Explaining Your
Reasoning
Learn to communicate mathematical thinking using words, diagrams, and equations so others can follow every step.
Pre-Algebra
What Does "Explaining Reasoning" Mean?
When you solve a math problem, getting the right answer is only half the job. Explaining your reasoning means showing how you got there and why each step makes sense: not just the final number, but the thinking behind it.
Your Three Reasoning Tools
Words
Describe your thinking in complete sentences: what you did and why it works.
Diagrams
Use pictures, number lines, or tables to make abstract ideas visible.
Equations
Write mathematical statements that show the relationships between quantities.
Why Explaining Matters
1
Deepens understanding: If you can explain it, you truly understand it.
2
Catches mistakes: Writing out your reasoning helps you spot errors before you finish.
3
Earns full credit: On tests and assignments, showing work is often worth more than the answer itself.
4
Helps others learn: A clear explanation lets classmates and teachers follow your thinking.
The Reasoning Framework
Connecting Words, Diagrams, and Equations
A complete explanation uses all three tools together. Your words tell the story, your diagram makes it visible, and your equation proves it mathematically. Each one reinforces the others.
Showing Work vs. Explaining Reasoning
| Showing Work | Explaining Reasoning | |
|---|---|---|
| Focus | The what | The what AND the why |
| Example | \(5 + 3 = 8\) | "I added 5 and 3 because combining the two groups gives the total." |
| Tools | Equations only | Words + diagrams + equations |
| Audience | Yourself | Anyone reading your work |
How to Explain Step by Step
1
Read the problem carefully. Identify what you know and what you need to find.
2
State your claim. Write a clear sentence with your answer or conclusion.
3
Show evidence. Write the equation or draw a diagram that supports your claim.
4
Explain your reasoning. Use words to connect each step: "I did ___ because ___."
5
Check and connect. Make sure your words, diagram, and equation all agree.
Example 1: Solve \(x + 5 = 12\) and Explain
Example 2: Explain Why \(3(x + 2) = 3x + 6\) Using a Diagram
Example 3: Is \(x = 4\) a Solution to \(2x - 3 = 5\)? Explain.
1
Claim: Yes, \(x = 4\) is a solution to \(2x - 3 = 5\).
2
Evidence (equation): Substitute \(x = 4\): \(2(4) - 3 = 8 - 3 = 5\). Both sides equal \(5\).
3
Reasoning (words): I substituted \(4\) for \(x\) because a solution must make the equation true. Since the left side simplifies to \(5\), which equals the right side, the equation balances.
Weak vs. Strong Explanations
✕ Weak
"I got 7 because I did the math."
✓ Strong
"I subtracted 5 from both sides of \(x + 5 = 12\) to isolate \(x\). Since \(12 - 5 = 7\), the answer is \(x = 7\). The number line confirms that jumping 7 from 5 lands on 12."
Useful Sentence Starters
Claim
"I believe the answer is ___ because ___."
"The solution is ___ ."
"The solution is ___ ."
Evidence
"The equation shows ___."
"The diagram illustrates ___."
"The diagram illustrates ___."
Reasoning
"This works because ___."
"I did this step because ___."
"I did this step because ___."
Common Mistakes to Avoid
✕ Answer Only
Writing just "\(x = 7\)" with no explanation of how or why.
✕ Vague Words
Saying "I just knew it" or "I did the math" without describing the steps.
✕ Missing Connections
Showing a diagram and an equation that don't clearly relate to each other.
Explanation Quality Checklist
| Element | Missing | Partial | Complete |
|---|---|---|---|
| Claim | No answer stated | Answer without context | Clear answer sentence |
| Evidence | No work shown | Equation OR diagram | Equation AND diagram |
| Reasoning | No words | "Because I solved it" | Step-by-step "why" |
| Connections | Tools unrelated | Some links | All three tools agree |
Question 1 of 5
A student writes: "The answer is \(x = 9\)." What is missing from this explanation?
The student stated a claim but provided no evidence (equation, diagram) or reasoning (explanation of why). A complete answer needs all three.
Question 2 of 5
Which sentence best shows reasoning (not just showing work)?
Choice B is correct because it explains what was done (added 4 and 6) AND why (the problem asks for the total). The other options lack the "why" connection.
Question 3 of 5
To verify that \(x = 3\) solves \(4x + 1 = 13\), which step shows the strongest evidence?
Substitution is the strongest evidence because it mathematically proves the left side equals the right side. The other choices lack concrete calculation.
Question 4 of 5
A student draws a number line for \(x - 4 = 6\) but writes no words. What should they add to complete the explanation?
The student has a diagram (the number line) but is missing words that explain the reasoning. A sentence connecting the diagram to the equation completes the explanation.
Question 5 of 5
Which of these explanations for "Why is \(3 \times 4 = 12\)?" best connects words, a diagram, and an equation?
Choice C is the strongest: it states the claim (\(3 \times 4 = 12\)), references a diagram (by describing rows of dots), provides an equation (\(4+4+4=12\)), and uses words to explain how they all connect. That's a complete explanation.