CCSS.Math.Content.HSN-RN.B.3
The standard
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Common Core State Standards for Mathematics · The Real Number System
What this standard means
Students need to reason about what happens when rational and irrational numbers are added or multiplied. They should use definitions, not just examples. A rational number can be written as a fraction. An irrational number cannot. Students should explain why operations stay inside or outside the rational numbers in specific cases.
Mastery looks like a short proof in words and symbols, such as using a/b and c/d for rational numbers. Students often get stuck thinking one example proves a rule. They also forget the nonzero condition for multiplying by a rational number, since 0 times any irrational number is 0.
Ways to teach it
- Hands-on activity: Sort number cards into rational and irrational, then pair cards and predict whether each sum or product is rational or irrational.
- Writing prompt: Explain why 3 plus square root of 2 cannot be rational, using a contradiction argument in three sentences.
- Quick assessment: Give four expressions, such as 2/5 times square root of 7, and ask students to label each with one reason.
- Real-world connection: Use calculator measurements like diagonal lengths, then discuss why rounded decimals can hide whether a number is irrational.
Plan a lesson for CCSS.Math.Content.HSN-RN.B.3
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