To examine Adi's use of algebra tiles for the expression [tex]\((-2x - 2)(2x - 1)\)[/tex], let's break down the process step-by-step:
1. Distributive Property:
Simplify the algebraic expression [tex]\((-2x - 2)(2x - 1)\)[/tex] using the distributive property:
[tex]\[
(-2x - 2)(2x - 1) = (-2x)(2x) + (-2x)(-1) + (-2)(2x) + (-2)(-1)
\][/tex]
2. Multiply Individual Terms:
Evaluate each multiplication step:
[tex]\[
(-2x)(2x) = -4x^2
\][/tex]
[tex]\[
(-2x)(-1) = 2x
\][/tex]
[tex]\[
(-2)(2x) = -4x
\][/tex]
[tex]\[
(-2)(-1) = 2
\][/tex]
3. Combine Like Terms:
Combine the results obtained from the multiplication:
[tex]\[
-4x^2 + 2x - 4x + 2
\][/tex]
4. Simplify:
Further simplify by combining like terms:
[tex]\[
-4x^2 - 2x + 2
\][/tex]
We have three aspects to consider in Adi's work with algebra tiles:
1. Representation of Original Factors:
Adi needed to assign algebra tiles properly for each term in the factors [tex]\((-2x - 2)\)[/tex] and [tex]\((2x - 1)\)[/tex]. If Adi did not represent the two original factors correctly on the headers, then this part of her work would be incorrect.
2. Signs of the Products:
It's essential that the multiplication and the resulting signs are handled correctly. Any mistake in the signs of the products, such as misinterpreting negative and positive multiplications, would indicate an error.
3. Powers of [tex]\(x\)[/tex]:
The result should properly reflect the degrees of [tex]\(x\)[/tex]. For example, products resulting in terms like [tex]\(x, x^2\)[/tex], etc., should be correctly represented. Given the problem, we see that [tex]\(x\)[/tex] and [tex]\(x^2\)[/tex] were appropriately accounted for.
Conclusion:
Given the provided results:
1. Adi did not represent the two original factors correctly on the headers.
2. The signs on some of the products are incorrect.
3. None of the products have incorrect powers of [tex]\(x\)[/tex].
So, the accurate interpretation is:
- She did not represent the two original factors correctly on the headers.
- The signs on some of the products are incorrect.
- All powers of [tex]\(x\)[/tex] in the products are correct.
Thus:
- Adi did not represent the two original factors correctly on the headers.
- The signs on some of the products are incorrect.
- All powers of [tex]\(x\)[/tex] in the products are correct.
By focusing on these specific errors, we confirm that Adi’s representation had issues with the original factor headers and signs but not with the powers of [tex]\(x\)[/tex].
