Add and subtract mixed numbers with like denominators using regrouping.
Teaching companion · Version 2.0.0
Exchange a whole for equal fractional pieces while preserving the amount.
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.
Rename 2 1/4 so that at least three fourths are available to remove. Does the amount change?
Look for: 2 1/4 = 1 5/4; the amount is unchanged.
If you observe: Writes 2 5/4 and keeps both original wholes.
Respond: Physically trade one whole strip for four fourths, moving the whole strip out of the model as the quarters enter.
Combine 1 3/4 and 2 2/4. How many complete wholes can you form?
Look for: 4 1/4: three original wholes plus five fourths, with four fourths exchanged for a whole.
If you observe: Leaves five fourths unrecognized or counts them as five wholes.
Respond: Group four of the five quarter pieces inside one whole outline and leave one quarter outside.
Check independently: Calculate 3 1/4 − 2/4 and check by addition.
2 3/4; adding 2/4 returns 3 1/4.
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 explains that a whole and four fourths have equal value, so the exchange preserves the starting amount.
Continue to the linked next lesson. Use repeated fractional groups and exchange complete groups for wholes.
Next lesson: Repeated Fraction Groupse1 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 explains that a whole and four fourths have equal value, so the exchange preserves the starting amount.
Ask the learner to demonstrate exactly which whole is exchanged in e1 and verify 2 1/4 = 1 5/4 before explaining the subtraction.
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: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one.
Try: 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.
Watch for: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one.
Try: 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.
Watch for: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one.
Try: 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.
Watch for: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one.
Try: 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.
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: 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: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit.
Try: 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.
Watch for: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit.
Try: 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.
Watch for: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit.
Try: 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.
Watch for: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit.
Try: 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.
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: Loses a complete whole when the fractional parts in a mixed-number sum reach or exceed one.
Try: 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.
Look for in the response:
Watch for: Chooses an operation from the numbers without identifying whether the task combines, takes a part, shares, or counts groups.
Try: 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.
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: 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:
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
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?
Represent a market purchase of 1 3/4 yards plus 2 2/4 yards with strips.
Combine whole strips and quarter pieces; trade four quarters for a whole.
Partners model a return of 2/4 yard from 2 1/4 yards by regrouping one whole.
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. Why is regrouping a whole allowed?
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.
Teacher answer key. Open responses include a rubric and one valid sample.
1 3/4 + 2 2/4 = 17/4. Use equal-sized parts before combining or removing them.
2 1/3 + 1 2/3 = 4. Use equal-sized parts before combining or removing them.
1 5/6 + 2 2/6 = 25/6. Use equal-sized parts before combining or removing them.
3 1/8 + 1 7/8 = 5. Use equal-sized parts before combining or removing them.
2 1/4 - 2/4 = 7/4. Use equal-sized parts before combining or removing them.
3 2/5 - 1 4/5 = 8/5. Use equal-sized parts before combining or removing them.
4 1/6 - 2 5/6 = 4/3. Use equal-sized parts before combining or removing them.
2 3/8 - 1 5/8 = 3/4. Use equal-sized parts before combining or removing them.
1 1/4 - 1/4 = 1. Use equal-sized parts before combining or removing them.
1 1/3 - 1/3 = 1. Use equal-sized parts before combining or removing them.
2 1/4 - 1 1/4 = 1. Use equal-sized parts before combining or removing them.
2 3/4 - 1/4 = 5/2. Use equal-sized parts before combining or removing them.
Regroup as 1 5/4, then remove 2/4 to leave 1 3/4.
After addition, each full group of four quarter pieces becomes one whole.
Combine lengths of ribbon; the total is 4 1/4 yards.
1 3/5 + 1 4/5 = 3 2/5.
2 1/4 - 3/4 = 3/2. Use equal-sized parts before combining or removing them.
One whole and four fourths have the same value, so regrouping preserves the total.
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 4. Add and subtract mixed numbers with like denominators using regrouping.
4.NF.B.3c curriculum reference
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.