##
* *Reflection: Checks for Understanding
Creating and Interpreting Exponential Functions - Section 1: Entry Ticket: Bears, Bacteria and Exponential Functions

In this Entry Ticket, students are asked to model and interpret a scenario that can be best modeled with a linear function and a second scenario that can be best modeled by an exponential function.

I like this activity because it gives students a chance to review concepts around linear functions AND further differentiate between linear and exponential functions. The **Student Work Interpreting Exponentials 1 **shows a solid understanding of both types of functions.

I like how the work sample** Student Work Interpreting Exponentials 2 **annotates the table by marking the pattern of change in the table (the first scenario increase by 10, the second by a common multiple). For these entry tickets I am looking to assess student current understanding of exponential functions. I also check in on students' understanding of linear functions through this entry ticket in an attempt to continue to spiral the curriculum and loop back to connected concepts throughout the year. In this way I hope to help students make connections that they may not otherwise make on their own.

*Checks for Understanding: Entry Ticket as Checks for Understanding*

# Creating and Interpreting Exponential Functions

Lesson 2 of 10

## Objective: SWBAT create and interpret exponential functions to model real world scenarios. SWBAT paraphrase complex arguments.

## Big Idea: Students work collaboratively to model different real world application of exponential functions!

*90 minutes*

For today’s **Entry Ticket: Creating and Interpreting Exponential Functions **I ask students to create functions to model two scenarios. I am flexible with respect to the form – I am fine with students developing a table of values, a graph or equations to model the scenario. The bear population in Problem 1 can be best modeled by a linear function. The bacteria population in Problem 2 can be best modeled by an exponential model.

I expect that my should be able to create the linear example. Even if they are not able to model the bacteria population correctly, I expect that they will be able to identify the need for a different approach than in modeling the bear population.

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In this section, the lesson builds off of the day’s entry ticket. We will start with the students' work and then discuss tools and strategies to create exponential models. I will base the flow of the conversation on the **Classnotes/Slides: Creating and Interpreting Exponential Functions**. I will be flexible to adjust the presentation based on students' prior knowledge, as shown on the Entry Ticket.

I plan to focus the instruction on two popular forms of exponential functions and include a number of **Turn and Talks **to provide students with time to process and construct their own meaning about the concepts. My goal for this section is for students to see the connection between two forms of exponential functions, namely y = ab^x and y = a(1+r)^x

The table addressing various values of the parameter b in the equation y = ab^x is a scaffolded support to help student inquire into how to interpret the structure of exponential functions. For example, it is important for students to realize that if b is less than 1 then they are dealing with exponential decay and not growth.

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For guided practice, I ask students to work in small groups on the **Guided Practice: Creating and Interpreting Exponential Functions. **For this assignment, students will complete a number of problems that ask them to interpret exponential functions.

After this practice work, I like to utilize the class textbook for sample problems. Today, I assign the whole class the same problem set to work on.

**Assignment (Algebra I McDougal-Littell, page 480, problems 1-5 and page 488, problems 1-3, 8-9)**

For the last 5-10 minutes of this section of the class, I will have different groups present their solutions to one of the problems to the class. Then, we will have a class discussion of the solution. I always try to give students the opportunity to express their thinking. As they speak about their work, it gives me an informal assessment of student understanding to guide instruction for the remainder of the class.

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After the presentations for the guided practice problems, I give students problems based on their current understanding. I **differentiate instruction **by providing more basic problems for students still struggling with the core concepts. I more complex scenarios for students who are thriving.

For this section I like to use technology to provide different problems to students. Some of the sample PARCC tasks at the Dana Center** **are excellent examples of exponential functions and use real-life scenarios that students can use and interact with online. The sample tasks can be fond here: **CCSSTOOLBOX Sample High School tasks**. I particularly like the **Cellular Growth **and the **Golf Ball** tasks for this lesson.

I typically give students the choice as to which activity they complete - in line with UDL, providing choice in lessons can increase student motivation and engagement. I picked the two activities form the CCSS Toolbox because I think they are high-quality tasks that has a nice balance of scaffolding and focus on interpreting exponential functions. In my class we have a set of iPads for a 2:1 student to iPad ratio so students would work in pairs for this section of class.

These tasks are also prototype items for the Common Core assessments, so I figure the more previewing and practice students can have on the type of tasks they may be asked to complete in the near future, the better off they will be. The prototype items also have a nice explanation as to how the items are aligned to the Common Core - I find the explanations a helpful resource and support for me as a teacher to continue increasing my understanding of the Common Core and its implementation in the classroom.

The NCTM illuminations also has a good problem-based lesson where students have to model **Trout Pond Populations**. There is also a good set of problems around interpreting exponential functions at the **NSA website **

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For the **Exit Ticket: Bears and Bacteria Revisited**, I assess student understanding with a formative assessment on interpreting exponential functions. I provide a similar structured problem to the day’s entry ticket on purpose so I can not only compare how students perform on the exit ticket relative to the standard and their peers but to also capture a growth measurement from the entry/exit ticket comparison for individual students and in the aggregate.

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For Homework I ask students to create their own exponential functions:

- One that involves exponential growth
- One that involves exponential decay

I encourage students to look online or at other resources to identify real-life scenarios modeled by exponential functions. I want students to begin to realize the relevance and importance of these functions as a tool to better understand their community and world.

*expand content*

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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
- UNIT 5: Exponential Functions
- UNIT 6: Operations on Polynomials
- UNIT 7: Interpret and Build Quadratic Functions and Equations
- UNIT 8: Our City Statistics: Who We Are and Where We are Going

- 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