Insights Into Algebra 1: Teaching for Learning
Quadratic Functions Lesson Plan 2: Bouncing Ball – Function Families
In this lesson, students explore quadratic functions by using a motion detector known as a Calculator Based Ranger (CBR) to examine the heights of the different bounces of a ball. Students will represent each bounce with a quadratic function of the form y = a (x – h)2 + k. This lesson plan is based on the activity Tremain Nelson uses in the video for Part II of this workshop.
Two 50-minute periods
Students will be able to:
- Generate a quadratic function that describes a given parabola by identifying the values of a, h, and k in vertex form.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, 2000
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers will need the following:
- Gridded flip-chart paper
- A set of “clean data” to give to students
- Calculator-Based Ranger (CBR) or other motion detector
Students will need the following:
- Notebook or journal
For each group of students, you will need:
- Program of “clean data“
Teachers Activities and Assignment
1. As a warm-up, allow students three to five minutes to complete the worksheet in which they graph the function y = – (x – 3)2 + 4 without using a calculator. They should also create an equation in vertex form for a graph that is given to them.
2.Bring students to the front of the room for a “table talk” session to discuss the solutions to the warm up activity. Several students should sit at the table with you and take notes while the other students gather around to participate in the review.
3.Discuss with the class the process for graphing the function y = – (x – 3)2 + 4.
4.Give students the parent function y = x2 as the basis for graphing quadratic functions in vertex form.
5.Conduct a brief discussion on how the parent function y = x2 could be used to help graph the function in vertex form: y = – (x – 3)2 + 4.
6.Select a student to plot the parent function y = x2 on a sheet of flip-chart paper.
7.Ask: “What effect does the value of a have on the graph of the function?” For the function y = – (x – 3)2 + 4, the negative sign in front of (x – 3)2 indicates that a = -1. Elicit from students that this value of a will cause the parabola to “flip” – that is, it will reflect over the x-axis and open downward. (Because the absolute value of a is 1, the function will have the same shape as y = x2.)
8.Ask: “What effect does the value of h have on the graph of the function?” In general, the value of h causes the parabola to shift right or left. Elicit from students that for the function y = -(x – 3)2 + 4, the parabola will shift 3 units to the right.
(Note: The vertex form of a quadratic equation is often stated as
y = a (x – h)2 + k. This implies that h is the x coordinate of the vertex. In Tremain Nelson’s class, the students look at the expression (x – h)2 and recognize that if it is written in this form, the graph shifts right h units. If the expression is written (x + h)2, then it shifts the graph h units to the left. Either explanation is appropriate for the classroom, and you may choose to use the explanation with which you are more comfortable or the one that coincides with your classroom textbook.)
9.Ask: “What effect does the value of k have on the graph of the function?” The value of k causes the parabola to shift up or down. A positive value of k causes the parabola to move up; a negative value of k causes the parabola to move down. For the function y = – (x – 3)2 + 4, the value of k = 4. Elicit from students that this will shift the parabola 4 units up.
10.Review the second problem from the warm up worksheet, which involves a graph. Students were to determine an equation in vertex form to match the graph.
11.Ask: “How can we determine the vertex form of this function?” Elicit from students that they should begin with the parent function y = x2, and then examine the graph to determine the values of h, k, and a.
12.Ask questions of students to have them identify the values of h, k, and a. As students suggest values, have them explain why they chose the numbers they did. Students should point out that h = 3 , k = -1, and a = 1. With the class, use the values of h, k, and a to generate the vertex form
y = (x – 3)2 – 1.
13.Ask what the numbers in the vertex form y = (x – 3)2 – 1 have in common with the graph of the parabola. Elicit from students that the vertex of the parabola occurs at the point (3, -1), the values of h and k in vertex form. Point out that the vertex form of a quadratic function allows for easy identification of the coordinates of the vertex.
14.Answer any other questions that students may have before continuing with the lesson.
1. Remind students about the ball throw in Lesson 1. Tell them to assume that the basket is now located in the middle of the floor and not against a wall. Ask what would happen if the shot were missed. Elicit that the ball would hit the floor and bounce. Draw this repetition of bounces on a coordinate graph to demonstrate.
2. Tell students that they are going to do an activity that measures the ball bouncing.
3. Conduct a demonstration of how the Calculator Based Ranger (CBR) collects data regarding motion. Explain the steps students will need to follow in order to use the calculator, but let them know that detailed instructions are also provided on the worksheet. This way, students can pay attention to you and not worry about taking notes. Drop a tennis ball or other type of ball and have the CBR collect the data for these bounces. Then, show the students the graph that the calculator will display using this data.
4. Have students collect data using the CBR in their groups. You may wish to have students use their own CBR-collected data for this lesson, or you may want them to use the CBR for practice and then download and use the clean data for the remainder of the lesson. (If you don’t have a CBR, you can download the clean data program using TI Connect and use that for the entire lesson.)
5. Give students time in their groups to create vertex form equations for each bounce of the ball. The questions on the worksheet will serve as a guide.
6. Bring students back for a group discussion to ensure that all students have reached the same conclusion.
7. Conduct a discussion about how students were able to create the functions to describe each bounce. Elicit from students that identifying the coordinates for the vertex (h, k) of each parabola is necessary, as is recognizing that the value of a must be constant for each bounce.
8. Identify the coordinates for the vertex of the parabola that represents the second bounce. Have students explain how those points will be used in the vertex form of the equation. Using the clean data from Tremain’s class, the coordinates are (0.734, 0.574).
9. Have students insert the vertex coordinates into the vertex form to generate the equation. In the above case, the equation would look like this: y = a (x – 0.734)2 + 0.574.
10. At this point, the value of a is not yet known. Have students explain how they can determine its value. A value of a = – 4 (or close to it) will generate a parabola that closely approximates the second bounce. For the time being, students should use a guess and check method to determine the value of a.
(Note: If students used the clean data to generate equations, you may wish to have them repeat the experiment, finding equations for the data they collected with the CBR.)
As a concluding activity, have students plan a presentation on how transformations occur in quadratic functions as a result of changes in the values of a, h, and k. Before students prepare these presentations, however, conduct a class discussion to determine how these presentations should be evaluated. Create a student generated list of criteria and use this list to evaluate the presentations.
Ask the class to take notes during the presentations. You may wish to have students take notes using the form provided. Inform students that they should be prepared to offer one of three types of comments at the end of the presentation:
- A question they would ask the presenting group
- A comment about something that the group did well
- An area in which the presentation could be improved.
After each presentation, allow the class to share their comments. Then, briefly discuss each group’s evaluation with them to offer immediate feedback.
Related Standardized Test Questions
The questions below dealing with quadratic functions have been selected from various state and national assessments. Although this lesson alone may not equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
- Taken from the National Assessment of Educational Progress (NAEP), Grade 12 (1992):
If f(x) = 4x2 – 7x + 5.7, what is the value of f(3.5)?
- Taken from the Ohio Grade 12 Proficiency Test (February 2001):
What is the standard form of the equation 2x2 + 6 = x2 – 7x – 4?
A. x2 + 6 = -7x – 4
B. x2 + 7x = -10
C. x2 + 7x + 10 = 0 (correct answer)
D. 2x2 = x2 – 7x – 10
- Taken from the Massachusetts Comprehensive Assessment, Grade 8 (2002):
Which equation states a rule for the pattern shown in the table below?
A. y = x2 – x + 1
B. y = x2 + x – 1 (correct answer)
C. y = x2 + 3
D. y = x2 + 1
- Taken from California High School Exit Examination (Spring 2001):
The length of the rectangle above is 6 units longer than the width. Which expression could be used to represent the area of the rectangle?
A. x2 + 6x (correct answer)
B. x2 – 36
C. x2 + 6x + 6
D. x2 + 12x + 36
- Taken from the National Assessment of Educational Progress (NAEP), Grade 12 (1992):
Student Work: Ball Bounce Activity
The class was given a warm-up activity to bring focus on their prior knowledge. Students were asked in the warm-up activity to graph the given vertex form of the quadratic function without using their calculator and to write the function that corresponds the given parabola. This student was able recall the effects of a, h, k, and the sign of a, to correctly flip the parent function, shift it 3 units to the right, and up four units. This understanding will be necessary for the student to complete the ball bounce activity and present the results to the class.
The major objectives of the ball bounce activity are for the students to use data collection to model parabolas and to use their prior knowledge to develop quadratic functions that will describe the motion of these parabolas. The students were expected to use the Calculator-Based Rangers (CBR’s) to collect height versus time data and to use their calculators to graph the data they collected. The students were then expected to use their knowledge of transformations to create several quadratic functions in vertex form that, when graphed, will trace over the ball bounce data collected with the CBR. This activity provides an opportunity for the students to apply their prior knowledge and for the teacher to assess the skills obtained from the previous lesson.
Once a student is able to get the first two bounces, the remaining bounce equations are determined from the same steps used in these first two bounces. Team 2 understood this process so well that they decided not to answer the questions for bounces 3 through 6, even though they were able to demonstrate these bounces during their presentation. While their answers to a few questions were incorrect, their presentation received high marks on their teacher and peer grading rubrics. Therefore, this team performed better in the oral communication than in the written form.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.