Solved Exercises on Explaining Mathematical Reasoning in Grade 3

Master explaining mathematical reasoning: using clear steps, justifying answers, and communicating mathematical thinking through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Explaining steps
Exercise 1
Sarah has 15 stickers. She gives 7 stickers to her friend. Then she finds 4 more stickers. How many stickers does Sarah have now? Explain your reasoning.
Definition:

Mathematical reasoning: The process of using logic and mathematical operations to solve problems. Explaining reasoning: Clearly describing the steps taken and why each step was necessary.

Explanation method:
  1. State the problem clearly
  2. Identify the first operation and explain why
  3. Perform the calculation
  4. Identify the next operation and explain why
  5. Perform the calculation
  6. State the final answer
  7. Verify your answer makes sense
Start
15
Give away
15 - 7 = 8
Find more
8 + 4 = 12
Step 1: Identify the starting amount

Sarah starts with 15 stickers

Step 2: Explain the first operation

She gives 7 stickers away, so I subtract: 15 - 7 = 8 stickers remaining

Step 3: Explain the second operation

Then she finds 4 more stickers, so I add: 8 + 4 = 12 stickers

Step 4: State the final answer

Sarah has 12 stickers now

Step 5: Verify the answer

15 - 7 + 4 = 12 ✓

Sarah has 12 stickers
Final answer:

Sarah has 12 stickers. I subtracted 7 from 15 to get 8, then added 4 to get 12.

Applied rules:

• Clear explanation: Describe each step in words

• Logical sequence: Follow the order of operations in the problem

• Verification: Check that your answer makes sense

2 Justifying answer
Exercise 2
Is 4 × 6 equal to 24? Explain why or why not using repeated addition.
Definition:

Justifying an answer: Providing evidence or reasoning to prove that your answer is correct. This involves showing your work and explaining your thinking.

Multiplication
4 × 6
Repeated addition
6 + 6 + 6 + 6
Result
24
Step 1: Understand the problem

I need to verify if 4 × 6 = 24 by using repeated addition

Step 2: Apply the definition of multiplication

4 × 6 means 4 groups of 6, or 6 + 6 + 6 + 6

Step 3: Perform the repeated addition

6 + 6 = 12, 12 + 6 = 18, 18 + 6 = 24

Step 4: Compare with the claimed answer

My result is 24, which matches the claimed answer

Step 5: State the conclusion

Yes, 4 × 6 = 24 because 6 + 6 + 6 + 6 = 24

Yes, 4 × 6 = 24
Final answer:

Yes, 4 × 6 = 24. I know this because multiplication is repeated addition: 4 × 6 means adding 6 four times (6 + 6 + 6 + 6 = 24).

Applied rules:

• Definition of multiplication: Multiplication is repeated addition

• Justification: Provide evidence to support your claim

• Verification: Confirm the result through an alternative method

3 Checking work
Exercise 3
Tom solved 18 - 9 + 3 and got 6. Is Tom correct? Explain your reasoning and show the correct way to solve it.
Definition:

Checking work: Reviewing calculations to identify errors and ensure accuracy. This involves understanding the order of operations and verifying each step.

Problem
18 - 9 + 3
Tom's answer
6
Correct answer
12
Step 1: Analyze Tom's error

Tom calculated 18 - 9 + 3 = 6, but this is incorrect

Step 2: Apply the correct order of operations

Addition and subtraction are performed from left to right

Step 3: Solve correctly

First: 18 - 9 = 9, then: 9 + 3 = 12

Step 4: Verify the correct answer

Working left to right: 18 - 9 + 3 = 9 + 3 = 12

Step 5: Explain Tom's mistake

Tom may have calculated 9 + 3 first, then subtracted from 18: 18 - 12 = 6

Tom is incorrect, answer is 12
Final answer:

No, Tom is not correct. The answer is 12. When adding and subtracting, we work from left to right: 18 - 9 = 9, then 9 + 3 = 12.

Applied rules:

• Order of operations: For addition and subtraction, work from left to right

• Error identification: Locate where the mistake occurred

• Verification: Show the correct process step by step

Rules and methods, laws,...
Reasoning = Steps + Logic + Verification
Mathematical Reasoning
Explanation Structure
What + How + Why
Explain what you did, how you did it, and why you chose that method
Verification Method
Work Backwards
Check your answer by reversing the operations
Error Analysis
Find + Explain + Correct
Locate errors, explain why they're wrong, show the correction
Key definitions:

Mathematical reasoning: The process of using logic, mathematical operations, and previously learned concepts to solve problems.

Explaining reasoning: Clearly describing the steps taken and the logic behind each step in solving a mathematical problem.

Justifying answers: Providing evidence or reasoning to prove that an answer is correct.

Verification: Checking work to ensure accuracy by using alternative methods or working backwards.

Order of operations: The sequence in which mathematical operations should be performed.

Mathematical reasoning methodology:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Decide what operations to use and in what order
  3. Solve: Perform the calculations step by step
  4. Explain: Describe your reasoning clearly
  5. Verify: Check your work for accuracy
Tip 1: Use transition words like "first," "then," "next," and "finally" to organize your explanation.
Tip 2: Always state why you chose a particular operation or method.
Tip 3: Check your answer by working backwards or using an alternative method.
Tip 4: When correcting others' work, be specific about the error and how to fix it.
Common errors: Not explaining the reasoning, skipping steps, using incorrect order of operations, not checking work, not stating why operations were chosen.
Real-world applications: Explaining solutions to others, making logical arguments, solving everyday problems systematically, critical thinking in all subjects.
Reasoning principles:

• Always explain what you did and why

• Use clear, sequential steps

• Justify your operations with mathematical reasoning

• Verify your answers for accuracy

• When correcting others, identify the specific error

• Use mathematical vocabulary correctly

Solution: Exercises 4 to 5
4 Logical argument
Exercise 4
Explain why 25 + 17 = 17 + 25 using a logical argument. What property does this demonstrate?
Definition:

Logical argument: A reasoned explanation that follows a sequence of statements to reach a conclusion. Commutative property: The property that states numbers can be added or multiplied in any order without changing the result.

Left side
25 + 17 = 42
Right side
17 + 25 = 42
Conclusion
Both equal 42
Step 1: Calculate both sides

25 + 17 = 42 and 17 + 25 = 42

Step 2: Compare the results

Both expressions equal 42

Step 3: Explain the underlying principle

When adding, the order of numbers doesn't matter because addition is commutative

Step 4: State the property

This demonstrates the Commutative Property of Addition

Step 5: Generalize the concept

For any numbers a and b: a + b = b + a

Demonstrates Commutative Property
Final answer:

25 + 17 = 17 + 25 because both equal 42. This demonstrates the Commutative Property of Addition, which states that numbers can be added in any order without changing the sum.

Applied rules:

• Commutative property: Order doesn't affect the result for addition

• Logical demonstration: Calculate both sides to show equality

• Generalization: Apply the principle to all similar cases

5 Communicating thinking
Exercise 5
You need to find how many cookies are in 6 bags if each bag has 8 cookies. Explain your mathematical thinking as if teaching a friend who doesn't know multiplication yet.
Definition:

Communicating mathematical thinking: Explaining mathematical concepts in clear, accessible language that others can understand. This involves connecting new concepts to familiar ideas.

Number of bags
6
Cookies per bag
8
Total cookies
6 × 8 = 48
Step 1: Relate to familiar concept

"I need to add 8 six times because there are 6 bags with 8 cookies each"

Step 2: Show repeated addition

"I can add: 8 + 8 + 8 + 8 + 8 + 8 = 48 cookies"

Step 3: Introduce multiplication

"Instead of adding 8 six times, I can write it as 6 × 8 = 48"

Step 4: Connect to real situation

"This means if I have 6 bags and each has 8 cookies, I have 48 cookies total"

Step 5: Verify the answer

"I can check by adding: 8 + 8 = 16, 16 + 8 = 24, 24 + 8 = 32, 32 + 8 = 40, 40 + 8 = 48"

6 bags × 8 cookies = 48 cookies
Final answer:

"To find the total cookies, I add 8 six times (once for each bag): 8 + 8 + 8 + 8 + 8 + 8 = 48. This is the same as multiplying 6 × 8 = 48. So there are 48 cookies in total."

Applied rules:

• Connect to prior knowledge: Use addition to explain multiplication

• Use concrete examples: Relate to real-world situations

• Verify understanding: Check with an alternative method

Complete Summary: Explaining Mathematical Reasoning
Reasoning = Steps + Logic + Verification
Mathematical Reasoning
Key definitions:

Mathematical reasoning: The process of using logic, mathematical operations, and previously learned concepts to solve problems and draw conclusions.

Explaining reasoning: Clearly describing the steps taken and the logic behind each step in solving a mathematical problem, using appropriate mathematical language.

Justifying answers: Providing evidence or reasoning to prove that an answer is correct, often by showing work or using alternative methods.

Verification: Checking work to ensure accuracy by using alternative methods, working backwards, or confirming with different approaches.

Mathematical communication: Expressing mathematical ideas clearly through words, symbols, diagrams, and logical arguments.

Order of operations: The sequence in which mathematical operations should be performed to ensure consistent results.

Complete mathematical reasoning methodology:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Decide what operations to use and in what order
  3. Solve: Perform the calculations step by step
  4. Explain: Describe your reasoning clearly using mathematical vocabulary
  5. Justify: Provide evidence that your answer is correct
  6. Verify: Check your work for accuracy
  7. Communicate: Present your solution in a clear, organized manner
Tip 1: Use transition words like "first," "then," "next," and "finally" to organize your explanation.
Tip 2: Always state why you chose a particular operation or method.
Tip 3: Check your answer by working backwards or using an alternative method.
Tip 4: When correcting others' work, be specific about the error and how to fix it.
Tip 5: Use concrete examples to explain abstract concepts to others.
Common errors: Not explaining the reasoning, skipping steps, using incorrect order of operations, not checking work, not stating why operations were chosen, using imprecise mathematical language.
Real-world applications: Explaining solutions to others, making logical arguments, solving everyday problems systematically, critical thinking in all subjects, communicating ideas clearly in academic and professional settings.
Key reasoning principles:

• Always explain what you did and why you chose that method

• Use clear, sequential steps with appropriate mathematical vocabulary

• Justify your operations with mathematical reasoning and properties

• Verify your answers using alternative methods or working backwards

• When correcting others, identify the specific error and provide the correct approach

• Use logical arguments to demonstrate mathematical properties

• Communicate mathematical thinking clearly to others

Visual Guide to Mathematical Reasoning

🧠
Explanation Structure

What: State the problem

How: Show the steps

Why: Explain the reasoning

Check: Verify the answer

Verification Methods

Work backwards

Alternative method

Estimation check

Reverse operations

Logical Argument

State the claim

Provide evidence

Apply properties

Draw conclusion

Communication Tips

Use transition words

Explain your thinking

Connect to prior knowledge

Be precise with math terms

Questions & Answers

Question: Why do I need to explain my math reasoning? Can't I just write the answer?

Answer: Explaining your reasoning is important for several reasons:

  • Understanding: It shows you truly understand the math, not just getting the right answer
  • Thinking: It helps you think through the problem more carefully
  • Communication: It allows others to understand your thought process
  • Learning: It helps you remember how to solve similar problems
  • Checking: It makes it easier to spot mistakes

Example: If you write "15 - 7 = 8" without explanation, I don't know if you understand subtraction. But if you say "I subtracted 7 from 15 because the problem says 'gives away,' and 15 - 7 = 8," I know you understand the concept.

Math is not just about answers - it's about thinking!

Question: I'm not sure how to explain my thinking. What words should I use?

Answer: Use these helpful transition words and phrases to organize your explanation:

  • Sequence: First, Next, Then, After that, Finally
  • Reason: Because, Since, This is because, I know this because
  • Operation: I added, I subtracted, I multiplied, I divided
  • Verification: To check, I verified, This proves, Let me make sure

Example: "First, I identified that Sarah had 15 stickers. Then, she gave away 7 stickers, so I subtracted: 15 - 7 = 8. Finally, she found 4 more stickers, so I added: 8 + 4 = 12. I know this is correct because 15 - 7 + 4 = 12."

These words help organize your thoughts and make your explanation clear!

Question: When I'm checking someone else's work, what should I look for?

Answer: When checking someone else's work, look for these key things:

  • Right operations: Did they use the correct operation (+, -, ×, ÷)?
  • Order: Did they follow the correct order of operations?
  • Calculations: Are the individual calculations correct?
  • Logic: Does their reasoning make sense?
  • Units: Did they include the correct units if needed?

Example: If someone writes "18 - 9 + 3 = 6," I notice they didn't follow left-to-right order. The correct way is: 18 - 9 = 9, then 9 + 3 = 12. The answer should be 12, not 6.

Always show the correct steps and explain what went wrong!