Getting Started 
Misconception/Error The student is unable to write expressions that represent the length and width of the pool region in terms of the border width. 
Examples of Student Work at this Level The student represents:
 The length as 48 and the width as 24 and writes the equation A = 48(24).
 The length as 48 + and the width as 24 + .
 The length as 48 + w and the width as 24 + w.
 The length as 48 + and the width as 24.

Questions Eliciting Thinking What are you being asked to do? What variables are given to you in the problem? What does each variable represent?
Can you explain how you determined your expressions for the length and width of the pool region? Can you show me how these expressions relate to the diagram?
What is the difference between 2w and ? 
Instructional Implications Review any vocabulary in the problem that may be unfamiliar to the student (e.g., uniform border). Model the process of writing the equation by first identifying the relevant area formula (A = lw). Then guide the student to use the defined variable (w, the width of the border) to write expressions for length and width. Have the student use a highlighter to trace the length and width of the pool area on the diagram. Emphasize that the width of the border serves as part of the total length in two locations, and the total length is the sum of the lengths of its component parts. Ask the student to rewrite an expression for the total length. Have the student also revise his or her expression for the width of the pool region. Next show the student how to substitute the expressions into the formula to develop the equation. Provide additional problems in geometric contexts in which the student must choose an appropriate formula to guide the structure of an equation, write expressions for unknown parameters, and substitute them into the formula to develop the equation.
Consider using the MFAS tasks Tech Repairs (ACED.1.2) or Tee It Up (ACED.1.2). 
Moving Forward 
Misconception/Error The student is unable to use expressions for the length and width to write an equation that represents the area of the pool region. 
Examples of Student Work at this Level The student represents the length of the pool region as 2w + 48 and the width of the pool region as 2w + 24. However, the student is unable to use these expressions to write an equation of the form Area = length Â width. For example, the student writes an expression or equation such as:
 A = (48 + 2w) + (24 + 2w) or A = 2(48 + 2w) + 2(24 + 2w)
 48 + 2w Â· 24 + 2w

Questions Eliciting Thinking What is the formula for the area of a rectangle? Did you use this formula to write the equation?
What is the difference between 48 + 2w Â· 24 + 2w and (48 + 2w)(24 + 2w)? Do they mean the same thing? 
Instructional Implications Model the process of writing the equation by first identifying the relevant area formula (A = lw). Then guide the student to substitute the expressions written for length and width into the formula to develop the equation. Provide additional problems in geometric contexts in which the student must choose an appropriate formula to guide the structure of an equation, write expressions for unknown parameters, and substitute them into the formula to develop the equation.
Give the student additional opportunities to write equations for increasingly more complex situations. Directly relate components of the equation to features of the description of the problem and the relationship among the variables. 
Almost There 
Misconception/Error The student writes an expression rather than an equation or makes an error rewriting a correctly written equation. 
Examples of Student Work at this Level The student represents the length of the pool region as 2w + 48 and the width of the pool region as 2w + 24 but:
 Writes an expression such as (48 + 2w) (24 + 2w) instead of an equation.
 Initially writes the equation correctly as A = (48 + 2w) (24 + 2w) but attempts to rewrite it in standard form and makes an error.

Questions Eliciting Thinking What is the difference between an expression and an equation? Did you write an equation?
How did you multiply the two binomial expressions?
It appears you made an error when you rewrote the expression for area in standard form. Can you find your error? 
Instructional Implications Provide feedback to the student regarding any errors made and allow the student to revise his or her work. If needed, review how to multiply two binomials.
Give the student additional opportunities to write equations for increasingly more complex situations. Directly relate components of the equation to features of the description of the problem and the relationship among the variables. 
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 represents the length of the pool region as 2w + 48 and the width of the pool region as 2w + 24 and writes the equation as A = (48 + 2w) (24 + 2w) orÂ

Questions Eliciting Thinking Are there any restrictions on the values of A and w?
What would the area be if the border were 3 feet wide?
How could you find the area of just the concrete border?
What type of equation did you write? Could you write the equation in another form? If you knew the total area of the pool and border, which form of the equation would make it easier to solve for w? 
Instructional Implications Ask the student to solve the equation and identify the viable solution(s).
Introduce the student to the concept of constraints on variables that are often a component of many real world models. Encourage the student to represent constraints with inequalities.
Consider using the MFAS task Quilts (ACED.1.1) which requires students to write and solve an equation for a similar type problem. 