How Many Servings: teaching companion

prepare. Before starting: Recognize a whole and count equal parts.

materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).

activity-1. Draw two whole liters and mark quarter-liter serving intervals.

activity-2. Count eight servings, then write 2 ÷ 1/4 = 8.

activity-3. Partners repeat with different unit fractions and check the total by multiplication.

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: Draw two whole liters and mark quarter-liter serving intervals.

Answer: Students: Follow the model; label the whole and the equal parts.

phase-3. 10 min · partner: Count eight servings, then write 2 ÷ 1/4 = 8.

Answer: Students: Work in pairs. One models, the other checks; switch roles.

phase-4. 7 min · discussion: Partners repeat with different unit fractions and check the total by multiplication.

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. Show a model or your reasoning. What unit belongs on the answer?

Answer: Students: Complete the two exit questions independently.

d1. Two liters are poured into quarter-liter servings. How many servings fit, and what is the answer unit?

Answer: Eight servings; four quarter-liter servings per liter.

Teacher response: If the learner answers 1/8 liter by treating the problem as equal sharing. Draw both liters and mark four quarter-liter portions in each, then count the portions rather than rename one portion.

d2. How is 1/4 ÷ 2 different from 2 ÷ 1/4?

Answer: The first finds one share of a quarter: 1/8. The second counts quarter-sized groups in two wholes: eight.

Teacher response: If the learner treats the operands as interchangeable because both expressions use two and one fourth. Act out two stories separately and label the unknown as “size of one share” or “number of groups” before calculating.

r1. How many half-liter servings fit into three liters? Check with multiplication.

Answer: Six servings; 6 × 1/2 liter = 3 liters.

e1. Calculate 2 ÷ 1/4. Show a model or your reasoning.

Answer: 8

Teacher response: 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.

e2. What unit belongs on the answer?

Answer: The answer counts servings, not liters per serving.

Look for:

• Name servings or groups as the answer unit.
• Identify 1/4 as the amount per serving in e1.
• Explain that eight servings, each 1/4 of a whole, use the two wholes.

Teacher response: Confuses a fraction of an identified whole with a length, area, or count. 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. 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.

teach-1. Label the available whole-number quantity and one serving’s fractional size.

teach-2. Mark that serving size repeatedly across every whole without changing it.

teach-3. Count servings and multiply count × serving size to check the original quantity.

next-1. 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.

Teacher response: Continue to the linked next lesson. Choose between counting pieces and sharing an amount in a multi-operation design. Next original lesson: /lessons/fraction-design-challenge/.

next-2. e1 does not yet meet this criterion: e1 gives 8 for 2 ÷ 1/4.

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/count-unit-fractions/.

next-3. e1 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.

Teacher response: Ask the learner to label all eight servings and a single quarter-liter serving in e1, then distinguish the units of those two numbers. Short intervention: /interventions/count-unit-fractions/.

c1. Calculate 2 ÷ 1/4. Show a model or your reasoning.

Answer: 8

Teacher response: 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.

c2. Calculate 3 ÷ 1/2. Show a model or your reasoning.

Answer: 6

Teacher response: 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.

c3. Calculate 4 ÷ 1/3. Show a model or your reasoning.

Answer: 12

Teacher response: 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.

c4. Calculate 2 ÷ 1/5. Show a model or your reasoning.

Answer: 10

Teacher response: 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.

c5. Calculate 5 ÷ 1/6. Show a model or your reasoning.

Answer: 30

Teacher response: 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.

c6. Calculate 3 ÷ 1/8. Show a model or your reasoning.

Answer: 24

Teacher response: 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.

c7. Calculate 6 ÷ 1/4. Show a model or your reasoning.

Answer: 24

Teacher response: 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.

c8. Calculate 1 ÷ 1/7. Show a model or your reasoning.

Answer: 7

Teacher response: 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.

s1. Calculate 1 ÷ 1/2. Show a model or your reasoning.

Answer: 2

Teacher response: 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.

s2. Calculate 1 ÷ 1/4. Show a model or your reasoning.

Answer: 4

Teacher response: 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.

s3. Calculate 2 ÷ 1/2. Show a model or your reasoning.

Answer: 4

Teacher response: 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.

s4. Calculate 2 ÷ 1/3. Show a model or your reasoning.

Answer: 6

Teacher response: 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.

h1. Model 2 ÷ 1/4 as servings.

Answer: Each whole contains four quarters; two wholes contain eight servings.

Look for:

• Show two equal wholes partitioned into quarter-sized servings.
• Count four servings in each whole and eight altogether.
• Label the quotient as eight servings, each of size 1/4 whole.

Teacher response: 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.

h2. Why does dividing by a fraction below one give more servings?

Answer: Each serving is smaller than a whole, so multiple servings fit in each whole.

Look for:

• Identify the divisor as the size of each serving.
• Explain that a serving smaller than one whole fits more than once into each whole.
• Relate the total serving count to the number of wholes without confusing count and serving size.

Teacher response: 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.

h3. Compare 2 ÷ 1/4 with 1/4 ÷ 2.

Answer: The first counts eight servings; the second shares a quarter into eighths.

Look for:

• Interpret 2 ÷ 1/4 as counting quarter-sized groups and obtain eight.
• Interpret 1/4 ÷ 2 as sharing a quarter between two and obtain 1/8.
• Distinguish the output units: number of servings versus amount in one share.

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 ÷ 1/2 by multiplication.

Answer: Six servings times 1/2 each recover three wholes.

Look for:

• Identify six as the number of half-sized servings.
• Multiply 6 × 1/2.
• Recover the original three wholes with the correct units.

Teacher response: 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.

e1. Calculate 2 ÷ 1/4. Show a model or your reasoning.

Answer: 8

Teacher response: 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.

e2. What unit belongs on the answer?

Answer: The answer counts servings, not liters per serving.

Look for:

• Name servings or groups as the answer unit.
• Identify 1/4 as the amount per serving in e1.
• Explain that eight servings, each 1/4 of a whole, use the two wholes.

Teacher response: Confuses a fraction of an identified whole with a length, area, or count. 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. 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.