All units / Unit 2
Equivalent Fractions & Defensible Comparisons
Preserve value while repartitioning, choose a comparison strategy, and justify decisions with equal wholes.
Learning 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.
Materials for your group
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
Teach in a connected sequence
Each complete lesson is 45 minutes. Use the pre-assessment to select a starting point; revisit a prerequisite whenever its observation is not yet secure. The short interventions in the response guide provide focused practice between lessons.
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Same Denominator Race
Goal: Compare counts of equal-sized units.
Observe: Does the explanation name the shared unit, not only the larger numerator?
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Same Numerator Picnic
Goal: Compare the size of parts when the counts agree.
Observe: Can the learner explain why three fifths exceed three eighths for equal wholes?
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Equivalent Strip Lab
Goal: Repartition without moving the fractional endpoint.
Observe: Look for the same scale factor applied to both numbers.
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Missing Number Bridges
Goal: Use multiplicative structure to fill equivalent-fraction equations.
Observe: Can the learner reverse a scale factor and verify with a model?
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Benchmark Sort
Goal: Use 1/2 and 1 to make meaningful estimates.
Observe: Can the learner say when a benchmark leaves the comparison undecided?
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Compare and Defend
Goal: Select a strategy and justify a comparison.
Observe: Check that both fractions refer to equal wholes and that the argument is conclusive.
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Tenths to Hundredths
Goal: Extend equivalent fractions to tenths and hundredths.
Observe: Does multiplying by ten change the part count while preserving the value?
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Decimal Number Line
Goal: Connect equivalent tenths and hundredths to positions.
Observe: Can two differently written numbers occupy the same location?
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Decimal Detective
Goal: Use place value as a special case of common fractional units.
Observe: Ask for a fraction model supporting a decimal comparison.
Before teaching: pre-assessment
Allow about 15–20 minutes. Ask for a model, explanation or equation. Keep access supports consistent between assessments. Items in the same position assess the same skill through a different task.
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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, scoring & response
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.
- 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.
Watch for: Changes only one count when repartitioning.
Respond: Mark two new parts inside every old part, including the unshaded part.
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Complete 3/4 = ?/12. Explain the operation on both numbers.
Answer, scoring & response
Answer: 9/12. Multiplying both numerator and denominator by 3 preserves the value.
- 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.
Watch for: Adds 8 to both numbers or scales only the denominator.
Respond: Show three twelfths inside each fourth and count three such groups.
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Compare 3/5 and 3/8 of the same whole. Explain without decimals.
Answer, scoring & response
Answer: 3/5 > 3/8. Fifths are larger than eighths, and each fraction counts three of its units.
- 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.
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.
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Two walkers travel 5/12 km and 7/12 km. Which distance is farther and by how much? Explain the unit you counted.
Answer, scoring & response
Answer: 7/12 km is farther by 2/12 km, or 1/6 km. Both distances count twelfths of a kilometer.
- 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.
Watch for: Compares denominators as distances or subtracts denominators.
Respond: Place both distances on one twelfths line and count the two-space gap.
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Use 1/2 to compare 2/5 and 5/8. Show why each lies on its side of 1/2.
Answer, scoring & response
Answer: 2/5 < 1/2 because 4 < 5; 5/8 > 1/2 because half of eight is four. Therefore 2/5 < 5/8.
- 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.
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.
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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, scoring & response
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.
- 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.
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.
After teaching: post-assessment
Allow about 15–20 minutes. Ask for a model, explanation or equation. Keep access supports consistent between assessments. Items in the same position assess the same skill through a different task.
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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, scoring & response
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.
- 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.
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.
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A label says 8/12 = 2/?. Find the missing denominator and describe how to undo the repartition.
Answer, scoring & response
Answer: The denominator is 3. Group the twelfths in fours: divide both 8 and 12 by 4 to obtain 2/3.
- 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.
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.
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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, scoring & response
Answer: 4/10 < 4/6. Tenths of a cup are smaller than sixths; taking four of each keeps that order.
- 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.
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.
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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, scoring & response
Answer: The runner is beyond the marker by 3/12 km, or 1/4 km, because eleven twelfths exceed eight twelfths by three twelfths.
- 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.
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.
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Sort 7/12 and 3/8 on either side of 1/2, then order them. Explain the halfway test.
Answer, scoring & response
Answer: 3/8 < 1/2 < 7/12. Twice 3 is less than 8; 7 is greater than half of 12, which is 6.
- 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.
Watch for: Uses the larger denominator alone to decide the order.
Respond: Compare twice each numerator with its denominator using equal-whole strips.
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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, scoring & response
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.
- 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.
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.
Read the three skill areas separately
Items 1–2, 3–4 and 5–6 form matching pre/post pairs by skill. Each item earns 0, 1 or 2 points using its own rubric. A blank or inaccessible response stays “not observed” until you collect evidence. Use a drawing, oral explanation or manipulatives consistently if that gives the learner a fair way to respond.
Equivalent names
Evidence: Preserves the whole and scales numerator and denominator together.
0–1 points — reteach: Split every part into the same number of smaller parts, keeping the shaded endpoint fixed. Open the intervention.
2–3 points — guided practice: Match strip partitions to multiplication equations and missing-number equations. Open the next lesson.
4 points — extend: Find two distinct common denominators for a pair and explain why each preserves value. Open the extension lesson.
Compare equal counts or units
Evidence: Explains the roles of numerator, denominator and same-sized whole.
0–1 points — reteach: Cut equal wholes into different numbers of equal parts and compare one part before counting several. Open the intervention.
2–3 points — guided practice: Sort comparisons into same-denominator and same-numerator cases and explain the strategy. Open the next lesson.
4 points — extend: Create one example where a larger denominator gives a smaller fraction and one where it does not. Open the extension lesson.
Choose and defend a benchmark
Evidence: Uses 1/2 or 1 when decisive and changes strategy when needed.
0–1 points — reteach: Mark the halfway point on equal bars; compare each fraction to that same benchmark. Open the intervention.
2–3 points — guided practice: Compare distances from 1 or rewrite using common units when both fractions lie on the same side of 1/2. Open the next lesson.
4 points — extend: Defend the same comparison with two strategies and explain which is more efficient. Open the extension lesson.
Record evidence & plan groups
Use codes S01–S40 and keep the same code for the pre- and post-assessment. Enter rubric scores only. Records stay in this browser; names and written responses are not collected. Download a CSV before clearing browser data.
Each skill has two items and a maximum of 4 points. Use 0–1: reteach, 2–3: guided practice, and 4: extend as planning prompts. An unobserved item keeps that skill out of a group until you collect evidence. These are instructional decisions, not standardized proficiency levels.
Groups for the selected assessment
No records yet. Score an assessment to see the next teaching steps.
Alignment & use
These original tasks follow the concepts in the linked standards. The pre- and post-assessments pair the same constructs with different contexts or representations; they are classroom planning tools, not a standardized test. Discuss unexpected score differences before changing a group.
Use the number line, fraction strips or area model to make reasoning visible.