Insights Into Algebra 1: Teaching for Learning
Quadratic Functions Lesson Plan 1: Up, Down, Right, Left – Function Families
This lesson requires students to explore quadratic functions by examining the family of functions described by y = a (x – h)2 + k. This lesson plan is based on the activity Tremain Nelson used in the video for Part I of this workshop.
Two 50-minute periods
Students will be able to:
- Describe how changing the values of a, h, and k in the equation
y = a (x – h)2 + k affects the graphical shape of the parabola and its position on the coordinate grid.
- Graph parabolas.
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers will need the following:
- Gridded flip-chart paper and markers
- Long (approx. 18″) pipe cleaners of five different colors for each group
For each group of four students, you will need to prepare a station that contains:
- A sheet of flip-chart paper (a chalkboard or dry-erase board will also work)
- At least four copies of the worksheet indicating the quadratic functions to be investigated
- Quadratic Function Grading Chart
Students will need the following:
- Notebook or journal
- Graphing calculators
Teachers Activities and Assignment
1.Introduce the lesson by explaining that students will investigate changes in quadratic functions that cause parabolas to shift up, down, right, left, and change their shape.
2.Divide the class into groups of three or four. Invite one member from each group to join you at the table for a “table talk” session. These students should take notes; they will serve as leaders when the groups begin their investigations of parabolas later in the lesson. All other students should gather around the table to listen to the discussion.
3.Have students at the table envision the following scenario: A student throws a ball from one side of the room and it lands in a basket on the other side of the room.
Let students suggest possible trajectories for the path of the ball. On a sheet of flip-chart paper, draw their suggested trajectories. They should include:
- A parabola that almost touches the ceiling
- A parabola that is somewhat flat and for which the height changes only a little
- Several parabolas in between.
(Rather than just discuss this scenario, you may wish to perform the activity with your students. Have one student lie on the floor and try to throw a ball – perhaps a tennis ball or a wadded piece of paper – across the room into a waste basket. To ensure that the trajectory resembles a parabola, instruct the student to throw the ball so that it comes as close to the ceiling as possible.)
4.Ask the students the name of the highest point of each parabola (Answer: the vertex). In addition, let them know that the vertical line that contains the vertex is called the axis of symmetry.
1. Discuss with students why it is necessary to consider the shapes of various parabolas. Reference the throwing of the ball into the basket, and point out that moving closer to the basket or further away would require the ball to trace a different parabola in order to make it into the basket.
2. Inform students that they will be investigating how parabolas can be transformed. Say: “The best way to see how parabolas move is to look at the vertex form of the quadratic equation.” On a piece of flip-chart paper or on the overhead projector, write the vertex form:
Tell students that the parent function, y = x2, can be modified to create other functions by adding or subtracting values from x2, or by multiplying x2 by a coefficient, or both. Explain that in the vertex form, x and y are variables, but a, h, and k are constants and can be positive or negative numbers.
3.Explain to students that they will be dividing into groups of four and moving through four different “stations” set up around the classroom. (Each station should have a folder containing a copy of the Quadratic Function Grading Chart, a sheet of flip-chart paper, and at least four copies of the worksheet.) At each station, the groups will be investigating a function with different values for a, h, and k. The functions for the different stations should be as follows:
- Station 1: y = ax2
- Station 2: y = x2
- Station 3: y = x2 + k
- Station 4: y = (x – h)2
4.Tell each team they will be graphing the function given at their first station. Remind students how to graph a function by plotting points. Explain that they will need to identify the following elements:
- axis of symmetry.
5.Before they begin, show students a copy of the Quadratic Function Station Grading Chart. Explain to students that, as they move from station to station, they will grade the work of the previous groups. Explain that they may award up to 25 points in each of four categories: neatness, knowledge, explanation, and completion.
6.Send each group to one of the stations set up around the room. There should be one station for each group of four students.
7.When they arrive at their first station, students should draw a large coordinate plane on the sheet of flip-chart paper. They should plot points on the graphs based on the functions at that station, and they should graph the parabolas using the pipe cleaners. (Students can graph with markers instead of pipe cleaners, but pipe cleaners make it easier for them to adjust their graphs if they make mistakes.) Allow the groups 12 minutes to complete the activity and worksheet at their first station. Circulate as the groups work, observe their investigations, and ask clarifying questions as necessary. During this time, note what students understand, what difficulties they have, and what may require clarification later. After 12 minutes, have students rotate to their next station.
8.At their second station, have students spend four to five minutes grading the previous group’s work. Remind students to give a grade for neatness, knowledge, explanation, and completion.
9.Have students rotate to the next station. Again, have them spend four to five minutes grading the original group’s work. Note that at this point, the group will be providing a second evaluation for the group that originally did the work; the work will already have an evaluation from a previous group.
10.Finally, have students rotate one last time and spend four to five minutes grading the work at a fourth station. When students complete this rotation, each group’s work will have been graded by three other groups. At the completion of this rotation, each group, after having prepared the work at one station, should have been exposed to the work at three other stations. Consequently, each group should have been exposed to all four different function types.
11.Reconvene the class at the front of the room. One member from each group should sit around the table for a table talk. (Note: Make sure this is a different student than the one who sat at the table talk at the beginning of the class).
12.Tell the class that they will now discuss how to prepare a presentation on this topic.
13.Review the effects of a, h, and k on the graph of the function. At this point, ask students to be specific in their responses. Students may use informal language during this activity; and they should be able to express the following:
- The parabola moves k units up if k is positive, and it moves k units down if k is negative.
- The parabola moves h units to the right if it is in written in the form (x – h)2, and it moves h units to the left if it is written in the form (x + h)2.
- The parabola opens upward if a is positive, but it opens downward if a is negative.
- The parabola gets “wider” as the value of a decreases (|a|<1), but the parabola gets “skinnier” as a increases (|a|>1 ).
Focus on the concept and do not get bogged down in formal language. Students should learn the precise language later.
14.Ask: “What are some of the things that you think would be very important for us to put in our presentation?” Through a brief discussion, elicit the important points from students, including the effects of a, h, and k, and how to graph a parabola in vertex form using the information about a, h, and k.
15.With the class, create a list of categories that states the elements an exceptional presentation should contain. The categories might contain the following:
- Group members demonstrate sufficient knowledge of the material.
- The presentation includes an explanation of the effects that a, h, and k have on a parabola.
- The presentation includes an explanation of how and why the shape of the parabola changes.
- Graphs provide a visual representation of the effects of a, h, and k.
- All members of the group participate.
- The written work, including graphs, is neat.
- The material is covered completely.
- Students recovered after making a mistake during the presentation.
16.After listing the criteria, have students determine how many points each of the criteria should be worth. Discuss with the class how important each element is.
17.Allow 20 minutes for each group to prepare a presentation. If you wish, students may use the presentation directions as a guide.
Allow each group to give a brief presentation on the material they have gathered. Inform other students to take notes during these presentations, focusing on three areas:
- Questions to ask the group after the presentation
- Positive feedback – things they thought the group did well
- Suggestions for improvement.
You may wish to have students take notes using the form provided. After each presentation, have students give feedback to the presenting group, allowing for at least two comments in each area listed above. In addition, offer your feedback to the group privately (perhaps at the back of the classroom) while the next group prepares to give their presentation.
Related Standardized Test Questions
The questions below dealing with quadratic functions have been selected from various state and national assessments. Although this lesson may not fully equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one eventually will develop the conceptual understanding needed to succeed on these and other state assessment questions.
- Taken from the Ohio Grade 12 Proficiency Test (February 2001):
If x2 + ax + a2/4 = 0, then x = ?
A. – a/4
B. – a/2 (correct answer)
- Taken from the Maryland High School Assessment in Algebra, Grade 11 (2002):
Doug makes a rectangular dog pen using 8 yards of fencing. The graph below shows the relationship between the width of the pen and the area of the pen. In the ordered pair (2, 4), what does the y-coordinate represent?
A. maximum area of the pen (correct answer)
B. maximum width of the pen
C. maximum length of the pen
D. maximum perimeter of the pen
Student Work: Graphing Quadratic Equations Assignment
The major objective of Station 4 is to understand the effects of “h” on the vertex form of the function. The students were asked to reflect their understanding on the worksheets as well as the wire and graph paper provided. Each station began with the parent function and was asked to perform the appropriate transformation and identify the key terms listed on the cover sheet for that station.
The team members of Station 4 seemed to have an understanding of the directions given. They were able to identify the y-intercept, x-intercept, axis of symmetry, vertex, and minimum for each parabola. The students were allowed to use their calculators to verify their information and to assist in summarizing the transformation taking place. The team members correctly summarized the effects of “h” on the parabola. The difficulty with the “h” transformation is that it appears to shift the parabola in the opposite direction from the students’ expectation. This team was able to explain the effects of the “h” transformation, however, did not answer questions 7, 8, or 9. These questions were intended to lead the students beyond a superficial understanding of this transformation and to begin to think about why this shifting was taking place. The students’ work shows they did not reach this level of understanding during the activity.
During the next table talk discussion and during their preparation for their presentations, they will have more opportunities to get this understanding. If this lesson was repeated, one change that might make this easier for the students to grasp would be to ask more leading questions to help them develop their explanation of why the graph appears to shift in the opposite direction from what they expect. Apparently, answering these open-ended questions was too much of a leap beyond their existing knowledge of transformations.
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.