Compare and Defend: teaching companion

prepare. Before starting: Recognize a whole and count equal parts.

materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).

activity-1. Choose 2/3 and 3/4 on identical wholes. Predict the larger amount.

activity-2. Partition both wholes into twelfths and locate their endpoints.

activity-3. Partners alternate between a common-denominator proof and a number-line proof.

phase-1. 5 min · warmup: Look at the first model or task. What is the whole? What do you notice?

Answer: Students: Sketch an initial prediction and explain it to a partner.

phase-2. 8 min · model: Choose 2/3 and 3/4 on identical wholes. Predict the larger amount.

Answer: Students: Follow the model; label the whole and the equal parts.

phase-3. 10 min · partner: Partition both wholes into twelfths and locate their endpoints.

Answer: Students: Work in pairs. One models, the other checks; switch roles.

phase-4. 7 min · discussion: Partners alternate between a common-denominator proof and a number-line proof.

Answer: Students: Compare two solutions and explain the relationship.

phase-5. 10 min · practice: Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.

Answer: Students: Solve independently, then compare the reasoning.

phase-6. 5 min · exit: Compare 2/3 and 3/4. Choose <, =, or >. The wholes are the same size. What makes a comparison proof convincing?

Answer: Students: Complete the two exit questions independently.

d1. Compare 2/3 and 3/4. Is comparing just the numerators enough?

Answer: 2/3 < 3/4 because 8/12 < 9/12; the numerators alone do not compare unlike units.

Teacher response: If the learner says 2 < 3 is the entire proof. Ask whether the same argument would compare 2/3 and 3/8 correctly, then build common-unit models.

d2. Compare 3/8 and 5/6 using one half. What evidence makes the benchmark decisive?

Answer: 3/8 < 1/2 < 5/6, so 3/8 < 5/6.

Teacher response: If the learner only states that one or both are near half. Verify 6 < 8 and 10 > 6 by doubling the respective numerators, then place the fractions on opposite sides of half.

r1. Compare 3/5 and 5/8 using a common partition.

Answer: 3/5 = 24/40 < 25/40 = 5/8.

e1. Compare 2/3 and 3/4. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

e2. What makes a comparison proof convincing?

Answer: It states the same whole and compares equal-sized units or distances.

Look for:

• Require a stated common whole or unit.
• Use evidence comparing equal-sized parts, equal-unit distances, or a separating benchmark.
• Explain how that evidence supports the inequality in question rather than only naming a rule.

Teacher response: Assumes equal fraction labels guarantee equal physical amounts even when the wholes differ. 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. 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.

teach-1. Check whole size before comparing physical amounts.

teach-2. Ask whether the denominators match, numerators match, or a benchmark separates the values.

teach-3. When none finishes the comparison, rename both in common units and explain the resulting count comparison.

next-1. 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.

Teacher response: Continue to the linked next lesson. Use common-unit reasoning to describe the missing fractional amount to one. Next original lesson: /lessons/compose-one/.

next-2. e1 does not yet meet this criterion: e1 gives < for 2/3 compared with 3/4.

Teacher response: 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. Short intervention: /interventions/benchmark-comparison/.

next-3. e1 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.

Teacher response: 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. Short intervention: /interventions/benchmark-comparison/.

c1. Compare 2/3 and 3/4. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

c2. Compare 3/5 and 5/8. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

c3. Compare 4/6 and 2/3. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. 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.

c4. Compare 7/10 and 2/3. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

c5. Compare 5/6 and 7/8. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

c6. Compare 2/5 and 3/7. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

c7. Compare 3/4 and 5/6. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

c8. Compare 7/12 and 4/9. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: Judges position relative to one half from numerator or denominator alone. 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.

s1. Compare 1/2 and 1/4. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: Assumes a larger denominator makes a larger piece when the whole and numerator are fixed. 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.

s2. Compare 2/4 and 3/4. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: Ignores how many equal-sized parts are selected when denominators match. 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.

s3. Compare 1/3 and 2/3. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: Ignores how many equal-sized parts are selected when denominators match. 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.

s4. Compare 1/2 and 2/4. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. 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.

h1. Defend 2/3 < 3/4 with common units.

Answer: Eight twelfths is less than nine twelfths.

Look for:

• Rename 2/3 as 8/12 and 3/4 as 9/12, or choose another valid common unit.
• Keep the whole the same for both quantities.
• Compare the equal-unit counts and conclude 2/3 < 3/4.

Teacher response: 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.

h2. Explain why bigger numerator alone is not enough.

Answer: 2/3 is greater than 3/8 even though two is less than three.

Look for:

• State that a numerator counts pieces whose sizes may differ.
• Give a valid counterexample, such as 2/3 > 3/8.
• Justify the counterexample with common units or aligned models.

Teacher response: 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.

h3. Show 3/5 and 5/8 on a common partition.

Answer: 24/40 is less than 25/40.

Look for:

• Use fortieths or another valid common partition for fifths and eighths.
• Represent 3/5 as 24/40 and 5/8 as 25/40, or equivalent counts.
• Compare the selected parts on the same whole.

Teacher response: 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.

h4. Invent a comparison that is easier with a benchmark than a common denominator.

Answer: 3/8 < 5/6 because one is below half and the other is above half.

Look for:

• Choose two fractions separated by a useful benchmark.
• Show one is below and the other above that benchmark.
• State the resulting order; a pair merely on the same side is insufficient.

Teacher response: Judges position relative to one half from numerator or denominator alone. 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.

e1. Compare 2/3 and 3/4. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

e2. What makes a comparison proof convincing?

Answer: It states the same whole and compares equal-sized units or distances.

Look for:

• Require a stated common whole or unit.
• Use evidence comparing equal-sized parts, equal-unit distances, or a separating benchmark.
• Explain how that evidence supports the inequality in question rather than only naming a rule.

Teacher response: Assumes equal fraction labels guarantee equal physical amounts even when the wholes differ. 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. 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.