Sums and Products of Rational and Irrational Numbers
I can explain and apply the closure rules for sums and products of rational and irrational numbers.
The problem this sheet solves
Closure is where students first learn that a rule can need a condition. They are taught that a rational times an irrational stays irrational, and the standard itself says NONZERO rational. Zero is the one exception, and a sheet that leaves it out is teaching a statement that a single counterexample destroys. This sheet treats the exception as the content rather than a footnote, and asks students to produce the counterexample themselves. It separates the four closure claims so a student can see which two are unconditional and which one is not, and the reasoning item asks WHY one counterexample is enough to sink a general claim, which is a habit that pays off well beyond this standard.
The key rule
Rational + rational and rational × rational always stay rational, closed. Rational + irrational is always irrational. A NONZERO rational × irrational is always irrational — zero is the one exception.
Test every claim by trying to break it — zero is the one number that does.
Watch out
The mistake: Marking 0 × √7 irrational — assuming rational times irrational is always irrational.
The fix: Needs NONZERO. 0 × √7 = 0, which is rational — the one exception.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.