Solved Exercises on Properties of Operations Activities for Grade 4

Master properties of operations through engaging activities: commutative, associative, distributive properties, and operational strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Commutative property activity
Exercise 1
Activity: Use the commutative property to make addition easier: 7 + 18 + 3
Definition:

Commutative property: Numbers can be added or multiplied in any order without changing the result. For addition: a + b = b + a. For multiplication: a × b = b × a.

Commutative property strategy:
  1. Look for numbers that add up to round numbers (like 10, 20, etc.)
  2. Rearrange numbers using the commutative property
  3. Add the round numbers first
  4. Complete the calculation
Original
7 + 18 + 3
Rearranged
7 + 3 + 18
Result
28
Step 1: Identify beneficial pairs

7 + 3 = 10 (round number), so we want to group these together

Step 2: Use commutative property

7 + 18 + 3 = 7 + 3 + 18 (we switched the order of 18 and 3)

Step 3: Add round numbers first

7 + 3 = 10

Step 4: Complete the calculation

10 + 18 = 28

7 + 18 + 3 = 28
Final answer:

By using the commutative property to rearrange numbers, we made the calculation easier: 7 + 18 + 3 = 7 + 3 + 18 = 10 + 18 = 28

Applied rules:

• Commutative property: Order doesn't matter in addition

• Strategic pairing: Look for numbers that make round numbers

• Mental math: Rearranging makes calculations easier

2 Associative property activity
Exercise 2
Activity: Use the associative property to make multiplication easier: 4 × 5 × 2
Definition:

Associative property: Numbers can be grouped differently when adding or multiplying without changing the result. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c).

Original
4 × 5 × 2
Regrouped
(4 × 5) × 2 or 4 × (5 × 2)
Result
40
Step 1: Identify beneficial grouping

5 × 2 = 10 (round number), so group these together

Step 2: Use associative property

4 × 5 × 2 = 4 × (5 × 2) = 4 × 10

We grouped 5 and 2 together first

Step 3: Complete the calculation

4 × 10 = 40

Step 4: Verify with different grouping

(4 × 5) × 2 = 20 × 2 = 40 ✓

4 × 5 × 2 = 40
Final answer:

Using the associative property to group 5 and 2 first: 4 × 5 × 2 = 4 × (5 × 2) = 4 × 10 = 40

Applied rules:

• Associative property: Grouping doesn't matter in multiplication

• Strategic grouping: Look for pairs that make round numbers

• Mental math: Regrouping simplifies calculations

3 Distributive property activity
Exercise 3
Activity: Use the distributive property to solve: 6 × 17 mentally
Definition:

Distributive property: Multiplication can be distributed over addition or subtraction: a × (b + c) = (a × b) + (a × c) and a × (b - c) = (a × b) - (a × c).

Original
6 × 17
Decomposed
6 × (10 + 7)
Distributed
(6 × 10) + (6 × 7)
Step 1: Decompose the difficult number

17 = 10 + 7 (breaking into easier parts)

Step 2: Rewrite using distributive property

6 × 17 = 6 × (10 + 7)

Step 3: Apply distributive property

6 × (10 + 7) = (6 × 10) + (6 × 7)

Step 4: Calculate easier multiplications

6 × 10 = 60 and 6 × 7 = 42

Step 5: Add the results

60 + 42 = 102

6 × 17 = 102
Final answer:

Using the distributive property: 6 × 17 = 6 × (10 + 7) = (6 × 10) + (6 × 7) = 60 + 42 = 102

Applied rules:

• Distributive property: Break down difficult multiplication

• Number decomposition: Split numbers into easier parts

• Mental math: Makes complex calculations manageable

Properties of Operations Guide
\(a × (b + c) = (a × b) + (a × c)\)
Distributive Property
Commutative
a + b = b + a
Order doesn't matter
Commutative
a × b = b × a
Order doesn't matter
Associative
(a + b) + c = a + (b + c)
Grouping doesn't matter
Associative
(a × b) × c = a × (b × c)
Grouping doesn't matter
Distributive
a × (b + c) = (a × b) + (a × c)
Distribute multiplication over addition
Activity Tip
Look for round numbers
10, 20, 100, etc.
Strategy
Decompose large numbers
Break into smaller parts
Solution: Exercises 4 to 5
4 Combined properties activity
Exercise 4
Activity: Solve 8 × 25 × 4 using multiple properties to make it easier
Definition:

Combined properties: Using multiple properties together to simplify complex calculations. Often involves both associative and commutative properties.

Original
8 × 25 × 4
Rearranged
8 × 4 × 25
Regrouped
(8 × 4) × 25
Result
800
Step 1: Use commutative property

8 × 25 × 4 = 8 × 4 × 25 (move 4 next to 8)

8 × 4 is easier to calculate than 25 × 4

Step 2: Use associative property

8 × 4 × 25 = (8 × 4) × 25

Group 8 and 4 together first

Step 3: Calculate the easier multiplication

8 × 4 = 32

Step 4: Complete the calculation

32 × 25 = 800

Method: 32 × 25 = 32 × (20 + 5) = (32 × 20) + (32 × 5) = 640 + 160 = 800

8 × 25 × 4 = 800
Final answer:

Using combined properties: 8 × 25 × 4 = 8 × 4 × 25 = (8 × 4) × 25 = 32 × 25 = 800

Applied rules:

• Commutative property: Rearrange for easier calculation

• Associative property: Group numbers strategically

• Multiple strategies: Combine properties for maximum efficiency

5 Word problem with properties
Exercise 5
Activity: Sarah has 3 boxes with 8 red marbles and 4 blue marbles each. How many marbles does she have in total?
Definition:

Word problems with properties: Real-world situations that can be solved using mathematical properties to simplify calculations.

Problem
3 × (8 + 4)
Distributed
(3 × 8) + (3 × 4)
Result
36
Step 1: Identify the structure

Each box has 8 red + 4 blue = 12 marbles

There are 3 boxes

Total = 3 × (8 + 4)

Step 2: Apply distributive property

3 × (8 + 4) = (3 × 8) + (3 × 4)

This gives us total red marbles + total blue marbles

Step 3: Calculate each part

Red marbles: 3 × 8 = 24

Blue marbles: 3 × 4 = 12

Step 4: Add the results

24 + 12 = 36 total marbles

3 × (8 + 4) = 36 marbles
Final answer:

Sarah has 36 marbles in total. Using the distributive property: 3 × (8 + 4) = (3 × 8) + (3 × 4) = 24 + 12 = 36

Applied rules:

• Real-world application: Properties apply to practical situations

• Problem translation: Convert words to mathematical expressions

• Distributive property: Useful for calculating totals in groups

Properties of Operations Activities Guide
\(a × (b + c) = (a × b) + (a × c)\)
Distributive Property Activity
Key definitions:

Commutative property: Numbers can be added or multiplied in any order without changing the result

Associative property: Numbers can be grouped differently when adding or multiplying without changing the result

Distributive property: Multiplication can be distributed over addition or subtraction

Mental math strategies: Techniques that use properties to simplify calculations

Activity-based learning methods:
  1. Identify patterns: Look for numbers that work well together
  2. Apply properties: Use commutative, associative, or distributive as needed
  3. Simplify: Make calculations easier by strategic rearrangement
  4. Verify: Check that the result is the same as the original
Tip 1: Look for pairs that make round numbers: 7 + 3 = 10, 6 + 4 = 10, 8 + 2 = 10.
Tip 2: In multiplication, look for pairs that make 10: 2 × 5 = 10, 4 × 25 = 100.
Tip 3: Break large numbers into smaller parts: 17 = 10 + 7, 24 = 20 + 4.
Tip 4: Practice with real-world examples to see how properties apply to daily life.
Common error: Applying properties to subtraction or division (they don't work!).
Key insight: Properties make mental math much faster and easier.
Properties to remember:

• Commutative: a + b = b + a and a × b = b × a

• Associative: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)

• Distributive: a × (b + c) = (a × b) + (a × c)

• Only work for addition and multiplication, not subtraction or division

Properties Effectiveness in Mental Math
Exercise 6: Strategy Efficiency
Compare the effectiveness of different properties in mental math:
Commutative, Associative, and Distributive properties

Analysis: The chart shows how different properties enhance mental math abilities.

  • Commutative property: Most effective for addition with friendly numbers
  • Associative property: Very effective for multiplication with round numbers
  • Distributive property: Highly effective for breaking down complex problems

Questions & Answers

Question: Why do we need to learn these properties? Can't I just solve problems the normal way?

Answer: These properties are incredibly useful because they make math faster and easier! Here's why:

  • They help you do mental math quickly (like 6 × 17 = 6 × (10 + 7) = 60 + 42 = 102)
  • They make complex problems simpler by breaking them down
  • They help you check your work by solving problems in different ways
  • They form the foundation for more advanced math in higher grades

For example, instead of calculating 7 + 18 + 3 the long way, you can use the commutative property to rearrange: 7 + 3 + 18 = 10 + 18 = 28. Much easier!

Question: Do these properties work for subtraction and division too?

Answer: No, these properties only work for addition and multiplication. Let me show you why:

  • Commutative doesn't work for subtraction: 8 - 3 = 5, but 3 - 8 = -5. These are different!
  • Commutative doesn't work for division: 12 ÷ 4 = 3, but 4 ÷ 12 = 1/3. These are different!
  • Associative doesn't work for subtraction: (10 - 5) - 2 = 5 - 2 = 3, but 10 - (5 - 2) = 10 - 3 = 7. Different results!
  • Associative doesn't work for division: (16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2, but 16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8. Different results!

So remember: properties of operations only apply to addition and multiplication!

Question: How do I know which property to use when solving a problem?

Answer: Here's a simple decision-making guide:

  • Commutative: Use when you can rearrange numbers to make friendlier pairs (like 7 + 3 = 10)
  • Associative: Use when you can group numbers to make round numbers (like 4 × 25 = 100)
  • Distributive: Use when you can break apart a large number into smaller parts (like 6 × 17 = 6 × (10 + 7))

Look for opportunities to make calculations easier. If you see 8 + 15 + 2, use commutative to get 8 + 2 + 15 = 10 + 15 = 25. If you see 4 × 7 × 25, use associative to get 4 × 25 × 7 = 100 × 7 = 700.