# Exponential Decay Functions F.LE

The learning progression will be based upon Exponential Decay Functions where the students will be able to understand the properties and used it to solve real-life problems. At this point, the students have prior knowledge of solving Exponential Growth Functions and will be able to build upon that knowledge and use it to understand how Exponential Decay Functions work. The CCSS.Math standards that are align with the learning progression are:

1. Math.HSF-LE.A.1: Construct and compare linear, quadratic, and exponential models and solve problems. Distinguish between situations that can be modeled with linear functions and with exponential functions.
2. Math.HSF-LE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.

3. Math.HSF-LE.B.2: Interpret the parameters in a linear or exponential function in terms of a context.

Learning Progression for Practice edTPA-1lavusr

# Exponential Growth and Decay Models F.LE.A

Learning Progression edTPA
High School: Algebra with Trigonometry
EXPONENTIAL GROWTH AND DECAY
The class I will be teaching is an algebra class with
trigonometry. The textbook used will be Algebra 2:
Equations, Graphs, applications by Ron Larson, Laurie
Boswell, Timothy D. Kanold, and Lee Stiff, published 2004
by McDougal Littell. This learning progression will follow
the student through meeting the Common Core State
Standards about exponential growth and decay. These
Common Core State Standards will be covered in chapter
8 of the textbook, sections 1 and 2. These sections cover
exponential growth and decay models. That is, to be able
to create an equation, graph it, and solve problems with
exponential models. The standards for these lessons will
cover the cluster involving exponential functions,
including HSF.LE.A.1.C, HSF.LE.A.3, and HSF. LE.B.5. The
math practices that are going to be used throughout the
learning progression are as follows: MP2, MP3 and MP4.
These lessons will span three days. The exponential
growth model lessons will take the first two days and the
exponential decay model lesson will take the third day.
On the fourth day, the students will have a worksheet
with practice problems covering 8.1 and 8.2. This gives
the students a chance to show me how well they are
understanding the material because I will collect the
worksheet at the end of class and will use it to see how
the students are doing. I can assess what they
understand and what I need to cover again.

CCSS.Math.Content.HSF.LE.A.1.C:
functions and interpreting situations that
need to be solved with an exponential model.
CCSS.Math.Content.HSF.LE.A.3:
Observe using graphs and tables that a
quantity increasing exponentially eventually
exceeds a quantity increasing linearly,
quadratically, or (more generally) as a
polynomial function.
CCSS.Math.Content.HSF.LE.B.5
Interpret the parameters in a linear or
exponential function in terms of a context.
COMMON CORE STATE STANDARDS
MATHEMATICAL PRACTICES
CCSS.Math.Practice.MP2:
Reason abstractly and quantitatively
CCSS.Math.Practice.MP3:
Construct viable arguments and critique the
reasoning of others.
CCSS.Math.Practice.MP4:
Model with mathematics.

Lesson 8.1 Exponential Growth Day 1
This lesson is the first day of a two-day lesson on
exponential growth model. Day-one is an introductory
lesson to the concept. I will start the lesson off by asking the
students what they remember from the prior year about
exponential growth. The students would have been
introduced to the concept in their algebra 1 course with a
very basic overview of the concept. I will then give the
students the equation for exponential growth for them to
get in their notes. The equation is y=bx where b>1 to create
a growth equation. I will tell the students that if the b is less
than 1 this is a decay function which we will get to later in
the chapter. I will also define an asymptote as it relates to
the graph, that is a horizontal asymptote on the x-axis or
y=0. I will then give the students four example problems in
which we will go through graphing together. Once I will as
though the students understand how to graph exponential
growth, I will show them a more complex equation which
includes transformations. They will now have the equation
y=abx-h+k where a represents stretching and shrinking, h is
the shift left or right and k is the shift up and down. I will
then give the students one equation to graph on their own
using transformations.
After we have completed the notes for this section, I will
move to a class activity. I will post graph paper sticky notes
on the front board with exponential equations on the top of
each paper. The students will come to the front of the class
and pick a sticky note and graph it individually, which is
modeling or MP4. They will put their names on the back and
post them back on the board when they are finished. They
will then look over other answers that have been posted. If
the students have and questions or comments, they can
look at the name on the back and find the person to discuss
the answer. The students will then have to explain their
answer and their thinking which will create great
conversations about the concepts, thus using MP3 in this
activity.

Learning Targets:
I know about the exponential growth
model.
I can graph exponential growth
equations
Common Core State Standard:
CCSS.Math.Content.HSF.LE.A.1.C:
functions and interpreting situations
that need to be solved with an
exponential model.
CCSS.Math.Content.HSF.LE.A.3:
Observe using graphs and tables that a
quantity increasing exponentially
eventually exceeds a quantity increasing
generally) as a polynomial function.
CCSS.Math.Practice.MP3:
Construct viable arguments and critique
the reasoning of others.
CCSS.Math.Practice.MP4:
Model with mathematics.

Lesson 8.1 Exponential Growth Models Day 2
This lesson is the second day of exponential growth model.
On day two, we will be working with applying the growth
model to word problems. I will be giving the students a new
formula they can use for application problems to create a
function as it relates to each word problem. The new growth
model is y=a(1+r)t where a is the initial amount, r is the
growth percentage as a decimal, t is the time, and y is the
end amount. I will also explain to the students that the
quantity (1+r) is known as the growth factor. After discussing
these notes, we will move to example problems to apply the
concepts. I will give the students a word problem and have
them tell me what information goes where in the formula.
Then once we have created a function, I will have them
graph it and have them solve for an end amount after a
given time. We will do this again with another problem to
practice. I will then ask the students how they are feeling
about the material by giving me a thumbs-up, thumbs-down,
or thumbs-sideways depending on how they feel. If the
students still seem to struggle with the concept, we will go
over another problem. If the students are understanding the
concepts then we will move to an activity.
For the activity for this lesson, I will write five different
exponential growth model functions on the board. The students
will choose one then graph the function and create a real-world
problem from the equation. The students will need to come up
with a scenario that will match the function. This will use MP4 for
the graphing and modeling and MP2 for reasoning abstractly and
creating a scenario from the equation. Once the students have
their scenario, they will share their word problem with their
neighbor and the neighbor must guess which function matches
the scenario. For example, student 1 and student 2 are paired up.
The students swap scenarios with the function covered or hidden.
Student 1 must guess which function student 2 choose and
explain their thinking, then they will repeat this with student 1’s
scenario. If either of them gets it wrong, they must guess again.
This gives the students a chance to show their understanding or
where they struggle. The students can help each other verify if
their answers are right and create great mathematic

Learning Targets:
I know about the exponential growth
model.
I can use the model to solve application
problems.
Common Core State Standard:
CCSS.Math.Content.HSF.LE.A.1.C:
functions and interpreting situations
that need to be solved with an
exponential model.
CCSS.Math.Content.HSF.LE.B.5
Interpret the parameters in a linear or
exponential function in terms of a
context.
CCSS.Math.Practice.MP2:
Reason abstractly and quantitatively
CCSS.Math.Practice.MP4:
Model with mathematics.

Lesson 8.2 Exponential Decay
For the final lesson in this learning progression, we will be
covering exponential decay. I will remind the students about
when I briefly mentioned how decay relates to growth on
the first day. I will ask the students to remind me of the
exponential function we used on the first day with the
transformations, which is y=ab(x-h)+k. I will then ask the
students to define each term in their own words. This gives
me a chance to assess whether or not the students were
taking notes and understanding the material. By having
them put it in their own words, it forces the students to
show an understanding of the concept rather than just
reading from their notes. I will also remind them that for the
function to be a decay model rather than a growth model,
the b value must be between 0 and 1. If the b value is above
1, we will have an exponential growth model. I will also
discuss the long term for the graph, as the t value gets large,
then the end amount or y value will get closer and closer to
zero but will never be zero, therefore having an asymptote
at y=0 or the x-axis. As long as the students do not have any
some practice problems of graphing three different
functions with the same b value of ½ but each has different
transformations.
Once we are done graphing each example, I will ask the
students to remind me of the exponential growth model
they had learned the day before, which was y=a(1+r)t. For
decay the equation is nearly the same except the growth
factor is now a decay factor and is 1-r instead of 1+r,
therefore making the quantity less than 1. All the terms are
the same as in the growth model, however the r value is
now a decay rate as a decimal instead of a growth rate.
Thus, the decay model being y=a(1-r)t.
After comparing the growth model to the decay model, I will
give the students a class activity. I will give the students a
word problem to apply the decay model to. This will be a
three-part problem where they will write the decay model
based on the problem, then graph the model, and finally solve for the end value after 3 years.
Students will work on this individually then discuss it with their neighbor once most people are
done. Once everybody seems to be done discussing the problem, I will give them the answer to
the end value after three years to check their work. If they got this correct then their decay
model should be correct and they should be able to find this point on their graph is they look at
what the value of y is when x is 3. This allows the students to check their work without me
going over each part and giving them all the answers.

Learning Targets:
I can graph the exponential decay
function.
I can solve problems using the
exponential decay functions.
Common Core State Standard:
CCSS.Math.Content.HSF.LE.A.1.C:
functions and interpreting situations
that need to be solved with an
exponential model.
CCSS.Math.Content.HSF.LE.A.3:
Observe using graphs and tables that a
quantity increasing exponentially
eventually exceeds a quantity increasing
generally) as a polynomial function.
CCSS.Math.Content.HSF.LE.B.5
Interpret the parameters in a linear or
exponential function in terms of a
context.
CCSS.Math.Practice.MP3:
Construct viable arguments and critique
the reasoning of others.
CCSS.Math.Practice.MP4:
Model with mathematics.

# What Does the Motion of a Rolling Object Look Like? HSF.IF.B.4

Students can get easily confused when it comes to understanding variables of a graph and visualizing what that information represents. This activity will allow students to see how the path of an object moving toward and away from a given point is modeled. With the use of their TI-84 Plus graphing calculators and the Vernier CBR-2, students in small groups will study the motion of a tennis ball and a toy car rolling up and then back down a ramp. This gives them an opportunity to practice interpreting  the motion of the objects through a hands on activity.

The graphs for the motion of either object should be similar in that they are parabolas, but students will be able to see what determines their properties. At first, they will be asked to visualize and sketch what they imagine the graphs would like. Then by using the technology explained, they can see how their original assumptions compare to the graphs obtained from the motion detector. This will encourage students to critically think about an object in motion and have a better understanding of how its distance from the starting point in relation to time is represented.

Activity Worksheet-Rolling Through Motion

Practice Standard:

CCSS.Math.Practice.MP5- Use appropriate tools strategically

This activity relates to the Common Core State Standard:

CCSS.Math.Content.HSF.IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

# How does the slope change over time for a ball thrown in the air? F.IF & F.BF

This will be taught in a calculus class. Using the software and tools as shown in the following links: https://www.vernier.com/ https://www.vernier.com/products/software/lp/, the students will use the equipment to throw a ball in the air and track its position as time goes on. The students will use the data to draw a graph of position vs time for the ball and create an equation which give the position as it is related to the time. The students will then take the slope at different points and use the data to draw a slope vs time graph. They will also create a function for this graph. This will be an introduction activity to derivatives. The students will create a functions of the position vs time as well as create a function for the slope vs time. We will then go into covering the material for derivatives. The students will understand the procedure of how to create a function from a graph or data. They will also understand the concept of what a derivative is and how it relates to the real world problem I have provided. Below you will find the handout I will give the students to guide them through the activity.

The CCSS for this lesson are as follows:

• CCSS.MATH.HSF-IF.B.4: Interpret functions that arise in applications in terms of context
• CCSS.MATH.HSF-CED.A.2: Create equations that describe numbers or relationships.
• CCSS.MATH.HSF-BF.A.1: Build a function that models a relationship between two quantities.

Ball Throwing Activity

Name:

Date:

1. Set up your iPad such that it is far enough away to get a video of the ball as you throw it straight up in the air. You should make sure the ball stays in the view of the iPad when it is thrown for you should not move the iPad during the experiment.
2. Open the Vernier Probes and Software program. Take a video of the ball as you lightly toss it straight up in the air.
3. Sketch a graph of a position vs time for the ball in the space below. Create a function for this graph.

1. Take the slope of the graph at t=0, at a point before the vertex, at the vertex, at a point after the vertex before the end, and finally at the point just before the ball hits the graph. What are the units of these slopes? What does this tell you about what the number means? Sketch a graph of the slopes as a time vs slope graph. Is it linear? What does that mean? Write an equation for the graph.

# Bridge Modeling F.LE.A.1

CCSS.MATH.CONTENT.HSF.LE.A.1

Distinguish between situations that can be modeled with linear functions and with exponential functions.

The Problem:

We begin the lesson with introducing (or reminding) students of the concept of a function. For the purposes of this lesson, we will define a function as an expression containing one or more variables in which each input has one and exactly one output.

It is also understood that before continuing with the lesson, students are aware of at least the basic shape of some basic graphs: linear, quadratic, polynomial, exponential, logarithmic, and sinusoidal.

Once the understanding has been established, we introduce the lesson objective to the students: for them to create a functional model for determining the breaking weight of paper bridges by predicting a best-fit graph.

To complete their task, students are given at least 10 strips of paper measuring 11”x4”. They will also require about 40 pennies, as well as two books of about the same thickness. Students will make a one inch fold along both of the long sides to create a bridge, then suspend their bridge between the two books, as shown below.

Students will then stack pennies on their bridge. Eventually their paper bridge will fall, and students will record the number of pennies their bridge was able to hold before falling. Once that number is recorded, that paper bridge is “retired”. Students will then take two strips of paper stacked on top of each other, create the same 1-inch fold along both of the longer sides, and suspend their new 2-layer bridge between the books. Students will again begin stacking pennies on their paper bridge, recording the number of pennies their bridge was able to hold when it falls. They will continue their process for 3 layers of paper, as well as at least 4 and 5 layers but continuing to as many layers as desired.

Once the data has been collected, students will use their calculator to input the values into the scatterplot and analyze their graph to determine a best-fit equation.

Our Approach:

Our experiment went well for our first three trials, that is the 1, 2, and 3 layer bridges. We found that for the 1-layer bridge we were able to hold 8 pennies. The 2-layer bridge held 16 pennies and the 3-layer bridge held 27. The difference between the first two values is 8, while the difference between the last two is 11. The differences are close enough that it could suggest a linear regression, although with three points it is hard to tell, and we concluded further testing was required.

Our fourth trial resulted in a paper bridge capable of withstanding 68 pennies. This was a difference of 41 from the 3-layer bridge, and threw our theory of a linear regression out the window. We went back to the differences between the first few data points, and although the differences were similar we realized the first difference is smaller than the second difference. We used a calculator to plot the four points, and to create an exponential regression.

However, we were hesitant to accept this result. Was our outcome for the 4-layer bridge a fluke, or a statistically significant event? We continued creating and testing 5, 6, and 7 layer bridges, and found that they were able to hold 45, 55, and 65 pennies, respectfully. The consistency of differences immediately told us that we are not, in fact, dealing with an exponential function. We went back to our original idea of a linear function, excluding our 4th trail’s data completely.

Because of our uncertainty, we decided to determine our margin of error. We used the equation for our linear function to “predict” what our outcome should have been for the 1-3 and 5-7 layer bridges, and we determined as follows:

 Trial # # of pennies bridge held # of pennies bridge should have held (following linear equation) Difference 1 8 7.5 .5 2 16 17 1 3 27 26.5 .5 5 45 45.5 .5 6 55 55 0 7 65 64.5 .5

By adding the differences and dividing by 6 (our number of data points), we get our average difference between the points and the linear regression line of 0.5. This means that we could estimate the number of pennies a 100-layer bridge could withstand, and our number would be accurate within ½ of a penny.

# Disney World needs a new Roller-coaster! by Emily Ivie

Learning Target – I will be able to use a motion detector to match and then create a time-distance graph. Represent two numerical variables on a scatter plot and describe any correlation and/or relationship between the two variables.

Common Core Standards: CCSS.Math.Content.HSF.IF.B.4, CCSS.Math.Content.HSF.IF.C.7.

Idea of the Lesson:

When students are introduced to the idea of slopes and rates, it is important to emphasize the application of slops and rates to the real world and the technology we can use to model this. The activity will help students connect the idea between slopes and roller-coasters. Then students will use a CBR, link cable, and graphing calculator with Easy data app downloaded to create a virtual roller-coaster. The benefit of this activity is that this is a real-world example of a job you can get with a math degree. The worksheet will be presented as a mathematician’s design of a roller-coaster. The students will also have to write up a conclusive “business proposal”.

Procedures:

• Teacher will remind students of a time distance graph.
• Create groups of 2 students each.
• Students will draw their ideal roller-coaster with no flips (each student draws their own, does not need to match their partner).
• Students will use CBR with calculator to create a simulation of their roller-coaster to match the slope while their partner runs the start/stop button.
• Write up their business proposal using proper vocabulary

Activity’s worksheet

# Create your own story using double variable equations F.LE.5

Alignment to Content Standards

CCSS.Math.HS.F-LE.5
Interpret the parameters in a linear or exponential function in terms of a context.

Students will be given the equation y=2x-10. Their job is to write a story that goes along with this equation. This story will include:

-A word problem that is equivalent to y=2x-10

-A picture/graph that narrates the answer.

Students will share their story with a peer and have the peer solve their story problem.

Commentary

Its common for students panic when trying to solve story problems and not think about the procedures that relate to the words in the problem. Therefore, I am going to challenge my algebra 1 students to create their own word problems to match a specific equation. This will require planning to make sure their parameters work out correctly. Plus they will have to think of a creative way as to how mathematics applies to the real world.

Solution

There will be MANY solutions to this problem. One solution could be:

Story: Sarah is running up the stairs in her house to get to the living room. She is 10 stairs below her living room. She can run up the stairs at a rate of two stairs per second. How long will it take Sarah to get to the living room?

Equation: y=2t-10

Parameters: y=how many stairs Sarah has climbed & t=how many seconds Sarah is running.

Answer: 0=2t-10 thus t=5. It will take Sarah 5 seconds to climb 10 stairs.

Picture: (include a graph of the function y=2x-10 and a picture of the scenario)

# Evaluating Functions in a Context: CCSS.Math.Content.HSF.IF.A.2

The basketball coach at Cashmere High School is planning on buying new jerseys for his team.  There is an initial design fee of 500 dollars plus a fee of 110 dollars per jersey.  The total cost of n jerseys can be written as the function

C(n)=500+110n

• Find C(16). Explain in words what this solution means.

• Find C(20). Explain in words what this solution means.

• If the coach has a budget of \$2,000, how many jerseys can he afford?

• The coach then went to a different brand to check their jersey prices. They told him there is a design fee of \$200 and each jersey is an additional \$150.  Write a new function, F(n), that represents the total price of these jerseys.

An assessment commentary and the solutions for this task can be found in the attached document.

Assessment Item (IM Format)-2jyo2l6

# What is a root of a quadratic equation? A.APR

CCSS.MATH.CONTENT.HSA.APR.B.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

CCSS.MATH.PRACTICE.MP4  Model with mathematics.

A ball is thrown up into the air at 6 feet per second from the ground.  At what point will the ball come back down and hit the ground?  (Hint: use 16 to represent gravity)  Write out all of your steps used in solving and graph the ball’s motion.

Commentary:

Learning how to use quadratics and how to find their roots can be very confusing to students at first.  When the vocabulary words roots and zeros of quadratic functions are first introduced students often do not create their own visualization of what these terms mean.  Students also do not realize that these words are interchangeable or what the purpose of finding them is.  This task gives them the opportunity to see one application of a quadratic function and also gives students the chance to visualize what the roots of a quadratic actually are.  Teachers may use this task while teaching students about the zeroes of polynomials and to show students how to model using mathematics.  This task should be given to students after they have been introduced to the methods used in finding the roots of a quadratic function.  This task is designed to give students a tangible scenario that will help them meet the common core standards.

Solution:

B (t) = -16t^2+6t

0 = t (-16t + 6)                 Factor out a t from both                                              terms and set equal to 0                                              to find the zeros of the                                                function.

0=t,  = t                             To find the t values, set                                               each factor equal to                                                     zero and solve for t

After (6/16) seconds the ball will fall back down and hit the ground.

# Limits of a Function – HSF.IF.C.8

At first glance, what would you expect the limit of the function to be as x approaches 2? Find the limit, if it exists. Why is predicting the values of this function at 2 difficult? What would the graph look like at this point?

An assessment commentary and the solution is given in the attached document.