##
* *Reflection: Developing a Conceptual Understanding
Understanding Domain and Range - Section 2: Investigation

In the opening activity for this lesson, the students had some interesting conversations. I was able to monitor the student discussions as they discussed the problem. Some students had misconceptions about what the graph represented. I tried not to "fix" any of the student thoughts at this point in time hoping that the scaffolded investigation would guide students to clearing up misconceptions.

Before giving students the investigation, I asked students to use the graph to picture what happened in this race. I had students turn and talk with their neighbor to share their ideas and create a verbal story about the race and how it unfolded. While students were constructing their story with their neighbor I listened and found two groups who could share their ideas with the whole class. The above activities helped students to process the question and understand it more deeply. This understanding helped them to create meaning around the abstract concepts such as f(5) and g(15) in the context of the question.

When students were investigating the domain and range of this scenario they made fewer mistakes on finding the range. This was a surprise to me as I anticipated more issues with students finding the range. Students did very well with question number 8. Even if they did not use the function notation of f(13)=g(13) they were able to say something about the race being tied at that point in time. You can see in the attached work that I commented on the student's answer for number 5. Many students said that you "cannot have negative time" which shows that they have a misunderstanding about the meaning of each variable (x and y). I will address this misunderstanding in the lesson on multiple representations of functions. In the opening students can explore what will happen if Adam spends more money than he has (the graph goes below the x-axis). They can make sense of this as being debt in the context of that particular question.

*Developing a Conceptual Understanding: Investigation Reflection*

# Understanding Domain and Range

Lesson 5 of 18

## Objective: SWBAT describe the domain and range of a function in terms of their relationship to a model.

#### Opening

*5 min*

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.

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#### Investigation

*20 min*

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).

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#### Debrief

*10 min*

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?

#### Resources

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#### Closing

*5 min*

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)?

#### Resources

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- LESSON 1: PRE-ALGEBRA: Evaluating Expressions
- LESSON 2: Defining Functions Recursively
- LESSON 3: Tower Task: Exploring Explicit Formulas
- LESSON 4: Function Notation
- LESSON 5: Understanding Domain and Range
- LESSON 6: Multiple Representation of Functions
- LESSON 7: Piecewise and Step Functions
- LESSON 8: Mirror Task: Understanding Equivalent Functions
- LESSON 9: Modeling with Functions
- LESSON 10: Functions Practice and Assessment
- LESSON 11: Introduction to Piecewise Functions: Dance-a-Thon Question
- LESSON 12: More with Piecewise Functions
- LESSON 13: Evaluating Functions Day 2
- LESSON 14: Transformation of Functions Day 1
- LESSON 15: Transformation of Functions Day 2
- LESSON 16: Transformations "How To" Guide
- LESSON 17: Functions Review Assignment
- LESSON 18: Functions Unit Assessment