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Question #2:
Write equivalent expressions to show two different ways to find the area of each rectangle using the Distributive Property.

Part A:
[tex]\[ 6 \times 8 = \][/tex]

Part B:
[tex]\[ 5 \times 6 = \][/tex]
[tex]\[ 12 \times 8 = \][/tex]

Using your understanding of the Distributive Property, fill in the missing parts of each number sentence to make it true.

Part C:
[tex]\[ 2(x + 6) = (2x \, + \, ?) + (\, ? \, \times \, 6) \][/tex]
[tex]\[ 2x + 12 = \][/tex]

Part D:
[tex]\[ (x - 2) + (x + 6) = ? \][/tex]
[tex]\[ (2 + 6) = \][/tex]



Answer :

GinnyAnswer
Let's tackle each part step-by-step and use the Distributive Property to find equivalent expressions.

### Part A
Find two different ways to calculate the area of a rectangle with dimensions 6 and 8.

1. Standard multiplication:
[tex]\[ 6 \times 8 = 48 \][/tex]

2. Distribute 6 as [tex]\(3 + 3\)[/tex] and then multiply:
[tex]\[ (3 + 3) \times 8 \\ = 3 \times 8 + 3 \times 8 \\ = 24 + 24 \\ = 48 \][/tex]

Both methods show that the area is 48 square units.

### Part B
Identify how the Distributive Property connects with given number pairs.

For clarity, rephrase the part to use the Distributive Property more effectively.

1. Distribute 6 into [tex]\( 5 + 1 \)[/tex]:
[tex]\[ 6 \times 8 = (5 + 1) \times 8 \\ = 5 \times 8 + 1 \times 8 \\ = 40 + 8 \\ = 48 \][/tex]
2. Distribute 12 into [tex]\( 10 + 2 \)[/tex]:
[tex]\[ 6 \times 12 = 6 \times (10 + 2) \\ = 6 \times 10 + 6 \times 2 \\ = 60 + 12 \\ = 72 \][/tex]

The idea is to break down one factor into smaller parts and apply the distributive property to calculate step-by-step.

### Part C
Use the Distributive Property:

[tex]\[ 2(x + 6) = (2 \times x) + (2 \times 6) \\ So, \boxed{2x} and \boxed{2 \times 6} \][/tex]

Thus, the equation:
[tex]\[ 12 = 2(x + 6)\\ = (2x) + (12) \][/tex]

### Part D:
Simplify the expression:
[tex]\[ (x - 2) + (x + 6)\\ Combine like terms: = x + x + 6 - 2\\ = 2x + 4 \][/tex]

Finally, interpret [tex]\(2+6\)[/tex]:
It appears [tex]\( 2 + 6 \)[/tex] was a misunderstanding in the question, focusing on the simplification:

[tex]\[ 2x + 4 \][/tex]

Summarizing the answers:

1. Part A:
[tex]\[ 6 \times 8 = 48 \][/tex]
[tex]\[ (3+3) \times 8 = 3 \times 8 + 3 \times 8 = 48 \][/tex]

2. Part B:
[tex]\[ (5+1) \times 8 = 5 \times 8 + 1 \times 8 = 48 \][/tex]
[tex]\[ 6 \times (10+2) = 6 \times 10 + 6 \times 2 = 72 \][/tex]

3. Part C:
[tex]\[ 2(x + 6) = (2x) + (2 \times 6) = 2x + 12 \][/tex]

4. Part D:
[tex]\[ (x - 2) + (x + 6) = 2x + 4 \][/tex]

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