Build a two-direction area model to connect fraction multiplication with overlap and square units.
The product counts the overlap of two selections, using the original whole as the reference area.
Materials: Paper and ruler; Two colored pencils, or different hatch patterns.
The second worked example, three unused practice tasks, and remaining transfer task are available for a follow-up session. Allow more time if the group needs to build the models.
Teaching move: Label the whole rectangle before adding a grid. Use vertical hatching for one factor and horizontal hatching for the other; count only the crosshatched cells for the product.
Open an interactive area model →1 · Quick diagnosis
Use the answer notes to interpret what you hear. Choose a response for each question to get a suggested teaching step. These choices describe this brief check and are not a mastery score.
Explain with a drawing, objects, or words.
Answer or sample response: 1/12, because the grid has 3 × 4 = 12 equal cells.
Listen for this difficulty: Adding row and column counts to get sevenths.
Try this response: Count three cells in each of four rows.
Look for:
Explain with a drawing, objects, or words.
Answer or sample response: 6 cells, in the 2-by-3 overlap.
Listen for this difficulty: Counting every cell with either shading instead of their overlap.
Try this response: Touch only cells crossed by both hatch directions.
Look for:
Explain with a drawing, objects, or words.
Answer or sample response: Smaller. Taking one half of two thirds gives 1/3.
Listen for this difficulty: Assuming multiplication must increase a positive number.
Try this response: Start with two thirds shaded and keep only half of that area.
Look for:
2 · Model together
Use one now. Ask the learner to explain each step before revealing the next part of your drawing.
Result: 6/12 = 1/2 of the original whole.
Listen for this difficulty: Counting only rows or using the shaded subregion as a new whole.
Try this response: Keep the outline of all twelve cells visible while counting the overlap.
Look for:
Result: 6/20 = 3/10 square meter.
Listen for this difficulty: Giving a length unit for an area or adding the side fractions.
Try this response: Point to the two dimensions of a single cell and then to the square-meter reference whole.
Look for:
3 · Guided practice
Mix a calculation or drawing with an explanation or error repair. Keep the other tasks for another short session.
Explain with a drawing, objects, or words.
Answer or sample response: 3/10 of the whole; three overlap cells out of ten.
Listen for this difficulty: Counting five total cells because only one partition direction is noticed.
Try this response: Trace both the two columns and five rows across the full rectangle.
Look for:
Explain with a drawing, objects, or words.
Answer or sample response: 2/4 × 1/3 = 2/12 = 1/6.
Listen for this difficulty: Treating two columns plus one row as three selected twelfths.
Try this response: Count the cells lying in both selections rather than adding the selection counts.
Look for:
Explain with a drawing, objects, or words.
Answer or sample response: A 3-by-4 grid has 12 cells and the 2-by-1 overlap has 2 cells, so the product is 2/12 = 1/6.
Listen for this difficulty: Applying addition to the factors instead of counting a rectangular overlap.
Try this response: Count equal cells in the whole and in the overlap separately.
Look for:
Explain with a drawing, objects, or words.
Answer or sample response: 1/3 square meter.
Listen for this difficulty: Reporting meters or interpreting multiplication as a perimeter calculation.
Try this response: Fit two of the six equal cells of a unit square into the rectangle.
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Explain with a drawing, objects, or words.
Answer or sample response: No. Rotating the rectangle exchanges rows and columns but keeps six overlap cells out of twelve.
Listen for this difficulty: Believing the first shading direction determines a different product.
Try this response: Rotate the paper a quarter turn and count the same cells.
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Explain with a drawing, objects, or words.
Answer or sample response: 6/20 = 3/10.
Listen for this difficulty: Using 3 + 2 for the numerator and 5 + 4 for the denominator.
Try this response: Count three cells in each of two overlap rows, then count all twenty cells.
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4 · Independent transfer
Choose one task without coaching. Check the explanation against both criteria in its teaching notes; use the other task for a later check.
Explain with a drawing, objects, or words.
Answer or sample response: 1/2 of the entire plot. The garden is 3/4 of the plot, and the herbs are 2/3 of that garden: 2/3 × 3/4 = 6/12.
Listen for this difficulty: Treating the garden and entire plot as the same reference whole.
Try this response: Outline the plot first, then take two thirds only within the selected garden region.
Look for:
Explain with a drawing, objects, or words.
Answer or sample response: 5/8 square meter, since 15/24 = 5/8. Its width is only 3/4 meter, so it fills less area than a 5/6-by-1-meter rectangle.
Listen for this difficulty: Assuming a product is larger than either positive factor.
Try this response: Compare the sign with the same-length rectangle of width 1 meter.
Look for:
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
Standards: 5.NF.B.4.a, 5.NF.B.4.b. Grade 5 alignment describes the mathematical work in this pack.
The tasks, worked examples, and teacher responses are original Fraction Classroom materials, version 2.0.0. The teaching approach draws on noticing the unit, moving between representations, and responding to student reasoning in the linked Illustrative Mathematics lesson. Its text and exercises are not reproduced here.