Divide a whole number by a unit fraction by counting servings.
Teaching companion · Version 2.0.0
Count how many unit-fraction-sized servings fit into a whole-number 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.
Two liters are poured into quarter-liter servings. How many servings fit, and what is the answer unit?
Look for: Eight servings; four quarter-liter servings per liter.
If you observe: Answers 1/8 liter by treating the problem as equal sharing.
Respond: Draw both liters and mark four quarter-liter portions in each, then count the portions rather than rename one portion.
How is 1/4 ÷ 2 different from 2 ÷ 1/4?
Look for: The first finds one share of a quarter: 1/8. The second counts quarter-sized groups in two wholes: eight.
If you observe: Treats the operands as interchangeable because both expressions use two and one fourth.
Respond: Act out two stories separately and label the unknown as “size of one share” or “number of groups” before calculating.
Check independently: How many half-liter servings fit into three liters? Check with multiplication.
Six servings; 6 × 1/2 liter = 3 liters.
Use the count unit fractions short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
e1 meets its criterion: e1 gives 8 for 2 ÷ 1/4. e2 also meets its criterion: e2 labels the answer as a number of servings or groups, not an amount per serving.
Continue to the linked next lesson. Choose between counting pieces and sharing an amount in a multi-operation design.
Next lesson: Fraction Design Challengee1 does not yet meet this criterion: e1 gives 8 for 2 ÷ 1/4.
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 labels the answer as a number of servings or groups, not an amount per serving.
Ask the learner to label all eight servings and a single quarter-liter serving in e1, then distinguish the units of those two numbers.
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: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Look for in the response:
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
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: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Look for in the response:
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Watch for: Confuses a fraction of an identified whole with a length, area, or count.
Try: Label the whole and the answer unit before counting. For a unit-square garden use square units; for a ribbon use meters; for cut pieces use the number of pieces. Check that the drawing and equation describe the same quantity.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
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?
Draw two whole liters and mark quarter-liter serving intervals.
Count eight servings, then write 2 ÷ 1/4 = 8.
Partners repeat with different unit fractions and check the total by multiplication.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Calculate 2 ÷ 1/4. Show a model or your reasoning. What unit belongs on the answer?
Treats a serving-count quotient as the size of a shared amount.
Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.
Teacher answer key. Open responses include a rubric and one valid sample.
2 ÷ 1/4 = 8. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
3 ÷ 1/2 = 6. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
4 ÷ 1/3 = 12. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
2 ÷ 1/5 = 10. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
5 ÷ 1/6 = 30. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
3 ÷ 1/8 = 24. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
6 ÷ 1/4 = 24. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
1 ÷ 1/7 = 7. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
1 ÷ 1/2 = 2. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
1 ÷ 1/4 = 4. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
2 ÷ 1/2 = 4. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
2 ÷ 1/3 = 6. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
Each whole contains four quarters; two wholes contain eight servings.
Each serving is smaller than a whole, so multiple servings fit in each whole.
The first counts eight servings; the second shares a quarter into eighths.
Six servings times 1/2 each recover three wholes.
2 ÷ 1/4 = 8. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
The answer counts servings, not liters per serving.
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. Divide a whole number by a unit fraction by counting servings.
5.NF.B.7b 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.