Hunting for Irrational Numbers
Students become number hunters, sorting decimals into rational and irrational families and pinning irrational numbers to their exact spot on a class number line.
Common Core standards covered
- 8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
- 8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Distinguish rational from irrational numbers, convert repeating decimals to fractions, and locate irrational numbers approximately on a number line.
Student target: “I can classify numbers as rational or irrational from their decimal expansions, convert a repeating decimal into a fraction, and place irrational numbers on a number line using rational approximations.”
Success criteria: I can sort decimals into terminating, repeating and non-repeating — and say which two kinds are rational. · I can convert a repeating decimal to a fraction with the shift-and-subtract method. · I can bracket an irrational number between tenths and place it on a number line. · I can use approximations to compare irrational and rational numbers.
Essential question
Every number has a decimal expansion. Some stop, some repeat forever, and some never do either. Which kind can hide from fractions?
Learning target: I can classify numbers as rational or irrational from their decimal expansions, convert a repeating decimal into a fraction, and place irrational numbers on a number line using rational approximations.
Success criteria
- I can sort decimals into terminating, repeating and non-repeating — and say which two kinds are rational.
- I can convert a repeating decimal to a fraction with the shift-and-subtract method.
- I can bracket an irrational number between tenths and place it on a number line.
- I can use approximations to compare irrational and rational numbers.
Why this matters
This is the lesson where the number line fills in. Students have lived on fractions; now they learn the line is mostly numbers no fraction can name — and that we handle them the way engineers do, by rational approximation to whatever precision the job needs. The shift-and-subtract conversion is also a beautiful early proof: it demonstrates WHY repeating decimals must be fractions, which is exactly what makes never-repeating decimals something genuinely new.
Students should already be able to
- Convert simple fractions to decimals by division (7.NS.A.2.d).
- Bracket square roots between whole numbers and tenths — previous lesson.
- Multiply a decimal by 10 or 100 (shift the point).
Vocabulary to define explicitly
- Rational number
A number expressible as a fraction of integers — its decimal terminates or repeats.
Example: 0.75 = 3/4 and 0.363636… = 4/11 are both rational.
Not an example: √2 is not rational — no fraction of integers squares to exactly 2.
🔒 3 more terms with examples and non-examples — Irrational number, Repeating block, Rational approximation — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The warm-up's three-way odd-one-out — listen for forever-versus-repeating.
- Machine runs during the hunt: shift size and tail cancellation observed.
- The trap ruling: the class vote before and after argument.
Summative — the graded evidence: The exit ticket (classify, convert, locate, compare), scored with the four-criterion rubric.
The task students work on
Your team runs a number-identification lab. Specimen cards carry decimal expansions; your job is to classify each specimen — terminating, repeating, or never-repeating — convert every repeater back to the fraction it came from, and pin the irrationals to the lab's number line with two-decimal precision. One specimen is engineered to fool you.
Specimen log.
| Specimen | Classification | Fraction (if rational) | Location on the line |
|---|---|---|---|
| 0.75 | |||
| 0.333… | |||
| 0.363636… | |||
| 1.41421356… (√2) | |||
| 3.14159265… (π) | |||
| 0.101001000100001… |
Anticipated student strategies
Sort, then convert, then locate. The conversion machine is the heart: once students see every repeater surrender a fraction, 'never repeats' stops being trivia and becomes the exact boundary of the rational world.
- Look for the block. The student scans the expansion for a chunk that cycles: 0.363636… has block 36; √2's digits never settle into one.
The sorting instinct. Share first — and note that 'never settles' can only be CLAIMED for specimens like √2 on authority; the trap card tests whether students confuse pattern with repetition.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
0.75 stops; 0.333… goes forever; 1.4142… goes forever WITHOUT repeating. All three cases are right — and the three behaviors are the day's whole taxonomy. Name the claim to be tested: the first two are secretly fractions; the third cannot be.
Listen for: No — it repeats, and repeating is the fingerprint of a fraction: it IS 1/3. Irrational needs forever AND patternless-forever.
Run it again for 0.777…: block of one digit, shift one place, 10x − x = 9x = 7, x = 7/9. Then the punchline: this machine works on EVERY repeating decimal — which is why a decimal that never repeats can never be a fraction.
- Block length picks the shift: one digit → 10x, two digits → 100x.
- The subtraction kills the tail — that is the whole trick.
Listen for: Both numbers have the SAME tail after the point — shifting by exactly one block aligned them. Identical tails cancel.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “It goes on forever, so it's irrational.” 0.333… goes on forever and equals 1/3. Forever is necessary but not sufficient — it must also never repeat.
- “0.101001000… has a pattern, so it's rational.” — the teacher response is in the full lesson.
- “π = 22/7.” — the teacher response is in the full lesson.
- “√2 eventually stops or loops if you compute far enough.” — the teacher response is in the full lesson.
- “Irrational numbers aren't really ON the number line.” — the teacher response is in the full lesson.
- “1/3 is 'approximately' 0.333.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 4 independent.
- I doConvert 0.777… to a fraction with the machine.
🔒 5 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Classify each, with one reason: 0.121212… and 0.101001000100001….
🔒 3 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Sort a stack of decimal cards into terminating, repeating, and never-repeating piles. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Model converting 0.363636… to a fraction using the multiply-and-subtract method. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Teams place √2, √5, π, and 2.7 on a shared class number line and defend each position. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Compare estimates and discuss why √2 lands between 1.4 and 1.5. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Provide a perfect-square reference chart for support; extend by comparing √50 and 7.1 without a calculator.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: Students place two irrational numbers on a number line and explain how they narrowed down each location.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.