Here are three ways of finding the area of a rectangle that is 24 units by 13 units.

Discuss with your partner:

1. How are the diagrams the same?

2. How are the diagrams different?

3. If you were to find the area of a rectangle that is 37 units by 19 units, which of the three ways of decomposing the rectangle would you use? Why?

Here are two ways to calculate 24 times 13.

Two vertical calculations of 24 times 13. Calculation A, 7 rows. First row: 24. Second row: multiplication symbol, 13. Horizontal line. Third row: 12. Fourth row: 60. Fifth row: 40. Sixth row: plus 200. Horizontal line. Seventh row: 312. Calculation B, 5 rows. First row: 24. Second row: multiplication symbol, 13. Horizontal line. Third row: 72. Fourth row: plus 240. Horizontal line. Fifth row: 312.

Discuss with your partner:

1. In Calculation A, where does each partial product—the 12, 60, 40, and 200—come from?

2. In Calculation B, where do 72 and 240 come from?

3. Which diagram in the first question corresponds to Calculation A? Which one corresponds to Calculation B? How do you know?

Find the product of 18 and 14 in two ways:

1. Calculate numerically.

   

2. Find the area, in square units, of this 18-by-14 rectangle. Show your reasoning.

   

Here is an area diagram that represents .

A rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 1, horizontal side, 2. Top right rectangle, vertical side, 1, horizontal side, 0 point 4. Bottom left rectangle, vertical side, 0 point 3, horizontal side, 2. Bottom right rectangle, vertical side, 0 point 3, horizontal side, 0 point 4.

1. Find the region that represents . Label it with its area of 0.12.

2. Label the other regions with their areas.

3. Find the value of . Show your reasoning.

Here are two ways of calculating .

Two vertical calculations of 2 point 4 times 1 point 3. Calculation A, 7 rows. First row: 2 point 4. Second row: multiplication symbol, 1 point 3. Horizontal line. Third row: 0 point 1 2. Fourth row: 0 point 6. Fifth row: 0 point 4. Sixth row: plus 2. Horizontal line. Seventh row: 3 point 1 2. Calculation B, 5 rows. First row: 2 point 4. Second row: multiplication symbol, 1 point 3. Horizontal line. Third row: 0 point 7 2. Fourth row: plus 2 point 4. Horizontal line. Fifth row: 3 point 1 2.

Analyze the calculations and discuss these questions with a partner:

1. In Calculation A, where do the 0.12 and other partial products come from?

2. In Calculation B, where do the 0.72 and 2.4 come from?

3. In each calculation, why are the numbers below the horizontal line aligned vertically the way they are?

Find the value of in two ways:

1. Draw and label a diagram. Show your reasoning.
2. Calculate numerically, without using a diagram. Be prepared to explain your reasoning.