Pre-assessment
Use an anonymous code S01–S40, not a name. Suggested time: 15–20 minutes. Show your reasoning with a model, words or an equation. Use the same access supports on both occasions.
c202-pre-1. A strip has 2 of 3 equal parts shaded. Split every third into two equal parts. Name the new shaded fraction and explain whether the amount changed.
Answer: 4/6 = 2/3. Each old part becomes two sixths; both shaded and total counts double while the whole and shaded length stay fixed.
Look for:
- 0 points: Changes only one count when repartitioning. No correct supporting model or explanation is present.
- 1 point: Writes 4/6 but does not explain the unchanged amount or common scale factor.
- 2 points: 4/6 = 2/3. Each old part becomes two sixths; both shaded and total counts double while the whole and shaded length stay fixed.
Teacher response: Watch for: Changes only one count when repartitioning. Respond: Mark two new parts inside every old part, including the unshaded part.
c202-pre-2. Complete 3/4 = ?/12. Explain the operation on both numbers.
Answer: 9/12. Multiplying both numerator and denominator by 3 preserves the value.
Look for:
- 0 points: Adds 8 to both numbers or scales only the denominator. No correct supporting model or explanation is present.
- 1 point: Gives 9 but does not explain that both numbers multiply by 3.
- 2 points: 9/12. Multiplying both numerator and denominator by 3 preserves the value.
Teacher response: Watch for: Adds 8 to both numbers or scales only the denominator. Respond: Show three twelfths inside each fourth and count three such groups.
c202-pre-3. Compare 3/5 and 3/8 of the same whole. Explain without decimals.
Answer: 3/5 > 3/8. Fifths are larger than eighths, and each fraction counts three of its units.
Look for:
- 0 points: Assumes the larger denominator makes the fraction larger. No correct supporting model or explanation is present.
- 1 point: Chooses the correct inequality but does not explain unit size and equal counts.
- 2 points: 3/5 > 3/8. Fifths are larger than eighths, and each fraction counts three of its units.
Teacher response: Watch for: Assumes the larger denominator makes the fraction larger. Respond: Compare one fifth and one eighth on equal strips, then select three of each.
c202-pre-4. Two walkers travel 5/12 km and 7/12 km. Which distance is farther and by how much? Explain the unit you counted.
Answer: 7/12 km is farther by 2/12 km, or 1/6 km. Both distances count twelfths of a kilometer.
Look for:
- 0 points: Compares denominators as distances or subtracts denominators. No correct supporting model or explanation is present.
- 1 point: Chooses 7/12 or finds the difference but omits the shared unit or another requested part.
- 2 points: 7/12 km is farther by 2/12 km, or 1/6 km. Both distances count twelfths of a kilometer.
Teacher response: Watch for: Compares denominators as distances or subtracts denominators. Respond: Place both distances on one twelfths line and count the two-space gap.
c202-pre-5. Use 1/2 to compare 2/5 and 5/8. Show why each lies on its side of 1/2.
Answer: 2/5 < 1/2 because 4 < 5; 5/8 > 1/2 because half of eight is four. Therefore 2/5 < 5/8.
Look for:
- 0 points: Compares numerators alone or uses a benchmark without checking both values. No correct supporting model or explanation is present.
- 1 point: Orders the fractions correctly but establishes only one relationship to 1/2.
- 2 points: 2/5 < 1/2 because 4 < 5; 5/8 > 1/2 because half of eight is four. Therefore 2/5 < 5/8.
Teacher response: Watch for: Compares numerators alone or uses a benchmark without checking both values. Respond: Double each numerator or mark half of each equal whole before deciding.
c202-pre-6. Both 3/4 and 5/6 are greater than 1/2. Is that enough to order them? Finish the comparison using distance from 1.
Answer: No. 3/4 is 1/4 below 1, while 5/6 is 1/6 below 1. Since 1/6 < 1/4, 5/6 is closer to 1 and 3/4 < 5/6.
Look for:
- 0 points: Treats fractions on the same side of a benchmark as equal. No correct supporting model or explanation is present.
- 1 point: Recognizes that being above 1/2 is inconclusive, or orders correctly without explaining the gaps.
- 2 points: No. 3/4 is 1/4 below 1, while 5/6 is 1/6 below 1. Since 1/6 < 1/4, 5/6 is closer to 1 and 3/4 < 5/6.
Teacher response: Watch for: Treats fractions on the same side of a benchmark as equal. Respond: Shade the missing portions up to 1 on equal strips and compare those portions.
Post-assessment
Use an anonymous code S01–S40, not a name. Suggested time: 15–20 minutes. Show your reasoning with a model, words or an equation. Use the same access supports on both occasions.
c202-post-1. A quilt diagram shades 3 of 4 equal rows. A student divides each row into three equal cells. How many cells are shaded, out of how many? Explain why the fraction is equivalent.
Answer: 9 of 12 cells, or 9/12 = 3/4. Every row becomes three cells, so both counts triple and the shaded area stays the same.
Look for:
- 0 points: Counts only new shaded parts while leaving the total unchanged. No correct supporting model or explanation is present.
- 1 point: Gets 9/12 but does not connect tripling both counts to unchanged shaded area.
- 2 points: 9 of 12 cells, or 9/12 = 3/4. Every row becomes three cells, so both counts triple and the shaded area stays the same.
Teacher response: Watch for: Counts only new shaded parts while leaving the total unchanged. Respond: Draw all three columns through every row, then count shaded and total cells.
c202-post-2. A label says 8/12 = 2/?. Find the missing denominator and describe how to undo the repartition.
Answer: The denominator is 3. Group the twelfths in fours: divide both 8 and 12 by 4 to obtain 2/3.
Look for:
- 0 points: Subtracts the same amount from both numbers to simplify. No correct supporting model or explanation is present.
- 1 point: Gives 3 without a valid grouping or common division explanation.
- 2 points: The denominator is 3. Group the twelfths in fours: divide both 8 and 12 by 4 to obtain 2/3.
Teacher response: Watch for: Subtracts the same amount from both numbers to simplify. Respond: Bundle four twelfths into each third and identify two shaded bundles out of three.
c202-post-3. A recipe uses either 4 scoops of size 1/6 cup or 4 scoops of size 1/10 cup. Which option uses less? Write and justify the comparison.
Answer: 4/10 < 4/6. Tenths of a cup are smaller than sixths; taking four of each keeps that order.
Look for:
- 0 points: Treats more partitions of a cup as larger scoops. No correct supporting model or explanation is present.
- 1 point: Chooses four tenth-cup scoops but gives no explanation about equal counts and unit size.
- 2 points: 4/10 < 4/6. Tenths of a cup are smaller than sixths; taking four of each keeps that order.
Teacher response: Watch for: Treats more partitions of a cup as larger scoops. Respond: Compare one scoop of each size using the same cup as the whole.
c202-post-4. A runner is at 11/12 km; a marker is at 8/12 km. Describe the runner’s position relative to the marker and the gap.
Answer: The runner is beyond the marker by 3/12 km, or 1/4 km, because eleven twelfths exceed eight twelfths by three twelfths.
Look for:
- 0 points: Subtracts denominators or treats a larger count of twelfths as smaller. No correct supporting model or explanation is present.
- 1 point: States the order or gap correctly but omits the other or the common-unit explanation.
- 2 points: The runner is beyond the marker by 3/12 km, or 1/4 km, because eleven twelfths exceed eight twelfths by three twelfths.
Teacher response: Watch for: Subtracts denominators or treats a larger count of twelfths as smaller. Respond: Mark both positions on one line and count intervals between them.
c202-post-5. Sort 7/12 and 3/8 on either side of 1/2, then order them. Explain the halfway test.
Answer: 3/8 < 1/2 < 7/12. Twice 3 is less than 8; 7 is greater than half of 12, which is 6.
Look for:
- 0 points: Uses the larger denominator alone to decide the order. No correct supporting model or explanation is present.
- 1 point: Orders correctly but shows evidence for only one fraction’s relationship to 1/2.
- 2 points: 3/8 < 1/2 < 7/12. Twice 3 is less than 8; 7 is greater than half of 12, which is 6.
Teacher response: Watch for: Uses the larger denominator alone to decide the order. Respond: Compare twice each numerator with its denominator using equal-whole strips.
c202-post-6. Two equal-capacity tanks are 7/8 full and 9/10 full. Both are almost full. Decide which is fuller by comparing the empty fractions.
Answer: The empty parts are 1/8 and 1/10. Because 1/10 is smaller, 9/10 is fuller and 7/8 < 9/10; the tanks have equal capacities.
Look for:
- 0 points: Assumes almost-full fractions are equal or orders their empty portions in the same direction. No correct supporting model or explanation is present.
- 1 point: Gives the correct fuller tank but does not explain the reversed order of the empty portions.
- 2 points: The empty parts are 1/8 and 1/10. Because 1/10 is smaller, 9/10 is fuller and 7/8 < 9/10; the tanks have equal capacities.
Teacher response: Watch for: Assumes almost-full fractions are equal or orders their empty portions in the same direction. Respond: Draw two equal-capacity tanks and compare the unfilled single parts.
Teaching sequence
Goal: Generate equivalent names and defend a comparison using unit size, common units or a benchmark. Prerequisite: Locate fractions on a number line and identify equal fractional units in the same whole.
c202-step-1. 1. Same Denominator Race: Compare counts of equal-sized units. Lesson: /lessons/same-denominator-race/
Answer: Does the explanation name the shared unit, not only the larger numerator?
c202-step-2. 2. Same Numerator Picnic: Compare the size of parts when the counts agree. Lesson: /lessons/same-numerator-picnic/
Answer: Can the learner explain why three fifths exceed three eighths for equal wholes?
c202-step-3. 3. Equivalent Strip Lab: Repartition without moving the fractional endpoint. Lesson: /lessons/equivalent-strip-lab/
Answer: Look for the same scale factor applied to both numbers.
c202-step-4. 4. Missing Number Bridges: Use multiplicative structure to fill equivalent-fraction equations. Lesson: /lessons/missing-number-bridges/
Answer: Can the learner reverse a scale factor and verify with a model?
c202-step-5. 5. Benchmark Sort: Use 1/2 and 1 to make meaningful estimates. Lesson: /lessons/benchmark-sort/
Answer: Can the learner say when a benchmark leaves the comparison undecided?
c202-step-6. 6. Compare and Defend: Select a strategy and justify a comparison. Lesson: /lessons/compare-and-defend/
Answer: Check that both fractions refer to equal wholes and that the argument is conclusive.
c202-step-7. 7. Tenths to Hundredths: Extend equivalent fractions to tenths and hundredths. Lesson: /lessons/tenths-to-hundredths/
Answer: Does multiplying by ten change the part count while preserving the value?
c202-step-8. 8. Decimal Number Line: Connect equivalent tenths and hundredths to positions. Lesson: /lessons/decimal-number-line/
Answer: Can two differently written numbers occupy the same location?
c202-step-9. 9. Decimal Detective: Use place value as a special case of common fractional units. Lesson: /lessons/decimal-detective/
Answer: Ask for a fraction model supporting a decimal comparison.
Skill scoring and next steps
Score each item 0–2 using its rubric. Keep unobserved items separate from zero. Each two-item skill has 4 points possible; group by skill, not by the overall total. Discuss the work before interpreting differences between the two occasions.
c202-skill-1. Equivalent names: items 1–2 on both assessments. Evidence: Preserves the whole and scales numerator and denominator together.
Answer: 0–1: reteach. Split every part into the same number of smaller parts, keeping the shaded endpoint fixed. /interventions/equivalent-factors/. 2–3: guided practice. Match strip partitions to multiplication equations and missing-number equations. /lessons/equivalent-strip-lab/. 4: extend. Find two distinct common denominators for a pair and explain why each preserves value. /lessons/missing-number-bridges/. Missing evidence: collect another response before grouping.
c202-skill-2. Compare equal counts or units: items 3–4 on both assessments. Evidence: Explains the roles of numerator, denominator and same-sized whole.
Answer: 0–1: reteach. Cut equal wholes into different numbers of equal parts and compare one part before counting several. /interventions/denominator-size/. 2–3: guided practice. Sort comparisons into same-denominator and same-numerator cases and explain the strategy. /lessons/same-numerator-picnic/. 4: extend. Create one example where a larger denominator gives a smaller fraction and one where it does not. /lessons/same-denominator-race/. Missing evidence: collect another response before grouping.
c202-skill-3. Choose and defend a benchmark: items 5–6 on both assessments. Evidence: Uses 1/2 or 1 when decisive and changes strategy when needed.
Answer: 0–1: reteach. Mark the halfway point on equal bars; compare each fraction to that same benchmark. /interventions/benchmark-comparison/. 2–3: guided practice. Compare distances from 1 or rewrite using common units when both fractions lie on the same side of 1/2. /lessons/benchmark-sort/. 4: extend. Defend the same comparison with two strategies and explain which is more efficient. /lessons/compare-and-defend/. Missing evidence: collect another response before grouping.