Solved Exercises on Properties of Operations Identity for Grade 4

Master identity properties: additive identity, multiplicative identity, and zero property through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Additive identity property
Exercise 1
Demonstrate the additive identity property with 15 + 0 and 0 + 23
Definition:

Additive identity property: Adding zero to any number leaves the number unchanged. For any number a: a + 0 = a and 0 + a = a. Zero is called the "additive identity."

Additive identity verification method:
  1. Take any number
  2. Add zero to it
  3. Observe that the result is the same as the original number
  4. Try adding zero first: 0 + number
Example 1
15 + 0 = 15
Example 2
0 + 23 = 23
Result
Identity preserved!
Step 1: Test first example

15 + 0 = 15

Adding zero to 15 gives back 15

Step 2: Test second example

0 + 23 = 23

Adding 23 to zero gives back 23

Step 3: Verify the pattern

In both cases, adding zero doesn't change the original number

15 + 0 = 15 and 0 + 23 = 23
Final answer:

The additive identity property holds: adding zero to any number preserves its value

Applied rules:

• Additive identity: Zero is the identity element for addition

• Preservation: The original number remains unchanged

• Commutative: Works both ways (a + 0 = 0 + a = a)

2 Multiplicative identity property
Exercise 2
Demonstrate the multiplicative identity property with 12 × 1 and 1 × 35
Definition:

Multiplicative identity property: Multiplying any number by one leaves the number unchanged. For any number a: a × 1 = a and 1 × a = a. One is called the "multiplicative identity."

Example 1
12 × 1 = 12
Example 2
1 × 35 = 35
Result
Identity preserved!
Step 1: Test first example

12 × 1 = 12

Multiplying 12 by 1 gives back 12

Step 2: Test second example

1 × 35 = 35

Multiplying 35 by 1 gives back 35

Step 3: Understand the concept

One is the "do nothing" number for multiplication

Step 4: Verify with other numbers

Try with any number: 100 × 1 = 100, 1 × 7 = 7

12 × 1 = 12 and 1 × 35 = 35
Final answer:

The multiplicative identity property holds: multiplying any number by one preserves its value

Applied rules:

• Multiplicative identity: One is the identity element for multiplication

• Preservation: The original number remains unchanged

• Commutative: Works both ways (a × 1 = 1 × a = a)

3 Zero property of multiplication
Exercise 3
Demonstrate the zero property with 8 × 0, 0 × 15, and 24 × 0 × 5
Definition:

Zero property of multiplication: Any number multiplied by zero equals zero. For any number a: a × 0 = 0 and 0 × a = 0. If zero appears anywhere in a multiplication, the result is zero.

Example 1
8 × 0 = 0
Example 2
0 × 15 = 0
Example 3
24 × 0 × 5 = 0
Step 1: Test simple cases

8 × 0 = 0 (any number times 0 is 0)

0 × 15 = 0 (0 times any number is 0)

Step 2: Test multiple numbers with zero

24 × 0 × 5 = 0

As soon as we multiply by 0, the whole result becomes 0

Step 3: Understand the concept

Zero is like a "reset button" in multiplication - once it appears, everything becomes 0

Step 4: Verify the pattern

No matter how many numbers are being multiplied, if one of them is 0, the result is 0

8 × 0 = 0, 0 × 15 = 0, 24 × 0 × 5 = 0
Final answer:

The zero property holds: any number multiplied by zero equals zero, regardless of other factors

Applied rules:

• Zero dominance: Zero "takes over" in multiplication

• Universal: Applies to any number multiplied by zero

• Chain rule: If zero appears anywhere in a multiplication chain, result is zero

Identity Properties Summary
\(a × 1 = a\)
Multiplicative Identity
Additive Identity
a + 0 = a
Adding zero doesn't change the number
Additive Identity
0 + a = a
Adding zero doesn't change the number
Multiplicative Identity
a × 1 = a
Multiplying by one doesn't change the number
Multiplicative Identity
1 × a = a
Multiplying by one doesn't change the number
Zero Property
a × 0 = 0
Any number times zero equals zero
Zero Property
0 × a = 0
Any number times zero equals zero
Special Case
a × 0 × b = 0
Zero in any position results in zero
Identity Pattern
a + 0 + b = a + b
Adding zero doesn't affect other numbers
Solution: Exercises 4 to 5
4 Mixed identity operations
Exercise 4
Solve: (18 + 0) × 1 + 5 × 0 and identify which identity properties apply
Definition:

Mixed identity operations: Problems that combine multiple identity properties. We solve step by step, applying the correct identity at each stage.

Original
(18 + 0) × 1 + 5 × 0
After identities
18 × 1 + 0
Result
18
Step 1: Apply additive identity

18 + 0 = 18 (adding zero doesn't change the number)

Step 2: Apply zero property

5 × 0 = 0 (any number times zero equals zero)

Step 3: Substitute simplified values

(18 + 0) × 1 + 5 × 0 = 18 × 1 + 0

Step 4: Apply multiplicative identity

18 × 1 = 18 (multiplying by one doesn't change the number)

Step 5: Complete the calculation

18 × 1 + 0 = 18 + 0 = 18

(18 + 0) × 1 + 5 × 0 = 18
Final answer:

Using identity properties: (18 + 0) × 1 + 5 × 0 = 18 × 1 + 0 = 18 + 0 = 18

Applied rules:

• Additive identity: 18 + 0 = 18

• Zero property: 5 × 0 = 0

• Multiplicative identity: 18 × 1 = 18

• Order of operations: Solve parentheses first

5 Identity properties in word problems
Exercise 5
Sarah has 12 apples. She doesn't buy any more apples (adds 0) and then triples her collection by multiplying by 1. Later, she tries to share with 0 friends. How many apples does she have?
Definition:

Word problems with identity properties: Real-world scenarios that demonstrate how identity properties work in practical situations.

Start
12 apples
Adds 0
12 + 0 = 12
Multiplies by 1
12 × 1 = 12
Shares with 0
12 × 0 = 0
Step 1: Initial amount

Sarah starts with 12 apples

Step 2: Adds 0 apples

12 + 0 = 12 (additive identity: adding zero doesn't change the amount)

Step 3: Multiplies by 1

12 × 1 = 12 (multiplicative identity: multiplying by one doesn't change the amount)

Step 4: Tries to share with 0 friends

12 × 0 = 0 (zero property: any number times zero equals zero)

Step 5: Interpret the result

Sharing with 0 friends means multiplying by 0, so Sarah ends up with 0 apples

Sarah ends up with 0 apples
Final answer:

Sarah has 0 apples after sharing with 0 friends. The zero property of multiplication makes the final result 0.

Applied rules:

• Real-world application: Identity properties appear in everyday situations

• Additive identity: Adding zero doesn't change quantity

• Multiplicative identity: Multiplying by one doesn't change quantity

• Zero property: Multiplying by zero results in zero

Properties of Operations Identity Guide
\(a × 1 = a\)
Multiplicative Identity Property
Key definitions:

Additive identity property: Adding zero to any number leaves it unchanged (a + 0 = a)

Multiplicative identity property: Multiplying any number by one leaves it unchanged (a × 1 = a)

Zero property of multiplication: Any number multiplied by zero equals zero (a × 0 = 0)

Identity element: A number that doesn't change other numbers when combined with them

Identity property recognition method:
  1. Look for zero in addition: If you see a + 0 or 0 + a, the answer is a
  2. Look for one in multiplication: If you see a × 1 or 1 × a, the answer is a
  3. Look for zero in multiplication: If you see a × 0 or 0 × a, the answer is 0
  4. Check mixed operations: Apply identities step by step
Tip 1: Zero is the "do nothing" number for addition.
Tip 2: One is the "do nothing" number for multiplication.
Tip 3: Zero "erases" everything in multiplication.
Tip 4: Practice with different numbers to see these patterns clearly.
Common error: Thinking that 0 × a = a (it's actually 0).
Memory aid: Remember: "Zero is a hero in addition but a zero in multiplication."
Properties to remember:

• Additive identity: a + 0 = 0 + a = a

• Multiplicative identity: a × 1 = 1 × a = a

• Zero property: a × 0 = 0 × a = 0

• Zero in multiplication chain: If any factor is 0, the product is 0

Identity Properties Impact on Calculations
Exercise 6: Identity Recognition Speed
Compare the speed of calculations with and without recognizing identity properties:
Regular calculation vs. identity property application

Analysis: The chart shows how recognizing identity properties speeds up calculations.

  • Additive identity: Saves time by avoiding unnecessary addition of zero
  • Multiplicative identity: Saves time by avoiding unnecessary multiplication by one
  • Zero property: Immediately gives result without full calculation

Questions & Answers

Question: Why is zero called the "additive identity" and one called the "multiplicative identity"? Why not call them something else?

Answer: The word "identity" means "who something is" or "what makes it itself." An "identity element" is a number that preserves the identity of another number when combined with it:

  • Zero is the "additive identity" because adding it to any number keeps that number's identity (unchanged)
  • One is the "multiplicative identity" because multiplying any number by it keeps that number's identity (unchanged)
  • These numbers don't change the original number's "personality" - they preserve its identity

It's like asking someone their identity and getting back the same person, not a transformed version!

Question: Does the zero property work if zero is in the middle of a multiplication, like 5 × 0 × 7?

Answer: Yes, absolutely! The zero property works no matter where zero appears in a multiplication. Here's why:

  • 5 × 0 × 7 = (5 × 0) × 7 = 0 × 7 = 0
  • Or you could think of it as: 5 × 0 × 7 = 5 × (0 × 7) = 5 × 0 = 0
  • As soon as you multiply by zero anywhere in the sequence, the entire result becomes zero

Zero is like a "contagion" in multiplication - if it touches any part of the multiplication, everything becomes zero!

Question: Are there identity properties for subtraction and division too?

Answer: There are identity elements for subtraction and division, but they work differently:

  • Subtraction identity: a - 0 = a (subtracting zero leaves the number unchanged)
  • Division identity: a ÷ 1 = a (dividing by one leaves the number unchanged)
  • However, note that 0 - a ≠ a and 1 ÷ a ≠ a (these don't preserve the original number)

For subtraction: zero works as an identity when it's the second number. For division: one works as an identity when it's the divisor.