Final answer:
By visualizing the equation 3(3+1+7)=3(3) +3(1) + 3(7) as a rectangular model with an area problem, we show that multiplying each part separately by 3 and then adding the results yields the same total area (33 units), confirming the equation is true using the associative property of multiplication.
Step-by-step explanation:
To show that the equation 3(3+1+7)=3(3) +3(1) + 3(7) is true using a rectangular model, we can visualize the equation as an area problem.
First, imagine a rectangle divided into three parts: one part with a width of 3 units, the second part with a width of 1 unit, and the third part with a width of 7 units. The height of the entire rectangle is 3 units, representing the multiplier in the equation.
According to the distributive property, when you multiply a sum by a number, you can multiply each addend separately and then add the products. So, the area of the entire rectangle is the sum of the areas of the three parts:
- Area of the first part: 3 units (height) × 3 units (width) = 9 square units
- Area of the second part: 3 units (height) × 1 unit (width) = 3 square units
- Area of the third part: 3 units (height) × 7 units (width) = 21 square units
The total area of the rectangle, which represents the left side of the equation, is the sum of these areas:
9 square units + 3 square units + 21 square units = 33 square units
Now, we calculate the right side of the equation:
- 3×(3) = 9
- 3×(1) = 3
- 3×(7) = 21
Adding these products together:
9 + 3 + 21 = 33
Therefore, both sides of the equation result in 33, showing that the equation is true. This demonstrates the associative property of multiplication over addition.