Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Student will be able to:
- Develop a linear model from constructed data
- Develop a exponential model from constructed data
- Distinguish between linear and exponential growth
Prior Knowledge: What prior knowledge should students have for this lesson?
- Students should be able to express powers of numbers using exponents. (This activity involves powers of 2.)
- Students should be able to distinguish between powers of a number and multiples of a number.
- Students should know how slope relates to linear growth.
- Students should have used formulas and understand how they can be applied to data.
- Students should be familiar with and be able to use function notation.
Guiding Questions: What are the guiding questions for this lesson?
Describe the kind of data changes you observe.
- How do you determine if y changes linearly or exponentially with x?
- What kind of predictions could you make from your data?
- How can you use algebra to help make predictions?
- When should you use a calculator and why?
- What are other examples of linear or exponential growth?
Teaching Phase: How will the teacher present the concept or skill to students?
This lesson is using the attached Piles of Paper activity to allow students to discover and process data that they create by answering questions in the activity. The end result should be that students understand by discovery the difference between linear and exponential growth and their algebraic representations.
The teacher's role is to help individuals or groups that are not able to answer questions successfully. Guided practice is part of students doing the activity without direct teacher instruction. After reading through the Piles of Paper activity, teachers may choose their own teaching/presentation method but these are some suggested steps:
- Short review of exponents and slope as suggested above.
- Assign small groups of 2-4 students.
- Each student should have copy of Piles of Paper activity, scientific calculator, one sheet of 8.5 X 11 inch paper.
- Give student groups 5 minutes to do Section 1 on Piles of Paper activity.
- Optional: Show YouTube video clip (3:54) minutes on paper folding (URL on teacher page of Piles of Paper activity sheet).
- Discuss Section 1 as a class. The teacher needs to make sure that students understand these exploratory questions and that they will have a variety of answers. The teacher's discussion should be lead by letting students give their answers to questions on the different sections of the Piles of Paper activity sheet. The teacher can then judge from answers what clarifications might be necessary.
- Have student groups work on Section 2 about 10 minutes.
- Discuss Section 2 as a class. The teacher should make sure that students are seeing that the paper piles have linear growth and a constant slope.
- Have student groups work on Section 3 for 15-20 minutes.
- Discuss Section 3 as a class. Teacher needed to emphasize that the paper growth is no longer linear and has changed to exponential.
- Have student groups work on Section 4 for 10 minutes.
- This section's discussion is important as it compares both types of growth and lets students evaluate and identify examples of linear and exponential growth in problem #17. Student responses can be discussed with class with teacher giving needed feedback. Help is given for teacher in teacher pages and in discussion of summative assessment.
- Collect student work.
- For homework the teacher should ask students to either
- Create 3 different types of functions that are 1) linear, 2) exponential, and c) something different as quadratic or cubic. They could do this with a table or algebraic equation.
- Research or create real world examples of linear and exponential growth. (Linear examples: number of days in x weeks, D = 7x or number of GPA points earned for x A's, P = 4x) (Exponential examples: Richter scale for earthquakes or light visibility in water)
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Guided practice is part of the Piles of Paper activity sheet. Piles of Paper contains questions that students will complete during the activity. Teacher will guide students with discussion at end of each section.
Teacher may also guide work by commenting on an individual's or a group's work as they are completing a section.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
As indicated in Teaching Phase, independent practice materials may include request for students to create their own linear, exponential, and other functions. They can do this with equations or table examples. Also students could research other real world examples of linear or exponential growth. Some examples are given above in Teaching Phase.
Students may be asked to find algebraic models or formulas from new data sets that could be given. These could include linear growth with a slope of -5 or exponential growth with a base of 1/2.
They also may be asked to graph data to view linear and exponential growth geometrically.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
At the end of the activity, it would be ideal if students and teacher discuss together Section 4 of the activity. It reviews the models found and their similarities and differences. Also real world examples of exponential and linear growth are given and can be discussed.
Teachers can also revisit the questions that have been discussed during the activity to emphasize major points of linear versus exponential growth. It should be stressed that linear functions have repeated addition and exponential functions have repeated multiplication.
Questions 16-18 on the Piles of Paper activity sheet will help check/review linear versus exponential understanding. This will either be checked by the end of class or if limited time, it should be collected and checked by teacher.
It is intended that class or group will talk about the kind of growth expected in examples in #17. The following function equations are given to help with this discussion.
- Ex. (years, simple interest) I(t) = prt
- (linear function where p = principle, r = yearly interest rate, and t = time in years, and I(t) = interest earned in t years at interest rate r)
- Ex. (years, compound interest) A(t) = p(1 + r)^t
- (Exponential function where p = principle, r = yearly interest rate, t = time in years, and A(t) = total of principle and interest after t years at interest rate r)
- Ex. (hours worked, wages) w(h) = 8.25h
- (linear equation where h = hours worked, 8.25 = hourly wage, and w(h)= wages earned at 8.25 an hour for h hours.)
- Ex. (days, new cell growth) g(d) = 2^d
- (exponential equation where d = number of days, 2= doubling growth base for each day, and g(d) = amount of cells in d days.)
- Ex. (seconds, height of a bouncing ball) h(b) = .6^b
- (Exponential equation where b = number of ball bounces, .6 = decay base of each bounce, and h(b) = height of ball after b bounces.)
- Ex. (radius, area of circle) A = pi* r^2
- (Quadratic equation where r = radius of circle and A = area of circle.)
- Ex. (radius, circumference of a circle) C = pi * d
- (Linear equation where d - diameter of circle and C = circumference of circle.)
Students are not expected to know these formulas but should be able to distinguish the type of growth with understanding of the function.
The teacher should have previously taught integral exponents and basic linear growth.
A short review of these concepts may be needed at the beginning of lesson. Students could be questioned on their understanding with a few warmup questions on exponents and slope.
- Find x: 32 = 2x
- 25 = ?
- If f(x) = -2x + 7 find f(4) and f(-5).
- If g(x) = -2(5)^x find g(0) and g(2)5.
- Given the linear function f(x) = -2x + 5, describe the change in f(x) (or y value) if x increases from 3 to 7.
- If p(x) = 3.40x represents a function where x is the number of gallons bought and p(x) is the price of gas for x gallons, interpret what the slope of this equations tells you.
Feedback to Students
When students are working on an activity section (see Piles of Paper activity attached), teacher should be monitoring and discussing with students/groups to check for understanding and on task behavior.
The teacher will question students' understanding after each of the 4 sections in the activity. Questioning may be with the whole class or with individual groups. Through this questioning, the teacher will confirm that students are ready to move to the next session. Teachers may choose to initial student or group work as it is checked. Some examples of questions that will provide meaningful feedback to the students are:
- What kind of function describes the growth shown in this data? Explain your answer and use math terms!
- What formula can you use to predict the height of the paper stack if F represents height and n represents the number of sheets of paper?