Getting Started

The student cannot effectively describe the relationship between two quantities related by a nonlinear function.

The student describes only one or two features of the graph such as it is (or is not) a function, it is nonlinear, or it is increasing. The student does not describe the relationship between the two quantities displayed in the graph.

What two quantities are modeled by this graph?

Can you describe the relationship the graph is modeling?

What kind of function is this? What do you know about this type of function?

Review the concept of a nonlinear function, providing examples that include both equations and graphs. Emphasize that the rate of change of nonlinearly related quantities is not constant and is often represented by a curve. Model describing the relationship between two quantities. For example, explain that over the course of a number of days, the number of bacteria increases slowly at first, then increases very quickly and, finally, levels off. Guide the student to consider whether the graph has any intercepts that can be represented as an ordered pair that include units of measure. Then have the student consider the context of the graph and explain the meaning of any intercepts.

Provide additional examples of functions, both linear and nonlinear, in context and ask the student to analyze and describe the relationship between the quantities. 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.

Making Progress

The student’s description of the relationship between the two quantities is incomplete or contains an incorrect feature.

The student does not clearly identify all phases of bacterial growth evident in the graph.

The student describes an aspect of the relationship not evident in the graph (e.g., the relationship is cyclic).

Do the bacteria increase at the same rate throughout the entire graph?

How do you know the growth took place over a couple of days?

Why does the graph stay the same? What feature of the graph indicates that the number of bacteria level off or stay fairly constant?

Where in the graph does it show that the process repeats? What would that look like? Does it happen?

Provide feedback to the student with regard to both the correct and incorrect aspects of his or her description. Guide the student to observe that the rate of change varies throughout the graph and that the graph can reasonably be separated into three phases. Ask the student to complete the description of the graph or revise the incorrect portions.

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.

Got It

The student provides complete and correct responses to all components of the task.

The student describes the relationship between the number of bacteria and time as nonlinear because the rate of change varies. The description includes the three phases evident in the graph: at first, the number of bacteria is steady or very slowly increasing; the next phase shows a very fast increase in the number of bacteria; finally, the increase slows and the number of bacteria remains steady.

Would it make sense to extend this graph into another quadrant? Why or why not?

Do you know what this kind of function is called?

Consider implementing the MFAS task Jet Fuel, which allows the student to investigate a linear function and describe the relationship between the quantities.

Consider implementing the MFAS tasks in this topic to help students interpret functions within a context.