Solved Exercises on Explaining Your Mathematical Reasoning in Grade 3

Master explaining your reasoning: logical thinking, justification, and communication skills through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Addition Reasoning
Exercise 1
Sarah has 24 stickers. Her friend gives her 17 more stickers. Sarah says she now has 41 stickers. Explain why Sarah is correct or incorrect.
Definition:

Explaining your reasoning: Clearly communicating the mathematical thinking and steps used to solve a problem, including why specific operations were chosen and how the solution was reached.

My Mathematical Thinking:

I need to check if 24 + 17 = 41.

1.
First, I will add the ones: 4 + 7 = 11. I write down 1 and carry over 1 ten.
2.
Next, I add the tens: 2 + 1 = 3, plus the carried over 1 = 4 tens.
3.
So I have 4 tens and 1 one, which equals 41.
Justification:

Sarah is correct because 24 + 17 = 41. I used addition to combine the two amounts of stickers, which is the right operation since she received more stickers.

How to Communicate:
I know this because... I added 24 and 17 and got 41.
First I... added the ones place (4 + 7 = 11).
Then I... added the tens place (2 + 1 + 1 = 4).
Final answer:

Sarah is correct. 24 + 17 = 41 stickers.

Applied rules:

• Addition: Combine quantities when items are added together

• Place value: Add ones with ones and tens with tens

• Regrouping: Carry over when a place value exceeds 9

2 Subtraction Reasoning
Exercise 2
Tom had 53 marbles. He gave 28 marbles to his sister. Tom says he has 35 marbles left. Explain why Tom is correct or incorrect.
Definition:

Mathematical reasoning: The process of using logical thinking to analyze problems, select appropriate strategies, and justify solutions through clear explanations.

My Mathematical Thinking:

I need to check if 53 - 28 = 35.

1.
I will subtract the ones first: 3 - 8. Since 3 is less than 8, I need to borrow from the tens.
2.
After borrowing: 13 - 8 = 5 ones. Now I have 4 tens left.
3.
Then I subtract the tens: 4 - 2 = 2 tens.
4.
So I have 2 tens and 5 ones, which equals 25.
Justification:

Tom is incorrect because 53 - 28 = 25, not 35. I used subtraction because marbles were taken away. Tom made a mistake in his calculation.

How to Communicate:
I know this because... 53 minus 28 equals 25, not 35.
First I... tried to subtract ones (3 - 8).
Then I... borrowed from tens and calculated 25.
Final answer:

Tom is incorrect. 53 - 28 = 25 marbles, not 35.

Applied rules:

• Subtraction: Remove quantities when items are taken away

• Borrowing: Regroup when a digit in the minuend is smaller than the subtrahend

• Place value: Subtract ones from ones and tens from tens

3 Geometry Reasoning
Exercise 3
A rectangle has 4 sides. A triangle has 3 sides. If I put a triangle on top of a rectangle, how many sides does the new shape have? Explain your reasoning.
Definition:

Spatial reasoning: The ability to visualize and manipulate objects in space, understanding how shapes relate to each other when combined or transformed.

My Mathematical Thinking:

I need to count the sides of the combined shape, considering that sides may be shared.

1.
A rectangle has 4 sides and a triangle has 3 sides, which is 7 sides total if separate.
2.
When I put the triangle on top of the rectangle, one side of the triangle connects to one side of the rectangle.
3.
The connecting sides become internal and are no longer part of the outside edge of the shape.
4.
So I subtract the 2 connected sides: 7 - 2 = 5 sides total on the outside.
Justification:

The new shape has 5 sides because when shapes share a side, that side is no longer counted as part of the perimeter of the combined shape.

How to Communicate:
I know this because... the shared side doesn't count toward the outside.
First I... counted sides of each shape separately.
Then I... subtracted the shared sides.
Final answer:

The new shape has 5 sides.

Applied rules:

• Perimeter: Only count the outside edges of a shape

• Shape combination: Shared sides are not part of the new perimeter

• Visualization: Draw or imagine the shapes to verify your reasoning

Mathematical Reasoning Process
Reasoning Steps
Think, Plan, Justify, Communicate
🤔
Think
📝
Plan
🔢
Calculate
💬
Explain
Think: Analyze the problem and decide what operations to use
Plan: Organize your approach and steps to solve the problem
Calculate: Perform the mathematical operations accurately
Explain: Communicate your thinking and justify your answer
Key Components of Mathematical Reasoning:
  • Understanding the problem
  • Selecting appropriate strategies
  • Performing calculations correctly
  • Justifying the solution
  • Communicating the process clearly
Tip 1: Always start by restating what the problem asks you to find.
Tip 2: Use words like "because," "since," and "therefore" to explain your thinking.
Tip 3: Show your work step by step so others can follow your reasoning.
Communication Strategies
Communication Tools
Words That Show Your Thinking
Sentence Starters for Reasoning:
I know this because... I performed the calculation and verified it.
First I... identified what the problem was asking.
Then I... chose the correct operation to solve it.
I checked my work by... using the inverse operation.
Therefore... my answer is reasonable.
Justification: Explain WHY you chose a particular method or operation
Sequence: Use words like "first," "next," "then," "finally" to show steps
Verification: Always check if your answer makes sense in context
Key concepts:

Mathematical communication: Using precise language to express mathematical thinking clearly

Logical reasoning: Making connections between mathematical concepts and procedures

Justification: Providing evidence and reasoning to support mathematical claims

Explanation framework:
  1. State your answer clearly
  2. Explain what strategy you used
  3. Describe each step of your process
  4. Justify why your method was appropriate
  5. Verify your solution makes sense
Tip 4: Practice explaining your thinking out loud before writing it down.
Tip 5: Ask yourself: "Would someone else understand my explanation?"
Comprehensive Summary and Key Concepts
Complete Guide to Explaining Your Mathematical Reasoning
📋
Definitions: Explaining your reasoning in mathematics means clearly communicating the logical thinking and steps you used to solve a problem. It includes stating your answer, describing your approach, justifying your methods, and verifying your solution. This skill demonstrates mathematical understanding beyond just getting the right answer.
📏
Core Rules:
  • Always explain WHY you chose a particular operation or strategy
  • Use precise mathematical vocabulary
  • Show your work step by step
  • Justify your answer with evidence
  • Check that your explanation makes sense
⚙️
Step-by-Step Methods:
  1. Restate the problem in your own words
  2. Identify what you know and what you need to find
  3. Choose an appropriate strategy or operation
  4. Solve the problem showing all work
  5. Explain your thinking using sentence starters
  6. Verify your answer makes sense
🎯
Examples:
  • Simple: "I added because the problem said 'total'"
  • Intermediate: "I used subtraction because items were taken away"
  • Advanced: "I multiplied because I needed to find groups of equal size"
⚠️
Tips and Common Mistakes:
  • Always use "because" to justify your steps
  • Don't skip steps in your explanation
  • Make sure your explanation matches your work
  • Practice using mathematical vocabulary
  • Check that someone else could follow your reasoning
💡
Key Notes for Memorization:
  • Use sentence starters: "I know this because..."
  • Follow the sequence: first, then, next, finally
  • Always verify your answer makes sense
  • Explain your strategy choice
  • Show all work and thinking

Student Questions

Question: I get the right answer, but I don't know how to explain my thinking. What should I say?

Answer: Great question Alex! Here's a simple way to explain your thinking:

  • Start with: "I know this because..." followed by your calculation
  • Use sequence words: "First I..., then I..., finally I..."
  • Explain why you chose addition/subtraction/multiplication
  • Check your answer by saying: "This makes sense because..."

For example: "I added because the problem said 'altogether.' First I added the ones, then the tens."

Question: How do I know if my explanation is good enough?

Answer: A good explanation should answer these questions:

  • What did you do? (The operations you used)
  • Why did you do it? (The reasoning behind your choices)
  • How did you do it? (The steps you took)
  • Does it make sense? (How you verified your answer)

If someone who didn't solve the problem could understand your explanation and follow your thinking, then it's good enough!

Question: Do I really need to explain my reasoning if I already got the right answer?

Answer: Yes, explaining your reasoning is very important! Getting the right answer shows you can calculate, but explaining your thinking shows you truly understand the math. It helps teachers see your thought process, helps you catch mistakes, and builds stronger mathematical thinking skills. Plus, it's an important skill for higher-level math and real-world problem solving.