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THIS RESOURCE IS ONLY AVAILABLE TO LOGGED IN USERS. PLEASE LOGIN AND TRY AGAIN. WE APOLOGIZE BUT THIS RESOURCE IS NOT AVAILABLE TO YOU. PLEASE READ BELOW FOR MORE INFORMATION. Resource ID#: 56664Primary Type: Formative Assessment

Students are given a graph of an exponential function and are asked to calculate and compare the average rate of change over two different intervals of time.

This task can be implemented individually, with small groups, or with the whole class.

The teacher asks the student to complete the problems on the Air Cannon worksheet.

The teacher asks follow-up questions, as needed.

TASK RUBRIC

Getting Started

Misconception/Error

The student does not understand the concept of average rate of change and how it is calculated.

Examples of Student Work at this Level

The student is unable to determine the one-second interval during which the rate of change is the greatest. In addition, the student is unable to calculate the average rates of change over the specified intervals. Instead, the student:

Visually inspects the graph and decides the rate of change is the greatest for the interval during which the graph is the â€ścurviest.â€ť

Divides the y-coordinates of the endpoints of intervals by their x-coordinates to calculate average rate of change.

Questions Eliciting Thinking

What is average rate of change? How does the average rate of change differ from the constant rate of change of a linear function?

What do you look for in a nonlinear graph to determine rate of change over an interval?

How do you calculate the rate of change of a linear function?

How do you calculate average rate of change of a non-linear function?

Can you identify the endpoints on the graph for the given intervals? Can you use these points to find the average rate of change for each interval?

Instructional Implications

Provide additional instruction on the concept of rate of change. Initially, consider linear relationships and relate the rate of change to the slope of the line that models a linear relationship between two variables. Ask the student to calculate the rate of change of a linear function using several different ordered pairs and guide the student to observe that the rate of change (like the slope) of a linear relationship is the same regardless of the ordered pairs used to calculate it. Remind the student that a defining attribute of linear relationships is that the rate of change is constant. Emphasize that the average rate of change over any interval will always be the same as the rate of change (or slope of the line) that represents the graph of the relationship.

Next, introduce the student to the concept of average rate of change in the context of nonlinear relationships. Begin with a relatively simple relationship such as . Ask the student to determine the change in y for several consecutive one unit intervals of x and to compare them. Relate the different rates of change in y to the steepness of the graph. Provide instruction on calculating the average rate of change over larger intervals.

Provide the student with a nonlinear graph that models the relationship between two variables. Ask the student to calculate the average rate of change over several different intervals. Ensure the student can identify the points on the graph represented by the endpoints of each interval. Have the student draw secant lines that contain the endpoints of the intervals. Relate the average rate of change calculation to the calculation of the slope of the secant lines.

Provide additional opportunities to calculate and interpret average rate of change over specified intervals for both linear and non-linear functions.

Moving Forward

Misconception/Error

The student demonstrates some understanding of the concept of average rate of change but makes major errors in calculating it.

Examples of Student Work at this Level

The student:

Calculates the reciprocals of the average rates of change.

Uses the wrong coordinates in the calculations.

Makes repeated calculation errors.

Disregards the scales of the axes and calculates the rates of change as if each axis were scaled by one.

Questions Eliciting Thinking

How is rate of change calculated?

Can you explain how you found the rate of change for these intervals?

How is calculating the average rate of change like calculating the slope of a line?

Instructional Implications

Provide the student with the graph of another nonlinear function. Describe average rate of change as a change in y over the corresponding change in x. Relate the average rate of change calculation to the calculation of the slope of a secant line that contains the endpoints of an interval. Ask the student to calculate the average rates of change for several intervals by calculating the slopes of the secant lines. Guide the student to observe any differences in the slopes of the secant lines.

Provide additional opportunities to calculate and interpret average rates of change over specified intervals for both linear and non-linear functions.

Almost There

Misconception/Error

The student makes a minor mathematical error or a minor error of omission.

Examples of Student Work at this Level

The student identifies the interval from 0 â€“ 1 second as the one-second interval with the greatest rate of change. However, the student makes a minor error in some aspect of his or her work. The student:

Does not indicate that the rates of change are negative.

Does not realize the coordinate (5, 17) was given in question #3 and uses an estimated y-coordinate to calculate the average rate of change.

Writes an unnecessary but incorrect statement in an explanation.

Makes a minor error in one of his or her calculations.

Does not provide a complete explanation for why the interval from 0 -1 second is the interval with the greatest rate of change.

Questions Eliciting Thinking

Is -110 the same as ?

I think you made a small mistake in your work. Can you try to find and correct it?

You described the rate of change of the interval from 0 â€“ 1 second but how did you know this was greater than the rates of change of the other intervals?

Instructional Implications

Provide the student with a few examples of common errors made when finding average rates of change and have him or her identify and correct those errors.

Provide additional opportunities to calculate and interpret average rate of change over specified intervals for both linear and nonlinear functions.

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:

Indicates that the greatest one-second rate of change occurs during the interval from 0 â€“ 1 seconds because this is the interval in which the graph is decreasing the most.

Calculates a rate of change of -50 fps for question #2.

Calculates a rate of change of -16.5 fps for question #3.

Questions Eliciting Thinking

Why are the average rates of change you calculated negative? What does that mean in the context of this problem?

How do the rates of change that you calculated for this function differ from the rates of change for two different intervals of a linear function?

Instructional Implications

Present the student with a similar problem that involves an air cannon with a velocity that may be represented by the equation . Have the student find the average rate of change from x = 0 to x = 1. Compare the average rate of change for this interval for the first souvenir and the second souvenir.

Ask the student to draw secant lines through the endpoints of the two intervals on the graph. Relate finding average rate of change over these intervals to finding the slope of the secant lines that contain the endpoints of the intervals. Guide the student to observe how the slopes of the secant lines relate to the steepness of the graph at each of the intervals.

Have the student calculate the average rate of change for intervals with the same left endpoint but of decreasing length (e.g. , then , then ) for the function . Have the student draw secant lines through the endpoints of each interval. Encourage the student to observe how the secant lines get progressively closer to the line tangent to the graph at (1, 1). If available, use a graphing utility to illustrate this concept.

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