Lesson Plan 1: Up, Down, Right, Left - Function Families
Teachers will need the following:
For each group of four students, you will need to prepare a station that contains:
Students will need the following:
- Gridded flip-chart paper and markers
- Long (approx. 18") pipe cleaners of five different colors for each group
- Notebook or journal
- Graphing calculators
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:
(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.)
- 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.
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:
y = a (x - h)2 + k
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:
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:
- Station 1: y = ax2
- Station 2: y = x2
- Station 3: y = x2 + k
- Station 4: y = (x - h)2
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.
- axis of symmetry.
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:
Focus on the concept and do not get bogged down in formal language. Students should learn the precise language later.
- 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 ).
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:
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.
- 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.
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:
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.
- Questions to ask the group after the presentation
- Positive feedback - things they thought the group did well
- Suggestions for improvement.