Learning Cycle for Quadratic Unit
Goal: Standard F.IF.6: Calculate and interpret the average rate of change of a
function (presented symbolically or as a table) over a specified interval. Estimate
the rate of change from a graph.
Develop Understanding Cycle
Develop Understanding
Surface Ideas
 Correlate rate of change to the slope of a line.
 Rates of change are dependent on intervals and specific functions.
 Rate of change can be interpreted as the y-unit divided x-unit (feet per
second).
 Different functions can have the same average rate of change.
Create Representations
 Show that slopes of a function change over different intervals.
 The slope of the line (speed) is directly related to the average rate of
change.
 Show
Task #1
Students will need access to calculator and graph paper.
Launch: Introduce the task as a challenge. Two people begin a race and one doesn’t
start until he is at his top speed. The second runner is faster and starts from a
standing position. Briefly discuss how the speed of the second person changes.
Explore: Let the groups work on the finding which car is fastest and what the
average speeds are for the three cars over each interval. Encourage students to use
multiple ways of analyzing and explaining their conclusions. This is a good time to
guide students to make the connection between slope and average speed. Leave the
vocabulary average rate of change for the next task unless a student brings it up.
1. Students will tend to confuse position with velocity when comparing the cars.
2. They may forget how to find the speed of the cars. Ask questions to help
them understand how speed is found. For example, what does miles per
hour mean?
3. See number 2.
Discussion:
1. Make sure students recognize the speeds aren’t constant for car two and car
three. Hopefully students will notice at various times each car is faster than
the other two.
2. The slope of the line over a specific interval is a representation of the average
speed of that car.
3. See number 2.
Task #1:
Given several tables, students will use the data to compare and contrast the speed of
different vehicles at different times.
Car 1
time (s)
1
2
3
4
5
Distance
(ft)
4
7
10
13
16
Car 2
Time (s) Distance
(ft)
1
2
2
5
3
10
4
17
5
26
Car 3
Time (s) Distance
(ft)
1
3
2
5
3
9
4
17
5
33
1. Using the above tables, compare the three cars and their positions after t seconds.
Which car is traveling the fastest? Justify your answer.
2. What are the average speeds for all three cars during the interval [0,2]?
3. What are the average speeds for all three cars during the interval [3,5]?
Solidify Understanding Cycle
Examine and extend:
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Velocity can be constant or it can change, depending on the function
You can find average rate of change of any of the functions
Analyze the meaning maximum and minimum of quadratics
Vertical and horizontal distances
Concept of time interval
Vocabulary terms: average rate of change, interval, secant line, ∆
Task #2:
The path of a pumpkin “chucked” into the air can be modeled by the function
-ℎ(𝑡) = −16𝑡 2 + 96𝑡 + 10 where h is the height of the pumpkin in feet and t is the
time in seconds after being “chucked”. Find and interpret the speed of the function
over the following intervals:
a) [0,2]
b) [3,6]
c) [2,4]
Task #3: Given the graph below, complete the table and create a function.
x
f(x)
f(x)=__________________________________________
Using your function, find the average rate of change (ARC) of the given interval:
[0,1.5].
Interpret your answer.
Find an interval from where the person stops decent to where the bungee has no
tension.
Interpret your answer (we are looking for [1.5,3]
Find an interval that will produce an ARC of 0. Why would this occur?