Getting Started 
Misconception/Error The student is unable to determine if a specific combination of values satisfies all constraints. 
Examples of Student Work at this Level The student:
 Provides an answer with no supporting work.
 Makes a decision based only on cost.
 Does not understand the results of calculations or how to determine if they satisfy the constraints.

Questions Eliciting Thinking How much money is spent on Food A? How much is spent on Food B?
Can you determine how much four servings of food A and three servings of food B will cost?
Can you determine how many grams of sugar are in four servings of food A and three servings of food B?
Can you determine how many grams of protein are in four servings of food A and three servings of food B?
What does â€śat most 30 grams of sugarâ€ť mean?
What does â€śat least 50 grams of proteinâ€ť mean? 
Instructional Implications Guide the student to calculate the cost of and the number of grams of sugar and protein in a meal that contains four servings of food A and three servings of food B. Help the student organize the information by asking the student to first determine the amounts of money, sugar, and protein for the given quantities of each food. Have the student write each component as an explicit product before performing the multiplication and then place each component in a table organized like the given table. Ask the student to find the total cost and the total number of grams of sugar and protein. Assist the student in understanding the constraints and determining if the meal will satisfy all of the constraints. Focus on the language used to describe the constraints: no more than, at most, and at least. Help the student understand the precise meaning of these phrases and how each is represented with an inequality symbol.
Next, using the table created to answer the first question as a guide, model how to represent each constraint with an inequality. Emphasize the similarity between the representations and the numerical example. If needed, provide additional review of the process of writing inequalities to represent the relationship among variable expressions and quantities.
Change some features of the problem (e.g., alter the values in the table and the constraints) and ask the student to complete the problem again. Provide additional opportunities to represent constraints with inequalities. 
Moving Forward 
Misconception/Error The student is unable to represent the constraints with inequalities. 
Examples of Student Work at this Level The student is able to determine that four servings of food A and three servings of food B will not satisfy the constraint that the lunch contains at most 30 grams of sugar. However, the student is unable to correctly write inequalities to represent the constraints. For example, the student writes the inequalities as:
 , ,
 Attempts to determine if another combination of foods A and B satisfy the constraints.
Note: The student may make a computational error when determining if the given numbers of servings of foods A and B satisfy the constraints. However, the student demonstrates that he or she understands how to make this determination.

Questions Eliciting Thinking What does â€śat most 30 grams of sugarâ€ť mean? What kind of a relationship (equality or inequality) can be used to express â€śat most 30 grams of sugar?'
What does â€śat least 50 grams of proteinâ€ť mean? What kind of a relationship (equality or inequality) can be used to express â€śat least 50 grams of protein?'
Are any of the requirements represented with an equation? Why or why not? 
Instructional Implications Ask the student to review the reasoning he or she used to answer the first question. Guide the student to write each component as an explicit product and then place each component in a table organized like the given table. Using this table as a guide, model how to represent each constraint with an inequality. Emphasize the similarity between the representations and the numerical example. Change some features of the problem (e.g., alter the values in the table and the constraints) and ask the student to complete the problem again.
Provide additional opportunities to represent constraints with equations, inequalities, and systems of equations and inequalities. Vary the language used when describing the constraints (e.g., no more than, at least, at most, fewer than, more than, greater than, less than, not less than, not greater than). 
Almost There 
Misconception/Error The studentâ€™s response contains a minor error. 
Examples of Student Work at this Level The student:
 Uses an equal sign in place of an inequality symbol when expressing a constraint.
 Reverses an inequality symbol in one of the constraints.
 Uses a strict inequality symbol (e.g., uses < in place of ).
 Makes a computational error when determining if the given numbers of servings of foods A and B satisfy the constraints.

Questions Eliciting Thinking How could you incorporate the meanings of â€śat leastâ€ť and â€śat most?'
What is the difference between â€ś=â€ť and â€śâ€ť? Which better represents â€śat least?' 
Instructional Implications Provide feedback to the student with regard to any errors made and allow the student to revise his or her response. Clarify the meaning of â€śat leastâ€ť and â€śat mostâ€ť and explain how to represent each relationship with an inequality symbol.
Provide additional opportunities to represent constraints with equations, inequalities, and systems of equations and inequalities. Vary the language used when describing the constraints (e.g., no more than, at least, at most, fewer than, more than, greater than, less than, not less than, not greater than). 
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 determines that it is not possible to satisfy all of the constraints with four servings of food A and three servings of food B due to an excess of sugar. The student shows work to support this conclusion. For example, the student writes:
 Total spent: (0.20)(4) + (0.40)(3) = $0.80 + $1.20 = $2.00
 Sugar amount: (7)(4) + (3)(3) = 37 grams (which is moreÂ than 30 grams)
 Protein amount: (6)(4) + (9)(3) = 51 grams (which is at least 50 grams)
The student represents the constraints as:
 20a + 40b 200 orÂ
 7a + 3b 30
 6a + 9b 50
 a 0, b 0

Questions Eliciting Thinking Can you identify a number of servings of food A and a number of servings of food B that will satisfy all of the constraints?
If the student omitted the nonnegativity constraints: Is it possible to have a negative number of servings? So what must be true of a and b? 
Instructional Implications Ask the student to graph the inequalities and identify a region of the plane that contains solutions of the system of inequalities. Have the student identify a solution and explain its meaning in context. 