Fraction Classroom · 2.0.0 · Teacher answers & responses
Same Denominator Race: teaching companion
Compare selected counts only after confirming equal unit sizes. Student pages include two diagnostic probes, a reteaching check, and the original exit ticket. The teacher copy includes the complete lesson preparation, teaching sequence, diagnostic responses, all original question guides, and updated rubrics.
Complete lesson preparation
Grade 3 · 45 minutes. Compare fractions with the same denominator and the same whole.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Draw identical strips partitioned into eighths. Mark racers at 3/8 and 5/8.
activity-2. Explain which racer is farther from zero without using whole-number guessing.
activity-3. Partners make four new same-denominator races and defend each result.
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: Draw identical strips partitioned into eighths. Mark racers at 3/8 and 5/8.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Explain which racer is farther from zero without using whole-number guessing.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners make four new same-denominator races and defend each result.
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 5/8 and 3/8. Choose <, =, or >. The wholes are the same size. What must be true about the wholes?
Answer: Students: Complete the two exit questions independently.
Diagnostic probes
Explain or draw your reasoning. Your teacher will use these responses to choose what to revisit.
d1. On identical wholes, compare 3/8 and 5/8. What stays the same and what changes?
Answer: 3/8 < 5/8; eighth-piece size stays the same and the selected count changes.
Teacher response: If the learner says they are equal because both denominators are eight. Match three shaded eighths on each strip, then point to the two extra eighths in the second.
d2. One strip is twice as long as another. Can comparing their shaded counts of eighths alone compare actual lengths?
Answer: No. Their eighth-sized pieces have different physical lengths.
Teacher response: If the learner uses numerator count alone without checking the wholes. Measure an eighth of each strip, then replace them with same-length strips before applying the count comparison.
Check after reteaching
Complete this check independently after discussing the model.
r1. Using the same-sized whole, compare 6/8 and 4/8 and state why.
Answer: 6/8 > 4/8 because six equal eighths exceed four equal eighths.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Compare 5/8 and 3/8. 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.
e2. What must be true about the wholes?
Answer: The wholes must be the same size.
Look for:
- State that the wholes must be the same size for comparing these physical amounts.
- Explain that a shared denominator then names equal-sized pieces.
- Relate this condition to the 5/8 and 3/8 comparison in e1.
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.
Focused reteach
Compare selected counts only after confirming equal unit sizes.
teach-1. Align two identical wholes and partition both into eighths.
teach-2. Pair the selected pieces of 3/8 and 5/8; locate the unmatched pieces.
teach-3. Change one whole length and ask which condition of that proof is now lost.
Use the exit evidence
A wrong answer invites a follow-up probe; it does not by itself establish a particular misconception. Check e1 and e2 separately.
next-1. e1 meets its criterion: e1 gives > for 5/8 compared with 3/8. e2 also meets its criterion: e2 states that the wholes must be the same size for comparing these amounts.
Teacher response: Continue to the linked next lesson. Contrast equal-sized pieces with equal counts of differently sized pieces. Next original lesson: /lessons/same-numerator-picnic/.
next-2. e1 does not yet meet this criterion: e1 gives > for 5/8 compared with 3/8.
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/whole-size/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 states that the wholes must be the same size for comparing these amounts.
Teacher response: Keep e1 visible while replacing one strip with a longer strip. Ask the learner to state the condition needed to restore the original proof. Short intervention: /interventions/whole-size/.
Original question guide and updated rubrics
Use these question-specific teaching responses alongside the original lesson. Accept other mathematically valid open responses that meet the criteria. The original question wording and mathematical answers are retained; model descriptions below make their references readable offline.
c1. Compare 3/8 and 5/8. 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.
c2. Compare 2/3 and 1/3. 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.
c3. Compare 3/4 and 2/4. 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.
c4. Compare 5/6 and 4/6. 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.
c5. Compare 1/2 and 2/2. 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.
c6. Compare 7/8 and 6/8. 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.
c7. Compare 2/6 and 2/6. 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.
c8. Compare 1/4 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.
s1. Compare 1/2 and 2/2. 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.
s2. 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.
s3. Compare 1/4 and 2/4. 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.
s4. Compare 2/6 and 3/6. 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.
h1. Explain why 5/8 is greater than 3/8.
Answer: Both have eighth-size parts; five parts exceed three parts.
Look for:
- Identify eighths as the shared unit for both fractions.
- Compare five selected eighths with three selected eighths.
- Conclude 5/8 > 3/8 using equal-sized wholes.
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.
h2. Invent two different fractions between 2/8 and 6/8.
Answer: 3/8 and 5/8 lie between them.
Look for:
- Give two distinct fractions strictly between 2/8 and 6/8.
- Verify each is above 2/8 and below 6/8, accepting equivalent fraction names.
- Use an ordered line or equal eighths to justify both placements.
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.
h3. Does this rule work for strips of different sizes?
Answer: The amounts cannot be compared from the labels alone without equal wholes.
Look for:
- Explain that the rule compares selected counts only when part sizes match.
- Identify different whole sizes as a reason eighth-sized physical pieces may differ.
- Require equal wholes or measured amounts before making a physical-amount comparison.
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.
h4. Make an eighths race that ends in a tie.
Answer: 4/8 compared with 4/8 ends at the same point.
Look for:
- Choose two eighths fractions with the same numerator.
- Use equal-sized wholes or the same unit line.
- Explain that equal eighth-counts reach the same endpoint and produce a tie.
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.
e1. Compare 5/8 and 3/8. 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.
e2. What must be true about the wholes?
Answer: The wholes must be the same size.
Look for:
- State that the wholes must be the same size for comparing these physical amounts.
- Explain that a shared denominator then names equal-sized pieces.
- Relate this condition to the 5/8 and 3/8 comparison in e1.
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.
Teaching notes
- Teaching supplement version 2.0.0. Original lesson questions and PDF materials remain version 1.0.0. Original PDFs contain their original teaching notes and rubrics; use this supplement for the updated diagnostic guidance.
- Short intervention: /interventions/whole-size/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/same-numerator-picnic/. Contrast equal-sized pieces with equal counts of differently sized pieces.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.