Fraction Classroom · 2.0.0 · Teacher answers & responses
Same Numerator Picnic: teaching companion
For equal numerators and a fixed whole, compare the size of the pieces. 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 same-numerator fractions by reasoning about part size.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Draw identical picnic sandwiches partitioned into thirds and fourths.
activity-2. Shade one part in each. Explain which serving is larger.
activity-3. Partners repeat with two thirds and two fourths and compare the same number of pieces.
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 picnic sandwiches partitioned into thirds and fourths.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Shade one part in each. Explain which serving is larger.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners repeat with two thirds and two fourths and compare the same number of pieces.
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 1/3 and 1/4. Choose <, =, or >. The wholes are the same size. How does dividing one whole into more pieces affect each piece?
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. Compare one third and one sixth of identical sandwiches. Which is larger?
Answer: 1/3 is larger; one third covers two sixths.
Teacher response: If the learner chooses 1/6 because six exceeds three. Overlay a third on a sixth-partitioned copy and count two sixths under it.
d2. Compare 2/4 and 2/8 of identical strips. Are two pieces always the same amount?
Answer: No. Two fourths cover more than two eighths.
Teacher response: If the learner says equal numerators guarantee equal amounts. Select two pieces from each strip and align endpoints, then ask which piece size was repeated.
Check after reteaching
Complete this check independently after discussing the model.
r1. Of the same whole, which is larger: 3/4 or 3/8? Explain without decimal division.
Answer: 3/4 is larger because three fourth-size pieces exceed three eighth-size pieces.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Compare 1/3 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.
e2. How does dividing one whole into more pieces affect each piece?
Answer: Each equal piece becomes smaller.
Look for:
- Keep one whole fixed.
- Specify that the added partitions make equal pieces.
- State that more equal pieces in that whole mean a smaller size for each piece.
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.
Focused reteach
For equal numerators and a fixed whole, compare the size of the pieces.
teach-1. Use two identical wholes; partition one into thirds and one into sixths.
teach-2. Compare one selected piece, then two selected pieces from each whole.
teach-3. Have the learner explain why more equal pieces per whole means a smaller piece, before comparing labels.
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 1/3 compared with 1/4. e2 also meets its criterion: e2 explains that more equal pieces in one fixed whole make each piece smaller.
Teacher response: Continue to the linked next lesson. Test why the equal-whole condition is necessary for physical comparisons. Next original lesson: /lessons/whole-size-matters/.
next-2. e1 does not yet meet this criterion: e1 gives > for 1/3 compared with 1/4.
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/denominator-size/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that more equal pieces in one fixed whole make each piece smaller.
Teacher response: Ask the learner to fold an identical strip once more and describe what happened to each part before restating the e2 explanation. Short intervention: /interventions/denominator-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 1/3 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.
c2. Compare 1/2 and 1/8. 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.
c3. Compare 2/3 and 2/6. 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.
c4. Compare 3/4 and 3/8. 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.
c5. Compare 1/6 and 1/8. 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.
c6. Compare 2/4 and 2/8. 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.
c7. Compare 1/3 and 1/6. 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.
c8. Compare 3/6 and 3/8. 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.
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 1/3 and 1/6. 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.
s3. Compare 1/4 and 1/8. 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.
s4. Compare 2/3 and 2/6. 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.
h1. Why is 1/3 greater than 1/4 of the same sandwich?
Answer: Thirds divide it into fewer equal pieces, so each piece is larger.
Look for:
- Use the same sandwich as the whole for both shares.
- Explain that dividing into three equal parts makes larger pieces than dividing into four.
- Conclude that one third is greater than one fourth.
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.
h2. Show 2/4 and 2/8 with identical strips.
Answer: Shade two quarters and two eighths; the two quarters cover more.
Look for:
- Draw identical-length strips with four and eight equal partitions.
- Shade exactly two pieces in each strip.
- Show or state that 2/4 covers more than 2/8.
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.
h3. Correct the claim that eighths are larger because eight is larger.
Answer: Eight counts more pieces of a fixed whole, so each eighth is smaller.
Look for:
- Reject the claim that a larger denominator means a larger piece.
- Hold the whole size fixed and explain that eight counts more equal pieces.
- Use a comparison such as 1/8 < 1/4 to demonstrate the smaller part size.
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.
h4. Describe when the same-numerator rule can be used.
Answer: Use the same numerator and same-sized wholes, then compare part sizes.
Look for:
- Require equal numerators.
- Require the same-sized whole for an amount comparison.
- Explain that with equal counts, the larger pieces produce the greater fraction.
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. 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.
e1. Compare 1/3 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.
e2. How does dividing one whole into more pieces affect each piece?
Answer: Each equal piece becomes smaller.
Look for:
- Keep one whole fixed.
- Specify that the added partitions make equal pieces.
- State that more equal pieces in that whole mean a smaller size for each piece.
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.
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/denominator-size/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/whole-size-matters/. Test why the equal-whole condition is necessary for physical comparisons.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.