Fraction Classroom · 2.0.0 · Teacher answers & responses
Mixed Number Market: teaching companion
Exchange a whole for equal fractional pieces while preserving the amount. 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 4 · 45 minutes. Add and subtract mixed numbers with like denominators using regrouping.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Represent a market purchase of 1 3/4 yards plus 2 2/4 yards with strips.
activity-2. Combine whole strips and quarter pieces; trade four quarters for a whole.
activity-3. Partners model a return of 2/4 yard from 2 1/4 yards by regrouping one whole.
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: Represent a market purchase of 1 3/4 yards plus 2 2/4 yards with strips.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Combine whole strips and quarter pieces; trade four quarters for a whole.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners model a return of 2/4 yard from 2 1/4 yards by regrouping one whole.
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: Calculate 2 1/4 - 3/4. Show a model or your reasoning. Why is regrouping a whole allowed?
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. Rename 2 1/4 so that at least three fourths are available to remove. Does the amount change?
Answer: 2 1/4 = 1 5/4; the amount is unchanged.
Teacher response: If the learner writes 2 5/4 and keeps both original wholes. Physically trade one whole strip for four fourths, moving the whole strip out of the model as the quarters enter.
d2. Combine 1 3/4 and 2 2/4. How many complete wholes can you form?
Answer: 4 1/4: three original wholes plus five fourths, with four fourths exchanged for a whole.
Teacher response: If the learner leaves five fourths unrecognized or counts them as five wholes. Group four of the five quarter pieces inside one whole outline and leave one quarter outside.
Check after reteaching
Complete this check independently after discussing the model.
r1. Calculate 3 1/4 − 2/4 and check by addition.
Answer: 2 3/4; adding 2/4 returns 3 1/4.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Calculate 2 1/4 - 3/4. Show a model or your reasoning.
Answer: 3/2
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
e2. Why is regrouping a whole allowed?
Answer: One whole and four fourths have the same value, so regrouping preserves the total.
Look for:
- State that a whole equals four fourths in this problem.
- Explain the exchange 2 1/4 = 1 5/4.
- Say that the total starting value stays unchanged even though its representation changes.
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
Focused reteach
Exchange a whole for equal fractional pieces while preserving the amount.
teach-1. Represent the whole parts and fourths separately.
teach-2. Before subtraction, exchange one whole for four fourths; before simplifying a sum, reverse the exchange.
teach-3. Write an equality on both sides of the exchange and add back a removed amount to check the result.
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 3/2 or 1 1/2 for 2 1/4 − 3/4. e2 also meets its criterion: e2 explains that a whole and four fourths have equal value, so the exchange preserves the starting amount.
Teacher response: Continue to the linked next lesson. Use repeated fractional groups and exchange complete groups for wholes. Next original lesson: /lessons/repeated-fraction-groups/.
next-2. e1 does not yet meet this criterion: e1 gives 3/2 or 1 1/2 for 2 1/4 − 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/mixed-regrouping/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that a whole and four fourths have equal value, so the exchange preserves the starting amount.
Teacher response: Ask the learner to demonstrate exactly which whole is exchanged in e1 and verify 2 1/4 = 1 5/4 before explaining the subtraction. Short intervention: /interventions/mixed-regrouping/.
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. Calculate 1 3/4 + 2 2/4. Show a model or your reasoning.
Answer: 17/4
Teacher response: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one. Combine the three whole strips and five quarter-pieces from 1 3/4 + 2 2/4. Place four quarter-pieces into one whole outline and exchange them for a whole strip. Count four whole strips and one remaining quarter: 4 1/4.
c2. Calculate 2 1/3 + 1 2/3. Show a model or your reasoning.
Answer: 4
Teacher response: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one. Combine the three whole strips and five quarter-pieces from 1 3/4 + 2 2/4. Place four quarter-pieces into one whole outline and exchange them for a whole strip. Count four whole strips and one remaining quarter: 4 1/4.
c3. Calculate 1 5/6 + 2 2/6. Show a model or your reasoning.
Answer: 25/6
Teacher response: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one. Combine the three whole strips and five quarter-pieces from 1 3/4 + 2 2/4. Place four quarter-pieces into one whole outline and exchange them for a whole strip. Count four whole strips and one remaining quarter: 4 1/4.
c4. Calculate 3 1/8 + 1 7/8. Show a model or your reasoning.
Answer: 5
Teacher response: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one. Combine the three whole strips and five quarter-pieces from 1 3/4 + 2 2/4. Place four quarter-pieces into one whole outline and exchange them for a whole strip. Count four whole strips and one remaining quarter: 4 1/4.
c5. Calculate 2 1/4 - 2/4. Show a model or your reasoning.
Answer: 7/4
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
c6. Calculate 3 2/5 - 1 4/5. Show a model or your reasoning.
Answer: 8/5
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
c7. Calculate 4 1/6 - 2 5/6. Show a model or your reasoning.
Answer: 4/3
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
c8. Calculate 2 3/8 - 1 5/8. Show a model or your reasoning.
Answer: 3/4
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
s1. Calculate 1 1/4 - 1/4. Show a model or your reasoning.
Answer: 1
Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.
s2. Calculate 1 1/3 - 1/3. Show a model or your reasoning.
Answer: 1
Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.
s3. Calculate 2 1/4 - 1 1/4. Show a model or your reasoning.
Answer: 1
Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.
s4. Calculate 2 3/4 - 1/4. Show a model or your reasoning.
Answer: 5/2
Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.
h1. Model 2 1/4 - 2/4 without changing the amount you start with.
Answer: Regroup as 1 5/4, then remove 2/4 to leave 1 3/4.
Look for:
- Represent 2 1/4 and exchange one whole for four fourths.
- Show 2 1/4 = 1 5/4 before removing 2/4.
- Obtain 1 3/4 and keep the starting amount unchanged during the exchange.
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
h2. Explain when four fourths must be traded for one whole.
Answer: After addition, each full group of four quarter pieces becomes one whole.
Look for:
- Identify a complete group of four fourth-size pieces.
- Exchange each such group for one whole without changing its value.
- Explain that an addition result may contain such a group, for example 5/4 = 1 1/4.
Teacher response: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one. Combine the three whole strips and five quarter-pieces from 1 3/4 + 2 2/4. Place four quarter-pieces into one whole outline and exchange them for a whole strip. Count four whole strips and one remaining quarter: 4 1/4.
h3. Write a market story for 1 3/4 + 2 2/4.
Answer: Combine lengths of ribbon; the total is 4 1/4 yards.
Look for:
- Write a context combining 1 3/4 and 2 2/4 of the same measurement unit.
- Use addition to represent the combination.
- Give a total of 4 1/4 with the story’s unit.
Teacher response: Chooses an operation from the numbers without identifying whether the task combines, takes a part, shares, or counts groups. Ask the learner to finish “I know …; I need …” with units. Act out the quantity: combine disjoint colors, overlap two fractional selections, share one amount, or count fixed-size pieces. Write an equation only after the action is identified.
h4. Check 3 2/5 - 1 4/5 by addition.
Answer: 1 3/5 + 1 4/5 = 3 2/5.
Look for:
- Identify the difference as 1 3/5.
- Add back 1 4/5 rather than reversing the subtraction.
- Show 1 3/5 + 1 4/5 = 3 2/5, recovering the original amount.
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
e1. Calculate 2 1/4 - 3/4. Show a model or your reasoning.
Answer: 3/2
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
e2. Why is regrouping a whole allowed?
Answer: One whole and four fourths have the same value, so regrouping preserves the total.
Look for:
- State that a whole equals four fourths in this problem.
- Explain the exchange 2 1/4 = 1 5/4.
- Say that the total starting value stays unchanged even though its representation changes.
Teacher response: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping. Build 2 1/4 with two whole strips and one quarter. Exchange one whole for four quarters, giving 1 5/4. Remove three quarters and recombine the remaining pieces; add the removed amount back to check.
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/mixed-regrouping/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/repeated-fraction-groups/. Use repeated fractional groups and exchange complete groups for wholes.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.