- Use the selected column to identify one third of the whole. - Look at how much of that column lies in the selected row. - Compare the single overlap cell with all the equal cells.

1. The overlap represents \(\frac{1}{2}\) of \(\frac{1}{3}\). 2. The grid has \(2\times3=6\) equal cells, and exactly \(1\) cell is in both selected regions. 3. Therefore \(\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}\).

\(\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}\), so the overlap covers \(\frac{1}{6}\) of the whole rectangle.

- Read the shaded fraction in panel a). - Read the shaded fraction in panel b). - In panel c), compare the overlap cell with all the equal cells in the grid.

1. Panel a) shades \(1\) of \(4\) columns, so it represents \(\frac{1}{4}\). 2. Panel b) shades \(1\) of \(3\) rows, so it represents \(\frac{1}{3}\). 3. Panel c) shows the one cell that lies in both bands. That is \(\frac{1}{12}\) of the whole. 4. Therefore choice A is correct: \(\frac{1}{4}\times\frac{1}{3}=\frac{1}{12}\).

A) \(\frac{1}{4}\times\frac{1}{3}=\frac{1}{12}\).

- Count all equal cells in the completed grid. - Count the cells in the shaded overlap. - Simplify the overlap fraction before writing the product.

1. The grid has \(6\) equal cells, and the overlap covers \(2\) of them. 2. The overlap is \(\frac{2}{6}=\frac{1}{3}\) of the whole. 3. Therefore \(\frac{2}{3}\times\frac{1}{2}=\frac{1}{3}\).

The overlap is \(\frac{1}{3}\) of the whole, so \(\frac{2}{3}\times\frac{1}{2}=\frac{1}{3}\).

- Read one factor from the vertical partition and the other from the horizontal partition. - The overlap is the part that satisfies both fractional conditions. - Compare the number of overlap cells with the total number of equal cells in the whole grid.

1. The vertical region covers \(5\) of \(8\) equal columns, so \(\frac{5}{8}\) of the animals are dogs. 2. The horizontal region covers \(2\) of \(5\) equal rows, so the model takes \(\frac{2}{5}\) of the dog region. 3. The overlap covers \(10\) of the \(40\) equal cells, so \(\frac{2}{5}\times\frac{5}{8}=\frac{10}{40}=\frac{1}{4}\).

\(\frac{2}{5}\times\frac{5}{8}=\frac{1}{4}\). Puppies make up \(\frac{1}{4}\) of all the animals.

- Read the width fraction by comparing green columns with all columns. - Read the height fraction by comparing green rows with all rows. - The green rectangle's area is the product of those two side fractions.

1. The pasture is \(5\) equal columns wide, and the green rectangle spans \(4\) columns, so its width is \(\frac{4}{5}\) of the pasture width. 2. The pasture is \(3\) equal rows tall, and the green rectangle spans \(2\) rows, so its height is \(\frac{2}{3}\) of the pasture height. 3. The area fraction is \(\frac{4}{5}\times\frac{2}{3}=\frac{8}{15}\).

a) \(\frac{4}{5}\) b) \(\frac{2}{3}\) c) \(\frac{4}{5}\times\frac{2}{3}=\frac{8}{15}\)

- One fraction controls columns and the other controls rows. - The cells that satisfy both conditions form the product area.

1. The overlap represents \(\frac{7}{8}\times\frac{3}{5}\). 2. Multiply: \(\frac{7}{8}\times\frac{3}{5}=\frac{21}{40}\). 3. The grid shows \(21\) overlap cells out of \(40\) total cells.

\(\frac{21}{40}\) of the rectangle lies in both regions.

- Read one factor from the fraction of columns in the vertical region. - Read the other factor from the fraction of rows in the horizontal region. - Compare the overlap cell count with all equal cells in the grid.

1. The vertical region covers \(5\) of \(9\) columns, so it represents \(\frac{5}{9}\). 2. The horizontal region covers \(3\) of \(5\) rows, so it represents \(\frac{3}{5}\). 3. Their overlap contains \(15\) of the \(45\) equal cells, so \(\frac{3}{5}\times\frac{5}{9}=\frac{15}{45}=\frac{1}{3}\).

The overlap is \(15\) of \(45\) cells, so \(\frac{3}{5}\times\frac{5}{9}=\frac{1}{3}\).

- In panel a), compare selected columns with all columns. - In panel b), compare selected rows with all rows. - The overlap panel must represent cells belonging to both factor regions.

1. Panel a) has \(5\) selected columns out of \(7\), so it represents \(\frac{5}{7}\). 2. Panel b) has \(3\) selected rows out of \(4\), so it represents \(\frac{3}{4}\). 3. Panel c) contains \(15\) shaded overlap cells out of \(28\) total cells. 4. Therefore \(\frac{5}{7}\times\frac{3}{4}=\frac{15}{28}\).

a) \(\frac{5}{7}\) and \(\frac{3}{4}\) b) \(15\) overlap cells out of \(28\) c) \(\frac{5}{7}\times\frac{3}{4}=\frac{15}{28}\)

- Treat orange as belonging to both shaded directions. - Count columns for the width fraction and rows for the height fraction. - Compare orange overlap cells with all cells to verify the product.

1. The grid has \(5\) columns, and the width region spans \(4\), so the width is \(\frac{4}{5}\) unit. 2. The grid has \(3\) rows, and the height region spans \(2\), so the height is \(\frac{2}{3}\) unit. 3. The orange overlap has \(8\) of the \(15\) cells, so \(\frac{4}{5}\times\frac{2}{3}=\frac{8}{15}\) square unit.

a) Width \(\frac{4}{5}\) unit; height \(\frac{2}{3}\) unit b) \(\frac{4}{5}\times\frac{2}{3}=\frac{8}{15}\) square unit

- Use one factor to select columns and the other to select rows. - Write the overlap cell count over the total number of grid cells. - Simplify the resulting fraction.

1. The grid has \(36\) equal cells. 2. Eight of the \(9\) columns and \(3\) of the \(4\) rows overlap in \(8\times3=24\) cells. 3. The overlap fraction is \(\frac{24}{36}\). 4. Simplify \(\frac{24}{36}=\frac{2}{3}\).

\(\frac{8}{9}\times\frac{3}{4}=\frac{2}{3}\).

- Read the vertical factor by comparing its shaded columns with all columns in panel a). - Read the horizontal factor by comparing its shaded rows with all rows in panel b). - Count the overlap cells in panel c) as a fraction of every cell, then simplify.

1. Panel a) covers \(2\) of \(4\) columns, so one factor is \(\frac{2}{4}=\frac{1}{2}\). 2. Panel b) covers \(3\) of \(5\) rows, so the other factor is \(\frac{3}{5}\). 3. Panel c) covers \(6\) of \(20\) cells, so the overlap is \(\frac{6}{20}=\frac{3}{10}\). 4. Therefore \(\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}\).

The factors are \(\frac{1}{2}\) and \(\frac{3}{5}\), and the overlap is \(\frac{3}{10}\) of the whole grid.

- In each model, read the first factor from the first grid and the second factor from the second grid. - The third grid shows only the cells that belong to both factor bands. - Simplify both overlap fractions before comparing the juice amounts.

1. In Model A, A1 shows \(\frac{4}{5}\) of the pitcher and A2 shows \(\frac{3}{8}\) of that amount. A3 shows \(12\) of \(40\) cells in the overlap, so \(\frac{3}{8}\times\frac{4}{5}=\frac{12}{40}=\frac{3}{10}\). 2. In Model B, B1 shows \(\frac{3}{5}\) of the pitcher and B2 shows \(\frac{1}{2}\) of that amount. B3 shows \(3\) of \(10\) cells in the overlap, so \(\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}\). 3. Both overlaps represent \(\frac{3}{10}\) of same-size pitchers, so the amounts are equal.

a) Model A: \(\frac{3}{8}\times\frac{4}{5}=\frac{3}{10}\) Model B: \(\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}\) b) The amounts are equal.

- Use where the regions are in the grid, not color alone, to identify them. - First compare the non-pond columns with all columns. - Then compare the grass rows within the non-pond region before writing the fraction-of-a-fraction product.

1. The pond occupies one of three equal columns, so \(\frac{2}{3}\) of the garden remains. 2. Within the two remaining columns, grass occupies two of five equal rows, so grass is \(\frac{2}{5}\) of the remaining region. 3. Therefore \(\frac{2}{5}\times\frac{2}{3}=\frac{4}{15}\). The four grass cells out of fifteen total cells confirm the product.

a) \(\frac{2}{3}\) b) \(\frac{2}{5}\) c) \(\frac{2}{5}\times\frac{2}{3}=\frac{4}{15}\)

- Read the vertical band's fraction from panel a). - Compare the horizontal band's height with the full grid height in panel b). - Use panel c) to check the overlap as a fraction of the whole grid.

1. Panel a) shows a vertical band covering \(3\) of \(5\) columns, so the band is \(\frac{3}{5}\) of the whole grid. 2. Panel b) shows a horizontal band covering \(2\) of \(4\) rows, so the overlap takes \(\frac{1}{2}\) of the vertical band. 3. Panel c) shows \(6\) of \(20\) cells, or \(\frac{3}{10}\) of the whole grid. 4. Therefore \(\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}\).

\(\frac{1}{2}\) of the vertical band lies in the overlap, and \(\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}\).

- In each panel, read one factor from columns and the other from rows. - Count the orange overlap cells and all cells in each panel. - Compare what changes and what stays the same when the two factors switch directions.

1. In model a), the vertical region spans \(3\) of \(4\) columns and the horizontal region spans \(2\) of \(3\) rows, so it represents \(\frac{3}{4}\times\frac{2}{3}\). The overlap is \(6\) of \(12\) cells, or \(\frac{1}{2}\). 2. In model b), the vertical region spans \(2\) of \(3\) columns and the horizontal region spans \(3\) of \(4\) rows, so it represents \(\frac{2}{3}\times\frac{3}{4}\). Its overlap is also \(6\) of \(12\) cells, or \(\frac{1}{2}\). 3. Reversing the factor directions does not change the size of the overlap.

a) \(\frac{3}{4}\times\frac{2}{3}=\frac{1}{2}\) b) \(\frac{2}{3}\times\frac{3}{4}=\frac{1}{2}\) Both models have \(6\) overlap cells out of \(12\), so the products are equal.

- In each panel, treat orange as part of both shaded factor regions. - Read one fraction from rows and the other from columns before counting the overlap. - Once both products are known, use a common denominator for their exact difference.

1. Model a) shows \(\frac{1}{2}\) of the rows and \(\frac{5}{6}\) of the columns. Its overlap is \(15\) of \(36\) cells, so the product is \(\frac{5}{12}\). 2. Model b) shows \(\frac{1}{3}\) of the rows and \(\frac{5}{6}\) of the columns. Its overlap is \(10\) of \(36\) cells, so the product is \(\frac{5}{18}\). 3. Using denominator \(36\), \(\frac{5}{12}-\frac{5}{18}=\frac{15}{36}-\frac{10}{36}=\frac{5}{36}\).

a) \(\frac{1}{2}\times\frac{5}{6}=\frac{5}{12}\) b) \(\frac{1}{3}\times\frac{5}{6}=\frac{5}{18}\) The product in a) is greater by \(\frac{5}{36}\).

- Read one factor from the number of selected columns and the other from the selected rows. - The orange cells belong to both regions. - Compare overlap cells with all cells instead of adding the factor numerators and denominators.

1. The vertical region spans \(3\) of \(4\) columns, so one factor is \(\frac{3}{4}\). The horizontal region spans \(2\) of \(5\) rows, so the other factor is \(\frac{2}{5}\). 2. Adding numerators and denominators does not describe the intersection of row and column regions. 3. The overlap has \(6\) cells out of \(20\), so \(\frac{2}{5}\times\frac{3}{4}=\frac{6}{20}=\frac{3}{10}\).

a) The factors are \(\frac{2}{5}\) and \(\frac{3}{4}\). b) The product is represented by the overlap, not by adding fraction parts. The overlap is \(\frac{6}{20}=\frac{3}{10}\).

- In panel a), compare selected columns with all columns. - In panel b), count how many rows the overlap occupies and compare that with the total number of rows. - Use the overlap dimensions to check the product after identifying both factors.

1. Panel a) has \(6\) selected columns out of \(8\), so the known factor is \(\frac{6}{8}=\frac{3}{4}\). 2. In panel b), the overlap is \(3\) rows high out of \(6\) total rows, so the missing factor is \(\frac{3}{6}=\frac{1}{2}\). 3. The overlap has \(3\times6=18\) cells out of \(48\), giving \(\frac{18}{48}=\frac{3}{8}\). 4. Thus \(\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\).

a) \(\frac{3}{4}\) b) \(\frac{3}{6}=\frac{1}{2}\) c) \(\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\)

- Compare the row and column partitions in each model with the denominators in the factors. - After choosing a model, use the numerators to decide how many rows and columns belong to the overlap. - Write the overlap cell count over the total cell count.

1. Model a) partitions one dimension into \(7\) equal parts and the other into \(9\) equal parts, matching the denominators. 2. Model b) uses \(4\) and \(5\) as partition counts, but those are the numerators, not the denominators. 3. Model a) has \(7\times9=63\) equal cells. Selecting \(4\) of the \(7\) rows and \(5\) of the \(9\) columns gives \(4\times5=20\) overlap cells. 4. Therefore \(\frac{4}{7}\times\frac{5}{9}=\frac{20}{63}\).

Model a) is appropriate because its \(7\) row parts and \(9\) column parts match the denominators. The product is \(\frac{20}{63}\).

- Read the vertical fraction from columns and the horizontal fraction from rows. - Count orange overlap cells relative to all cells. - Compare the overlap count with half the total cell count.

1. The vertical factor spans \(7\) of \(9\) columns, and the horizontal factor spans \(2\) of \(3\) rows, so the model represents \(\frac{7}{9}\times\frac{2}{3}\). 2. The orange overlap contains \(14\) of \(27\) cells, so the product is \(\frac{14}{27}\). 3. Half of \(27\) cells is \(13\frac{1}{2}\), so \(14\) cells is more than half. Thus \(\frac{14}{27}>\frac{1}{2}\).

The model represents \(\frac{7}{9}\times\frac{2}{3}=\frac{14}{27}\), and the overlap is greater than \(\frac{1}{2}\).

- Count only the orange cells for part a). - A fraction of a whole requires a numerator count and a total equal-part count. - Read the factor fractions from selected columns and rows, including orange in both regions.

1. There are \(20\) orange overlap cells. 2. A product represented as a fraction of the whole needs the overlap count compared with all equal cells, not just the count alone. 3. The model shows \(\frac{4}{9}\) of the columns and \(\frac{5}{6}\) of the rows. The whole grid has \(54\) cells, so the product is \(\frac{20}{54}=\frac{10}{27}\).

a) \(20\) cells b) \(20\) is only a count; it must be compared with all \(54\) equal cells. c) \(\frac{4}{9}\times\frac{5}{6}=\frac{20}{54}=\frac{10}{27}\).

- Include orange cells when reading both the selected row region and selected column region. - Compare orange cells with all cells for the product. - Convert the benchmark fraction into a cell count before finding the gap.

1. The model shows \(7\) of \(8\) columns and \(5\) of \(6\) rows, so the expression is \(\frac{7}{8}\times\frac{5}{6}\). 2. The orange overlap has \(35\) of \(48\) cells, so the product is \(\frac{35}{48}\). 3. Three fourths of \(48\) cells is \(36\) cells, so one additional overlap cell is needed.

a) \(\frac{7}{8}\times\frac{5}{6}\) b) \(\frac{35}{48}\) c) \(1\) additional cell

- Read one factor from columns and one from rows, counting orange as part of both shaded regions. - Express orange cells over all equal cells and simplify. - Translate \(\frac{1}{4}\) of the whole grid into a target number of cells.

1. The model shows \(5\) of \(8\) columns and \(2\) of \(7\) rows, so the expression is \(\frac{5}{8}\times\frac{2}{7}\). 2. The orange overlap has \(10\) cells out of \(56\), so the product is \(\frac{10}{56}=\frac{5}{28}\). 3. One fourth of \(56\) cells is \(14\), so the overlap needs \(14-10=4\) more cells.

a) \(\frac{5}{8}\times\frac{2}{7}\) b) \(10\) cells, representing \(\frac{5}{28}\) of the grid c) \(4\) more cells

- Use the horizontal band's height to identify its fraction of the total rows. - The product is represented only by cells that belong to both bands. - Compare the number of overlap cells with the total number of cells in the grid.

1. The horizontal band covers \(2\) of \(5\) rows, so the second factor is \(\frac{2}{5}\). 2. The overlap covers \(2\times4=8\) cells out of \(5\times7=35\) cells. 3. Therefore the overlap is \(\frac{8}{35}\), matching \(\frac{4}{7}\times\frac{2}{5}=\frac{8}{35}\).

The second factor is \(\frac{2}{5}\), and the product is \(\frac{8}{35}\). The grid verifies this because \(8\) of its \(35\) cells lie in both factor regions.