Lesson 2 — Rational and Irrational Numbers

Summary

Learners learn to distinguish rational from irrational numbers. A rational number can be written as a fraction a/b (b ≠ 0); its decimal either terminates or repeats. An irrational number cannot — its decimal goes on forever without repeating. Learners sort a mixed set and apply the a/b test.

Objectives

• Distinguish rational from irrational numbers and explain the test: rational numbers can be written as a/b and have terminating or repeating decimals, while irrational numbers cannot. (D05.S1.08.01)

Connection

Split three equal flatbreads among five people: each gets 3/5, and 3/5 = 0.6 — a clean, terminating decimal. Split one flatbread among three: each gets 1/3 = 0.333…, repeating forever but still a fraction. Now measure a square’s diagonal with its side as the ruler — the diagonal is about 1.414… times the side, and it never settles into a repeating pattern. That is the difference: some numbers are fractions, and some are not.

Materials

• Number-sort cards
• Decimal-deciding page

Preparation

• Copy the number-sort cards and decimal-deciding page.
• Have worked examples ready: 0.75 = 3/4 (terminating → rational); 0.333… = 1/3 (repeating → rational); √2 ≈ 1.41421… (neither → irrational); π ≈ 3.14159… (neither → irrational).
• Recall from Lesson 1: the four strands and the number strand’s Grade 8 extension.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a rational number can be written as a/b where a and b are integers and b ≠ 0; its decimal terminates or repeats; (2) an irrational number cannot be written as a/b; its decimal never terminates or repeats; (3) √2 and π are irrational. Because the a/b test is a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of calling every long decimal “irrational” (0.333… is rational — it repeats) and of thinking √9 is irrational (√9 = 3, rational). The critical-thinking lens: “irrational” is a mathematical fact about the number, not a slur — many learners first meet the word in everyday English, so name that clearly. The global lens: the realization that the diagonal is incommensurable with the side was made, as legend tells it, in the ancient Greek circle (Hippasus, 5th century BCE — S-353), but the number line is a shared human discovery, held side by side with other traditions (philosophy §11). The egalitarian lens: deciding “fraction or not” is a skill anyone can do with one clean test — mathematics as sense-making open to all.

Procedure

1. Gather (5 min). Last time you mapped the year. Today you meet a question: is every number a fraction?
2. Meet the test (10 min). A rational number can be written as a/b (a and b integers, b ≠ 0). Its decimal terminates (0.75) or repeats (0.333…). An irrational number cannot be written as a/b; its decimal never terminates or repeats.
3. Worked examples (10 min). Decide together: 0.75 = 3/4 → rational. 0.333… = 1/3 → rational. √2 ≈ 1.41421356… → goes on, no repeating block → irrational. π ≈ 3.14159… → irrational. √9 = 3 → rational (be careful!).
4. Sort the cards (15 min). With a partner, sort each number into rational or irrational, and say why using the a/b test. Trade with another pair and check each other’s reasons.
5. Close (10 min). Share the hardest number to decide and how you settled it. Remember: the number line is fuller than the fractions alone — between the fractions, the irrationals fill in the gaps.

Differentiation

• Support: Use a smaller card set (clear terminating and repeating decimals first); say each decimal aloud and ask “does it stop, or does it repeat?” so learners with low vision can follow.
• Accessibility: Offer a raised-line or string number line with large, raised numerals so learners who are blind or have low vision can feel where each number sits; for learners with dyscalculia, provide the decimal expansions pre-printed and keep the focus on the terminates-or-repeats decision rather than on arithmetic.
• Extension: Write 0.121212… as a fraction to prove it is rational, and explain in words why √2 cannot be written as a/b.

Assessment

• Formative (observation/self): Can the learner sort numbers and justify each with the a/b test (terminates/repeats → rational; never → irrational)?
• Self-check: The learner asks, “Did I check whether the decimal terminates or repeats? Did I try to write it as a/b? Am I sure √9 and √4 are rational?”

Home connection

At home, find three numbers — a price, a measurement, a ratio — and decide whether each is rational or irrational, then explain your reasoning to someone.

Resources

• Rational and irrational numbers are standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019) (S-026). The story that the irrationality of √2 was noticed in the ancient Greek circle is a hedged legend (MacTutor, Hippasus, S-353); that √2 and π are irrational is a proved mathematical fact. Treating all mathematical traditions as worthy is a value (docs/philosophy.md §4).