Getting Started 
Misconception/Error The student does not recognize the trinomial as the product of two binomials. 
Examples of Student Work at this Level The student attempts a mathematically incorrect approach to finding the area. For example, the student:
 Writes the equation + 5x â€“ 24 = x + 8 + w; then subtracts x + 8 from both sides and identifies the width as + 4x â€“ 32.Â
 Ignores the variable in the factor by writing, â€śThe width would be half of the length, so if the length is 8, the width is 4.â€ť
 Substitutes eight for x into the expression that represents the area to find the width of the rectangle.Â

Questions Eliciting Thinking What type of figure is given? Can you tell me the formula used to determine the area of this figure?
What kind of expression is used to describe the area? What kind of expression is used to describe the length? How should these expressions be related? 
Instructional Implications Review the area formula for a rectangle using the symbols A for area, l for length, and w for width. Tell the student to write this formula on his or her paper. Ask the student to identify the two parts of the area formula that are given and to write them underneath the corresponding parts of the formula. Discuss the operation that relates the length and the width in the area formula and how the length and the width can be considered factors of the area. Remind the student that since the area of the rectangle is given in trinomial form, the binomial factors of the trinomial represent the rectangleâ€™s length and width. Since one binomial factor is given, he or she only need determine the missing binomial factor.
Provide the student with more practice and experience multiplying and factoring polynomials. Guide the student to recognize factorable polynomials based on their structure. Be sure the student understands that the factored and expanded forms of a given polynomial are equivalent, but one form might be more useful in a given problem setting. Also, be sure the student understands that not all polynomials are factorable. 
Making Progress 
Misconception/Error The student recognizes the trinomial as the product of two binomials but is unable to correctly rewrite it in factored form. 
Examples of Student Work at this Level The student:
 Attempts to solve for w by dividing the trinomial + 5x â€“ 24 by (x + 8) but is unable to correctly complete the division.
 Attempts to factor the trinomial but makes an error.

Questions Eliciting Thinking Is there a way other than division to determine the missing factor for a trinomial?
If you use a set of parentheses instead of w in your equation, could you figure out what goes in the parentheses?
Did you check your answer by multiplying the two factors? Did you get back to your original trinomial? Are your signs correct?
When given a trinomial representing the area of a rectangle, how does factoring help you find the rectangleâ€™s width? 
Instructional Implications Provide instruction on factoring trinomials of the form +bx + c when a = 1. Ask the student to check his or her factorization by finding the product of the factors and then comparing the result to the original trinomial.
Provide the student with more practice and experience multiplying and factoring polynomials. Guide the student to recognize factorable polynomials based on their structure. Be sure the student understands that the factored and expanded forms of a given polynomial are equivalent, but one form might be more useful in a given problem setting. Also, be sure the student understands that not all polynomials are factorable. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student recognizes the trinomial + 5x â€“ 24 as factorable and realizes that x + 8 is one of its factors. The student correctly factors + 5x â€“ 24 into (x + 8)(x â€“ 3) and identifies (x â€“ 3) as the width of the rectangle.

Questions Eliciting Thinking How would you have found the width if (x + 8) was not a factor of + 5x â€“ 24? 
Instructional Implications Challenge the student to write an expression for the width of the rectangle if the given trinomial represented the perimeter of the rectangle.
Provide additional opportunities to rewrite factorable polynomials in factored form. Challenge the student to factor polynomials that are quadratic in form (e.g., + â€“ 24 or â€“ 16). 