Fraction Classroom · 2.0.0 · Teacher answers & responses

Repeated Fraction Groups: teaching companion

Interpret a whole-number factor as the number of equal fractional groups. Student pages include two diagnostic probes, a reteaching check, and the original exit ticket. The teacher copy includes the complete lesson preparation, teaching sequence, diagnostic responses, all original question guides, and updated rubrics.

Complete lesson preparation

Grade 4 · 45 minutes. Interpret a whole number times a fraction as repeated groups.

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 three groups of 2/5 on strips. Count all fifth-size pieces.

activity-2. Write both repeated addition and multiplication for the same model.

activity-3. Partners model four groups of 3/4 and explain how groups can exceed one.

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 three groups of 2/5 on strips. Count all fifth-size pieces.

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

phase-3. 10 min · partner: Write both repeated addition and multiplication for the same model.

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

phase-4. 7 min · discussion: Partners model four groups of 3/4 and explain how groups can exceed one.

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 × 2/5. Show a model or your reasoning. What does the whole-number factor count?

Answer: Students: Complete the two exit questions independently.

Diagnostic probes

Explain or draw your reasoning. Your teacher will use these responses to choose what to revisit.

d1. Three bags each hold 2/5 cup. How many cups altogether, and what does the three count?

Answer: 6/5 cups; three counts the bags or equal groups.

Teacher response: If the learner treats the three as an extra part of the denominator. Draw three bag outlines with two fifth-cup pieces in each; count all six pieces without recutting them.

d2. A learner writes 3 × 2/5 = 6/15. What changed incorrectly?

Answer: The denominator changed even though the pieces are still fifths; the total is 6/5.

Teacher response: If the learner multiplies numerator and denominator by the whole-number factor as if making an equivalent fraction. Contrast renaming one 2/5 amount with collecting three copies of that amount; the latter triples the shaded quantity.

Check after reteaching

Complete this check independently after discussing the model.

r1. Model four groups of 3/4 and name the total.

Answer: Twelve fourths make three wholes.

Exit ticket

Complete both original exit questions independently. Use a drawing or explanation when asked.

e1. Calculate 3 × 2/5. Show a model or your reasoning.

Answer: 6/5

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.

e2. What does the whole-number factor count?

Answer: It counts the number of equal fractional groups.

Look for:

Identify the whole-number factor as the number of groups.
Identify the fraction as the amount in each group.
Connect 3 × 2/5 to three groups each containing two fifths.

Focused reteach

Interpret a whole-number factor as the number of equal fractional groups.

teach-1. Write “number of groups” above the whole number and “amount per group” above the fraction.

teach-2. Draw or build every group and translate the picture into repeated addition.

teach-3. Count the fixed-size pieces and regroup complete wholes only after finding the total.

Use the exit evidence

A wrong answer invites a follow-up probe; it does not by itself establish a particular misconception. Check e1 and e2 separately.

next-1. e1 meets its criterion: e1 gives 6/5 for 3 × 2/5. e2 also meets its criterion: e2 says the whole-number factor counts equal fractional groups.

Teacher response: Continue to the linked next lesson. Extend from whole-number groups to taking a fractional part of a fractional amount. Next original lesson: /lessons/area-product-garden/.

next-2. e1 does not yet meet this criterion: e1 gives 6/5 for 3 × 2/5.

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/scaling/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 says the whole-number factor counts equal fractional groups.

Teacher response: Ask the learner to label three separate groups and the 2/5 inside each, then use those labels to explain what the factor three counts. Short intervention: /interventions/scaling/.

Original question guide and updated rubrics

Use these question-specific teaching responses alongside the original lesson. Accept other mathematically valid open responses that meet the criteria. The original question wording and mathematical answers are retained; model descriptions below make their references readable offline.

c1. Calculate 3 × 2/5. Show a model or your reasoning.

Answer: 6/5

c2. Calculate 4 × 3/4. Show a model or your reasoning.

Answer: 3

c3. Calculate 2 × 5/6. Show a model or your reasoning.

Answer: 5/3

c4. Calculate 5 × 1/3. Show a model or your reasoning.

Answer: 5/3

c5. Calculate 6 × 2/8. Show a model or your reasoning.

Answer: 3/2

c6. Calculate 3 × 3/7. Show a model or your reasoning.

Answer: 9/7

c7. Calculate 4 × 2/9. Show a model or your reasoning.

Answer: 8/9

c8. Calculate 2 × 7/10. Show a model or your reasoning.

Answer: 7/5

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

Answer: 1

s2. Calculate 3 × 1/4. Show a model or your reasoning.

Answer: 3/4

s3. Calculate 2 × 1/3. Show a model or your reasoning.

Answer: 2/3

s4. Calculate 4 × 1/4. Show a model or your reasoning.

Answer: 1

h1. Draw 3 × 2/5 as repeated groups.

Answer: Three groups each show two fifths; six fifths is 1 1/5.

Look for:

Draw three distinct groups each containing 2/5.
Count six fifth-sized pieces altogether.
Label the total 6/5 or 1 1/5 relative to the original whole.

h2. Explain why 4 × 3/4 equals three wholes.

Answer: Twelve fourth-size pieces regroup into three wholes.

Look for:

Interpret the expression as four groups of 3/4.
Count twelve fourths.
Regroup four fourths per whole to obtain three wholes.

Teacher response: Does not recognize that b copies of 1/b make one whole. Fill one outlined whole with b equal pieces and label the completed whole b/b. Exchange the complete collection for a card labeled 1; do the exchange again with a different partition.

h3. Create two stories for 5 × 1/3.

Answer: Five bags each hold a third cup, or five walks each cover a third mile.

Look for:

Give two different contexts with five equal groups.
Make each group exactly 1/3 of a stated unit.
Relate each story to the total 5/3 of that unit.

h4. Correct 3 × 2/5 = 6/15.

Answer: Three groups add fifths, so the result is 6/5; the fifth-size unit stays fixed.

Look for:

Correct the result to 6/5.
Explain that three groups of two fifth-pieces give six fifth-pieces.
State that the fifth-sized unit remains fixed; 6/15 would change the piece size.

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.

e1. Calculate 3 × 2/5. Show a model or your reasoning.

Answer: 6/5

e2. What does the whole-number factor count?

Answer: It counts the number of equal fractional groups.

Look for:

Identify the whole-number factor as the number of groups.
Identify the fraction as the amount in each group.
Connect 3 × 2/5 to three groups each containing two fifths.

Teaching notes

Teaching supplement version 2.0.0. Original lesson questions and PDF materials remain version 1.0.0. Original PDFs contain their original teaching notes and rubrics; use this supplement for the updated diagnostic guidance.
Short intervention: /interventions/scaling/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
Next original lesson: /lessons/area-product-garden/. Extend from whole-number groups to taking a fractional part of a fractional amount.
Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.