Fraction Classroom · 2.0.0 · Teacher answers & responses
Mixed Number Repair: teaching companion
Match fractional units, then exchange a whole without altering the starting quantity. 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 5 · 45 minutes. Regroup mixed numbers to subtract unlike fractional parts.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Build 2 1/4 with whole and quarter strips. Remove 3/4 by trading a whole for quarters.
activity-2. Rebuild with sixths and fourths; decide a common unit before regrouping.
activity-3. Partners identify and repair a worked subtraction that reverses the fractional subtraction.
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: Build 2 1/4 with whole and quarter strips. Remove 3/4 by trading a whole for quarters.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Rebuild with sixths and fourths; decide a common unit before regrouping.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners identify and repair a worked subtraction that reverses the fractional subtraction.
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. What value does regrouping preserve?
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. A learner claims 2 1/4 − 3/4 = 2 2/4 by subtracting the smaller numerator from the larger. Repair the reasoning.
Answer: Exchange 2 1/4 for 1 5/4, then remove 3/4 to get 1 1/2.
Teacher response: If the learner reverses the fractional subtraction and keeps both wholes. Act out the original removal using quarters; trade one whole when one quarter is insufficient.
d2. To find 3 1/3 − 1 3/4, which pieces should the exchanged whole supply?
Answer: Twelfths: 3 4/12 = 2 16/12, then subtract 1 9/12 to get 1 7/12.
Teacher response: If the learner exchanges into thirds while trying to remove fourths. Name both amounts in twelfths before trading a whole for twelve twelfths.
Check after reteaching
Complete this check independently after discussing the model.
r1. Find 2 1/5 − 1 1/2 after making tenths.
Answer: 2 2/10 = 1 12/10; subtracting 1 5/10 leaves 7/10.
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. What value does regrouping preserve?
Answer: The total starting amount stays unchanged.
Look for:
- Identify the total starting amount as the preserved value.
- Explain that one whole is exchanged for an equal set of fractional parts.
- Connect to e1 with 2 1/4 = 1 5/4 rather than adding extra value.
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
Match fractional units, then exchange a whole without altering the starting quantity.
teach-1. Rename the fractional parts in a common unit when needed.
teach-2. Exchange exactly one whole for one denominator’s worth of those pieces, reducing the whole count by one.
teach-3. Subtract in the original order, then add the removed quantity 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 states that regrouping preserves the total starting amount.
Teacher response: Continue to the linked next lesson. Use size predictions and inverse checks to reject implausible mixed or unlike-fraction results. Next original lesson: /lessons/estimate-before-exact/.
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 states that regrouping preserves the total starting amount.
Teacher response: Require the equality 2 1/4 = 1 5/4 beside the model, then ask the learner to explain why one fewer whole and four more fourths preserve the total. 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 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.
c2. Calculate 3 1/3 - 1 3/4. Show a model or your reasoning.
Answer: 19/12
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
c3. Calculate 2 1/5 - 1 1/2. Show a model or your reasoning.
Answer: 7/10
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
c4. Calculate 4 1/6 - 2 2/3. 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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
c5. Calculate 3 1/8 - 1 5/6. Show a model or your reasoning.
Answer: 31/24
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
c6. Calculate 2 1/2 - 1 3/5. Show a model or your reasoning.
Answer: 9/10
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
c7. Calculate 5 1/4 - 2 2/3. Show a model or your reasoning.
Answer: 31/12
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
c8. Calculate 3 2/5 - 1 7/10. Show a model or your reasoning.
Answer: 17/10
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
s1. Calculate 1 1/4 - 1/2. 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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
s2. Calculate 2 1/3 - 2/3. Show a model or your reasoning.
Answer: 5/3
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.
s3. Calculate 1 1/5 - 2/5. Show a model or your reasoning.
Answer: 4/5
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.
s4. Calculate 2 1/2 - 3/4. 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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
h1. Explain 2 1/4 - 3/4 using regrouping.
Answer: 2 1/4 = 1 5/4; removing 3/4 leaves 1 2/4 = 1 1/2.
Look for:
- Exchange 2 1/4 for the equal amount 1 5/4.
- Remove 3/4 from five fourths while retaining the one whole.
- Obtain 1 2/4 = 1 1/2.
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.
h2. Repair 2 1/4 - 3/4 = 2 2/4.
Answer: The subtraction was reversed; regroup before removing to get 1 1/2.
Look for:
- Identify that reversing 1/4 − 3/4 changes the subtraction.
- Regroup 2 1/4 as 1 5/4.
- Correct the result to 1 1/2 and explain the exchange of one whole.
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.
h3. Check a regrouped result by addition.
Answer: 1 1/2 + 3/4 = 2 1/4.
Look for:
- State the subtraction and the proposed difference being checked.
- Add the removed amount back to that difference.
- Recover the original starting amount, for example 1 1/2 + 3/4 = 2 1/4.
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.
h4. Why should common units be found before regrouping unlike fractions?
Answer: Then the whole can be exchanged into exactly the parts being removed.
Look for:
- Explain that unlike fractional pieces must first be named in one unit.
- State that the exchanged whole supplies exactly one denominator’s worth of that unit.
- Give an example such as exchanging one whole for twelve twelfths when subtracting thirds and fourths.
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. Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.
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. What value does regrouping preserve?
Answer: The total starting amount stays unchanged.
Look for:
- Identify the total starting amount as the preserved value.
- Explain that one whole is exchanged for an equal set of fractional parts.
- Connect to e1 with 2 1/4 = 1 5/4 rather than adding extra value.
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/estimate-before-exact/. Use size predictions and inverse checks to reject implausible mixed or unlike-fraction results.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.