Getting Started 
Misconception/Error The student is unable to identify the slope from either the graph or the equation. 
Examples of Student Work at this Level For Jackâ€™s equation the student:
 Identifies the yintercept as the slope.
 Identifies the slope as five rather than .
 Writes the slope as an ordered pair (5, 4).
For Jillâ€™s graph the student:
 Describes the line rather than identifying the slope (e.g., â€śstraightâ€ť or â€śincreasingâ€ť or â€śpositiveâ€ť).
 Reverses the ratio, giving Â rather than Â as the slope.
 Lists the slope as an ordered pair (5, 4).
 Estimates the slope using an approximation of a point that does not have integer coordinates, e.g., using (4, 3) and gives a slope of .

Questions Eliciting Thinking How did you determine that the slope is five? What do each of the numbers and variables mean in Jackâ€™s equation?
I agree that the line is increasing, but can you tell by how much the price changes for each gallon?
Is there a difference between Â and ? How would you determine which describes the slope of this line?
What is the difference between a slope and an ordered pair? 
Instructional Implications Provide explicit instruction on slope. Include both how to use the graphed line to find the slope (by counting the â€śriseâ€ť and the â€śrunâ€ť) and how to identify the slope from the equation of a proportional relationship. Point out that the slope of the line is the same as the rate of change (change in y over the change in x). Graph a line with a slope of Â and a line with a slope of Â on the same set of axes so the student can see the difference. Make explicit the difference in the interpretations of a slope of Â and a slope of . Encourage the student to describe the slope in context. 
Moving Forward 
Misconception/Error The student is unable to explain the meaning of slope within the context of the problem. 
Examples of Student Work at this Level When describing the meaning of slope in context, the student:
 Says, â€śIt is increasing,â€ť rather than explaining the meaning of the slope.
 Explains in general terms that as the volume increases, the price also increases.
 Explains a method to find the slope (e.g., â€śrise over runâ€ť or â€ścount up then overâ€ť).
 Reverses the meaning of the slope, explaining for Jackâ€™s equation, â€śGallons go up by five and price goes up by four.â€ťÂ
 Misinterprets the meaning of the slope, explaining for Jill, â€śEvery pail costs $4 and holds five gallons.â€ť
 Gives numbers without units (e.g., 4 to 5).

Questions Eliciting Thinking I agree that the graphed line is increasing but specifically, what is increasing? How does that relate to the units for slope?
You described how to find the slope of the graphed line. Can you use the labels on the graph to assign units to the slope and describe what it means?
Can you explain how you know it would cost $4 for a fivegallon pail? Do all sizes cost the same? How could we determine the cost for a different size of pail? How can you reword your statement to account for all sizes? 
Instructional Implications Have the student assign units of measure to each part of the slope ratio. Guide the student to explain the meaning of slope as an amount of the dependent variable (e.g., $5.00) associated with a corresponding amount of the independent variable (e.g., 4 gallons). Ask the student to convert the slope to a unit rate including the units of measure (e.g., $1.25 for every one gallon). Then guide the student to interpret the slope in terms of the context of the problem. Model explaining, â€śEach gallon of water costs $1.25.â€ť
Consider using other MFAS tasks such as Interpreting Slope and Proportional Paint. 
Almost There 
Misconception/Error The student is unable to compare the slopes within the context of the problem. 
Examples of Student Work at this Level When comparing the slopes of the graph, the student:
 Compares the graphs, rather than the slopes of the graphs, explaining, â€śThey both increaseâ€ť or â€śJackâ€™s increases faster.â€ť
 Compares the numbers, rather than the slopes in context, explaining, â€śThey are both fractionsâ€ť or â€śThey are reciprocalsâ€ť or gives Â without any contextual reference.
 Gives a comparison of one of the variables, rather than comparing the ratio of one value â€śperâ€ť the other, explaining, â€śJack sold more pailsâ€ť or â€śJill made less money.â€ť
 Only explains how they are alike, rather than different, explaining, â€śThey both are linear, meaning the rate of change is constant.â€ť
 Makes a comparison statement that is unclear or misleading.

Questions Eliciting Thinking You described the slopes as â€śincreasing;â€ť what would it mean for the slope to increase? How is that different from the values on the graph increasing?
You are correct that the slopes are fractions, but within the given context, how do those fractions compare?
Does the graph or equation tell who made the most money or who sold more pails? Is there any other information you would need to make those decisions? 
Instructional Implications Have the student make a table of values for each of the relationships using the same values of w. Then have the student compare the corresponding values of p and draw conclusions based on the patterns observed.
Have the student change each ratio (slope) to a unit rate and compare the unit rate to the slope in the context of the problem.
Provide additional opportunities to find and compare the slopes of lines that represent proportional relationships from tables of values, equations, and graphs. 
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 identifies and interprets the slope in each representation. The student says that the slope of the graph of Jackâ€™s equation is Â which means that Jack charges $5 for every 4 gallons of water the pail holds, or $1.25 per gallon and the slope of Jillâ€™s graph is Â which means that Jill charges $4 for every 5 gallons of water the pail holds or $0.80 per gallon. When comparing the slopes, the student says:
 If we graphed Jackâ€™s equation, the slope would be greater than the slope of Jillâ€™s graph (or Jillâ€™s slope is smaller than Jackâ€™s).
 Jack charges more per gallon for his pails (or Jill charges less per gallon for her pails).
 You will have to pay more for the same size pail if you buy it from Jack (or youâ€™ll have to pay less if you buy from Jill).
 Jack has a bigger slope, so his is more expensive.

Questions Eliciting Thinking What does the point (10, 8) on Jillâ€™s graph represent in the context of the problem?
Suppose you graphed Jackâ€™s equation. What would the graphs have in common?
Do Jackâ€™s and Jillâ€™s graphs intersect? If so, where?
Is there a reason why Jillâ€™s graph is only drawn in quadrant one? 
Instructional Implications If the student did not already calculate a price per gallon (unit rate) for each representation, have the student do so and describe how the unit rate and slope are related.
Ask the student to compare linear graphs and equations that are not proportional. 