Find the overlapping part of a whole

d1. A whole rectangle has 3 equal columns and 4 equal rows. What fraction of the whole is one cell?

Answer: 1/12, because the grid has 3 × 4 = 12 equal cells.

Look for:

• Identifies 12 equal cells.
• Names one cell as 1/12 of the whole.

Teacher response: Likely difficulty: Adding row and column counts to get sevenths. Response: Count three cells in each of four rows.

d2. In a 3-column by 4-row grid, shade 2 columns vertically and 3 rows horizontally. How many cells have both shadings?

Answer: 6 cells, in the 2-by-3 overlap.

Look for:

• Counts 6 overlap cells.
• Distinguishes overlap from the union of shaded regions.

Teacher response: Likely difficulty: Counting every cell with either shading instead of their overlap. Response: Touch only cells crossed by both hatch directions.

d3. Should 1/2 × 2/3 be greater or smaller than 2/3? Explain the meaning of the first factor.

Answer: Smaller. Taking one half of two thirds gives 1/3.

Look for:

• Predicts smaller.
• Interprets the first factor as taking a part of the second.

Teacher response: Likely difficulty: Assuming multiplication must increase a positive number. Response: Start with two thirds shaded and keep only half of that area.

w1. Use an area model for 2/3 × 3/4.

1. Draw one rectangle and label it one whole.
2. Split it into three columns and select two.
3. Split the same rectangle into four rows and select three.
4. The overlap has 2 × 3 = 6 cells out of 3 × 4 = 12 equal cells; six twelfths is one half.

Worked example: 6/12 = 1/2 of the original whole.

Look for:

• Counts six overlap cells out of twelve.
• Names the original rectangle as the whole.

Teacher response: Likely difficulty: Counting only rows or using the shaded subregion as a new whole. Response: Keep the outline of all twelve cells visible while counting the overlap.

w2. A rectangular mat is 3/4 meter long and 2/5 meter wide. Find its area with a unit-square model.

1. Draw a 1 meter by 1 meter reference square.
2. Divide its length into fourths and width into fifths, making twenty equal cells.
3. Select three fourths in one direction and two fifths in the other.
4. The mat occupies six cells, each 1/20 square meter, for an area of 3/10 square meter.

Worked example: 6/20 = 3/10 square meter.

Look for:

• Uses multiplication for rectangular area.
• Reports 3/10 square meter and explains twentieth-sized cells.

Teacher response: Likely difficulty: Giving a length unit for an area or adding the side fractions. Response: Point to the two dimensions of a single cell and then to the square-meter reference whole.

p1. Draw a grid showing 1/2 × 3/5. Label the whole and count the overlap.

Answer: 3/10 of the whole; three overlap cells out of ten.

Look for:

• Draws ten equal cells.
• Identifies three overlapping cells.

Teacher response: Likely difficulty: Counting five total cells because only one partition direction is noticed. Response: Trace both the two columns and five rows across the full rectangle.

p2. A rectangle is divided into 4 columns and 3 rows. Two columns and one row are selected. Write the product and its value.

Answer: 2/4 × 1/3 = 2/12 = 1/6.

Look for:

• Writes factors that match the selected dimensions.
• Finds 1/6 from two overlap cells.

Teacher response: Likely difficulty: Treating two columns plus one row as three selected twelfths. Response: Count the cells lying in both selections rather than adding the selection counts.

p3. A student says 2/3 × 1/4 = 3/7 by adding both pairs of numbers. Explain using a grid.

Answer: 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.

Look for:

• Uses a 12-cell whole.
• Finds a 2-cell overlap and 1/6.

Teacher response: Likely difficulty: Applying addition to the factors instead of counting a rectangular overlap. Response: Count equal cells in the whole and in the overlap separately.

p4. Find the area of a rectangle measuring 2/3 meter by 1/2 meter. State the unit.

Answer: 1/3 square meter.

Look for:

• Finds 1/3.
• Uses square meter as the unit.

Teacher response: Likely difficulty: Reporting meters or interpreting multiplication as a perimeter calculation. Response: Fit two of the six equal cells of a unit square into the rectangle.

p5. Does switching 2/3 × 3/4 to 3/4 × 2/3 change the area? Explain by turning a grid.

Answer: No. Rotating the rectangle exchanges rows and columns but keeps six overlap cells out of twelve.

Look for:

• States equal products.
• Explains the unchanged overlap under rotation.

Teacher response: Likely difficulty: Believing the first shading direction determines a different product. Response: Rotate the paper a quarter turn and count the same cells.

p6. A rectangle has 5 columns and 4 rows. An overlap covers 3 columns and 2 rows. Give its fraction of the whole in simplest form.

Answer: 6/20 = 3/10.

Look for:

• Counts 6 of 20 cells.
• Simplifies to 3/10.

Teacher response: Likely difficulty: Using 3 + 2 for the numerator and 5 + 4 for the denominator. Response: Count three cells in each of two overlap rows, then count all twenty cells.

t1. A garden uses 3/4 of a rectangular plot. Herbs occupy 2/3 of the garden. What fraction of the entire plot contains herbs? Name both wholes in the story.

Answer: 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.

Look for:

• Finds 1/2 of the plot.
• Distinguishes the plot whole from the garden subregion.

Teacher response: Likely difficulty: Treating the garden and entire plot as the same reference whole. Response: Outline the plot first, then take two thirds only within the selected garden region.

t2. A sign has side lengths 5/6 meter and 3/4 meter. Find its area, then explain why the result is less than 5/6 square meter.

Answer: 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.

Look for:

• Finds 5/8 square meter.
• Uses a geometric comparison to explain the smaller area.

Teacher response: Likely difficulty: Assuming a product is larger than either positive factor. Response: Compare the sign with the same-length rectangle of width 1 meter.