2. You can use a rectangle to represent \( (0.3) \cdot(0.5) \). a. What must the side length of each square represent for the rectangl correctly represent \( (0.3) \cdot(0.5) \) ? b. What area is represented by each square? c. What is \( (0.3) \cdot(0.5) \) ? Show your reasoning.

Certainly! Let's break down each part of the problem to understand how to represent the multiplication \( (0.3) \times (0.5) \) using a rectangle and an area model. ### a. **Side Length Representation** **Question:** *What must the side length of each square represent for the rectangle to correctly represent \( (0.3) \times (0.5) \)?* **Answer:** Each side length of the squares should represent the **tenths place**. **Explanation:** - **0.3** can be thought of as 3 tenths. - **0.5** can be thought of as 5 tenths. To accurately represent these decimals on a grid: - **One side of the rectangle** (representing 0.3) should be divided into **10 equal parts**, with each part representing **0.1** (one tenth). - **The other side of the rectangle** (representing 0.5) should also be divided into **10 equal parts**, with each part representing **0.1**. Therefore, each square within the grid represents **0.1 (from one side) multiplied by 0.1 (from the other side)**. ### b. **Area Represented by Each Square** **Question:** *What area is represented by each square?* **Answer:** Each square represents an **area of 0.01**. **Explanation:** Since each side of a square is **0.1**, the area of one square is: \[ \text{Area} = \text{Side length} \times \text{Side length} = 0.1 \times 0.1 = 0.01 \] So, each small square in the grid corresponds to **0.01**. ### c. **Calculating \( (0.3) \times (0.5) \) Using the Area Model** **Question:** *What is \( (0.3) \times (0.5) \)? Show your reasoning.* **Answer:** \( (0.3) \times (0.5) = 0.15 \) **Explanation:** 1. **Draw the Rectangle:** - **Length:** Divide one side into **10 parts**, representing **0.1** each, and shade **3 parts** to represent **0.3**. - **Width:** Divide the adjacent side into **10 parts**, representing **0.1** each, and shade **5 parts** to represent **0.5**. 2. **Determine the Number of Squares Overlapping:** - **3 parts (from 0.3) × 5 parts (from 0.5) = 15 squares** 3. **Calculate the Total Area:** - **15 squares × 0.01 (area per square) = 0.15** Therefore, using the area model, multiplying 0.3 by 0.5 yields **0.15**. ### **Visual Representation** Here's a simple visual to aid understanding: ``` |---|---|---|---|---|---|---|---|---|---| | X | X | X | | | | | | | | | X | X | X | | | | | | | | | X | X | X | | | | | | | | |---|---|---|---|---|---|---|---|---|---| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |---|---|---|---|---|---|---|---|---|---| ``` - **X's** represent the shaded area for \( 0.3 \times 0.5 \). - Total shaded squares = **15** - Each square = **0.01** - Total area = **15 × 0.01 = 0.15** Therefore, \( (0.3) \times (0.5) = 0.15 \).