Question

A rectangular area model is divided vertically and horizontally into four sections. The left side of the model is labeled 10 and 4; the top side of the model is labeled 40 and 5.
Show how to use the area model and the Distributive Property to find 14×45
.

(2 points)

Answer 1

To find \( 14 \times 45 \) using an area model and the Distributive Property, we can break down both numbers into parts that are easier to multiply.

### Step 1: Break Down the Numbers
We can break down \( 14 \) and \( 45 \) as follows:
- \( 14 \) can be broken down into \( 10 + 4 \).
- \( 45 \) can be broken down into \( 40 + 5 \).

### Step 2: Set Up the Area Model
We can create a rectangular area model that has four sections, divided by the horizontal and vertical breakdowns we just performed:

```
40 5
+------------------+
10| (10 × 40) | (10 × 5)
| |
+------------------+
4 | (4 × 40) | (4 × 5)
| |
+------------------+
```

### Step 3: Calculate the Areas of Each Section
Now, we calculate the area of each section:

1. \( 10 \times 40 = 400 \)
2. \( 10 \times 5 = 50 \)
3. \( 4 \times 40 = 160 \)
4. \( 4 \times 5 = 20 \)

### Step 4: Add All the Areas Together
Next, we add the areas of all four sections to get the total area:

\[
400 + 50 + 160 + 20
\]

### Step 5: Perform the Addition
Now we add the values:

1. \( 400 + 50 = 450 \)
2. \( 450 + 160 = 610 \)
3. \( 610 + 20 = 630 \)

### Conclusion
Thus, the calculation using the area model shows that:
\[
14 \times 45 = 630
\]

This method uses the Distributive Property effectively by breaking down both multiplicands into sums that are easier to work with.