Getting Started 
Misconception/Error The student cannot write a meaningful inequality to represent the given constraints. 
Examples of Student Work at this Level The student:
 Uses a direct numerical approach to solve rather than writing an inequality.
 Writes an expression or equation instead of an inequality.
 Does not use a variable in the inequality statement, writing Â + 32 = 109.3.
 Uses the given value of 109.3 twice or in the wrong place in the inequality, writing (109.3) + 32 = 109.3.
 Reverses the positions of the Fahrenheit temperature and Celsius variable, writing (109.3) + 32 = C.

Questions Eliciting Thinking What does it mean to write an inequality?
Is the problem going to have one or more than one solution? What symbol can you use to show this?
Do you know the Celsius temperature or do you know the Fahrenheit temperature? Which are you solving for? What variable should you use and where does it go? 
Instructional Implications Work with the student on modeling relationships among quantities with inequalities. Begin with situations that can be modeled by onestep inequalities. Then progress to situations leading to twostep inequalities. Ask the student to explicitly describe the meaning of any variables used in the inequality. Emphasize the relationship between algebraic expressions and the quantities they represent in the context of the situations in which they arise. For example, if x represents the number of degrees Celsius, then xÂ + 32 represents the degrees Fahrenheit.
Use the studentâ€™s numerical work, if possible, as a starting point for the development of an inequality. For example, if the student completed a series of computations, such as 109.3 â€“ 32 = 141.3; 141.3(5) = 706.5; 706.5 Ă· 9 = 78.5, guide the student to replace the final answer, 78.5, with a variable and work backwards to develop the inequality.
If necessary, review solving onestep and twostep equations and inequalities. Provide additional opportunities to solve word problems by writing, solving, and graphing inequalities involving rational numbers. 
Moving Forward 
Misconception/Error The student is unable to solve the inequality. 
Examples of Student Work at this Level The student correctly writes an inequality to represent the problem situation but is unable to correctly solve the inequality. The student:
 Changes the inequality to an equation in some step of the solution process.
 Makes one or more errors, including errors with rational number or integer operations.
 â€śDropsâ€ť the variable C rather than solving for it.
 Uses an incorrect method to solve the inequality (e.g., dividing only some of the terms by Â in the first step).
 Uses a â€śguessandcheckâ€ť or a workbackwards numerical approach to solve the inequality, rather than solving the inequality algebraically.

Questions Eliciting Thinking What does the inequality symbol mean? Does an equal sign have the same meaning?
What steps did you take to solve this? Are they the same steps you would take to solve a twostep equation?
How should you add/subtract decimals? How do you multiply/divide decimals? 
Instructional Implications Provide direct feedback on any errors that the student might have made and allow the student to correct them. Review solving onestep and twostep equations and inequalities. Provide additional opportunities to solve word problems by writing, solving, and graphing inequalities involving rational numbers. Remind the student to use substitution to check solutions in the original inequality.
Review operations with rational numbers and provide frequent opportunities to add, subtract, multiply, and divide rational numbers in a variety of forms. Explain that dividing by a fraction is equivalent to multiplying by the reciprocal of the fraction. Using the inequality in this task, demonstrate how this idea can be used in the process of solving. Suggest using two steps to multiply both sides of CÂ 141.3 by (e.g., by first multiplying both sides of the inequality by five, then dividing each side of the inequality by nine).
Consider implementing the MFAS tasks Rational Addition and Subtraction (7.NS.1.1) or Applying Rational Number Properties (7.NS.1.2) to provide additional review for students struggling with rational number operations. 
Making Progress 
Misconception/Error The student is unable to correctly graph the inequality. 
Examples of Student Work at this Level The student correctly writes and solves the inequality but makes an error when graphing the solution by:
 Scaling the number line improperly.
 Giving the positive and negative numbers different scales.
 Using an open rather than a closed circle (or uses no circle at all).
 Graphing x 109.3 rather than x 78.5.
 Not shading the number line to indicate all possible solution values.
 Not giving the number line a numerical scale at all.

Questions Eliciting Thinking On the number line, do the negative numbers get larger as you move to the right or to the left?
What does an open or closed circle indicate on a graph? How do you choose which one to use?
How do you decide which direction to shade? What does the arrow mean on the graph of an inequality? Is there a way to use a specific value to check if your shading is correct?
What does the inequality symbol mean? Does it include numbers that are greater than or less than the given number? Does it also include the exact value of the given number? How can you show this on the graph? 
Instructional Implications Provide instruction on graphing inequalities on the number line. Be sure the student understands the conventions in graphing inequalities and their meaning (e.g., the use of an open versus closed dot, the direction of shading, and the arrow). If necessary, provide instruction on the meaning of the inequality symbols. Have the student graph a variety of inequalities (including some with the variable written to the right of the inequality symbol) and write inequalities to match given graphs.
Review what it means for a number to be a solution of an inequality. Give an example of an inequality and provide a set of numbers, some of which are not solutions. Demonstrate how to use substitution to test numbers to determine whether or not they are solutions. Guide the student to graph â€śall solutionsâ€ť by shading those in the solution set that make the inequality true. 
Almost There 
Misconception/Error The student makes minor errors in writing, solving, or graphing the inequality. 
Examples of Student Work at this Level The student makes one minor error but all other work is correct given the error. For example, the student:
 Makes a calculation error in some step of the problem.
 Changes Â to a decimal incorrectly.
 Drops the negative symbol during some step of the problem.
 Misinterprets the answer as having only one solution, writing â€śthe temperature must be 78.5.â€ť
 Reverses the inequality symbol in the initial equation.
 Partially solves the inequality using a numerical approach before writing it.

Questions Eliciting Thinking Can you review your work and look for any errors with your calculations?
What does the inequality mean? How many solutions does it represent?
Can you write the inequality without first solving for C? 
Instructional Implications Provide direct feedback to the student concerning any error made and allow the student to revise his or her work accordingly. If necessary, ask the student to use the graph to find an example of a solution and to substitute this solution into the original inequality to determine if it satisfies the inequality. Provide additional opportunities to solve inequalities with rational coefficients and remind the student to check solutions. 
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 writes the inequality CÂ + 32 109.3, solves it to get CÂ 78.5Â° , scales the number line appropriately, and graphs the solutions using a closed dot at 78.5 and shading to the left.

Questions Eliciting Thinking What does the solution mean?
Is there only one solution? If not, what are some examples of other solutions?
Is 78 a solution? Is 79 a solution? Is 78.5 a solution? 
Instructional Implications Introduce the student to compound inequalities. Give the student a statement such as, â€śThe various routes that Kelvin can drive to work range from 8.2 miles to 9.7 miles in length,â€ť and ask the student to represent the lengths, m, as a compound inequality (e.g., in the form 8.2 m 9.7 or in the form m 8.2 and m 9.7). Guide the student to graph the inequality on the number line and verbally describe the values that satisfy it. Gradually increase the complexity of the problem so the student must first write and solve a one or twostep inequality for a given context. 