6.3(E) — Number and Operations: Represent rational numbers as terminating or repeating decimals, fractions, and percents on a number line
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Generate a lesson like this 6th Grade · Math · pre-filled for youProject a blank number line from 0 to 1. Show three markers: 0.33, 1/3, and 33%. Ask students: "Which of these is largest? Which is smallest? Are any of them equal?" Students hold up 1, 2, or 3 fingers to vote.
Students write a quick prediction in their notebook. Then they explore using a decimal-fraction-percent conversion chart. Class discussion: What did you discover? Who was surprised by their answer?
The hook should create productive confusion — if students are certain 1/3 = 0.33, that's the misconception you need to address. Ask: "Is 0.33 the same as 1/3, or is 0.33 a rounded number? What does rounding mean?" This primes the distinction between terminating and repeating decimals.
Introduce the rule: if the denominator (in simplest form) has only 2s and 5s as prime factors, the decimal terminates. Any other prime factor creates a repeating decimal. Model 5 examples: 1/2, 3/5, 7/8 (terminating) vs. 1/3, 2/7 (repeating).
Students test 8 fractions using the 2s-and-5s rule, then verify with a calculator. They sort fractions into two columns on their desk: "Terminates" and "Repeats." Check as a class and discuss any disagreements.
The prime factorization rule is the key insight — but students need the reasoning, not just the rule. Ask: "Why does 2 and 5 make the decimal stop? What's special about base 10?" If time allows, explain base 10's connection to tenths, hundredths, thousandths. If not, hold it as a "bonus question" for advanced students.
Students place 9 rational numbers (a mix of fractions, decimals, and percents) on a number line from 0 to 1: 1/4, 0.6, 3/5, 75%, 1/3, 0.25, 0.9, 2/5, 50%. Numbers appear in random order — students must convert to order them.
Pairs compare number lines and resolve any ordering conflicts by converting to decimals. Teacher circulates and selects 2 pairs to present: one with no errors, one who made a correction during partner check. The second pair's before/after shows the value of the conversion check.
Watch for students who place 0.25 and 1/4 on different points — they're not recognizing equivalent forms. Ask: "If 0.25 and 1/4 are the same number, can they be in two different places?" The visual contradiction will do the teaching — you just name it.
Students receive 12 rational numbers in mixed formats and must: (1) convert all to decimals, (2) order from least to greatest, (3) write one sentence explaining their conversion strategy for the most challenging pair.
Students work independently. The most challenging pair is deliberately chosen to have close values (e.g., 5/8 = 0.625 vs. 0.63) — this forces careful comparison, not just quick conversion. Students self-check using a partner's ordered list for the first 6 items before continuing.
Circulate and note which students are converting everything to decimals, even when a fraction comparison would be faster. For those students, ask: "Could you compare 3/5 and 5/8 without converting? What would you need to do?" Plant the idea of common denominators — it's the algebraic thinking you want to develop.
Three comparison problems, each requiring conversion to the same form: (1) 7/12 vs. 0.58, (2) 125% vs. 1.25, (3) 2/9 vs. 0.22. Students must show their conversion for each.
Exit ticket on index cards. Students show all conversions; partial credit for correct reasoning even if final answer is wrong.
Problem 2 is a trap — 125% and 1.25 are equal. Students who answer "<" or ">" missed the equivalence. If multiple students fall for this, use it as the next day's warm-up: "Yesterday, 2/3 of you said 125% < 1.25. Let's find out together." The productive struggle builds number sense better than a lecture.
Provide a pre-completed conversion chart for the first two activities. Use visual number lines with fractions already placed — students only add the decimal and percent equivalents. For the exit ticket, reduce to 2 comparison problems. Allow calculator use throughout.
Challenge: Find a fraction between 0.3 and 0.4 with a denominator of 7. Then find all fractions between 0.3 and 0.4 that have a denominator of 7. Order them from least to greatest. This problem requires understanding repeating decimals and equivalence — more rigorous than it appears.
Pre-teach: terminate, repeat, denominator, numerator, equivalent. Provide a bilingual conversion table with Spanish number names for fractions (medios, tercios, cuartos). Use a physical number line manipulative — students physically place cards with fractions, decimals, and percents. Allow verbal comparison before written work.
Exit ticket scored on 3-point rubric per problem: 3 = correct comparison + shown work + reasoning clear; 2 = correct comparison with incomplete work shown; 1 = incorrect comparison. Total 9 points possible. Students scoring below 5 receive small-group reteach on the conversion method before the next class. Identifying the 125% = 1.25 trap is extra credit (not required).
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