Before:
1. Play "What's My Rule": Write a simple equation on a card, for example, (y=x^2-3). The teacher should select 4 students to provide an input value from 1 to 10. The teacher will use mental calculations to provide an output value. Working in pairs, have students attempt to guess the rule. Allow each pair of students to write down their function and prove by substitution that it works.
2. Using computers, direct students to the Function Machine Activity on the Math Playground website. Students will need to select the advanced level. Allow them to practice writing a function rule when given domain and range values. The difficulty of the activity can be adjusted by changing the level (1,2, or 3 found at the bottom of the screen) or by allowing students to select the domain values themselves. This activity will provide an association to a function machine which will be used in the explain portion of the lesson. Allow 5 - 10 minutes for this activity. You may wish to pair students to provide peer tutoring. This could be done as a whole group activity if individual computers are not available.
During: Using the PowerPoint presentation teachers will explain the process of finding a composition of functions.
1. Teachers will display the slide showing a composition of functions using the analogy of a function machine. Show students how an input value is dropped into the machine and an output value falls out into another machine thus producing a composition.
2. Working in pairs students will be prompted to complete the Composition of Functions Notes page for the function machine. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.
3. Teachers will display the slide showing a composition of functions from a table of values. Explain to students how to find the output associated with the given input for the first function and then use that output as the input for the second function to obtain the value for the composition.
4. Working in pairs, students will be prompted to complete the Composition of Functions Notes page for the table of values. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.
5. Teachers will display the slide showing a composition of functions using graphs. Explain to students how to find the output (y-value) associated with the given input (x-value) on the first graph and then allow that output (y-value) to become the input (x-value) for the second function to find the value of the composition.
6. Working in pairs, students will be prompted to complete the Composition of Functions Notes page for graphs. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.
7. Teachers will display the slide showing a composition of functions using function rules. They need to emphasize that this time the function rule itself will be the input for the second function.
8. Working in pairs, students will be prompted to complete the Composition of Functions Notes page for function rules. First, they will explain how the example from the slideshow worked. Then they will create their own problem and will explain how to work it. While students are completing notes, the teacher will be able to circulate the room and address misunderstandings. Students will also be able to provide peer tutoring.
The notes can be used as an assessment tool to determine if additional examples or instruction is needed. Allow 30 - 40 minutes for instruction.
After: Students will complete a Frayer Model for the composition of functions. Teachers will ask students to complete the form by writing a definition in their own words for the composition of functions. Next, they will provide an example of any kind for the composition of two functions. They will be asked to describe the different representations of functions that were discussed, and finally, they will be allowed to express any problems or concerns that they may have. This will be their ticket out the door. Allow 5 minutes for students to complete.
