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:

which is about modeling exponential

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:

which is about modeling exponential

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.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

conversations about exponential growth models.

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:

which is about modeling exponential

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

questions about this equation, we will move onto doing

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:

which is about modeling exponential

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.

CCSS.Math.Practice.MP3:

Construct viable arguments and critique

the reasoning of others.

CCSS.Math.Practice.MP4:

Model with mathematics.