Interpret a whole number times a fraction as repeated groups.
Teaching companion · Version 2.0.0
Interpret a whole-number factor as the number of equal fractional groups.
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.
Three bags each hold 2/5 cup. How many cups altogether, and what does the three count?
Look for: 6/5 cups; three counts the bags or equal groups.
If you observe: Treats the three as an extra part of the denominator.
Respond: Draw three bag outlines with two fifth-cup pieces in each; count all six pieces without recutting them.
A learner writes 3 × 2/5 = 6/15. What changed incorrectly?
Look for: The denominator changed even though the pieces are still fifths; the total is 6/5.
If you observe: Multiplies numerator and denominator by the whole-number factor as if making an equivalent fraction.
Respond: Contrast renaming one 2/5 amount with collecting three copies of that amount; the latter triples the shaded quantity.
Check independently: Model four groups of 3/4 and name the total.
Twelve fourths make three wholes.
Use the scaling short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
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.
Continue to the linked next lesson. Extend from whole-number groups to taking a fractional part of a fractional amount.
Next lesson: Area Product Gardene1 does not yet meet this criterion: e1 gives 6/5 for 3 × 2/5.
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 says the whole-number factor counts equal fractional groups.
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.
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: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Look for in the response:
Watch for: Does not recognize that b copies of 1/b make one whole.
Try: 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.
Look for in the response:
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Look for in the response:
Watch for: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit.
Try: 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.
Look for in the response:
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
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 three groups of 2/5 on strips. Count all fifth-size pieces.
Write both repeated addition and multiplication for the same model.
Partners model four groups of 3/4 and explain how groups can exceed one.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Calculate 3 × 2/5. Show a model or your reasoning. What does the whole-number factor count?
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.
Teacher answer key. Open responses include a rubric and one valid sample.
3 × 2/5 = 6/5. Multiply the numerators and denominators; interpret the groups or overlap.
4 × 3/4 = 3. Multiply the numerators and denominators; interpret the groups or overlap.
2 × 5/6 = 5/3. Multiply the numerators and denominators; interpret the groups or overlap.
5 × 1/3 = 5/3. Multiply the numerators and denominators; interpret the groups or overlap.
6 × 2/8 = 3/2. Multiply the numerators and denominators; interpret the groups or overlap.
3 × 3/7 = 9/7. Multiply the numerators and denominators; interpret the groups or overlap.
4 × 2/9 = 8/9. Multiply the numerators and denominators; interpret the groups or overlap.
2 × 7/10 = 7/5. Multiply the numerators and denominators; interpret the groups or overlap.
2 × 1/2 = 1. Multiply the numerators and denominators; interpret the groups or overlap.
3 × 1/4 = 3/4. Multiply the numerators and denominators; interpret the groups or overlap.
2 × 1/3 = 2/3. Multiply the numerators and denominators; interpret the groups or overlap.
4 × 1/4 = 1. Multiply the numerators and denominators; interpret the groups or overlap.
Three groups each show two fifths; six fifths is 1 1/5.
Twelve fourth-size pieces regroup into three wholes.
Five bags each hold a third cup, or five walks each cover a third mile.
Three groups add fifths, so the result is 6/5; the fifth-size unit stays fixed.
3 × 2/5 = 6/5. Multiply the numerators and denominators; interpret the groups or overlap.
It counts the number of equal fractional groups.
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 4. Interpret a whole number times a fraction as repeated groups.
4.NF.B.4b 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.