Week 8 Mini Unit

Introduction

This unit is a bit of a kludge because I am using it to catch students up on calculator use. My goal is to use an introduction to exponential functions as a means to introduce the more advanced uses of a TI-84. My students know how to use the basic functions such as adding, subtracting, multiplying, etc, but none of the list, table, or graphing capabilities. This seems to be a good opportunity to introduce it as we move from something they’re familiar with (linear functions) to something they aren’t. My students are low functioning. They either have an average intelligence, but have a very poor background in math along with severe emotional problems, or they have an IQ in the low 80s. I have to make sure I don’t introduce too many variations and give them lots of practice.

The book is Algebra I from McDougal Littell (2007).

Outcomes

  1. Develop ability to graph exponential functions by hand and using a TI-84.
  2. Recognize and explain the difference between an exponential growth and a decay function.
  3. Develop ability to model exponential functions with rules in the form y=a(bx).
  4. Develop ability to use reasoning, estimation, and curve-fitting utilities to model data patterns exhibiting exponential growth patterns.

MME Content Expectations

  1. A1.1.1 Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.
  2. A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.
  3. A2.5.1 Write the symbolic form and sketch the graph of an exponential function given appropriate information (e.g., given an initial value of 4 and a rate of growth of 1.5, write f(x) = 4 (1.5)x).
  4. A2.5.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and how base affects the rate of growth or decay.
  5. A3.2.5 Relate exponential functions to real phenomena, including half-life and doubling time.

Five-Day Plan

  1. Day One
  2. Introduce the lesson by discussing the difficulties that people have had graphing equations. Some of those difficulties can’t be avoided, but we can make some things easier by learning how to use our TI-84s better.
  3. The first day’s lesson is a variation of the Scavenger Hunt activity. Students will be assigned a “skill” to master along with either directions on where it is in the TI-84 tutorial, or if computers aren’t available, a handout explaining how to do it. The scavenger hunt begins after 10 minutes becoming “experts”. A lesson follows that reviews how to use graphs and tables on the calculator, first by graphing linear functions and then exponential functions. Short discussion to follow on why the graphs and tables might be different.
  4. Day two consists of the Pay It Forward lesson (see attached).
  5. On day three I plan to reinforce day two by having them complete the Exponential Stations activity using their calculators. Students will be divided into pairs with each pair assigned a station. A discussion will follow on what each pair “discovered”. (
  6. Day four consists of the Surviving a Zombie Attack, which will be a review of exponential growth functions and an introduction to exponential decay functions. This lesson may spill over into the next day. The lesson may be found at (
  7. Review and practice graphing of exponential functions. Students will be asked to graph the functions by hand and then check the results with the calculator.

Formative Assessments

After day one, have students write directions on how to graph a function as a “ticket out the door”. This can also be done the following day as a refresher.

In addition, each day, students are given a warm-up activity similar to what they did in class the previous day. I make these up on the spot and they typically consist of two problems similar to whatever they had the most difficulty with the previous day.

Summative Assessments

Upon completion of the mini-unit they will take the quiz at the end of this document and will be allowed (and encouraged) to use their calculators to check their work. We will spend a couple of days in review and they will take the chapter 8 test from the book, which covers the above and applying the properties of exponents.

Algebra for All Online

Lesson Plan

Mathematics Professional Development

This lesson was originally created by Emily Burkett and may be found at: . It has been only minimally modified by me.

Title: Pay It Forward - Exponents

Content Expectation(s):

Strand: Algebra and Functions

Standard: A3: Families of Functions

Topic: A3.2 Exponential and Logarithmic Functions

Content Expectation: A3.2.5 Relate exponential functions to real phenomena, including half-life and doubling time.

Lesson Outcome:

  1. Graph and use exponential functions
  2. Use vocabulary to support mathematical reasoning
  3. Develop disposition to look for and ability to recognize exponential growth patterns and phenomena
  4. Develop ability to model exponential functions with rules in the form y=a(bx) where a > 0 and b>1
  5. Develop ability to use tables and graphs to solve problems about exponential growth
  6. Develop ability to use reasoning, estimation, and curve-fitting utilities to model data patterns exhibiting exponential growth patterns.
  7. Develop ability to model exponential functions with rules in the form y=abx-h+k

Materials and Resources:

  1. White boards, markers, and erasers.
  2. Handout (at end of plan)

Procedures:

  1. Students come into class and receive synopsis of Pay It Forward and watch a clip from the movie ( (3-4 minutes)
  2. Students individually make predictions from the “Pay It Forward” picture to answer question #1 on Pay It Forward Essential Question page.
  3. Students have the opportunity to write their answer on the white board to show me their choice (Graph 1, 2 or 3) and their predictions on what stage 25,000 good deeds will be done.
  4. Students can engage in conversation about their results. Ask them to list a few other things that grow quickly. If zombies don’t come up, bring them up and say we’ll be looking at them in a couple of days.
  5. Students will get in “partner formation” to work on making their data tables and graph. (7-10 minutes).
  1. Students will make their graphs and data tables on the white boards and I will come around and check their work.
  2. At any point, student groups may hold up their whiteboards.
  3. Once all groups are finished, 1-2 groups can volunteer to share their data, or if more than two groups volunteer they can take turns.
  1. Students then will complete the rest of the activity (approx. 10 minutes)
  1. Students then can share their findings with the class
  1. Direct teacher instruction (approximately 20 minutes)
  1. Use paper-folding activity from A4A to lead students to discover how to write exponential functions. This can be followed up with the notes in the book (McDougal Little – Algebra 1 (8.5)). If there is not enough time, the book material can be given/reviewed on the following day.
  2. Points to stress:
  3. A missing “a” value is 1.
  4. For a vertical shift up you can say that the 1st individual is an over-achiever and helps 5 people instead of 3 so you so you have the function f(x)=3^x+2
  5. We translate graphs to understand how small changes in the parent function affect the end result.
  1. Homework

Formative Assessment:

Work on white boards.

Attachments:

(See below).

Pay It Forward

In the popular book and movie, Pay It Forward, 12-year-old Trevor McKinney gets a challenging assignment from his social studies teacher.

Think of an idea for world change, and put it into practice! Trevor came up with an idea that fascinated his mother, his teacher, and his classmates.

He suggested that he would do something really good for three people. Then when they would ask how they can pay him back for the good deeds, he would tell them to “pay it forward” – each doing something good for three other people.

Trevor figured that those three people would do something good for a total of nine others. Those nine would do something good for 27 others, and so on. He was sure that before long there would be good things happening to billions of people all around the world.

PAY IT FORWARD ESSENTIAL QUESTIONS

Is it possible for one idea to change the world?

  1. Which of the graphs below do you think is most likely to represent the pattern by which the number of people receiving Pay It Forward good deeds increase as the process continues over time?
  1. What is your best guess about the number of people who would receive Pay It Forward good deeds at the tenth stage?
  1. How many people would receive a Pay It Forward good deed at each of the next several stages?
  2. Make a table on your white board that shows the number of people who will receive good deeds at each of the next seven stages of the Pay It Forward process. Then plot the data on a graph. Make sure you have accurate axes labels and scales.

Stage of Process / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Number of Good Deeds / 3 / 9 / 27
  1. On a separate sheet of paper answer the following question: How does the number of good deeds at each stage grow as the tree progresses? How is that pattern change shown in the plot of the data?
  2. How many stages of the Pay It Forward process will be needed before a total of at least 25,000 good deeds will be done?
  3. Write a rule relating Number of Good Deeds (N) to the Stage of Process (x). This rule could be used to model the Pay It Forward Process in which each person does good deeds for three other new people. Show work below:

Mini-unit Summative Assessment

Graph the function.

1.

2.Write an exponential function to model the situation. Then estimate the value of the function after 5 years (to the nearest whole number).

A population of 290 animals that increases at an annual rate of 9%.

3.Writing: Explain the difference between a linear function and an exponential function. Give an example of each function type.

Write a rule for the function.

4.

Graph the function.

____5.y = (1/3)x

a. / / c. /
b. / / d. /

6.Graph the function and label as exponential growth or decay.

7.Choose the equation that represents exponential decay.

a. b.

8.The enrollment at Alpha-Beta School District has been declining 4% each year from 1994 to 2000. If the enrollment in 1994 was 1575, find the 2000 enrollment.

Mini-unit Summative Assessment

Answer Section

1.ANS:

PTS:1DIF:Level BREF:MALG1201

TOP:Lesson 8.5 Write and Graph Exponential Growth Functions

KEY:graph | exponentialBLM:KnowledgeNOT:978-0-618-65612-7

2.ANS:

f(x) = ; 446

PTS:1DIF:Level BREF:MAL21010

TOP:Lesson 8.5 Write and Graph Exponential Growth Functions

KEY:exponential | growth | decay | write | equation | wordBLM:Application

NOT:978-0-618-65612-7

3.ANS:

Sample answer: A linear function in one in which the growth is constant. Its graph is a straight line; example: y = 2x + 3. An exponential function in one in which the rate of growth is not constant. Its graph is a curve; example: .

PTS:1DIF:Level BREF:MAL21025NAT:NCTM 9-12.ALG.1.e

TOP:Lesson 8.5 Write and Graph Exponential Growth Functions

KEY:exponential | function | linearBLM:AnalysisNOT:978-0-618-65612-7

4.ANS:

PTS:1DIF:Level BREF:7ef48cae-cdbb-11db-b502-0011258082f7

TOP:Lesson 8.5 Write and Graph Exponential Growth Functions

KEY:Exponential growthBLM:KnowledgeNOT:978-0-618-65612-7

5.ANS:DPTS:1DIF:Level BREF:MALG1206

TOP:Lesson 8.6 Write and Graph Exponential Decay Functions

KEY:graph | exponential | functionBLM:KnowledgeNOT:978-0-618-65612-7

6.ANS:

exponential growth

PTS:1DIF:Level BREF:MALG1207NAT:NCTM 9-12.ALG.1.c

TOP:Lesson 8.6 Write and Graph Exponential Decay Functions

KEY:graph | exponential | growth | decayBLM:Comprehension

NOT:978-0-618-65612-7

7.ANS:

a.

PTS:1DIF:Level AREF:MALG1209NAT:NCTM 9-12.ALG.1.c

TOP:Lesson 8.6 Write and Graph Exponential Decay Functions

KEY:determine | exponential decayBLM:KnowledgeNOT:978-0-618-65612-7

8.ANS:

1233

PTS:1DIF:Level BREF:MALG1216

TOP:Lesson 8.6 Write and Graph Exponential Decay Functions

KEY:solve | exponential equation | growth | writeBLM:Comprehension

NOT:978-0-618-65612-7