Solved Exercises on Checking Work with Inverse Operations in Grade 3

Master inverse operations: addition/subtraction, multiplication/division, and verification methods through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition and Subtraction Inverses
Exercise 1
Solve: \(25 + 17 = ?\) Then check your answer using inverse operations.
Definition:

Inverse operations: Operations that undo each other (addition and subtraction are inverse operations)

Method for checking with inverse operations:
  1. Solve the original problem
  2. Take the result and perform the inverse operation with one of the original numbers
  3. If you get back to the other original number, your answer is correct
Original
\(25 + 17 = 42\)
Check 1
\(42 - 25 = 17\)
Check 2
\(42 - 17 = 25\)
Result
✓ Correct
Operation Result Verification
\(25 + 17\) 42 \(42 - 17 = 25\) ✓
\(42 - 25\) 17 \(17 = 17\) ✓
Step 1: Solve the original problem

25 + 17 = 42

Step 2: Check using inverse operation

Take the result (42) and subtract one of the original numbers (25)

42 - 25 = 17

Since we get back to the other original number (17), our answer is correct

Step 3: Double-check

We can also check: 42 - 17 = 25

Since we get back to the other original number (25), our answer is confirmed

\(25 + 17 = 42\)
Final answer:

\(25 + 17 = 42\) (verified by inverse operations)

Applied rules:

• Inverse relationship: Addition and subtraction are inverse operations

• Verification principle: If \(a + b = c\), then \(c - b = a\) and \(c - a = b\)

• Checking method: Use inverse operations to verify answers

Tip 1: Always check your addition by subtracting one of the addends from the sum.
Tip 2: Inverse operations "undo" each other, bringing you back to where you started.
2 Subtraction and Addition Inverses
Exercise 2
Solve: \(56 - 23 = ?\) Then check your answer using inverse operations.
Definition:

Subtraction inverse: Adding the result to the subtracted number returns the original number

Original
\(56 - 23 = 33\)
Check
\(33 + 23 = 56\)
Alternative Check
\(56 - 33 = 23\)
Result
✓ Correct
Operation Result Verification
\(56 - 23\) 33 \(33 + 23 = 56\) ✓
\(33 + 23\) 56 \(56 = 56\) ✓
Step 1: Solve the original problem

56 - 23 = 33

Step 2: Check using inverse operation

Take the result (33) and add the subtracted number (23)

33 + 23 = 56

Since we get back to the original number (56), our answer is correct

Step 3: Alternative check

We can also check: 56 - 33 = 23

Since we get back to the subtracted number (23), our answer is confirmed

\(56 - 23 = 33\)
Final answer:

\(56 - 23 = 33\) (verified by inverse operations)

Applied rules:

• Inverse relationship: Subtraction and addition are inverse operations

• Verification principle: If \(a - b = c\), then \(c + b = a\) and \(a - c = b\)

• Checking method: Use inverse operations to verify answers

Tip 1: Always check your subtraction by adding the subtracted number to the result.
Tip 2: The inverse operation should bring you back to your starting point.
3 Multiplication and Division Inverses
Exercise 3
Solve: \(7 × 8 = ?\) Then check your answer using inverse operations.
Definition:

Multiplication inverse: Dividing the product by one of the factors returns the other factor

Original
\(7 × 8 = 56\)
Check 1
\(56 ÷ 7 = 8\)
Check 2
\(56 ÷ 8 = 7\)
Result
✓ Correct
Operation Result Verification
\(7 × 8\) 56 \(56 ÷ 8 = 7\) ✓
\(56 ÷ 7\) 8 \(8 = 8\) ✓
Step 1: Solve the original problem

7 × 8 = 56

Step 2: Check using inverse operation

Take the result (56) and divide by one of the original numbers (7)

56 ÷ 7 = 8

Since we get back to the other original number (8), our answer is correct

Step 3: Double-check

We can also check: 56 ÷ 8 = 7

Since we get back to the other original number (7), our answer is confirmed

\(7 × 8 = 56\)
Final answer:

\(7 × 8 = 56\) (verified by inverse operations)

Applied rules:

• Inverse relationship: Multiplication and division are inverse operations

• Verification principle: If \(a × b = c\), then \(c ÷ b = a\) and \(c ÷ a = b\)

• Checking method: Use inverse operations to verify answers

Tip 1: Always check your multiplication by dividing the product by one of the factors.
Tip 2: Inverse operations work the same way for multiplication and division.
Rules and methods, laws,...
\(a + b = c \Rightarrow c - b = a\)
Addition/Subtraction Inverse
\(a × b = c \Rightarrow c ÷ b = a\)
Multiplication/Division Inverse
Pair 1
\(\text{Addition/ Subtraction}\)
Inverse pair
Pair 2
\(\text{Multiplication/ Division}\)
Inverse pair
Property
\(\text{Verification}\)
Undo operations
Key definitions:

Inverse operations: Operations that undo each other (addition and subtraction, multiplication and division)

Addition and subtraction: Inverse operations that undo each other

Multiplication and division: Inverse operations that undo each other

Verification: Using inverse operations to check if an answer is correct

Undo: Reversing an operation to return to the original value

Fact family: Related operations using the same numbers

Reciprocal relationship: Operations that reverse each other's effects

Self-checking: Using inverse operations to verify your own work

Complete methodology for checking with inverse operations:
  1. Solve the original problem - Perform the initial operation
  2. Identify the inverse operation - Determine which operation undoes the original
  3. Apply the inverse operation - Use the result and one of the original numbers
  4. Verify the result - Check if you get back to the other original number
  5. Confirm correctness - If the verification works, the original answer is correct
Tip 1: Addition and subtraction are inverse operations that undo each other.
Tip 2: Multiplication and division are inverse operations that undo each other.
Tip 3: Always verify your answer using the inverse operation.
Tip 4: The inverse operation should bring you back to where you started.
Common errors: Forgetting to use the inverse operation, applying the wrong inverse, not checking the verification result, misidentifying the inverse operation.
Memory aid: "Add and subtract undo each other; multiply and divide undo each other."
Essential rules:

• Addition/subtraction inverse: If \(a + b = c\), then \(c - b = a\) and \(c - a = b\)

• Multiplication/division inverse: If \(a × b = c\), then \(c ÷ b = a\) and \(c ÷ a = b\)

• Verification rule: Use inverse operations to check your work

• Undo principle: Inverse operations reverse the effect of the original operation

Solution: Exercises 4 to 5
4 Division and Multiplication Inverses
Exercise 4
Solve: \(48 ÷ 6 = ?\) Then check your answer using inverse operations.
Definition:

Division inverse: Multiplying the quotient by the divisor returns the dividend

Original
\(48 ÷ 6 = 8\)
Check
\(8 × 6 = 48\)
Alternative Check
\(48 ÷ 8 = 6\)
Result
✓ Correct
Operation Result Verification
\(48 ÷ 6\) 8 \(8 × 6 = 48\) ✓
\(8 × 6\) 48 \(48 = 48\) ✓
Step 1: Solve the original problem

48 ÷ 6 = 8

Step 2: Check using inverse operation

Take the result (8) and multiply by the divisor (6)

8 × 6 = 48

Since we get back to the original number (48), our answer is correct

Step 3: Alternative check

We can also check: 48 ÷ 8 = 6

Since we get back to the divisor (6), our answer is confirmed

\(48 ÷ 6 = 8\)
Final answer:

\(48 ÷ 6 = 8\) (verified by inverse operations)

Applied rules:

• Inverse relationship: Division and multiplication are inverse operations

• Verification principle: If \(a ÷ b = c\), then \(c × b = a\) and \(a ÷ c = b\)

• Checking method: Use inverse operations to verify answers

Tip 1: Always check your division by multiplying the quotient by the divisor.
Tip 2: The inverse operation should return you to the original dividend.
5 Multi-Step Verification
Exercise 5
Solve: \(15 + 24 = ?\) Then check using inverse operations. Next, subtract 10 from your answer and check that result too.
Definition:

Multi-step verification: Using inverse operations to check multiple calculations in sequence

Step 1
\(15 + 24 = 39\)
Step 1 Check
\(39 - 24 = 15\)
Step 2
\(39 - 10 = 29\)
Step 2 Check
\(29 + 10 = 39\)
Step Operation Result Verification
1 \(15 + 24\) 39 \(39 - 24 = 15\) ✓
2 \(39 - 10\) 29 \(29 + 10 = 39\) ✓
Step 1: Solve the first operation

15 + 24 = 39

Step 2: Check the first operation

Take the result (39) and subtract one of the original numbers (24)

39 - 24 = 15 ✓ (We get back to the other original number)

Step 3: Solve the second operation

39 - 10 = 29

Step 4: Check the second operation

Take the result (29) and add the subtracted number (10)

29 + 10 = 39 ✓ (We get back to the previous result)

Step 5: Final verification

Both operations have been verified using inverse operations

\(15 + 24 = 39\), \(39 - 10 = 29\)
Final answer:

\(15 + 24 = 39\) and \(39 - 10 = 29\) (both verified by inverse operations)

Applied rules:

• Sequential verification: Check each step individually using inverse operations

• Multi-step principle: Apply inverse operations to each calculation in sequence

• Chain verification: Verify the connection between multiple calculations

Tip 1: When solving multi-step problems, verify each step individually.
Tip 2: Use inverse operations after each step to catch errors early.
Key Concepts, Laws, Methods, and Definitions for Inverse Operations
\(a + b = c \Rightarrow c - b = a \text{ and } c - a = b\)
Addition/Subtraction Inverse Relationship
\(a × b = c \Rightarrow c ÷ b = a \text{ and } c ÷ a = b\)
Multiplication/Division Inverse Relationship
Key definitions:

Inverse operations: Operations that undo each other (addition and subtraction, multiplication and division)

Addition and subtraction: Inverse operations that cancel each other out

Multiplication and division: Inverse operations that cancel each other out

Verification: Using inverse operations to check if an answer is correct

Undo: Reversing an operation to return to the original value

Fact family: Related operations using the same numbers

Reciprocal relationship: Operations that reverse each other's effects

Self-checking: Using inverse operations to verify your own work

Verification principle: If an operation is correct, its inverse should return the original values

Complete methodology for inverse operations:
  1. Step 1: Perform the original operation - Calculate the result
  2. Step 2: Identify the inverse operation - Determine which operation undoes the original
  3. Step 3: Apply the inverse operation - Use the result and one of the original numbers
  4. Step 4: Verify the result - Check if you get back to the other original number
  5. Step 5: Confirm correctness - If the verification works, the original answer is correct
  6. Step 6: State the conclusion - Confirm the answer is verified
Tip 1: Addition and subtraction are inverse operations that undo each other.
Tip 2: Multiplication and division are inverse operations that undo each other.
Tip 3: Always verify your answer using the inverse operation.
Tip 4: The inverse operation should bring you back to your starting point.
Tip 5: For multi-step problems, verify each step individually.
Common errors to avoid: Forgetting to use inverse operations, applying the wrong inverse, not checking the verification result, misidentifying which numbers to use in the inverse operation.
Exam preparation: Practice using inverse operations for all four basic operations, work on multi-step verification, and develop the habit of checking your work.
Essential formulas and rules:

• Addition/subtraction inverse: If \(a + b = c\), then \(c - b = a\) and \(c - a = b\)

• Multiplication/division inverse: If \(a × b = c\), then \(c ÷ b = a\) and \(c ÷ a = b\)

• Verification rule: Use inverse operations to check your work

• Undo principle: Inverse operations reverse the effect of the original operation

• Self-checking: Always verify your answers using inverse operations

Detailed Summary: Checking Work with Inverse Operations

Definitions of Key Concepts:
Inverse operations: Operations that undo each other (e.g., 5 + 3 = 8, and 8 - 3 = 5).
Addition and subtraction: Inverse operations that cancel each other out.
Multiplication and division: Inverse operations that cancel each other out.
Verification: Using inverse operations to check if an answer is correct.
Undo: Reversing an operation to return to the original value.
Fact family: Related operations using the same numbers (e.g., 3, 4, 7: 3+4=7, 4+3=7, 7-3=4, 7-4=3).
Reciprocal relationship: Operations that reverse each other's effects.
Self-checking: Using inverse operations to verify your own work.
Core Rules and Principles:
Addition/Subtraction Rule: If a + b = c, then c - b = a and c - a = b (e.g., if 5 + 3 = 8, then 8 - 3 = 5 and 8 - 5 = 3).
Multiplication/Division Rule: If a × b = c, then c ÷ b = a and c ÷ a = b (e.g., if 4 × 5 = 20, then 20 ÷ 5 = 4 and 20 ÷ 4 = 5).
Verification Principle: The inverse operation should bring you back to your original number.
Undo Principle: Inverse operations reverse the effect of the original operation.
Checking Method: Always verify your answer using the inverse operation.
Step-by-Step Methods and Procedures:
Basic Verification Method: 1) Perform the original operation, 2) Take the result, 3) Apply the inverse operation with one of the original numbers, 4) Verify you get back to the other original number.
Addition Verification: 1) Calculate a + b = c, 2) Check: c - b = a or c - a = b.
Subtraction Verification: 1) Calculate a - b = c, 2) Check: c + b = a.
Multiplication Verification: 1) Calculate a × b = c, 2) Check: c ÷ b = a or c ÷ a = b.
Division Verification: 1) Calculate a ÷ b = c, 2) Check: c × b = a.
Concrete Examples:
Simple Example: 6 + 4 = 10, check: 10 - 4 = 6 ✓.
Intermediate Example: 7 × 8 = 56, check: 56 ÷ 8 = 7 ✓.
Advanced Example: 45 - 18 = 27, check: 27 + 18 = 45 ✓.
Tips, Tricks, and Common Pitfalls:
Tip 1: Always check your work using inverse operations.
Tip 2: The inverse operation should return you to your starting number.
Tip 3: For multiplication, divide by either factor to check.
Tip 4: For division, multiply by the divisor to check.
Pitfall 1: Forgetting to verify your answer using inverse operations.
Pitfall 2: Using the wrong number in the inverse operation.
Pitfall 3: Not confirming that the verification actually returns the original number.
Key Notes for Memorization:
Mnemonic: "Add and subtract undo each other; multiply and divide undo each other."
Mnemonic: "To check your work, do the opposite operation."
Key Point: Inverse operations always bring you back to where you started.
Key Point: Verification with inverse operations is a powerful tool for catching errors.
Explanation for Students:
Inverse operations are like opposites that cancel each other out! If you add 3 to a number, you can subtract 3 to get back to where you started. If you multiply by 4, you can divide by 4 to get back to where you started. This is super helpful for checking your work. When you solve a math problem, you can use the inverse operation to see if your answer is right. For example, if you calculate 6 + 4 = 10, you can check by doing 10 - 4 = 6. Since you get back to your original number (6), you know your answer was correct! It's like a built-in error checker for your math.