Model a fraction of a fraction as overlapping parts of a unit area.
Teaching companion · Version 2.0.0
Interpret a fraction product as overlap in a specified unit area.
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.
A unit square has three equal rows and four equal columns. Two rows and three columns are selected. How many cells are in both selections?
Look for: Six of the twelve equal cells; the product is 6/12 = 1/2.
If you observe: Counts all cells selected by either color instead of the overlap.
Respond: Trace one of the two selected rows and count its three doubly selected cells, then repeat in the second row.
Does the denominator twelve count only the shaded cells or all cells in the whole unit square?
Look for: All twelve equal cells in the whole unit square.
If you observe: Uses the singly shaded region as the new whole.
Respond: Outline the entire unit square in a third color and count all three-by-four cells before identifying the overlap.
Check independently: Show 1/2 × 2/3 on a unit square. What fraction overlaps?
Two of six equal cells overlap: 2/6 = 1/3.
Use the area product short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
e1 meets its criterion: e1 gives 1/2 for 2/3 × 3/4. e2 also meets its criterion: e2 identifies twelve total equal cells and six overlap cells as the product denominator and numerator.
Continue to the linked next lesson. Predict how the product changes relative to the starting amount from the factor’s size.
Next lesson: Scaling Storiese1 does not yet meet this criterion: e1 gives 1/2 for 2/3 × 3/4.
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 identifies twelve total equal cells and six overlap cells as the product denominator and numerator.
Ask the learner to point to all twelve cells and then only the six overlap cells, connecting those two counts to the written product.
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.
Area model: one unit rectangle has 3 equal rows and 4 equal columns. 2 rows and 3 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 2 equal rows and 5 equal columns. 1 rows and 2 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 4 equal rows and 3 equal columns. 3 rows and 2 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 5 equal rows and 7 equal columns. 2 rows and 4 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 5 equal rows and 2 equal columns. 3 rows and 1 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 6 equal rows and 4 equal columns. 5 rows and 3 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 7 equal rows and 5 equal columns. 2 rows and 3 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Area model: one unit rectangle has 8 equal rows and 5 equal columns. 3 rows and 4 columns are selected; look for their overlap.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Look for in the response:
Watch for: Confuses a fraction of an identified whole with a length, area, or count.
Try: 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.
Look for in the response:
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Look for in the response:
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Look for in the response:
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
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 one unit garden rectangle with thirds as rows and fourths as columns.
Shade two of three rows one way and three of four columns another.
Count the overlap and relate six of twelve cells to 2/3 × 3/4.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Calculate 2/3 × 3/4. Show a model or your reasoning. What do the product denominator and numerator count?
Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.
Teacher answer key. Open responses include a rubric and one valid sample.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
1/2 × 2/5 = 1/5. Multiply the numerators and denominators; interpret the groups or overlap.
3/4 × 2/3 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
2/5 × 4/7 = 8/35. Multiply the numerators and denominators; interpret the groups or overlap.
3/5 × 1/2 = 3/10. Multiply the numerators and denominators; interpret the groups or overlap.
5/6 × 3/4 = 5/8. Multiply the numerators and denominators; interpret the groups or overlap.
2/7 × 3/5 = 6/35. Multiply the numerators and denominators; interpret the groups or overlap.
3/8 × 4/5 = 3/10. Multiply the numerators and denominators; interpret the groups or overlap.
1/2 × 1/2 = 1/4. Multiply the numerators and denominators; interpret the groups or overlap.
1/3 × 1/2 = 1/6. Multiply the numerators and denominators; interpret the groups or overlap.
1/4 × 1/2 = 1/8. Multiply the numerators and denominators; interpret the groups or overlap.
2/3 × 1/2 = 1/3. Multiply the numerators and denominators; interpret the groups or overlap.
Two of three rows and three of four columns overlap in six of twelve equal cells, or one half.
The product is a fraction of that unit area, not a free-standing length.
No, the same six of twelve cells overlap.
Shade one half in each direction; one of four equal cells overlaps.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
The denominator counts all equal grid cells; the numerator counts overlap cells.
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 5. Model a fraction of a fraction as overlapping parts of a unit area.
5.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.