Getting Started 
Misconception/Error The student is unable to write an equation that models the problem. 
Examples of Student Work at this Level The student:
 Writes an expression that does not model the problem.
 Uses a guess and check approach and incorrectly states that the border is one foot wide because five times six equals 30.

Questions Eliciting Thinking What is this word problem about? What are you being asked to find?
How did you start to solve this problem? Explain to me what you did.
What is unknown in this problem? How can you represent this unknown quantity?
How would you represent the length and the width with expressions? 
Instructional Implications Provide instruction on writing onestep linear equations that model simple problem situations. Gradually increase the complexity of the problems to those modeled by twostep linear and quadratic equations. Guide the student to explicitly identify the unknown and to create a variable to represent it. Ask the student to justify his or her equation by relating each term and operation to a specific feature of the problem. If necessary, provide instruction on solving quadratic equations, and encourage the student to always assess the reasonableness of solutions. Guide the student to use the solutions of the equation to explicitly answer any questions asked in the problem.
Provide additional examples of representing two related quantities in terms of the same variable. Give the student additional opportunities to write representations of related quantities in terms of the same variable. 
Moving Forward 
Misconception/Error The student attempts to write an equation but makes significant errors. 
Examples of Student Work at this Level The student:
 Writes expressions for the length and width of the quilt but sets their product equal to zero [e.g., (2x + 4)(2x + 5) = 0].
 Represents the length and width incorrectly [e.g., (x + 5) and (x + 4)].

Questions Eliciting Thinking Can you explain the problem in your own words? What important information were you given in the problem?
What is the unknown in the problem? What is known in the problem? What variable did you use? What does your variable represent?
How did you decide what to include in your equation?
Are the width and the length of the border the same? How do you know?
Show me, in the diagram, the width of the quilt. Show me the width of the border. How would you write an expression for the total width of the quilt and border?
Show me, in the diagram, the length of the quilt. Show me the length of the border. How would you write an expression for the total length of the quilt and border? 
Instructional Implications Ask the student to justify his or her equation by relating each term and operation to a specific feature of the problem. Provide feedback and encourage the student to revise the equation as needed. Provide additional opportunities to write and solve equations that model problems. Encourage the student to selfassess by justifying each component of his or her equation and relating it to features of the original description of the problem. Encourage the student to always assess the reasonableness of solutions. Guide the student to use the solution of the equation to explicitly answer any questions asked in the problem.
Provide additional examples of representing two related quantities in terms of the same variable. Give the student additional opportunities to write representations of related quantities in terms of the same variable. 
Making Progress 
Misconception/Error The student correctly writes the equation but is unable to solve it. 
Examples of Student Work at this Level The student writes the equation 30 = (4 +2x)(5 + 2x), but:
 Cannot correctly use algebraic properties to solve the equation (e.g., does not multiply the binomials correctly, does not combine like terms correctly, or does not rewrite the equation in standard form).
 Stops and indicates that he or she cannot solve the equation.

Questions Eliciting Thinking What type of equation have you written? What methods have you learned to solve a quadratic equation?
How must the equation be written before you can start factoring? 
Instructional Implications Provide the student with additional instruction and practice in solving quadratic equations. Require the student to neatly show all work, so if an error occurs, it is easier to find. Provide feedback and encourage the student to revise his or her work as needed. Remind the student to always check the reasonableness of solutions in the context of the problem. Encourage the student to always go back to the problem and make sure the question asked was answered. 
Almost There 
Misconception/Error The student correctly writes the equation but makes minor errors solving the equation or does not answer the question being asked. 
Examples of Student Work at this Level The student writes the equation 30 = (4 +2x)(5 + 2x), but:
 Makes an error factoring.
 Solves the equation correctly but chooses the wrong solution.
 Solves the equation correctly but chooses both solutions.

Questions Eliciting Thinking There is a minor mistake in your work. Can you find it?
What was your first step in trying to solve the equation?
Did you check your factoring?
What were you being asked to find? Does your answer make sense? Why did you eliminate as an answer? 
Instructional Implications Provide specific feedback to the student concerning any errors made and allow the student to revise his or her work. Remind the student to always check the reasonableness of solutions in the context of the problem. Encourage the student to always go back to the problem and make sure the question asked was answered.
Pair the student with another Almost There student to compare equations and solutions. Ask the students to reconcile any differences in their work. 
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 assigns the variable x to represent the width of the border and writes the equation 30 = (4 +2x)(5 + 2x). The student solves the equation and determines that x = or x = 5. The student then states that the width of the border is ft. When asked, the student can explain that 5 is not a reasonable answer in the context of this problem because the width cannot have a negative value.

Questions Eliciting Thinking Can you determine the area of just the border?
If you did not know that the total area was 30 square feet, how would you write an expression for just the area of the border? 
Instructional Implications If the student solved the equation without initially factoring out the common factor from the terms, remind the student that, if there is a common factor, factoring it out first will make the resulting trinomial easier to factor.
Provide the student with more complex problem situations to model with equations and solve.
Consider using the MFAS task Follow Me (ACED.1.1). 