Getting Started 
Misconception/Error The student is unable to represent the constraint on the domain with an inequality. 
Examples of Student Work at this Level The student:
 Writes an inequality that does not correctly represent the constraint.
 Calculates p(500).

Questions Eliciting Thinking What does s represent in this problem? Can s stand for a negative number? Can s be zero?
What does the problem say about the number of footballs the committee bought? Can the committee sell more footballs than they bought?
What kinds of numbers can s represent? What is the least that sÂ can be? What is the most that s can be?
What mathematical symbol is used to show â€śless than?' What symbol is used to show â€śless than or equal to?' 
Instructional Implications Guide the student to identify precisely the values that s represents and initially write these values using set notation, for example, as {0, 1, 2, 3, 4, â€¦, 500}. Then, model how to represent the constraint in this problem using an inequality. Make explicit that the appropriate variable to use is s as given in the problem.
Provide the student with verbal descriptions expressing constraints such as:
 Textbooks can cost no more than $120 per student per semester.
 The rate of pay is either $10 per hour or $12 per hour.
 At least 100 tickets must be sold to offset the cost of production.
 The length of a side or the diagonal of the square must be strictly between two units and three units.
Ask the student to represent eachÂ constraint with an equation or inequality. Encourage the student to also consider the kinds of values that the quantities in each problem context represent (e.g., whole numbers, integers, rationals, or reals).

Moving Forward 
Misconception/Error The student represents the constraint on the domain with an inequality, but is unable to determine the viability of solutions. 
Examples of Student Work at this Level The student represents the constraint on the domain as Â orÂ .Â The student may use a variable other than the given one, s. However, the student does not understand how to determine the viability of solutions. The student:
 Makes a decision without considering the profit equation.
 Makes a decision without showing any supporting work.

Questions Eliciting Thinking What kinds of numbers does s represent? Can s = 2.5?
How would you calculate the profit if 200 footballs were sold?
How do you think you could find the maximum profit possible?
Does the 1500 in the second question represent a profit or a number of footballs? If you substituted 1500 into the equation for p and solved, what do you think the solution indicates? 
Instructional Implications Model for the student how to determine the viability of a profit that is exactly 1500 in the context of this problem. Clarify for the student the significance of the solution of the equation 1500 = 5s â€“ 128 and what it represents in the context of this problem. Discuss with the student that s must be a whole number. After solving the equation, ask the student to find the profits associated with the whole number values of s that are just above and just below s = 325.6 (i.e.,Â s = 325 and s = 326). Discuss that a profit very close to 1500 can be achieved, but not one that is exactly 1500. Ask the student to complete the third problem and provide feedback.
If the student represented the constraint using a variable other than s, explain why s is the appropriate variable to use and allow the student to correct his or her work. Also, guide the student to describe the lower limit on s if this was not included in his or her answer.
Provide additional opportunities for the student to determine the viability of solutions in realworld and mathematical problems. Encourage the student to always consider the kinds of values that the quantities in each problem context represent (e.g., whole numbers, integers, rationals, or reals). 
Almost There 
Misconception/Error The student does not correctly interpret the results of calculations to determine the viability of solutions. 
Examples of Student Work at this Level The student writes and solves appropriate equations to determine the viability of the given solutions. However, the student:
 Concludes that a profit of $1500 is possible if 325.6 footballs are sold or since 325.6 is less than 500.
 Concludes that profit of at least $2400 is not possible since a fraction of a football cannot be sold.
 Makes a computational error that results in drawing the wrong conclusion.
 Does not understand the meaning of the variable s and misinterprets the results.

Questions Eliciting Thinking Are the words exactly equal (as used in #2) and at least (as used in #3) important?Â
Can you calculate the profit that would be realized if 326 footballs are sold? Is this exactly $1500?
Can you sell sixtenths of a football? Does that make sense?
I think you may have made a mistake in your work for the second (or third) question. Can you check your work again and see if you can find it? 
Instructional Implications Guide the student to discover and correct any errors in his or her work. Discuss with the student the significance of the solutions of the equations written for the second and third questions and what they represent in the context of this problem. Encourage the student to always consider the kinds of values that the quantities in a problem context represent (e.g., whole number, integer, rational, or real).
Ask the student to explain the difference between the phrases â€śat least $1500,â€ť â€śequal to $1500,â€ť and â€śat most $1500â€ť and to represent each with an inequality. Ask the student to determine and describe the viability of the possible values of s associated with each.
If the student represented the constraint using a variable other than s, explain why s is the appropriate variable to use and allow the student to correct his or her work. Also, guide the student to describe the lower limit on s if this was not included in his or her answer.
Provide additional opportunities for the student to determine the viability of solutions in realworld and mathematical problems. Encourage the student to always consider the kinds of values that the quantities in each problem context represent (e.g., whole numbers, integers, rationals, or reals). 
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 constraint on the domain as Â orÂ .Â The student writes, solves, and interprets equations necessary to determine the viability of the given solutions. The student says that a profit of exactly $1500 is not possible because it would require selling exactly 325.6 footballs which is not possible since a fraction of a football cannot be sold. The student says that a profit of at least $2400 is not possible since it would require selling more than 500 footballs.

Questions Eliciting Thinking What do you think the 128 represents in this equation? Why do you suppose it is being subtracted from 5s?
How many footballs would have to be sold to make a profit?
Is it possible to break even? 
Instructional Implications Ask the student to determine the set of profits associated with the domain of the profit function (i.e.,Â ) and to represent it with an inequality. Ask the student to explain the meaning of both the rate of change and initial value in the equation p = 5s â€“ 128 in the context of this problem.
Discuss the difference between continuous and discrete variables. Provide the student with examples of each. Encourage the student to generate his or her own realworld examples of continuous and discrete variables. 