Question
Adi used algebra tiles to represent the product (negative 2 x minus 1)(2 x minus 1).
An algebra tile configuration. 4 tiles are in the Factor 1 spot: 2 are labeled negative x and 2 are labeled negative. 3 tiles are in the Factor 2 spot: 2 are labeled + x and 1 is labeled negative. 12 tiles are in the Product spot: 4 are labeled negative x squared, 4 are labeled negative x, 2 are labeled + x, and 2 are labeled +.
Which is true regarding Adi’s use of algebra tiles?
She used the algebra tiles correctly.
She did not represent the two original factors correctly on the headers.
The signs on some of the products are incorrect.
Some of the products do not show the correct powers of x.
An algebra tile configuration. 4 tiles are in the Factor 1 spot: 2 are labeled negative x and 2 are labeled negative. 3 tiles are in the Factor 2 spot: 2 are labeled + x and 1 is labeled negative. 12 tiles are in the Product spot: 4 are labeled negative x squared, 4 are labeled negative x, 2 are labeled + x, and 2 are labeled +.
Which is true regarding Adi’s use of algebra tiles?
She used the algebra tiles correctly.
She did not represent the two original factors correctly on the headers.
The signs on some of the products are incorrect.
Some of the products do not show the correct powers of x.
Answers
GPT-4o mini
Let's analyze the expression \(( -2x - 1)(2x - 1)\) and the given algebra tile configuration to determine the correctness of Adi's setup.
1. **Factors Representation**:
- The first factor is \(-2x - 1\) which corresponds to:
- 2 tiles for \(-x\) (which we can denote as being represented by 2 tiles labeled \(-x\))
- 1 tile for \(-1\) (1 tile labeled \(-\))
- The second factor is \(2x - 1\) which corresponds to:
- 2 tiles for \(+x\) (which we can denote as being represented by 2 tiles labeled \(+x\))
- 1 tile for \(-1\) (1 tile labeled \(-\))
2. **Tiles Summary**:
- Factor 1 has 4 tiles: 2 labeled \(-x\) and 2 labeled \(-\) which is correct for the factor \(-2x - 1\).
- Factor 2 has 3 tiles: 2 labeled \(+x\) and 1 labeled \(-\), which is correct for the factor \(2x - 1\).
3. **Product Representation**:
- Now, let's calculate the product \(( -2x - 1)(2x - 1)\) using distribution:
\[
-2x(2x) + -2x(-1) + -1(2x) + -1(-1)
\]
\[
= -4x^2 + 2x + 2x + 1
\]
\[
= -4x^2 + 4x + 1
\]
- In the algebra tile product configuration:
- 12 tiles are shown: 4 are labeled \(-x^2\), 4 are labeled \(-x\), 2 are labeled \(+x\), and 2 are labeled \(+\).
- Here, the representation is incorrect because the correct terms after the product are:
- \(-4x^2\) (correct)
- \(+4x\) (we have \(-x\) labeled) which represents a discrepancy
- \(+1\) which represents an additional discrepancy as we had 2 labeled \(+\).
4. **Conclusion**:
- Both factors are represented correctly, but regarding the product, the signs of the products are incorrect. We therefore would conclude that Adi "did not represent the signs on some of the products correctly".
**Final Answer: The signs on some of the products are incorrect.**
1. **Factors Representation**:
- The first factor is \(-2x - 1\) which corresponds to:
- 2 tiles for \(-x\) (which we can denote as being represented by 2 tiles labeled \(-x\))
- 1 tile for \(-1\) (1 tile labeled \(-\))
- The second factor is \(2x - 1\) which corresponds to:
- 2 tiles for \(+x\) (which we can denote as being represented by 2 tiles labeled \(+x\))
- 1 tile for \(-1\) (1 tile labeled \(-\))
2. **Tiles Summary**:
- Factor 1 has 4 tiles: 2 labeled \(-x\) and 2 labeled \(-\) which is correct for the factor \(-2x - 1\).
- Factor 2 has 3 tiles: 2 labeled \(+x\) and 1 labeled \(-\), which is correct for the factor \(2x - 1\).
3. **Product Representation**:
- Now, let's calculate the product \(( -2x - 1)(2x - 1)\) using distribution:
\[
-2x(2x) + -2x(-1) + -1(2x) + -1(-1)
\]
\[
= -4x^2 + 2x + 2x + 1
\]
\[
= -4x^2 + 4x + 1
\]
- In the algebra tile product configuration:
- 12 tiles are shown: 4 are labeled \(-x^2\), 4 are labeled \(-x\), 2 are labeled \(+x\), and 2 are labeled \(+\).
- Here, the representation is incorrect because the correct terms after the product are:
- \(-4x^2\) (correct)
- \(+4x\) (we have \(-x\) labeled) which represents a discrepancy
- \(+1\) which represents an additional discrepancy as we had 2 labeled \(+\).
4. **Conclusion**:
- Both factors are represented correctly, but regarding the product, the signs of the products are incorrect. We therefore would conclude that Adi "did not represent the signs on some of the products correctly".
**Final Answer: The signs on some of the products are incorrect.**