Properties of Multiplication

Florida B.E.S.T. Standard

MA.3.AR.1.1

Apply the distributive property to multiply a one-digit number and two-digit number. Apply properties of multiplication to find a product of one-digit whole numbers.

Key Points:

Distributive property is the PRIMARY focus for breaking apart larger problems
Students should also know commutative, associative, identity, and zero properties
Properties are STRATEGIES for solving - not just rules to memorize
Use arrays, area models, and number lines to visualize properties

The Five Properties of Multiplication

1. Commutative Property

You can multiply numbers in any order. The answer stays the same.

a × b = b × a

Example: 6 × 4 = 24 AND 4 × 6 = 24
Why it helps: If you forget 6 × 8, but know 8 × 6, you have the answer!

2. Associative Property

When multiplying three or more numbers, you can group them differently. The answer stays the same.

(a × b) × c = a × (b × c)

Example: (3 × 5) × 2 = 15 × 2 = 30 OR 3 × (5 × 2) = 3 × 10 = 30
Why it helps: Group to make easier calculations (multiply by 10 is easy!)

3. Distributive Property (MOST IMPORTANT)

You can break apart one factor and multiply the parts separately, then add.

a × (b + c) = (a × b) + (a × c)

Example: 7 × 14 = 7 × (10 + 4) = (7 × 10) + (7 × 4) = 70 + 28 = 98
Why it helps: Makes hard problems like 6 × 13 easier by breaking into 6 × 10 + 6 × 3

4. Identity Property

Any number times 1 equals itself.

a × 1 = a

Example: 7 × 1 = 7, 1 × 25 = 25
Why it helps: One group of 7 is still 7. Quick to calculate!

5. Zero Property

Any number times 0 equals 0.

a × 0 = 0

Example: 8 × 0 = 0, 0 × 100 = 0
Why it helps: Zero groups of anything is nothing!

Common Misconceptions & Fixes

Misconception: Commutative property works for subtraction/division

Students think 8 - 3 = 3 - 8 or 8 ÷ 2 = 2 ÷ 8.

Fix: Show counterexamples. "Does 8 - 3 equal 3 - 8? Let's check: 5 ≠ -5." Commutative works ONLY for addition and multiplication.

Misconception: Confusing distributive with "distributing" equally

Students don't understand how breaking apart works mathematically.

Fix: Use arrays! Draw 7 × 14 as 7 rows. Show how you can split the 14 columns into 10 + 4. Count to verify both methods give the same answer.

Misconception: Adding instead of multiplying after distributing

Student writes 7 × 14 = 7 × 10 + 4 = 70 + 4 = 74 (forgot to multiply the 4).

Fix: Emphasize "multiply BOTH parts" with arrows. Use the phrase "7 times 10 AND 7 times 4."

Misconception: Thinking 0 × 5 = 5 (confusing with identity property)

Students mix up × 1 and × 0 rules.

Fix: Use concrete examples. "If you have 0 bags with 5 cookies each, how many cookies? Zero bags means zero cookies!" vs "1 bag with 5 cookies = 5 cookies."

5-Day Lesson Sequence

Day 1: Commutative and Identity Properties

Use arrays to show 3 × 4 = 4 × 3 (rotate the array!)
Practice "turn-around facts" - if you know one, you know two
Introduce identity property: 1 row of n = n
Connect to fact fluency - doubles work!

Day 2: Zero Property and Associative Property

Zero property with real objects: "0 groups of 4 pennies = ?"
Associative property: multiply 3 numbers different ways
Strategy: regroup to make 10s (e.g., 5 × 4 × 2 = 5 × 8 or 20 × 2)
Practice choosing helpful groupings

Day 3: Introduction to Distributive Property

Use arrays: split 6 × 7 into 6 × 5 + 6 × 2
Area models with rectangles
Practice with single-digit × single-digit first
Emphasize "break apart, multiply both, add"

Day 4: Distributive Property with Two-Digit Numbers

Break apart two-digit numbers into tens and ones
Example: 4 × 13 = 4 × 10 + 4 × 3 = 40 + 12 = 52
Use place value language
Practice with various two-digit numbers (12-19 first, then larger)

Day 5: Mixed Practice & Problem Solving

Identify which property is being used
Choose the best property for a given problem
Word problems that benefit from properties
FAST-format practice questions

Differentiation Strategies

For Struggling Learners

Use manipulatives (counters, tiles) for every property
Focus on commutative and identity first - most concrete
For distributive: always draw the array or area model
Use graph paper to keep arrays organized
Post anchor charts with each property and an example

For Advanced Learners

Distributive with larger numbers (e.g., 7 × 23)
Use properties to prove equivalent expressions
Find multiple ways to solve the same problem using different properties
Create their own word problems that use properties

FAST Test Tip:

FAST problems often show a multiplication equation and ask "Which property is shown?" or give an expression like 6 × 15 and ask students to rewrite it using the distributive property. Students need to recognize AND apply properties. Focus on the distributive property as it appears most frequently.

Teacher Guide: Properties of Multiplication