- Look at the columns of each color separately. - What has the same size in both parts?

1. The left part is \(2\) columns wide and the right part is \(3\) columns wide. 2. The expression is \(3\times(2+3)\). The factors inside the parentheses count the two widths.

\(3\times(2+3)\). The \(2\) counts columns in the left part; the \(3\) counts columns in the right part.

- Count grid rows and columns from the picture before writing any expression. - The vertical boundary divides the full width into two smaller widths while keeping the same row count. - Each smaller product should represent one complete section of the rectangle.

1. The model has \(8\) rows and \(8\) columns. The vertical line splits the columns into \(5\) on the left and \(3\) on the right. 2. The whole rectangle is \(8 \times 8\), and the two sections are \(8 \times 5+8 \times 3\). 3. \(8 \times 5+8 \times 3=40+24=64\), so \(8 \times 8=64\). 4. The distributive property lets us split the rectangle into the two smaller rectangles and add their products.

a) \(8\) rows; \(8\) columns; widths \(5\) and \(3\) b) \(8 \times 8=8 \times 5+8 \times 3=64\) c) Distributive property

- Use the picture to find how many columns are in each row. - Find the multiplication fact shown by the first \(5\) rows. - Find how many squares the bottom row adds.

1. The first \(5\) rows each have \(7\) squares, so they represent \(5 \times 7=35\). 2. The additional bottom row adds \(1 \times 7=7\) unit squares. 3. Therefore, \(6 \times 7=5 \times 7+1 \times 7=35+7=42\).

\(6 \times 7=5 \times 7+1 \times 7=35+7=42\). The additional bottom row adds \(7\) unit squares.

- Use the image to count the rows and columns in the first part. - Find how many squares are in the extra rightmost column. - Add the two parts to describe the whole grid.

1. The first \(5\) columns have \(6\) rows, so they represent \(6 \times 5=30\). 2. The extra rightmost column adds \(6 \times 1=6\) unit squares. 3. Therefore, \(6 \times 6=6 \times 5+6 \times 1=30+6=36\). The known fact is \(6 \times 5\).

\(6 \times 6=6 \times 5+6 \times 1=30+6=36\)

- Use the vertical boundary to find the two widths. - Look for a familiar multiplication fact in one section. - Explain both your calculation and why the split helps.

1. The vertical boundary splits the \(9\) columns into a left section of \(5\) columns and a right section of \(4\) columns. 2. Use \(6 \times (5+4)=6 \times 5+6 \times 4\). 3. Calculate \(30+24=54\). This is easier because it begins with the familiar fact \(6 \times 5\).

One way is \(6 \times (5+4)=6 \times 5+6 \times 4=30+24=54\). It uses the familiar fact \(6 \times 5\).

- Use the picture to find the total number of rows and columns. - Compare the total rows with the \(5\) rows in the upper section. - Use the number of columns to find the number of squares in the lower rows.

1. The image shows a \(7\)-row by \(8\)-column grid. 2. The upper section has \(5\) rows, so the lower section has \(7-5=2\) rows. 3. The lower section contains \(2\times8=16\) unit squares. 4. The whole grid is \(7\times8=5\times8+2\times8=40+16=56\).

The lower section contains \(16\) unit squares. \(7 \times 8=5 \times 8+2 \times 8=40+16=56\).

- Count the rows and the columns in each section. - Write one multiplication fact for the left section and one for the right section. - Add those two products to find the whole grid.

1. The left section has \(6\) rows and \(5\) columns, so it represents \(6\times5=30\). 2. The right section has \(6\) rows and \(3\) columns, so it represents \(6\times3=18\). 3. Add the two smaller products: \(6\times8=6\times5+6\times3=30+18=48\).

\(6\times8=(6\times5)+(6\times3)=30+18=48\)

- Use the picture to find the number of rows and total lights in each row. - The right section shows the same number of broken lights in every row. - Subtract the broken rectangular section from the whole array.

1. The image shows \(9\) rows with \(7\) lights in each row. 2. Since \(2\) lights in each row are broken, \(7 - 2 = 5\) lights work in each row. 3. Thus, \(9 \times 5 = 45\) lights are on. 4. The same result is \(9 \times 7 - 9 \times 2 = 63 - 18 = 45\).

\(9 \times 7 - 9 \times 2 = 63 - 18 = 45\). There are \(45\) working lights.

- Use the two images to compare the numbers of rows. - The number of spaces in each row stays the same. - Write the enlarged rectangle as the original rectangle plus only the new rows.

1. Diagram a) shows \(6\) rows of \(9\), so the original bed has \(6\times9=54\) spaces. 2. Diagram b) shows \(8\) rows of \(9\), so the enlarged bed has \(8\times9=72\) spaces. 3. The added rectangle has \(2\) rows of \(9\), or \(18\) spaces. 4. The relationship is \(8\times9=6\times9+2\times9=54+18=72\).

Original bed: \(6\times9=54\) Enlarged bed: \(8\times9=72\) \(8\times9=6\times9+2\times9\), so \(18\) new spaces were added.

- Read the row count and total column count from the grid. - The vertical boundary splits only the columns; both sections keep all the rows. - Write one smaller product for each visible section.

1. The rectangle has \(4\) rows and \(7\) columns, so the whole is \(4 \times 7\). 2. The vertical split divides the \(7\) columns into \(5\) and \(2\). 3. The two smaller rectangles are \(4 \times 5\) and \(4 \times 2\). 4. Therefore, \(4 \times 7=4 \times 5+4 \times 2=20+8=28\).

a) \(4 \times 7=28\) b) \(4 \times 7=4 \times 5+4 \times 2\) c) \(28\)

- Imagine covering one color. What shape is left uncovered? - How can you check that your two smaller products count every square once?

1. The left part is \(2\times3=6\) squares. 2. The right part is \(2\times2=4\) squares. 3. Together they make \(6+4=10\) squares, the same as \(2\times5\).

Left: \(2\times3=6\). Right: \(2\times2=4\). Together: \(6+4=10\). Whole grid: \(2\times5=10\). The two parts cover every square once.

- How many rows does each color cover? - How can two smaller products count the same squares as one whole-grid product?

1. The top part has \(1\) row of \(3\) squares; the bottom has \(3\) rows of \(3\) squares. 2. Their total is \((1\times3)+(3\times3)=3+9=12\). 3. The whole grid is \(4\times3=12\). Both methods count the same squares.

Top: \(1\times3=3\). Bottom: \(3\times3=9\). Together: \(3+9=12\). Whole grid: \(4\times3=12\). Both count every square once.

- Compare the full number of columns with the columns that will remain. - What does one colored square in each row add up to over all the rows?

1. The grid has \(6\) rows and \(10\) columns. The colored strip is one column of \(6\) squares. 2. Subtract the strip: \(60-6=54\). 3. The remaining grid has \(6\) rows and \(9\) columns: \(6\times9=54\).

\(60-6=54\) and \(6\times9=54\). Removing one column of six squares leaves nine columns of six.

- Check Toni's calculation, then compare it with the colored parts of the grid. - How could you describe the size of each colored rectangle using its sides?

1. The top part has \(2\) rows of \(4\) squares. 2. The bottom part has \(4\) rows of \(4\) squares, not three rows. 3. Toni's two groups of three rows give the same total, but not the pictured split. The pictured equation is \(6\times4=(2\times4)+(4\times4)=8+16=24\).

No. Toni used two groups of \(3\) rows, but the picture has parts of \(2\) and \(4\) rows. The matching equation is \(6\times4=(2\times4)+(4\times4)=24\).

- Find whether each boundary separates rows or columns. - In both smaller products, \(5\) is the number of rows.

1. In array A, the boundary is vertical. Each of the \(5\) rows is split into \(4\) unit squares and \(3\) unit squares. 2. Therefore, array A represents \(5 \times 4 + 5 \times 3\). 3. In array B, the boundary separates complete rows, so it represents a different split.

Array A, because each row is split into \(4\) unit squares and \(3\) unit squares.

- Read the whole rectangle’s rows and columns from the grid. - The marked strip is exactly one column wide. - Your subtraction equation should show the whole product and the strip product. - For the final equation, express the whole rectangle as the sum of its two rectangular parts.

1. The whole grid has \(7\) rows and \(10\) columns, so its area is \(7 \times 10 = 70\) square units. 2. The marked strip has \(7\) rows and \(1\) column, so its area is \(7 \times 1 = 7\) square units. 3. Subtract the strip from the whole: \(7 \times 10 - 7 \times 1 = 70 - 7 = 63\). 4. The same partition can be written \(7 \times 10 = 7 \times 9 + 7 \times 1\). Thus the unmarked rectangle has area \(7 \times 9 = 63\) square units.

a) \(7 \times 10 = 70\) b) \(7 \times 1 = 7\) c) \(7 \times 10 - 7 \times 1 = 70 - 7 = 63\) d) \(7 \times 10 = 7 \times 9 + 7 \times 1\)

- Do the two colors cover the same number of rows? - How can you show each color's amount without counting square by square?

1. The left part has \(4\) rows and \(6\) columns: \(4\times6=24\). 2. The right part has \(4\) rows and \(3\) columns: \(4\times3=12\). 3. The left has \(24-12=12\) more squares. Together they make \(24+12=36\), the whole \(4\times9\) grid.

Left: \(4\times6=24\). Right: \(4\times3=12\). Difference: \(24-12=12\). Whole: \(4\times9=24+12=36\).

- Keep one side length the same in each part. - Make the two other side lengths add to the whole side. - Check that the partial products add to the whole product.

1. The whole rectangle has \(5\) rows and \(7\) columns. 2. For example, splitting the columns into three and four gives \(5\times3=15\) and \(5\times4=20\). 3. Then \(5\times7=15+20=35\). Other splits are correct if both parts exactly cover the rectangle.

Example: Split \(7\) columns into \(3\) and \(4\). Then \(5\times7=(5\times3)+(5\times4)=15+20=35\). Other correct partitions are accepted.

- Place each dividing line between squares, not through a square. - Can you use the rows and columns of each new rectangle instead of counting each square? - Check that your parts leave no squares out and count no square twice.

1. The grid has \(4\) rows and \(9\) columns, so the total is \(4\times9=36\) squares. 2. One horizontal split is \(1\) row and \(3\) rows: \(1\times9+3\times9=9+27=36\). 3. One vertical split is \(5\) columns and \(4\) columns: \(4\times5+4\times4=20+16=36\). 4. Both partitions cover the same rectangle without overlap. Other valid splits are accepted.

Example horizontal split: \(4\times9=(1\times9)+(3\times9)=36\). Example vertical split: \(4\times9=(4\times5)+(4\times4)=36\). Both count the same \(36\) squares. Other valid splits are accepted.

- What do you know about the height and total tiles of each part? - How could the tile count and row count help you find one part's width? - What happens to the widths when the rectangles sit side by side?

1. The left width is \(30\div6=5\) columns; the right width is \(24\div6=4\) columns. 2. The parts have \(6\times5=30\) and \(6\times4=24\) tiles. 3. Together they are \(5+4=9\) columns wide and have \(30+24=54\) tiles. 4. This gives \(6\times9=(6\times5)+(6\times4)=54\), because the two full-height parts cover the whole wall.

Left width: \(5\) columns; right width: \(4\) columns. Left: \(6\times5=30\). Right: \(6\times4=24\). Whole wall: \(5+4=9\) columns and \(30+24=54\) tiles, so \(6\times9=(6\times5)+(6\times4)\).