a monthly subscription fee of $15 automatically debited (\(-$15\)) from their account. If this fee was charged for 5 consecutive months , what was the total change in the account balance?
Equation:
Part 5: Error Detective — Critique Reasoning
21. Marcus solved \( (-7) \times (-6) \) and wrote −42 . He stated: "A negative times a negative has two minus signs, so the answer must stay negative."
What error did Marcus make?
Correct Product:
22. Maya solved \( (-3) \times (-2) \times (-4) \) and wrote +24 . Explain what rule Maya forgot regarding an odd number of negative factors.
Explanation:
Correct Product:
Unit: Rational Numbers & Integers • Standard 7.NS.A.2 Page 2 of 2
Integer Multiplication Answer Key Teacher Resource Answer Key & Scoring Guide
Multiplying Integers — Solutions
Total Points: 30 pts
Curriculum: Common Core 7.NS.A.2
Scoring Standard:
Same Signs: (+) × (+) = + | (−) × (−) = + Different Signs: (+) × (−) = − | (−) × (+) = −
Part 1: Sign Diagnostic Solutions (1 pt each • 6 pts total) Sign Accuracy Focus
\( (-5) \times 8 \)
− (Negative)
Different signs
\( (-9) \times (-3) \)
+ (Positive)
Same signs
\( 12 \times (-4) \)
− (Negative)
Different signs
\( (-7) \times (-7) \)
+ (Positive)
Same signs
\( (-11) \times 6 \)
− (Negative)
Different signs
\( 15 \times 4 \)
+ (Positive)
Same signs
Part 2: Two-Factor Multiplication Solutions (1 pt each • 12 pts total) Check both sign and absolute value product.
\( 6 \times (-8) \) = −48
Opposite signs → Negative
\( (-5) \times (-9) \) = +45 (45)
Same signs → Positive
\( (-12) \times 4 \) = −48
Opposite signs → Negative
\( (-7) \times (-8) \) = +56 (56)
Same signs → Positive
\( (-10) \times 11 \) = −110
Opposite signs → Negative
\( (-4) \times (-15) \) = +60 (60)
Same signs → Positive
\( 13 \times (-3) \) = −39
Opposite signs → Negative
\( (-25) \times (-4) \) = +100 (100)
Same signs → Positive
\( 0 \times (-18) \) = 0
Zero property: 0 has no sign (+ or −)
\( (-14) \times (-2) \) = +28 (28)
Same signs → Positive
\( (-9) \times 12 \) = −108
Opposite signs → Negative
\( (-1) \times (-56) \) = +56 (56)
Multiplying by −1 gives opposite
Teacher Key • Multiplying Integers Mastery Page 1 of 2
Part 2 of 2
Extended Solutions & Rubrics
Section C: 6 pts • Section D: 4 pts • Section E: 2 pts
Part 3 Solutions: Multiple Factors & Powers (1 pt each • 6 pts total)
\( (-2) \times (-3) \times (-4) \) = −24
3 negative signs (odd count) → Negative
\( (-5) \times 2 \times (-3) \) = +30 (30)
2 negative signs (even count) → Positive
\( (-1) \times (-1) \times (-1) \times (-1) \) = +1 (1)
4 negative signs (even count) → Positive
\( (-6)^2 = (-6) \times (-6) \) = +36 (36)
Squaring a negative always yields positive
\( (-2)^3 = (-2) \times (-2) \times (-2) \) = −8
Odd exponent with negative base → Negative
\( 4 \times (-3) \times (-2) \times (-1) \) = −24
3 negative signs (odd count) → Negative
Part 4 Solutions: Word Problems (2 pts each • 4 pts total)
19. Deep Sea Submarine Descent: 2 Points
Equation (1 pt): \( 6 \times (-18) = -108 \) (or \( -18 \times 6 = -108 \))
Interpretation/Position (1 pt): The submarine is at −108 meters (or 108 meters below sea level ).
Teacher note: Accept either "-108 m" or "108 m below sea level". If student writes "-108 m below sea level", gently correct the double negative.
20. Budget Deficit: 2 Points
Equation (1 pt): \( 5 \times (-15) = -75 \) (or \( -15 \times 5 = -75 \))
Total Change (1 pt): The account balance changed by −$75 (or decreased by $75 ).
Part 5 Solutions: Error Detective (1 pt explanation + 1 pt correct answer • 4 pts total)
21. Marcus's Error:
Exemplary Explanation: Marcus confused addition with multiplication. When adding two negatives, the sum is negative (e.g., \( -7 + -6 = -13 \)), but when multiplying two numbers with the same negative sign, the two negatives cancel out to give a positive product .
Correct Product: +42 (or 42)
22. Maya's Error:
Exemplary Explanation: Maya assumed having multiple negative signs makes the answer positive. However, an odd number of negative factors (3 negatives) results in a negative product because \( (-3) \times (-2) = +6 \), and then \( (+6) \times (-4) = \mathbf{-24} \).
Correct Product: −24
Scoring Rubric: 28-30 (Exceeds) • 24-27 (Proficient) • 20-23 (Developing) • <20 (Needs Intervention) Page 2 of 2
