Compare unlike denominators using common units and defend the result.
Teaching companion · Version 2.0.0
Select a valid comparison strategy and state the common unit behind the proof.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
Compare 2/3 and 3/4. Is comparing just the numerators enough?
Look for: 2/3 < 3/4 because 8/12 < 9/12; the numerators alone do not compare unlike units.
If you observe: Says 2 < 3 is the entire proof.
Respond: Ask whether the same argument would compare 2/3 and 3/8 correctly, then build common-unit models.
Compare 3/8 and 5/6 using one half. What evidence makes the benchmark decisive?
Look for: 3/8 < 1/2 < 5/6, so 3/8 < 5/6.
If you observe: Only states that one or both are near half.
Respond: Verify 6 < 8 and 10 > 6 by doubling the respective numerators, then place the fractions on opposite sides of half.
Check independently: Compare 3/5 and 5/8 using a common partition.
3/5 = 24/40 < 25/40 = 5/8.
Use the benchmark comparison short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
e1 meets its criterion: e1 gives < for 2/3 compared with 3/4. e2 also meets its criterion: e2 identifies a common whole and evidence comparing equal units or distances, rather than an unsupported rule.
Continue to the linked next lesson. Use common-unit reasoning to describe the missing fractional amount to one.
Next lesson: Compose Onee1 does not yet meet this criterion: e1 gives < for 2/3 compared with 3/4.
Ask the learner to explain the attempt, then use the matching diagnostic probe to identify the difficulty. Work through the focused reteach below and its check. Use the linked short intervention if the difficulty persists; retry e1 and e2 before moving on.
Open the linked short interventione1 meets its criterion, but e2 does not yet show this evidence: e2 identifies a common whole and evidence comparing equal units or distances, rather than an unsupported rule.
Ask the learner to attach the e2 proof criteria to e1 explicitly: identify the whole, write 8/12 and 9/12, and state why the counts can now be compared.
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Watch for: Assumes a larger denominator makes a larger piece when the whole and numerator are fixed.
Try: Fold identical strips into thirds and sixths. Place one third over two sixths, then compare one third with one sixth. For equal numerators, select the same number of pieces from each strip.
Watch for: Ignores how many equal-sized parts are selected when denominators match.
Try: Align two equal-length strips with the same partition. Pair their selected pieces one by one. The strip with extra pieces has the greater amount; equal counts end together.
Watch for: Ignores how many equal-sized parts are selected when denominators match.
Try: Align two equal-length strips with the same partition. Pair their selected pieces one by one. The strip with extra pieces has the greater amount; equal counts end together.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Look for in the response:
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Look for in the response:
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Look for in the response:
Watch for: Judges position relative to one half from numerator or denominator alone.
Try: For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Look for in the response:
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Watch for: Assumes equal fraction labels guarantee equal physical amounts even when the wholes differ.
Try: Compare half of a 12 cm strip with half of a 6 cm strip. Measure 6 cm and 3 cm. Keep the fraction label visible and ask what changed; then repeat the comparison using two equal-length strips.
Watch for: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Try: Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Look for in the response:
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
Eight core questions and a two-question exit ticket. Print one set per student.
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Choose 2/3 and 3/4 on identical wholes. Predict the larger amount.
Partition both wholes into twelfths and locate their endpoints.
Partners alternate between a common-denominator proof and a number-line proof.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Compare 2/3 and 3/4. Choose <, =, or >. The wholes are the same size. What makes a comparison proof convincing?
Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark.
Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
Teacher answer key. Open responses include a rubric and one valid sample.
2/3 < 3/4. Rename them as equal-sized parts or place them on the same unit number line.
3/5 < 5/8. Rename them as equal-sized parts or place them on the same unit number line.
4/6 = 2/3. Rename them as equal-sized parts or place them on the same unit number line.
7/10 > 2/3. Rename them as equal-sized parts or place them on the same unit number line.
5/6 < 7/8. Rename them as equal-sized parts or place them on the same unit number line.
2/5 < 3/7. Rename them as equal-sized parts or place them on the same unit number line.
3/4 < 5/6. Rename them as equal-sized parts or place them on the same unit number line.
7/12 > 4/9. Rename them as equal-sized parts or place them on the same unit number line.
1/2 > 1/4. Rename them as equal-sized parts or place them on the same unit number line.
2/4 < 3/4. Rename them as equal-sized parts or place them on the same unit number line.
1/3 < 2/3. Rename them as equal-sized parts or place them on the same unit number line.
1/2 = 2/4. Rename them as equal-sized parts or place them on the same unit number line.
Eight twelfths is less than nine twelfths.
2/3 is greater than 3/8 even though two is less than three.
24/40 is less than 25/40.
3/8 < 5/6 because one is below half and the other is above half.
2/3 < 3/4. Rename them as equal-sized parts or place them on the same unit number line.
It states the same whole and compares equal-sized units or distances.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Designed for Grade 4. Compare unlike denominators using common units and defend the result.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.