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* *Reflection: Checks for Understanding
The Luckiest Man in the World: Graphing Exponential and Linear Functions - Section 4: Exit Ticket: The Luckiest Man on Earth (revisited)

I like to have an entry ticket and exit ticket for many of my classes as a classroom routine to check student understanding before and after class.

For the lesson on graphing linear and exponential functions, I focused the entry and exit tickets on comparing and contrasting linear and exponential functions with two different lottery scenarios.

For the first student sample, I was pleased with the level of understanding in the **Entry Ticket Sample 1, **as well as the **Exit Ticket Sample 1**. I was especially pleased with the level of complexity demonstrated in the **Exit Ticket Written Response Sample 1**, as writing about math is a skill that many of my students struggle with and that I focus on improving in the classroom.

The **Entry Ticket Sample 2** and the **Exit Ticket Sample 2** again demonstrates a solid beginning understanding of the concepts as well as growth from the beginning to the end of the lesson.

Formative assessments with a pre-post design can be powerful tools to not only assess student work, but also can help teachers focus their lesson on the priority concepts and skills to be taught.

*Checks for Understanding: Pre and Post Formative Assessments*

# The Luckiest Man in the World: Graphing Exponential and Linear Functions

Lesson 6 of 10

## Objective: SWBAT graph and interpret exponential and linear functions. SWBAT paraphrase complex arguments.

To begin the lesson I use the hook of a lottery problem as an **Entry Ticket (Entry Ticket: The Luckiest Man on Earth (Graphing Exponential and Linear Functions))**. For this example I keep the two scenarios as a linear and exponential function because I feel one of the more important concepts in Algebra I for students to master is the difference between linear and exponential functions.

This particular entry ticket has a lot of flexibility built into it. Teachers could focus on the difference in rate between the two functions (one is increasing by a constant number, the other by a constant factor). Another slice of this problem is to focus on generating multiple representations of functions and to provide students with more practice graphing linear and exponential functions. As a third option, teachers can focus on solving systems of equations for this entry ticket.

The beauty in problems that are open-ended is it allows teachers to take the problem where the students begin. Now this also means teachers need to have an idea of possible trajectories to the problem so that it is used effectively, but it is a nice way to **Differentiate Instruction **and meet students at the different levels of understanding that they are at.

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After the entry ticket, I go through the P**owerPoint Slides: Graphing Exponential and Logarithmic Functions** with the class. It is important to note that students are active participants during this section, not passive listeners. In addition to taking **Two-Column Notes** in this section, there are a number of **Turn and Talks **where students are provided with an opportunity to engage in speaking and listening with a partner to grapple with the novel concepts being presented.

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After the presentation and note-taking, I provide students with time to work in groups on graphing exponential and logarithmic functions (**Group Practice: Graphing Exponential and Linear Functions**). For each pair of functions I include a critical thinking question to support students' continued exploration of the lesson concepts.

This particular activity looks different in my class depending on the group of students and how the day is going. Sometimes I have groups complete all of the problems. As an alternative I might assign one set of problems to each group, and have each group put their tables, graphs and responses on different white boards around the room. To wrap up the activity I would have a gallery walk where student make observations and constructively critique the reasoning of the other groups.

**Note: **For groups that are struggling with graphing exponential and logarithmic functions, I tend to provide them with additional practice (see homework section of this lesson for more details on suggestions on implementing in the classroom).

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To conclude the lesson, students are presented with the second lottery that the man won (hence the luckiest man on earth!): **Exit Ticket: The Luckiest Man on Earth (Revisited)**

In this example, I present students with a scenario that can be modeled by an exponential function and a second with a linear function. The prompts are strategically set up so that the man would receive a bigger payout under the linear scenario after five years, but after ten years would get a larger payout going with the exponential scenario. I set up this problem to again show students that exponential growth functions will always eventually exceed linear functions with a positive slope.

Teachers can use this **Exit Ticket: The Luckiest Man on Earth (Revisited) ** in a number of different ways. The assignment can be used as a formative assessment/quiz grade where students work on it independently. Students could also work on the assignment in groups and present their work on different white boards throughout the room.

This assignment is also an excellent homework assignment if time runs out for this lesson.

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For homework, I have students watch one video example from Khan Academy on graphing exponential functions. I also ask students to complete the **Homework: Graphing Exponential and Linear Functions, **where they create their own lottery scenario.

I want to balance review and pushing students to extend their thinking on this homework assignment. They watch the video as a way to activate their prior knowledge of what we learned in class today. The create a scenario problem encourages students to think about the application of exponential functions.

**Teaching Note**: I may use these videos in class for additional practice and note-taking on these skills. For example, I have a high-support class that meets every day for a 90 minute block. I might utilize these videos more in class for that particular section, and would tend to assign the videos as homework for my honors class.

Khan Academy Videos for Homework, Review or Extension of the Lesson:

**1. Example of Graphing an Exponential Function**

2. **Graphing Logarithmic Functions**

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- UNIT 1: Thinking Like a Mathematician: Modeling with Functions
- UNIT 2: Its Not Always a Straight Answer: Linear Equations and Inequalities in 1 Variable
- UNIT 3: Everything is Relative: Linear Functions
- UNIT 4: Making Informed Decisions with Systems of Equations
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- LESSON 1: Rewriting Radical and Rational Exponents (Plus Exponents Review)
- LESSON 2: Creating and Interpreting Exponential Functions
- LESSON 3: Constructing Linear and Exponential Functions
- LESSON 4: Comparing and Contrasting Linear and Exponential Functions
- LESSON 5: Pizza, Hot Chocolate and Newton's Law of Cooling: Adding Constants to Exponential Functions
- LESSON 6: The Luckiest Man in the World: Graphing Exponential and Linear Functions
- LESSON 7: Formative Assessment: Modeling Population Growth (A Math Assessment Project Classroom Challenge)
- LESSON 8: Marketing Exponential Functions: A Group Performance Assessment Task
- LESSON 9: Review Lesson on Exponential Functions
- LESSON 10: Writing in Math Classroom, Part 3: Comparing and Contrasting Arithmetic and Geometric Sequences