Understanding Solutions for Grade 1 - Mathematics - Verification and Problem-Solving Guide

Master understanding solutions: verification, checking answers, accuracy, and problem-solving strategies through comprehensive exercises with detailed explanations.

Solutions: Exercises 1 to 10
1 Basic Verification
Exercise 1
You solved 5 - 2 = 3. How do you know your answer is correct?
Difficulty: Beginner Time: ~2 minutes Skills: Solution Verification
Definition:

Verification: Checking if your answer is correct

Substitution test: Adding the answer back to the subtracted number to see if you get the original number

Step-by-step verification method:
  1. Take your answer (3) and add it to the subtracted number (2)
  2. Check if the sum equals the original number (5)
  3. If yes, your answer is correct
Original Problem
5 - 2 = 3
Verification
3 + 2 = 5
Result
✓ Correct
Step 1: Identify your answer

Your answer is 3

Step 2: Add back the subtracted number

3 + 2 = 5

Step 3: Compare to original number

5 = 5, so your answer is correct

3 + 2 = 5 ✓
Final answer:

Your answer is correct because 3 + 2 = 5, which matches the original number.

Applied rules:

• Verification principle: Always check your answer

• Substitution test: Add your answer to the subtracted number

• Accuracy: Verification ensures correct solutions

• Practice Tip: Check every answer to build accuracy habits

Related Examples:
  • 4 - 1 = 3 → 3 + 1 = 4 ✓
  • 6 - 2 = 4 → 4 + 2 = 6 ✓
  • 7 - 3 = 4 → 4 + 3 = 7 ✓
Quick Tips:
  • Always add your answer back to check
  • If you get the original number, your answer is correct
  • This method works for all subtraction problems
Frequently Asked Questions:

Q: Why do I need to check my answer?
A: Checking helps you catch mistakes and makes sure you're right.

Q: Do I need to check every problem?
A: Yes, checking every answer builds good habits and accuracy.

2 Solution Checking
Exercise 2
Tom says 8 - 3 = 6. Is he right? Show how to check his answer.
Difficulty: Beginner Time: ~2 minutes Skills: Solution Verification
Definition:

Solution checking: Verifying if someone's answer is correct

Error detection: Finding incorrect answers through verification

Step-by-step checking method:
  1. Take Tom's answer (6) and add the subtracted number (3)
  2. Check if the sum equals the original number (8)
  3. If no, Tom's answer is incorrect
Claimed Answer
8 - 3 = 6
Verification
6 + 3 = 9
Result
✗ Incorrect
Step 1: Identify Tom's claim

Tom says: 8 - 3 = 6

Step 2: Apply verification

Add answer to subtracted number: 6 + 3 = 9

Step 3: Compare to original

9 ≠ 8, so Tom's answer is incorrect

Step 4: Find correct answer

8 - 3 = 5 (since 5 + 3 = 8)

6 + 3 = 9 ≠ 8 ✗
Final answer:

No, Tom is wrong. 8 - 3 = 5, not 6.

Applied rules:

• Verification: Always check claimed answers

• Error detection: If sum ≠ original number, answer is wrong

• Correction: Find the correct answer that verifies properly

• Practice Tip: Help others by checking their work too

Related Examples:
  • 7 - 4 = 4 → 4 + 4 = 8 ≠ 7 (incorrect)
  • 7 - 4 = 3 → 3 + 4 = 7 ✓ (correct)
  • 6 - 2 = 5 → 5 + 2 = 7 ≠ 6 (incorrect)
Quick Tips:
  • Always check if the sum equals the original number
  • If verification fails, find the correct answer
  • Help friends by checking their work
Frequently Asked Questions:

Q: What if verification doesn't work?
A: The answer is incorrect. Find the right answer that does verify.

Q: Should I check my friend's work?
A: Yes, checking helps everyone learn and avoid mistakes.

3 Accuracy Verification
Exercise 3
Verify that 9 - 4 = 5 is accurate by using addition to check.
Difficulty: Beginner Time: ~2 minutes Skills: Solution Verification
Definition:

Accuracy verification: Using addition to confirm a subtraction answer is correct

Inverse relationship: Addition and subtraction are opposite operations

Step-by-step verification method:
  1. Take the subtraction result (5) and add the subtracted number (4)
  2. Check if the sum equals the original number (9)
  3. If yes, the subtraction is verified as accurate
Subtraction
9 - 4 = 5
Addition Check
5 + 4 = 9
Verification
✓ Accurate
Step 1: Note the subtraction

9 - 4 = 5

Step 2: Perform addition check

Add result to subtrahend: 5 + 4 = 9

Step 3: Verify equality

9 = 9, so the subtraction is accurate

5 + 4 = 9 ✓
Final answer:

Yes, 9 - 4 = 5 is accurate. Verification: 5 + 4 = 9.

Applied rules:

• Inverse operations: Addition undoes subtraction

• Verification principle: Adding back should restore original number

• Relationship: Subtraction and addition are opposite operations

• Practice Tip: Develop habit of checking all calculations

Related Examples:
  • 7 - 2 = 5 → 5 + 2 = 7 ✓
  • 8 - 3 = 5 → 5 + 3 = 8 ✓
  • 6 - 1 = 5 → 5 + 1 = 6 ✓
Quick Tips:
  • Always add your answer to the number you subtracted
  • The sum should equal the starting number
  • This method works for all subtraction problems
Frequently Asked Questions:

Q: Why does addition check subtraction?
A: Because they are inverse operations that undo each other.

Q: Can I use subtraction to check addition?
A: Yes! For a + b = c, check with c - b = a.

Solutions: Exercises 4 to 5
4 Word Problem Solutions
Exercise 4
Sarah had 6 apples. She ate 2 apples. She says she has 4 apples left. Verify her solution by checking if the answer makes sense.
Definition:

Word problem verification: Using the verification method to check answers to story problems

Context checking: Verifying that the answer makes sense in the problem's context

Step-by-step verification method:
  1. Translate the word problem to a subtraction equation
  2. Take the claimed answer (4) and add the subtracted amount (2)
  3. Check if the sum equals the original amount (6)
  4. If yes, the solution is correct
Word Problem
6 - 2 = 4?
Verification
4 + 2 = 6
Result
✓ Correct
Step 1: Translate to equation

6 - 2 = 4 (Sarah's claim)

Step 2: Apply verification

Add answer to subtracted amount: 4 + 2 = 6

Step 3: Verify against original

6 = 6, so Sarah's answer is correct

Step 4: Confirm context

Starting with 6 apples, eating 2 leaves 4 apples - this makes sense

4 + 2 = 6 ✓
Final answer:

Yes, Sarah is correct. She has 4 apples left.

Applied rules:

• Problem translation: Convert words to mathematical equations

• Verification: Check answer using inverse operation

• Context awareness: Ensure answer fits the problem's scenario

• Practice Tip: Always verify word problem answers with verification

Related Examples:
  • 5 birds, 2 fly away, says 3 left → 3 + 2 = 5 ✓ (correct)
  • 7 toys, 3 lost, says 4 left → 4 + 3 = 7 ✓ (correct)
  • 8 cookies, 1 eaten, says 7 left → 7 + 1 = 8 ✓ (correct)
Quick Tips:
  • Convert word problems to equations first
  • Always check that your answer makes sense in the story
  • Use verification to confirm the mathematical part
Frequently Asked Questions:

Q: How do I check word problem answers?
A: Convert to equation and use verification method.

Q: Should I check if the answer makes sense?
A: Yes, verify both mathematically and in context.

5 Missing Number Solutions
Exercise 5
Find the missing number in 7 - ? = 3. Then verify your solution by checking if it works in the original equation.
Definition:

Missing number solutions: Equations where one number is unknown and needs to be found

Verification of unknowns: Using verification to check if the found value is correct

Step-by-step solution method:
  1. Find the missing number (7 - ? = 3, so ? = 4)
  2. Substitute the found value back into the original equation
  3. Verify using the verification method: 3 + 4 = 7
Original Problem
7 - ? = 3
Found Value
? = 4
Verification
3 + 4 = 7 ✓
Step 1: Find missing number

7 - ? = 3, so ? = 7 - 3 = 4

Step 2: Verify with substitution

Check: 7 - 4 = 3

Step 3: Apply verification method

Add answer to subtrahend: 3 + 4 = 7

Step 4: Verify equality

7 = 7, so the answer is correct

7 - 4 = 3, and 3 + 4 = 7 ✓
Final answer:

The missing number is 4. Verification: 7 - 4 = 3, and 3 + 4 = 7.

Applied rules:

• Missing number strategy: Use inverse operations to find unknowns

• Verification principle: Always check found values with verification

• Completeness: Ensure the full equation is satisfied

• Practice Tip: Check all unknown values with verification method

Related Examples:
  • 6 - ? = 4 → ? = 2 → 4 + 2 = 6 ✓
  • 8 - ? = 3 → ? = 5 → 3 + 5 = 8 ✓
  • ? - 4 = 3 → ? = 7 → 3 + 4 = 7 ✓
Quick Tips:
  • Always check missing number answers with verification
  • For 7 - ? = 3, think: "What added to 3 equals 7?"
  • Verification works for all missing number problems
Frequently Asked Questions:

Q: How do I find missing numbers?
A: Use inverse operations: if 7 - ? = 3, then ? = 7 - 3.

Q: Do I need to check missing number answers?
A: Yes, always verify with verification method to ensure correctness.

Solutions: Exercises 6 to 10
6 Multi-Step Solutions
Exercise 6
Solve 10 - 3 - 2 = 5. Then verify your solution by adding back step by step.
Definition:

Multi-step verification: Checking complex calculations with multiple operations

Sequential verification: Verifying each step in a multi-step problem

Step-by-step solution method:
  1. Solve the sequential subtraction: 10 - 3 - 2
  2. Verify each step using verification principle
  3. Add back in reverse order to check final result
Original Problem
10 - 3 - 2 = 5
Step 1
10 - 3 = 7
Step 2
7 - 2 = 5
Verification
5 + 2 + 3 = 10 ✓
Step 1: First subtraction

10 - 3 = 7

Step 2: Second subtraction

7 - 2 = 5

Step 3: Multi-step verification

Add back in reverse order: 5 + 2 = 7, then 7 + 3 = 10

Step 4: Final verification

5 + 2 + 3 = 10, which matches the original number

10 - 3 - 2 = 5, and 5 + 2 + 3 = 10 ✓
Final answer:

Yes, 10 - 3 - 2 = 5 is correct. Verification: 5 + 2 + 3 = 10.

Applied rules:

• Sequential operations: Perform subtractions from left to right

• Multi-step verification: Check each step individually

• Reverse operations: Add back in reverse order

• Practice Tip: Break complex problems into simpler steps

Related Examples:
  • 8 - 3 - 2 = 3 → 3 + 2 + 3 = 8 ✓
  • 9 - 4 - 1 = 4 → 4 + 1 + 4 = 9 ✓
  • 7 - 2 - 3 = 2 → 2 + 3 + 2 = 7 ✓
Quick Tips:
  • Solve one step at a time
  • Add back in reverse order for verification
  • Check each intermediate step
Frequently Asked Questions:

Q: How do I check multi-step subtraction?
A: Add back in reverse order of the subtractions.

Q: Should I check each step?
A: Yes, checking each step helps catch errors early.

7 Visual Solutions
Exercise 7
You drew 8 circles and crossed out 3 circles. You say 5 circles remain. Verify your solution by counting all circles (crossed out + remaining).
Definition:

Visual verification: Using physical or drawn objects to verify subtraction results

Concrete verification: Checking with tangible representations

Step-by-step verification method:
  1. Count the crossed-out circles (3)
  2. Count the remaining circles (5)
  3. Add them together to verify: 3 + 5 = 8
Total Circles
8
Crossed Out
3
Remaining
5
Verification
3 + 5 = 8 ✓
Step 1: Identify parts

Total = 8, subtracted = 3, remaining = 5

Step 2: Apply verification

Add subtracted and remaining: 3 + 5 = 8

Step 3: Verify total

8 = 8, so the subtraction is correct

3 + 5 = 8 ✓
Final answer:

Yes, if you cross out 3 from 8, 5 remain. Verification: 3 + 5 = 8.

Applied rules:

• Visual verification: Use physical objects to confirm calculations

• Part-whole relationship: Parts should sum to the whole

• Concrete checking: Tangible methods enhance understanding

• Practice Tip: Visual verification is excellent for understanding concepts

Related Examples:
  • 6 blocks, remove 2, say 4 remain → 2 + 4 = 6 ✓
  • 7 stars, cross out 4, say 3 remain → 4 + 3 = 7 ✓
  • 9 hearts, color 3, say 6 remain → 3 + 6 = 9 ✓
Quick Tips:
  • Draw or use physical objects to visualize
  • Count all parts to verify the whole
  • Visual verification builds conceptual understanding
Frequently Asked Questions:

Q: How does visual verification work?
A: Count all parts (subtracted + remaining) to ensure they equal the total.

Q: Is this the same as verification?
A: Yes, it's a visual representation of the verification principle.

8 Story Solutions
Exercise 8
Create a story for 8 - 3 = 5, then verify your story is accurate using the verification method.
Definition:

Story verification: Creating a narrative that matches the mathematical equation and verifying it with verification

Contextual checking: Ensuring the story accurately represents the math

Step-by-step creation and verification method:
  1. Create a story that matches 8 - 3 = 5
  2. Identify the starting amount (8), subtracted amount (3), and result (5)
  3. Apply verification method: 5 + 3 = 8
  4. Confirm the story matches the verified math
Story Created
"8 birds, 3 fly away, 5 stay"
Verification
5 + 3 = 8
Result
✓ Accurate
Step 1: Create matching story

"There were 8 birds on a tree. 3 birds flew away. 5 birds remained."

Step 2: Identify components

Start: 8 birds, subtracted: 3 birds, result: 5 birds

Step 3: Apply verification method

Add result to subtracted: 5 + 3 = 8

Step 4: Verify story accuracy

Math checks out, so the story accurately represents the equation

5 + 3 = 8 ✓
Final answer:

Story: "There were 8 birds on a tree. 3 birds flew away. 5 birds remained." Verification: 5 + 3 = 8.

Applied rules:

• Story creation: Match numbers to realistic scenarios

• Verification: Check that the story reflects accurate math

• Context accuracy: Ensure the narrative aligns with the equation

• Practice Tip: Stories help connect math to real-world understanding

Related Examples:
  • 6 - 2 = 4: "6 apples, eat 2, have 4 left" → 4 + 2 = 6 ✓
  • 7 - 3 = 4: "7 toys, lose 3, have 4 left" → 4 + 3 = 7 ✓
  • 5 - 1 = 4: "5 cookies, eat 1, have 4 left" → 4 + 1 = 5 ✓
Quick Tips:
  • Create stories that make sense with the numbers
  • Always verify stories with verification method
  • Stories help understand mathematical concepts
Frequently Asked Questions:

Q: How do I know if my story is accurate?
A: Use verification method to check the math in your story.

Q: Do stories help with math understanding?
A: Yes, stories connect abstract math to real-world situations.

9 Inverse Solutions
Exercise 9
If 6 - 2 = 4, verify this by starting with the answer and adding back to reach the original number.
Definition:

Inverse verification: Starting with the answer and performing the inverse operation to return to the original number

Backward checking: Working backwards to confirm the solution

Step-by-step verification method:
  1. Start with the answer (4)
  2. Add the subtracted number (2)
  3. Check if you reach the original number (6)
Original
6 - 2 = 4
Inverse
4 + 2 = 6
Result
✓ Correct
Step 1: Begin with answer

Start with 4 (the result)

Step 2: Perform inverse operation

Add the subtracted amount: 4 + 2 = 6

Step 3: Verify return to original

6 = 6, so the subtraction is verified

4 + 2 = 6 ✓
Final answer:

Yes, 6 - 2 = 4 is correct. Verification: 4 + 2 = 6.

Applied rules:

• Inverse verification: Start with the answer and work back

• Inverse operations: Addition reverses subtraction

• Cycle completion: Should return to original number

• Practice Tip: This method confirms the completeness of your solution

Related Examples:
  • 7 - 3 = 4 → 4 + 3 = 7 ✓
  • 8 - 5 = 3 → 3 + 5 = 8 ✓
  • 9 - 1 = 8 → 8 + 1 = 9 ✓
Quick Tips:
  • Start with your answer
  • Add back what you subtracted
  • Should return to where you started
Frequently Asked Questions:

Q: What is inverse checking?
A: Starting with the answer and adding back to see if you return to the original number.

Q: Is this the same as verification?
A: Yes, it's another way to describe the verification method.

10 Mixed Review
Exercise 10
Solve and verify: 11 - 4 - 3 = 4. Show your verification work.
Definition:

Mixed verification: Applying the verification method to complex problems with multiple operations

Comprehensive checking: Ensuring all parts of a problem are accurate

Step-by-step solution method:
  1. Solve the sequential subtraction: 11 - 4 - 3
  2. Apply the verification method by adding back in reverse order
  3. Verify that the sum equals the original number
Original Problem
11 - 4 - 3 = 4
Step 1
11 - 4 = 7
Step 2
7 - 3 = 4
Verification
4 + 3 + 4 = 11 ✓
Step 1: First subtraction

11 - 4 = 7

Step 2: Second subtraction

7 - 3 = 4

Step 3: Apply verification method

Add back in reverse order: 4 + 3 + 4 = 11

Step 4: Final verification

11 = 11, so the solution is correct

11 - 4 - 3 = 4, and 4 + 3 + 4 = 11 ✓
Final answer:

Yes, 11 - 4 - 3 = 4 is correct. Verification: 4 + 3 + 4 = 11.

Applied rules:

• Sequential operations: Perform subtractions from left to right

• Multi-step verification: Check the entire calculation chain

• Reverse addition: Add back in reverse order of subtraction

• Practice Tip: Consistent verification builds computational accuracy

Related Examples:
  • 13 - 5 - 3 = 5 → 5 + 3 + 5 = 13 ✓
  • 14 - 6 - 2 = 6 → 6 + 2 + 6 = 14 ✓
  • 12 - 3 - 4 = 5 → 5 + 4 + 3 = 12 ✓
Quick Tips:
  • Break complex problems into steps
  • Verify each step when possible
  • Add back in reverse order for final check
Frequently Asked Questions:

Q: How do I check multi-operation problems?
A: Add back in reverse order of the operations performed.

Q: Should I verify every complex problem?
A: Yes, verification is especially important for complex calculations.

Understanding Solutions
If a - b = c, then c + b = a
Verification Principle
Solution understanding components:

Verification: Checking if a calculation is correct

Inverse operations: Addition and subtraction as opposite operations

Accuracy: Ensuring answers are mathematically correct

Confidence: Building trust in mathematical computations

Solution verification method:
  1. Take the answer: The result of your subtraction
  2. Add the subtracted number: Add the number you subtracted
  3. Check equality: See if you get back to the original number
  4. Verify correctness: If equal, your subtraction was correct
Tip 1: Always add your answer to the number you subtracted.
Tip 2: The sum should equal your original number.
Tip 3: Use this test for every subtraction to build accuracy.
Tip 4: This works for all subtraction problems.
Common errors: Forgetting to check answers, adding incorrectly during verification, skipping the test.
Success strategies: Consistent verification, understanding inverse operations, building checking habits.
Solution rules:

• Inverse relationship: Addition undoes subtraction

• Verification principle: Adding back should restore original number

• Universal application: Works for all subtraction problems

• Accuracy building: Regular use improves computational accuracy

• Confidence booster: Verification builds trust in mathematical abilities

Rules and Methods for Understanding Solutions
a - b = c → c + b = a
Verification Formula
Principle
Add Answer
Take result and add subtracted number
Verification
Check Equality
Sum should equal original number
Result
Confirm Accuracy
If equal, subtraction is correct

Key Takeaways

  • Verification checks subtraction by adding the answer back
  • Always add your result to the number you subtracted
  • The sum should equal your original number
  • This method works for all subtraction problems
  • Regular verification builds computational accuracy

Questions & Answers

Question: Why do I need to check my subtraction? Isn't solving it enough?

Answer: Checking your work is super important! Here's why:

  • Catch mistakes: Everyone makes errors sometimes
  • Build confidence: Knowing your answer is right feels great
  • Learn better: Checking helps you understand the math
  • Develop good habits: Checking is a skill used in higher math
  • Get better grades: Teachers appreciate careful work

Verification is like having a friend double-check your work!

It takes just a few seconds and ensures your answer is correct!

Question: My child is reluctant to check their work. How can I encourage this habit?

Answer: Here are effective ways to encourage checking:

  • Make it fun: Call it the "math detective" game
  • Lead by example: Show them you check your own work
  • Start small: Begin with simple problems
  • Provide incentives: Praise and rewards for checking
  • Explain benefits: Show how it prevents mistakes

Example activity: "Let's play teacher and check this problem together!"

Make checking a regular part of homework routine!

Question: What if the verification method doesn't work? Does that mean I'm bad at math?

Answer: Not at all! When the verification method doesn't work, it just means you made a small mistake:

This is completely normal and happens to everyone learning math!

Use it as a tool to find and fix errors:

  • Go back and try the subtraction again
  • Use objects or fingers to help visualize
  • Ask for help if needed

Verification is your friend - it helps you catch mistakes before turning in your work!

Great mathematicians check their work too!

Solution Understanding Glossary

Verification
The process of checking if a calculation is correct by using an alternative method.
Inverse Operations
Operations that undo each other; subtraction and addition are inverse operations.
Subtrahend
The number that is subtracted from the minuend in a subtraction problem.
Minuend
The number from which another number is subtracted in a subtraction problem.
Solution
The answer to a mathematical problem or equation.

Math Excellence Educational Team

Certified Elementary Mathematics Educators & Curriculum Specialists

Our team of experienced elementary math teachers and curriculum developers creates research-based, student-friendly solution understanding resources. All content is aligned with Common Core State Standards and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness. We specialize in helping young learners develop accuracy and confidence in mathematical computations through verification techniques.

Comprehensive Summary for Learning and Revision

  • Definition of Solution Verification: Solution verification is a method where you add the result of a subtraction problem back to the number that was subtracted to check if it equals the original number. If a - b = c, then c + b = a.
  • Key Concepts:
    • Verification: Checking if a calculation is correct using an alternative method
    • Inverse Operations: Addition and subtraction are opposite operations that undo each other
    • Accuracy: Ensuring mathematical computations are correct
    • Confidence: Building trust in mathematical abilities through verification
  • Core Rules:
    • Always add the answer back to the number you subtracted
    • The sum should equal the original number
    • This method works for all subtraction problems
    • Use verification method for every subtraction to build accuracy
    • If the verification fails, rework the problem to find the error
  • Step-by-Step Method:
    1. Complete the original subtraction problem
    2. Take the result (answer) of the subtraction
    3. Add it to the number you subtracted
    4. Check if the sum equals the original number
    5. If equal, the subtraction was correct; if not, recheck the work
  • Examples (Simple to Advanced):
    • Simple: 5 - 2 = 3 → 3 + 2 = 5 ✓
    • Moderate: 8 - 3 = 5 → 5 + 3 = 8 ✓
    • Advanced: 12 - 5 - 2 = 5 → 5 + 2 + 5 = 12 ✓
  • Tips and Tricks:
    • Develop the habit of checking every subtraction problem
    • Use the verification method immediately after solving
    • For multiple operations, add back in reverse order
    • Visualize the process with objects or drawings
  • Common Pitfalls:
    • Forgetting to check answers
    • Adding incorrectly during verification
    • Skipping the verification step
    • Getting discouraged when tests fail initially
    • Mixing up which numbers to add
  • Memorization Aids:
    • "Add back what you took away"
    • "Check your work, don't be a jerk"
    • "Subtract and verify, stay ahead"
    • Remember: subtraction and addition are opposites
  • Application Strategies:
    • Apply verification method to all subtraction problems
    • Use it for word problems after translating to equations
    • Check missing number problems with verification
    • Verify complex multi-step problems step by step