Getting Started 
Misconception/Error The student is unable to sketch a graph that modelsl the relationship between two quantities. 
Examples of Student Work at this Level The studentâ€™s graph does not show variations that correspond to the different phases of bacterial growth over time. For example, the studentâ€™s graph displays:
 A separate bar for each of the four phases.
 A sketch of a line with no other information.
 Variations that do not correspond to the description.

Questions Eliciting Thinking What are the two quantities described in this problem?
Can you explain what each section of your graph represents? What parts of the description correspond to each section of your graph?
What is the meaning of exponential growth? How would that appear on a graph?
How would the graph look during periods of no growth? 
Instructional Implications Guide the student to identify and describe the two related quantities in the problem description and to label the axes accordingly. Then ask the student to analyze the problem description by identifying each phase. Guide the student to sketch segments of the graph that correspond to each phase paying careful attention to the length of each phase in hours.
Provide examples of completed graphs for the student to analyze as well as descriptions of the relationship between two quantities that the student can graph. Guide the student to address features of the graph such as intercepts, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums, and rate of change. Assist the student in learning and using mathematical terminology to describe these features. 
Moving Forward 
Misconception/Error The student misinterprets or omits some portion of the description when sketching a graph. 
Examples of Student Work at this Level Some portions of the studentâ€™s graph are correct, but others are incorrect or missing. For example, the student:
 Depicts a linear rather than exponential relationship for the second phase.
 Is missing the first constant section (lag phase) and begins at (1, 50).
 Labels the xaxis with the names of the phases rather than number of hours.
 Begins at zero rather than 50 or ends at 50 rather than zero.
 Shows an instantaneous increase and/or decrease of rate rather than exponential or gradual.

Questions Eliciting Thinking Can you explain what each section of your graph represents? What parts of the description correspond to each section of your graph?
What are the two quantities described in this problem?
What is the initial number of bacteria according to the description? How can you show that on your graph?
What does â€śexponentialâ€ť mean? How quickly will the bacteria grow? How can your scale be numbered so you have a possibility of recording those larger numbers?
What is described in the final phase? How can you show on your graph that all bacteria gradually die off over four hours?
How would you sketch a gradual decline? Would the line be vertical? 
Instructional Implications Provide feedback to the student with regard to both the correct and incorrect parts of his or her graph. Address any misconceptions the student might have about how the context relates to the graph (e.g., exponential growth is shown by a curve rather than a line; the final phase lasts four hours, so the graph decreases to zero at eight hours). Ask the student to complete the graph or revise the incorrect portions.
Provide additional descriptions of the relationship between two quantities that the student can graph. Guide the student to relate features of the graph such as intercepts, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums, and rate of change to particular aspects of the description. Assist the student in learning and using mathematical terminology to describe these features. 
Almost There 
Misconception/Error The student makes minor errors when labeling or scaling the graph. 
Examples of Student Work at this Level The student draws an accurate graph but:
 Scales an axis inappropriately and does not provide a title.
 Does not scale an axis.
 Uses unequal intervals on one or both axes.

Questions Eliciting Thinking Are your axes scaled correctly? What must be true of the intervals?
How will the reader know what relationship your graph portrays? Does your graph have a title?
How is each quantity measured? Did you include the unit of measure when you labeled the axes? 
Instructional Implications Ask the student to provide a title for the graph if it was omitted. Also, guide the student to label axes with both a term that describes the quantity represented as well as its unit of measure [e.g., label the xaxis as â€śTime (hours)â€ť and the yaxis as â€śNumber of Bacteriaâ€ť].
Provide the student with additional opportunities to draw graphs from verbal descriptions. Consider using MFAS task Graph the Ride. Also, provide opportunities to interpret a given graph by describing it verbally. Consider implementing MFAS tasks Jet Fuel and Population Trend. 
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 the two related quantities and labels the axes appropriately (e.g., Time or Hours on the xaxis and Number of Bacteria on the yaxis). The graph shows 50 bacteria at time zero remaining constant until hour one. The graph shows an exponential increase over hours two and three followed by a period of no growth from hour three to four. The graph decreases gradually from hour four until reaching zero at hour eight. 
Questions Eliciting Thinking What part(s) of the graph would change if the number of bacteria started at 10?
How would the graph change if the growth phase was geometric rather than exponential?
How would the graph change if each phase lasted for double the stated time? 
Instructional Implications Provide the student with additional opportunities to draw graphs from verbal descriptions. Consider using MFAS task Graph the Ride. Also, provide opportunities to interpret a given graph by describing it verbally. Consider implementing MFAS tasks Jet Fuel and Population Trend. 