Regroup mixed numbers to subtract unlike fractional parts.
Teaching companion · Version 2.0.0
Match fractional units, then exchange a whole without altering the starting quantity.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
A learner claims 2 1/4 − 3/4 = 2 2/4 by subtracting the smaller numerator from the larger. Repair the reasoning.
Look for: Exchange 2 1/4 for 1 5/4, then remove 3/4 to get 1 1/2.
If you observe: Reverses the fractional subtraction and keeps both wholes.
Respond: Act out the original removal using quarters; trade one whole when one quarter is insufficient.
To find 3 1/3 − 1 3/4, which pieces should the exchanged whole supply?
Look for: Twelfths: 3 4/12 = 2 16/12, then subtract 1 9/12 to get 1 7/12.
If you observe: Exchanges into thirds while trying to remove fourths.
Respond: Name both amounts in twelfths before trading a whole for twelve twelfths.
Check independently: Find 2 1/5 − 1 1/2 after making tenths.
2 2/10 = 1 12/10; subtracting 1 5/10 leaves 7/10.
Use the mixed regrouping short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
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.
Continue to the linked next lesson. Use size predictions and inverse checks to reject implausible mixed or unlike-fraction results.
Next lesson: Estimate Before Exacte1 does not yet meet this criterion: e1 gives 3/2 or 1 1/2 for 2 1/4 − 3/4.
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.
Open the linked short interventione1 meets its criterion, but e2 does not yet show this evidence: e2 states that regrouping preserves the total starting amount.
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.
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Look for in the response:
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Look for in the response:
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Look for in the response:
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Look for in the response:
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Watch for: Changes the starting amount when exchanging a whole, or reverses a subtraction to avoid regrouping.
Try: 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.
Look for in the response:
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Eight core questions and a two-question exit ticket. Print one set per student.
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Build 2 1/4 with whole and quarter strips. Remove 3/4 by trading a whole for quarters.
Rebuild with sixths and fourths; decide a common unit before regrouping.
Partners identify and repair a worked subtraction that reverses the fractional subtraction.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Calculate 2 1/4 - 3/4. Show a model or your reasoning. What value does regrouping preserve?
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.
Teacher answer key. Open responses include a rubric and one valid sample.
2 1/4 - 3/4 = 3/2. Use equal-sized parts before combining or removing them.
3 1/3 - 1 3/4 = 19/12. Use equal-sized parts before combining or removing them.
2 1/5 - 1 1/2 = 7/10. Use equal-sized parts before combining or removing them.
4 1/6 - 2 2/3 = 3/2. Use equal-sized parts before combining or removing them.
3 1/8 - 1 5/6 = 31/24. Use equal-sized parts before combining or removing them.
2 1/2 - 1 3/5 = 9/10. Use equal-sized parts before combining or removing them.
5 1/4 - 2 2/3 = 31/12. Use equal-sized parts before combining or removing them.
3 2/5 - 1 7/10 = 17/10. Use equal-sized parts before combining or removing them.
1 1/4 - 1/2 = 3/4. Use equal-sized parts before combining or removing them.
2 1/3 - 2/3 = 5/3. Use equal-sized parts before combining or removing them.
1 1/5 - 2/5 = 4/5. Use equal-sized parts before combining or removing them.
2 1/2 - 3/4 = 7/4. Use equal-sized parts before combining or removing them.
2 1/4 = 1 5/4; removing 3/4 leaves 1 2/4 = 1 1/2.
The subtraction was reversed; regroup before removing to get 1 1/2.
1 1/2 + 3/4 = 2 1/4.
Then the whole can be exchanged into exactly the parts being removed.
2 1/4 - 3/4 = 3/2. Use equal-sized parts before combining or removing them.
The total starting amount stays unchanged.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Designed for Grade 5. Regroup mixed numbers to subtract unlike fractional parts.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.