Fraction as Division: 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. Share three paper rectangles equally among four partners. Cut each into fourths.

activity-2. Give every partner one quarter from each rectangle and total the share.

activity-3. Write 3 ÷ 4 = 3/4 and check that four shares recover three wholes.

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: Share three paper rectangles equally among four partners. Cut each into fourths.

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

phase-3. 10 min · partner: Give every partner one quarter from each rectangle and total the share.

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

phase-4. 7 min · discussion: Write 3 ÷ 4 = 3/4 and check that four shares recover three wholes.

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 3 ÷ 4. Show a model or your reasoning. What does the denominator count in an equal-sharing story?

Answer: Students: Complete the two exit questions independently.

d1. Three same-sized loaves are shared equally among four people. How much does each person receive?

Answer: 3/4 loaf per person.

Teacher response: If the learner writes 4/3 by reversing recipients and loaves. Cut each loaf into four equal pieces and distribute one piece from each loaf to each of four person spaces.

d2. Four loaves are shared among three people. Must each person get less than one loaf?

Answer: No. Each gets 4/3 or 1 1/3 loaves.

Teacher response: If the learner assumes every fraction or share is less than one. Give each person one whole loaf first, then share the remaining loaf into thirds.

r1. Share two loaves equally among three people and check one share.

Answer: Each gets 2/3 loaf; 3 × 2/3 = 2 loaves.

e1. Calculate 3 ÷ 4. Show a model or your reasoning.

Answer: 3/4

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

e2. What does the denominator count in an equal-sharing story?

Answer: It counts the number of equal shares or people.

Look for:

• Identify the denominator as the number of equal shares or people.
• Explain that this determines each whole’s partition in the sharing model.
• Connect to 3 ÷ 4 by naming four recipients and 3/4 per person.

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

teach-1. Label the total number of same-sized wholes and the number of recipients.

teach-2. Divide every whole into the same number of recipient-sized shares and distribute equally.

teach-3. Count each recipient’s pieces and multiply one share by the number of recipients to recover the total.

next-1. e1 meets its criterion: e1 gives 3/4 for 3 ÷ 4. e2 also meets its criterion: e2 says the denominator counts equal shares or people in the sharing story.

Teacher response: Continue to the linked next lesson. Compare sharing several wholes with sharing a unit-fraction amount among people. Next original lesson: /lessons/unit-fraction-sharing/.

next-2. e1 does not yet meet this criterion: e1 gives 3/4 for 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/unit-fraction-sharing/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 says the denominator counts equal shares or people in the sharing story.

Teacher response: Use the sharing intervention to emphasize equal recipients, then return to three whole loaves and have the learner label the four people represented by the denominator. Short intervention: /interventions/unit-fraction-sharing/.

c1. 3 identical loaves are shared equally by 4 people. How many loaves does each person get?

Answer: 3/4

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

c2. 2 identical loaves are shared equally by 3 people. How many loaves does each person get?

Answer: 2/3

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

c3. 5 identical loaves are shared equally by 6 people. How many loaves does each person get?

Answer: 5/6

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

c4. 4 identical loaves are shared equally by 3 people. How many loaves does each person get?

Answer: 4/3

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves. Restarts the fraction count or changes the denominator after passing one. Place two equal-length unit strips end to end, both split into fourths. Label 0/4, 1/4, …, 8/4 continuously. Circle 4/4 at one and take one more unchanged quarter-step to 5/4.

c5. 7 identical loaves are shared equally by 4 people. How many loaves does each person get?

Answer: 7/4

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves. Restarts the fraction count or changes the denominator after passing one. Place two equal-length unit strips end to end, both split into fourths. Label 0/4, 1/4, …, 8/4 continuously. Circle 4/4 at one and take one more unchanged quarter-step to 5/4.

c6. 1 identical loaves are shared equally by 5 people. How many loaves does each person get?

Answer: 1/5

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

c7. 5 identical loaves are shared equally by 8 people. How many loaves does each person get?

Answer: 5/8

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

c8. 6 identical loaves are shared equally by 5 people. How many loaves does each person get?

Answer: 6/5

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves. Restarts the fraction count or changes the denominator after passing one. Place two equal-length unit strips end to end, both split into fourths. Label 0/4, 1/4, …, 8/4 continuously. Circle 4/4 at one and take one more unchanged quarter-step to 5/4.

s1. Share 1 same-sized wholes equally among 2 people. Find one share.

Answer: 1/2

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

s2. Share 1 same-sized wholes equally among 3 people. Find one share.

Answer: 1/3

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

s3. Share 2 same-sized wholes equally among 4 people. Find one share.

Answer: 1/2

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

s4. Share 3 same-sized wholes equally among 3 people. Find one share.

Answer: 1

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

h1. Draw the equal sharing of three wholes among four people.

Answer: Each gets three fourth-size pieces, or 3/4.

Look for:

• Show three same-sized wholes and four recipients.
• Partition and distribute so every recipient receives an equal amount.
• Identify each share as three fourth-pieces, or 3/4 of one whole.

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

h2. Explain why a/b can represent a ÷ b.

Answer: Divide each of a wholes into b equal shares; each person receives a parts of size 1/b.

Look for:

• Interpret a as the number of wholes and b as a positive number of equal shares.
• Explain that each whole contributes a 1/b piece to every share.
• Conclude that each share receives a such pieces, giving a/b = a ÷ b.

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

h3. Give a sharing story with a share greater than one whole.

Answer: Five loaves shared by four people gives 5/4 loaf each.

Look for:

• Create a story with more wholes than recipients.
• Share equally and identify the measurement unit.
• Calculate a share greater than one, such as 5/4 loaf for five loaves and four people.

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves. Restarts the fraction count or changes the denominator after passing one. Place two equal-length unit strips end to end, both split into fourths. Label 0/4, 1/4, …, 8/4 continuously. Circle 4/4 at one and take one more unchanged quarter-step to 5/4.

h4. Check a 2/3 share for three people.

Answer: Three times 2/3 equals the two wholes that were shared.

Look for:

• Interpret 2/3 as the amount per person.
• Multiply by the three people.
• Show 3 × 2/3 = 2, recovering the original two wholes.

Teacher response: Confuses the number of equal groups with the size of each group. Draw three separate bags, each holding 2/5 cup. Label “3 groups” and “2/5 cup per group.” Count six fifth-cup pieces in total and connect the drawing to 2/5 + 2/5 + 2/5.

e1. Calculate 3 ÷ 4. Show a model or your reasoning.

Answer: 3/4

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.

e2. What does the denominator count in an equal-sharing story?

Answer: It counts the number of equal shares or people.

Look for:

• Identify the denominator as the number of equal shares or people.
• Explain that this determines each whole’s partition in the sharing model.
• Connect to 3 ÷ 4 by naming four recipients and 3/4 per person.

Teacher response: Reverses wholes and recipients in an equal-sharing quotient. Draw three loaves and four recipient spaces. Cut every loaf into fourths and give one fourth from each loaf to every person. Each receives 3/4 loaf; four shares recover the original three loaves.