🎯 Properties - Grade 3
Understanding Math Properties!
Properties are special rules in mathematics that make solving problems easier! They help us understand how numbers work together.
📚 Addition, Subtraction, Multiplication and Division Terms
Addition Terms:
\(\text{Addend} + \text{Addend} = \text{Sum}\)
Example: \(5 + 3 = 8\)
5 and 3 are addends
8 is the sum
Subtraction Terms:
\(\text{Minuend} - \text{Subtrahend} = \text{Difference}\)
Example: \(9 - 4 = 5\)
9 is the minuend (starting number)
4 is the subtrahend (number being subtracted)
5 is the difference
Multiplication Terms:
\(\text{Factor} \times \text{Factor} = \text{Product}\)
Example: \(6 \times 7 = 42\)
6 and 7 are factors
42 is the product
Division Terms:
\(\text{Dividend} \div \text{Divisor} = \text{Quotient}\)
Example: \(24 \div 6 = 4\)
24 is the dividend (number being divided)
6 is the divisor (number dividing by)
4 is the quotient (answer)
( ) Understanding Parentheses
What are Parentheses?
Parentheses ( ) are symbols that group numbers together. They tell us which operation to do FIRST!
RULE: Always solve what's inside the parentheses first!
Examples:
Example 1: Addition
\((3 + 2) + 5\)
Step 1: Solve inside parentheses: \(3 + 2 = 5\)
Step 2: Now add: \(5 + 5 = 10\)
Answer: 10 ✓
Example 2: Multiplication
\(4 \times (5 + 2)\)
Step 1: Solve inside parentheses: \(5 + 2 = 7\)
Step 2: Now multiply: \(4 \times 7 = 28\)
Answer: 28 ✓
💡 Remember: Parentheses = "Do me first!"
➕ Properties of Addition
1. Commutative Property of Addition
You can add numbers in any order, and the sum stays the same!
\(a + b = b + a\)
Example:
\(7 + 5 = 12\)
\(5 + 7 = 12\)
Both give the same answer! ✓
2. Associative Property of Addition
When adding three or more numbers, you can group them in any way, and the sum stays the same!
\((a + b) + c = a + (b + c)\)
Example:
\((2 + 3) + 4 = 5 + 4 = 9\)
\(2 + (3 + 4) = 2 + 7 = 9\)
Both ways give the same answer! ✓
3. Identity Property of Addition
When you add zero to any number, you get the same number!
\(a + 0 = a\)
Example:
\(15 + 0 = 15\)
\(0 + 23 = 23\)
Zero doesn't change the number! ✓
🧮 Solve Using Properties of Addition
How to Use Properties to Solve:
Properties help us make addition easier by rearranging numbers!
Strategy 1: Using Commutative Property
Problem: \(28 + 15 + 2\)
Strategy: Rearrange to make it easier!
\(28 + 2 + 15\) (Commutative Property)
\(30 + 15 = 45\) ✓
Why? 28 + 2 = 30, which is easier!
Strategy 2: Using Associative Property
Problem: \(6 + (4 + 17)\)
Strategy: Regroup to make tens!
\((6 + 4) + 17\) (Associative Property)
\(10 + 17 = 27\) ✓
Why? 6 + 4 = 10, which is easier!
✖️ Properties of Multiplication
1. Commutative Property of Multiplication
You can multiply numbers in any order, and the product stays the same!
\(a \times b = b \times a\)
Example:
\(4 \times 6 = 24\)
\(6 \times 4 = 24\)
Both give the same answer! ✓
2. Associative Property of Multiplication
When multiplying three or more numbers, you can group them in any way, and the product stays the same!
\((a \times b) \times c = a \times (b \times c)\)
Example:
\((2 \times 3) \times 4 = 6 \times 4 = 24\)
\(2 \times (3 \times 4) = 2 \times 12 = 24\)
Both ways give the same answer! ✓
3. Identity Property of Multiplication
When you multiply any number by one, you get the same number!
\(a \times 1 = a\)
Example:
\(12 \times 1 = 12\)
\(1 \times 45 = 45\)
One doesn't change the number! ✓
4. Zero Property of Multiplication
When you multiply any number by zero, you always get zero!
\(a \times 0 = 0\)
Example:
\(25 \times 0 = 0\)
\(0 \times 100 = 0\)
Zero makes everything zero! ✓
🎲 Solve Using Properties of Multiplication
How to Use Properties to Solve:
Properties help us make multiplication easier by rearranging numbers!
Strategy 1: Using Commutative Property
Problem: \(2 \times 8 \times 5\)
Strategy: Rearrange to make it easier!
\(2 \times 5 \times 8\) (Commutative Property)
\(10 \times 8 = 80\) ✓
Why? 2 × 5 = 10, which is easier!
Strategy 2: Using Associative Property
Problem: \(4 \times (25 \times 3)\)
Strategy: Regroup to make hundreds!
\((4 \times 25) \times 3\) (Associative Property)
\(100 \times 3 = 300\) ✓
Why? 4 × 25 = 100, which is easier!
🔀 Distributive Property
What is the Distributive Property?
The distributive property lets us break apart multiplication problems to make them easier!
We can "distribute" (share) the multiplication to each number inside the parentheses!
Distributive Property Formula:
\(a \times (b + c) = (a \times b) + (a \times c)\)
OR
\(a \times (b - c) = (a \times b) - (a \times c)\)
Example with Addition:
Problem: \(3 \times (4 + 2)\)
Method 1: Add first, then multiply
\(3 \times (4 + 2) = 3 \times 6 = 18\) ✓
Method 2: Use Distributive Property
\(3 \times (4 + 2) = (3 \times 4) + (3 \times 2)\)
\(= 12 + 6 = 18\) ✓
Both methods give the same answer!
Example with Subtraction:
Problem: \(5 \times (8 - 3)\)
Method 1: Subtract first, then multiply
\(5 \times (8 - 3) = 5 \times 5 = 25\) ✓
Method 2: Use Distributive Property
\(5 \times (8 - 3) = (5 \times 8) - (5 \times 3)\)
\(= 40 - 15 = 25\) ✓
Both methods give the same answer!
❓ Distributive Property: Find the Missing Factor
How to Find the Missing Factor:
Sometimes we need to find the missing number in a distributive property equation!
Example 1: Find the Missing Outside Factor
Problem: \(? \times (3 + 5) = 24\)
Step 1: Simplify inside parentheses: \(3 + 5 = 8\)
Step 2: Now we have: \(? \times 8 = 24\)
Step 3: Divide: \(24 \div 8 = 3\)
Answer: The missing factor is 3 ✓
Check: \(3 \times (3 + 5) = 3 \times 8 = 24\) ✓
Example 2: Find the Missing Addend
Problem: \(4 \times (6 + ?) = 32\)
Step 1: Divide both sides by 4: \(32 \div 4 = 8\)
Step 2: Now: \(6 + ? = 8\)
Step 3: Subtract: \(8 - 6 = 2\)
Answer: The missing addend is 2 ✓
Check: \(4 \times (6 + 2) = 4 \times 8 = 32\) ✓
✖️ Multiply Using the Distributive Property
Breaking Apart Numbers:
The distributive property helps us multiply big numbers by breaking them into smaller parts!
Example 1: Using Place Value
Problem: \(7 \times 13\)
Step 1: Break 13 into 10 + 3
\(7 \times 13 = 7 \times (10 + 3)\)
Step 2: Distribute the 7
\(= (7 \times 10) + (7 \times 3)\)
Step 3: Multiply
\(= 70 + 21\)
Step 4: Add
\(= 91\) ✓
Example 2: Breaking into Easier Parts
Problem: \(6 \times 18\)
Step 1: Break 18 into 10 + 8
\(6 \times 18 = 6 \times (10 + 8)\)
Step 2: Distribute the 6
\(= (6 \times 10) + (6 \times 8)\)
Step 3: Multiply
\(= 60 + 48\)
Step 4: Add
\(= 108\) ✓
🔗 Relate Addition and Multiplication
How are They Related?
Multiplication is repeated addition! When you add the same number over and over, that's multiplication!
\(\text{Multiplication} = \text{Repeated Addition}\)
Examples:
Example 1:
Addition: \(4 + 4 + 4 + 4 + 4 = 20\)
(Adding 4 five times)
Multiplication: \(5 \times 4 = 20\)
(5 groups of 4)
Same answer! ✓
Example 2:
Addition: \(7 + 7 + 7 = 21\)
(Adding 7 three times)
Multiplication: \(3 \times 7 = 21\)
(3 groups of 7)
Same answer! ✓
Conversion Formula:
\(a \times b = \underbrace{b + b + b + ... + b}_{a \text{ times}}\)
🔗 Relate Multiplication and Division
How are They Related?
Multiplication and division are opposite operations! They undo each other!
We call them inverse operations or fact families!
Fact Family Relationship:
If \(a \times b = c\)
Then \(c \div a = b\)
And \(c \div b = a\)
Example: Fact Family for 3, 4, and 12
Multiplication Facts:
\(3 \times 4 = 12\)
\(4 \times 3 = 12\)
Division Facts:
\(12 \div 3 = 4\)
\(12 \div 4 = 3\)
All four equations use the same three numbers!
Using Multiplication to Check Division:
Division Problem: \(35 \div 7 = ?\)
Think: What times 7 equals 35?
\(? \times 7 = 35\)
\(5 \times 7 = 35\)
Answer: \(35 \div 7 = 5\) ✓
Check: \(5 \times 7 = 35\) ✓
Using Division to Check Multiplication:
Multiplication Problem: \(6 \times 8 = 48\)
Check with division:
\(48 \div 6 = 8\) ✓
OR
\(48 \div 8 = 6\) ✓
If division gives the other factor, multiplication is correct!
📝 Important Properties Summary
Addition Properties:
Commutative: \(a + b = b + a\)
Associative: \((a + b) + c = a + (b + c)\)
Identity: \(a + 0 = a\)
Multiplication Properties:
Commutative: \(a \times b = b \times a\)
Associative: \((a \times b) \times c = a \times (b \times c)\)
Identity: \(a \times 1 = a\)
Zero Property: \(a \times 0 = 0\)
Distributive Property:
\(a \times (b + c) = (a \times b) + (a \times c)\)
\(a \times (b - c) = (a \times b) - (a \times c)\)
Relationships:
Multiplication = Repeated Addition
Division = Inverse of Multiplication
💡 Quick Learning Tips
- ✓ Addends add up to a sum
- ✓ Factors multiply to a product
- ✓ Dividend ÷ Divisor = Quotient
- ✓ Parentheses ( ) mean "solve me first!"
- ✓ Commutative: You can change the order
- ✓ Associative: You can change the grouping
- ✓ Identity for addition: +0 doesn't change the number
- ✓ Identity for multiplication: ×1 doesn't change the number
- ✓ Zero Property: Any number × 0 = 0
- ✓ Distributive Property breaks apart multiplication
- ✓ Use properties to make problems easier!
- ✓ Multiplication is repeated addition
- ✓ Division is the opposite of multiplication
- ✓ Fact families use the same three numbers
- ✓ Check division with multiplication!
