--- title: "Building and critiquing a mathematical argument" description: "Construct and present logical mathematical arguments involving multiple steps; critique others' reasoning about fractions, angles, or calculations and clearly explain errors or alternative methods" canonical: https://lightmysky.com/learn/mathematics/building-and-critiquing-a-mathematical-argument-mt_2ESZh70NyS source: https://lightmysky.com/learn/mathematics/building-and-critiquing-a-mathematical-argument-mt_2ESZh70NyS.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Building and critiquing a mathematical argument Construct and present logical mathematical arguments involving multiple steps; critique others' reasoning about fractions, angles, or calculations and clearly explain errors or alternative methods Subject: Mathematics · Area: Mathematical Thinking · Ages 9 to 10 Page: https://lightmysky.com/learn/mathematics/building-and-critiquing-a-mathematical-argument-mt_2ESZh70NyS ## Ready when they can - Prove that 3/4 > 2/3 using a common denominator argument and a visual model - Find and explain an error in a long multiplication where a partial product was misaligned - Present a chain of reasoning to show that angles in a triangle sum to 180° by tearing and arranging ## Lesson: Building a Strong Math Argument A math argument is a chain of steps that leads to a true answer. Each step must follow from the step before it. If you can walk someone through every step, you have proven your point, not just guessed it. That same skill lets you check someone else's work: find the one step where their chain breaks. **Example.** Which is bigger, 3/4 or 2/3? Step 1: give both fractions the same denominator. 12 works for both. Step 2: 3/4 becomes 9/12. Step 3: 2/3 becomes 8/12. Step 4: compare the new numerators. 9 is more than 8, so 9/12 is more than 8/12. Step 5: that means 3/4 is greater than 2/3. *(drawing: Tear the three corners off a triangle, then line them up along a straight edge.)* **Example.** Here is that argument as a chain. Step 1: tear the three corners off any triangle. Step 2: line them up side by side along a straight edge. Step 3: they fit exactly, with no gap and no overlap. Step 4: a straight line measures 180 degrees. Step 5: so the three angles of any triangle add up to 180 degrees. **Example.** Ben multiplied 23 by 14. He wrote 23 x 4 = 92 on the first line. For the tens digit, he wrote 23 x 1 = 23, then added 92 + 23 to get 115. But the 1 in 14 stands for 10, not 1, so that line should be 23 x 10 = 230, shifted one place left. The correct total is 92 + 230 = 322. Ben's mistake was forgetting to shift the second partial product. **Tip.** When you check someone's work, don't just say it's wrong. Point to the exact step where the argument breaks, and explain what should happen instead. That helps far more than just saying 'wrong.' **Recap.** A good argument moves step by step, and a good check finds the exact step that breaks. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Angles that add up on a diagram](https://lightmysky.com/learn/mathematics/angles-that-add-up-on-a-diagram-mt_89riIKwGYp) - [Justifying mathematical reasoning (age 8+)](https://lightmysky.com/learn/mathematics/justifying-mathematical-reasoning-age-8-mt_a3dov8CZkq) - [Comparing fractions (age 9+)](https://lightmysky.com/learn/mathematics/comparing-fractions-age-9-mt_SXbZ3bC9z7) - [Analysing Your Own Errors](https://lightmysky.com/learn/learning-to-learn/analysing-your-own-errors-mt_TDUpy57QVM) ## Opens up - [Justifying Views About Texts](https://lightmysky.com/learn/english/justifying-views-about-texts-mt_IX37F4rNed) - [Constructing mathematical arguments](https://lightmysky.com/learn/mathematics/constructing-mathematical-arguments-mt_j5YqQnN6xe)