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Question

3 rectangles, each labeled x, 2 rectangles, each labeled negative x, 5 squares, each labeled 1, and 2 squares, each labeled negative 1
























Which models represent algebraic expressions equivalent to the algebraic expression represented by the model shown?

Select the two correct answers. 

(1 point)
Responses

2 rectangles, each labeled x, 2 rectangles, each labeled negative x, 2 squares, each labeled 1, and 2 squares, each labeled negative 1
Image with alt text: 2 rectangles, each labeled x, 2 rectangles, each labeled negative x, 2 squares, each labeled 1, and 2 squares, each labeled negative 1

2 rectangles, each labeled x, 1 rectangle labeled negative x, 4 squares, each labeled 1, and 1 square labeled negative 1
Image with alt text: 2 rectangles, each labeled x, 1 rectangle labeled negative x, 4 squares, each labeled 1, and 1 square labeled negative 1

3 rectangles, each labeled x, 1 rectangle labeled negative x, 5 squares, each labeled 1, and 1 square labeled negative 1
Image with alt text: 3 rectangles, each labeled x, 1 rectangle labeled negative x, 5 squares, each labeled 1, and 1 square labeled negative 1

1 rectangle labeled x and 1 square labeled 1
Image with alt text: 1 rectangle labeled x and 1 square labeled 1

1 rectangle, each labeled x and 3 squares, each labeled 1
Image with alt text: 1 rectangle, each labeled x and 3 squares, each labeled 1
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Answer 1

To evaluate which models represent algebraic expressions equivalent to the original model, let's analyze the components of the first model you described:

• 3 rectangles labeled x
• 2 rectangles labeled -x
• 5 squares labeled 1
• 2 squares labeled -1

Now, let's translate these into an algebraic expression:

1. The total contribution from the rectangles:

   • \(3x\) for the \(3\) rectangles labeled \(x\)
   • \(2(-x)\) or \(-2x\) for the \(2\) rectangles labeled \(-x\)

   Combining these gives: \[ 3x - 2x = x \]

2. The total contribution from the squares:

   • \(5(1)\) or \(5\) for the \(5\) squares labeled \(1\)
   • \(2(-1)\) or \(-2\) for the \(2\) squares labeled \(-1\)

   Combining these gives: \[ 5 - 2 = 3 \]

Thus, the overall expression represented by the model is: \[ x + 3 \]

Now we need to find equivalent expressions among the provided options:

1. Option: \(2 rectangles, each labeled x, 2 rectangles, each labeled -x, 2 squares, each labeled 1, and 2 squares, each labeled -1\)

   • This translates to:
     • \(2x - 2x + 2 - 2 = 0\)
   • Not equivalent to \(x + 3\).

2. Option: \(2 rectangles, each labeled x, 1 rectangle labeled -x, 4 squares, each labeled 1, and 1 square labeled -1\)

   • This translates to:
     • \(2x - x + 4 - 1 = x + 3\)
   • This is equivalent to \(x + 3\).

3. Option: \(3 rectangles, each labeled x, 1 rectangle labeled -x, 5 squares, each labeled 1, and 1 square labeled -1\)

   • This translates to:
     • \(3x - x + 5 - 1 = 2x + 4\)
   • Not equivalent to \(x + 3\).

4. Option: \(1 rectangle labeled x and 1 square labeled 1\)

   • This translates to:
     • \(x + 1\)
   • Not equivalent to \(x + 3\).

5. Option: \(1 rectangle labeled x and 3 squares, each labeled 1\)

   • This translates to:
     • \(x + 3\)
   • This is also equivalent to \(x + 3\).

Thus, the two correct answers are:

• Option 2: \(2 rectangles, each labeled x, 1 rectangle labeled -x, 4 squares, each labeled 1, and 1 square labeled -1\)
• Option 5: \(1 rectangle labeled x and 3 squares, each labeled 1\)