What Counts as a Mathematical Explanation
A free Logic lesson from the “Proof Readiness” unit, with a worked example and practice problems including step-by-step solutions.
Begin with arbitrary allowed objects and their definitions. Use justified algebra to obtain the target form, then explain why it proves the claim. For parity, specify that the multiplier is an integer. A numerical example illustrates a proof but cannot replace the arbitrary-object argument.
What you'll learn
- Describe the parts of a clear mathematical explanation
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Let a=2r and b=2s for integers r,s. Which line proves a+b is even?
- Substitute the definitions and factor 2.
- Closure of integers under addition supplies an integer multiplier.
Answer: a+b=2(r+s), and r+s is an integer
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Let a=2r and b=2s for integers r,s. Which line proves a+b is even?
Choices: a+b is even because it looks even · Take a=2 and b=4 only · a+b=2(r+s), and r+s is an integer · a+b=r+s
Show solution
- Substitute the definitions and factor 2.
- Closure of integers under addition supplies an integer multiplier.
Answer: a+b=2(r+s), and r+s is an integer
2. For odd integers a=2r+1 and b=2s+1, simplify a+b into even form.
Choices: r+s+2 · 2(r+s+1) · 2(r+s)+1 · 2rs+1
Show solution
- Add the two expressions: 2r+2s+2.
- Factor out 2.
Answer: 2(r+s+1)
3. To prove the square of any even integer is even, write n=2k. Which next calculation establishes the result?
Choices: n²=4k²=2(2k²) · n²=2k²+1 · n²=2k · Check only n=4
Show solution
- Square both factors, then display an integer multiple of 2.
Answer: n²=4k²=2(2k²)
4. A proof says n=2k, then n+6=2(k+3). Which missing fact completes its parity conclusion?
Choices: k must be positive · n must equal 6 · Every real number is even · k+3 is an integer
Show solution
- The definition of even requires an integer multiplier, not just a factor of 2.
Answer: k+3 is an integer
5. To prove that a multiple of 12 is divisible by 3, start n=12k. What finishes the proof?
Choices: n must be prime · n=3k+1 · n=3(4k), with 4k an integer · n=12+k
Show solution
- Re-express the arbitrary multiple as an integer multiple of 3.
Answer: n=3(4k), with 4k an integer
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Write a direct proof
Prove that the sum of an even integer and an odd integer is odd. Write a proof for arbitrary integers rather than a list of examples.
Review rubric and solution
- Represent the integers as 2r and 2s+1 with r,s integers.
- Derive 2(r+s)+1.
- State that r+s is an integer and invoke the definition of odd.
- Let a=2r and b=2s+1 for integers r,s.
- Then a+b=2r+2s+1=2(r+s)+1.
- Because r+s is an integer, the sum is odd.
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