Question

armando used algebra tiles to represent the product (3x(2x - 1)).

which is true regarding armandos use of algebra tiles?

• he used the algebra tiles correctly.
• some of the products do not show the correct powers of (x).
• the signs on some of the products are incorrect.
• he did not represent the two original factors correctly.

Explanation:

Analyze the intended product

The problem asks us to evaluate Armando's algebra tile representation of the product:

$$ 3x(2x - 1) $$

This represents multiplying a factor of \(3x\) by a factor of \((2x - 1)\).

Examine the represented factors

Let's look at the factors placed along the grid's headers:

• Top header (horizontal factor): There are three tiles labeled \(+x\). This represents:

$$ +x + x + x = 3x $$

• Left header (vertical factor): There are two tiles labeled \(+x\) and one tile labeled \(-\) (which represents \(-1\)). This represents:

$$ +x + x - 1 = 2x - 1 $$

Thus, the factors on the headers correctly represent \(3x\) and \(2x - 1\).

Examine the area model products

Let's check the multiplication of each intersecting tile:

• First column (\(+x\)) and first row (\(+x\)):

$$ (+x) \cdot (+x) = +x^2 $$

The tile shown is labeled \(+x^2\). This is correct.

• Second column (\(+x\)) and first row (\(+x\)):

$$ (+x) \cdot (+x) = +x^2 $$

However, the tile shown in the grid is labeled \(+x\). This is incorrect because the product of two linear terms \(x \cdot x\) must be a quadratic term \(x^2\).

• Third column (\(+x\)) and first row (\(+x\)):

$$ (+x) \cdot (+x) = +x^2 $$

The tiles shown in the remaining columns of this row are labeled \(+x\), which are also incorrect.

Identify the error

Since several of the product tiles representing \(x \cdot x\) are labeled as \(+x\) instead of \(+x^2\), some of the products do not show the correct powers of \(x\).

Match with the given options

Let's evaluate the choices:

• He used the algebra tiles correctly. (False, because \(x \cdot x\) should yield \(x^2\), not \(x\)).
• Some of the products do not show the correct powers of \(x\). (True, as \(+x\) is shown instead of \(+x^2\)).
• The signs on some of the products are incorrect. (False, the signs of the products match the sign rules).
• He did not represent the two original factors correctly. (False, the factors \(3x\) and \(2x - 1\) are represented correctly on the headers).

Answer:

• He used the algebra tiles correctly.
• Some of the products do not show the correct powers of \(x\). (Correct answer)
• The signs on some of the products are incorrect.
• He did not represent the two original factors correctly.