Compose One: 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. Shade 3/8 of a strip. Identify the amount needed to finish the whole.

activity-2. Pair cards that sum to one and verify each with a strip.

activity-3. Write two different decompositions of one using eighths.

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: Shade 3/8 of a strip. Identify the amount needed to finish the whole.

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

phase-3. 10 min · partner: Pair cards that sum to one and verify each with a strip.

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

phase-4. 7 min · discussion: Write two different decompositions of one using eighths.

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: How much completes 3/8 to one? Why does the denominator stay fixed?

Answer: Students: Complete the two exit questions independently.

d1. A whole is 8/8 and 3/8 is shaded. What amount is missing?

Answer: 5/8; three eighths plus five eighths fill eight eighths.

Teacher response: If the learner answers 5 without the fractional unit. Point to each of the five missing pieces and ask for the size of one piece before naming the total.

d2. A learner writes 3/8 + 5/8 = 8/16. Did the pieces become sixteenths?

Answer: No. They remain eighths, so the sum is 8/8 = 1.

Teacher response: If the learner adds denominators when combining same-sized pieces. Combine the physical eighth pieces without cutting them and ask what action would actually create sixteenths.

r1. Complete 7/10 + ? = 1 and explain the denominator.

Answer: 3/10; seven tenths and three tenths make ten tenths. The pieces stay tenths.

e1. How much completes 3/8 to one?

Answer: 5/8

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

e2. Why does the denominator stay fixed?

Answer: The size of each part stays the same while the counts combine.

Look for:

• Name the denominator as the size of the counted unit.
• Explain that combining pieces does not repartition them.
• Use e1 to show three eighths plus five eighths still count eighths.

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. 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.

teach-1. Write one whole as b/b for the partition in use.

teach-2. Count the uncovered parts and retain their part-size name.

teach-3. Combine the selected and missing pieces to check the total, explaining why their denominator stays fixed.

next-1. e1 meets its criterion: e1 gives 5/8 as the amount completing 3/8 to one. e2 also meets its criterion: e2 explains that the denominator stays fixed because the size of each piece stays the same.

Teacher response: Continue to the linked next lesson. Use a full collection of fractional pieces as the basis for exchanging one whole. Next original lesson: /lessons/mixed-number-market/.

next-2. e1 does not yet meet this criterion: e1 gives 5/8 as the amount completing 3/8 to one.

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/unlike-estimation/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that the denominator stays fixed because the size of each piece stays the same.

Teacher response: Have the learner act out 3/8 + 5/8 with eighth pieces and name each piece while explaining why combining counts does not change sizes. Short intervention: /interventions/unlike-estimation/.

c1. How much must you add to 3/8 to make one whole?

Answer: 5/8

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c2. How much must you add to 1/4 to make one whole?

Answer: 3/4

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c3. How much must you add to 2/5 to make one whole?

Answer: 3/5

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c4. How much must you add to 5/6 to make one whole?

Answer: 1/6

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c5. How much must you add to 7/10 to make one whole?

Answer: 3/10

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c6. How much must you add to 2/3 to make one whole?

Answer: 1/3

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c7. How much must you add to 1/8 to make one whole?

Answer: 7/8

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c8. How much must you add to 3/4 to make one whole?

Answer: 1/4

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

s1. Find the missing share: 1/2 + ? = 1.

Answer: 1/2

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

s2. Find the missing share: 1/3 + ? = 1.

Answer: 2/3

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

s3. Find the missing share: 1/4 + ? = 1.

Answer: 3/4

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

s4. Find the missing share: 2/4 + ? = 1.

Answer: 1/2

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

h1. Decompose one into three positive eighths fractions.

Answer: 1/8 + 2/8 + 5/8 = 8/8 = 1.

Look for:

• Give three positive fractions all expressed in eighths.
• Make their numerators sum to eight.
• Show the sum is 8/8 = 1.

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

h2. Why is 3/8 + 5/8 equal to 1 rather than 8/16?

Answer: We count eight eighth-size pieces; their size did not change.

Look for:

• Combine three eighths and five eighths to count eight eighths.
• Explain that the piece size stays 1/8.
• Reject 8/16 because it changes the named unit; conclude 8/8 = 1.

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. 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.

h3. Create a one-whole sum with quarters using a repeated addend.

Answer: 1/4 + 1/4 + 2/4 = 1.

Look for:

• Use quarter-sized addends and include a repeated addend.
• Make the selected quarter-counts total four.
• Write an equation showing the sum equals one.

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

h4. Explain the complement of 7/10 using a model.

Answer: Three more tenths fill the ten equal parts.

Look for:

• Represent a whole divided into ten equal parts with seven selected.
• Identify three tenths as the uncovered amount.
• Connect the model to 7/10 + 3/10 = 10/10 = 1.

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

e1. How much completes 3/8 to one?

Answer: 5/8

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

e2. Why does the denominator stay fixed?

Answer: The size of each part stays the same while the counts combine.

Look for:

• Name the denominator as the size of the counted unit.
• Explain that combining pieces does not repartition them.
• Use e1 to show three eighths plus five eighths still count eighths.

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. 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.