Students should be working in pairs on this opening activity. For this lesson students can be grouped heterogeneously as all students will have access to the content. Some students will show greater or less level of sophistication in their answers based on their background knowledge with reading line graphs.
Project the attached picture up for the students and explain to them that the graph represents a 100 meter race between John and his Father. Ask the students to write down as many things as they can about the graph and what is being shown by themselves (MP4). Then have them pair up and try to expand their list by at least two more things. Have them pay attention to any items that both they and their partner have in common and write down any addtional things that they hear that are new. At this time student pairs will not share their ideas with the whole class.
Explain to students that they are going to spend a portion of the class analyzing this particular graph from the perspective of functions.
In this investigation students begin to think about a basic line graph from the perspective of functions. While students are working on this investigation with their partners the teacher can facilitate the conversations. Try not to give too much information but instead ask questions that will help students make their own discoveries about the math.
What to watch for: (1) In question #8, many students will say that there are lots of times when John's distance from the starting line is the same as his father's distance. However, this is a misundstanding that when f(x)=g(x) it means that they are the same (same height or y value) for the same value of x, not for two different values. As students grapple witht this question, they will make meaning around the notation which will help them later in systems of equations.
(2) In question #5 and #6 students will be making meaning about the domain and range of the two functions. Watch for students that are having trouble noticing that f and g have the same range but have two different domains. You can question students about this by having them explain how long it took John and his father to finish the race, respectively (MP3.MOV).
Here we bring the class back together to go over important points from this lesson. The points of emphasis will depend on the class, however, here are some starting points. For each of these questions give students plenty of wait time. You can even use a Think-Pair-Share strategy for questions that may require time for students to process.
Teaching point: I like to use think-pair-share when a question is asked and you can see that only a handful of students have an idea of the answer. The strategy allows extra processing time for all students and allows those that have an idea to share what they are thinking.
With all of the questions below, when accepting multiple answers, I like to have each subsequent answer build on the one before. If you use this strategy often, it forces students to listen to each other rather than just focusing on what they will say next.
1) Now that you are starting to learn about the domain and the range of a function, how is the answer for question #1 related to the domain. ?
2) Questions #2 and #3 have strictly numerical answers. Ask students the meaning of f(5) (or another value) in the context of the problem. (After 5 seconds, John's Father had run about 25 meters)
3) Have a few groups share out the answers to #6 and #7. Ask, why are the domains different but the ranges are the same?
4) What is the meaning of question #8 in the context of the problem?
Project the image up for students and ask them to record their responses to the following on paper (stop and jot). Note: for students that have difficulty with auditory directions you can put these question into a powerpoint or other medium beforehand with the graph at the top. Explain that this is another race, but this race takes place in a pool. The two girls swim to then end of the pool and back. Just to keep students from getting too locked in to f and g notation write the following on the board:
C(x)=Cindy's distance from the starting point
K(x)=Kelly's distance from the starting point
1) Write the domain and the range for both C(x) and K(x).
2) Who won the race? How does this connect to the domain?
3) What are the three values of x (times) when C(x)=K(x)?