Author: johnsonkatie

Exponential Growth and Decay Models F.LE.A

johnsonkatie CCSS-Math Learning Progressions, HS Functions, HS Modeling

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.

Triangle Similarity G.SRT.B

johnsonkatie CCSS-Math Learning Progressions, HS Geometry, ML Ratios & Proportional Relationships


High School: Geometry
SIMILARITY WITH TRIANGLES
The class I will be teaching will be a 9th grade high school
Geometry class. The textbook used will be Geometry:
Integration, Application, Connection by Glencoe and McGraw-
Hill Companies published in 2001. This learning progression
will follow the student through meeting the Common Core
State Standards about proving theorems involving similarities
with triangles. These Common Core State Standards will be
covered in chapter 7 of the textbook, sections 3, 4, and 5. The
Standards for this learning progression will be the cluster
involving proving theorems involving similarity, which are
CCSS.Math.Content.HSG.SRT.B.4 and
CCSS.Math.Content.HSG.SRT.B.5. The math practices that are
going to be used throughout the learning progression are as
follows: MP3, MP4 and MP5.
CCSS.Math.Content.HSG.SRT.B.4:
Prove theorems about triangles. Theorems
include: a line parallel to one side of a
triangle divides the other two proportionally,
and conversely.
CCSS.Math.Content.HSG.SRT.B.5:
Use congruence and similarity criteria for
triangles to solve problems and to prove
relationships in geometric figures.
COMMON CORE STATE STANDARDS
MATHEMATICAL PRACTICES
CCSS.Math.Practice.MP3:
Construct viable arguments and critique the
reasoning of others.
CCSS.Math.Practice.MP4:
Model with mathematics.
CCSS.Math.Practice.MP5:
Use appropriate tools strategically.


Lesson 7-3 Identifying Similar Triangles
For this lesson I will start out by putting a picture under the
document camera of the pyramids in Egypt and explain how
Greek mathematician Thales used geometry for the first
time to solve for the height of the Great Pyramids. I will
write the hinge question on the white board on the side for
the students to consider throughout the lesson. The
students should be able to answer the hinge question by the
end of the lesson. The CCSS covered in this lesson is the first
part of the math standard HSG.SRT.B.5 about solving
problems for triangles with similarity and congruence.
I will follow the introduction with notes for the class in
which I will go through some example of the concepts under
the document camera for the student to copy down. I will
go through three similarities and examples for each to show
the students how they can use the similarity to solve
problems. The similarities are angle-angle (AA), side-sideside
(SSS), and side-angle-side (SAS). While going through
examples, I will be very student involved and ask students
questions as often as possible to have them solving the
problems with me. Once we have finished that, I will ask the
student if they have any questions on anything we have
done so far.
Next, we will do a hand on activity. The students will need a
ruler and protractor. Since the students will be using the
tools and drawing the triangles they will be using MP4 and
MP5. The students will draw a triangle and measure all the
sides of the triangle. They will then draw another triangle
with a scale factor of ½ of 2. They measure the angles of the
triangles to compare. They will answer the questions: Are
these triangles similar? Why? Which triangle similarity is
this? Answer: Since the sides are proportionate the triangles
are similar and this is the SSS similarity. The students
checked their answer by measuring the angles which could
be AA similarity. The students will be assigned homework in
which some problems will be basic problems directly using
the formulas while some of the problems will be real world
problems where the students will have to apply the material
and make connections to solve the problem. Therefore,
being able to answer the hinge question.
Hinge Question:
How can you use similar triangles to
solve problems?
Common Core State Standard:
CCSS.Math.Content.HSG.SRT.B.5:
Use congruence and similarity criteria
for triangles to solve problems and to
prove relationships in geometric
figures.
Angle-Angle (AA) Similarity:
If two angles of one triangle are
congruent to two angles of another
triangle, then the triangles are similar.
Side-Side-Side (SSS) Similarity:
If the measures of the corresponding
sides of two triangles are proportional,
then the triangles are similar.
Side-Angle-Side (SAS) Similarity:
If the measures of two sides of a
triangle are proportional to the
measures of two corresponding sides of
another triangle and the included
angles are congruent, then the triangles
are similar.
CCSS.Math.Practice.MP4:
Model with mathematics.
CCSS.Math.Practice.MP5:
Use appropriate tools strategically.


Lesson 7-4 Parallel Lines and Proportional Parts
I will start this activity with a warm-up related to the last
activity. I will give the students two triangles with two sides
labeled and the angle between the sides labeled as well. The
students will have to show these are similar triangles using
SAS similarity. This will be used as a way to review the
material from the day before.
I will then move onto the new material. The students will
have to prove the two theorems, 7-4 and 7-5. Similar to the
last lesson, I will walk the students through the proofs for
the theorems under the document camera. I will ask
questions to get the students involved in the proofs and
have them assisting me to solve the proof. Once the proofs
are done, I will make sure the students understand the
material and see if anybody has any questions.
I will use the rest of the class to give the students an activity.
Prior to class, I will print out an assortment of triangles with
lines through them, some parallel and some not. The
students will use rulers and protractors to make
measurements based on the theorems to determine if the
line is parallel or not. The students will put their name on
the back and tape it to the board. Once everybody has done
one problem and taped it to the front, then the students will
look at other students’ answers and discuss each other’s
answers and critique their answers and give reasons, which
is using the practice MP3.
Again, the students will be given practice problems for
homework in which some of them will be simple while
others are more challenging and will cause the students to
need to make connections and apply the concepts.
Hinge Question:
Are these lines parallel?
Common Core State Standard:
CCSS.Math.Content.HSG.SRT.B.4:
Prove theorems about triangles.
Theorems include: a line parallel to one
side of a triangle divides the other two
proportionally, and conversely.
Theorem 7-4:
If a line is parallel to one side of a
triangle and intersects the other two
sides in two distinct points, then it
separates these sides into segments of
proportional lengths.
Theorem 7-5:
If a line intersects two sides of a triangle
and separates the sides into
corresponding segments of
proportional lengths, then the line is
parallel to the third side.
CCSS.Math.Practice.MP3:
Construct viable arguments and critique
the reasoning of others.

Lesson 7-5
This lesson, I will start with a warm-up from the first lesson. I
will give the students two triangles with different sides and
angles labeled as it relates to each similarity. The warm up
will have three problems each about one of the following
similarities: AA, SSS, SAS. I will then have a follow up
question to solve for the other side using proportions. The
CCSS covered by this lesson is the second part of
HSG.SRT.B.5 about proving relationships in triangles from
similarity.
I will use the warm up to move into the new concepts. In
this lesson, the students will learn about four theorems
which come from triangle similarity. Like the other lessons, I
will use the document camera to guide the students through
examples for each theorem. We will go through the
theorems 7-7, 7-8, 7-9, and 7-10 as listed on the right. I will
go over one example for each theorem and when I am
finished I will see if the students have any questions about
any of the material covered.
We will then move to an activity where the students will
have a chance to model the concepts in a problem. The
students will use a ruler to make a diagram for the problem
45 on page 376 of the book. Two similar triangular jogging
paths are laid out in a park with one path inside the other.
The dimensions of the inner path are 300, 350, and 550
meters. The shortest side of the outer path is 600 meters.
Will a jogger on the inner path run half as far as the one on
the outer path? Explain. The students will be allowed to
work with their neighbor on this problem but they are not
allowed to move around the room. This give the students
the chance to use math practices MP4 and MP5 by using a
ruler to move the problem. The students will turn this in at
the end of class.
For the students’ homework this time they will be given a
take home quiz. There will be three matching problems to
start for the similarities AA, SSS, and SAS with three pairs of triangles and different sides or angles labeled. The next section will have theorems 7-4 and 7-5
and the students will need to show an example for each like we did in class. The final section
will have two triangles and the students will need to explain two of the theorems from today’s
lesson, that is theorems 7-7,7-8,7-9, and 7-10. This should be an easy take home quiz if the
students took notes. They will not be allowed to use the same examples as in class so they will
be showing they understand the notes.
Hinge Question:
What can similar triangles tell us about
other characteristics of the triangle?
Common Core State Standard:
CCSS.Math.Content.HSG.SRT.B.5:
Use congruence and similarity criteria
for triangles to solve problems and to
prove relationships in geometric figures.
Theorem 7-7
If two triangles are similar, then the
perimeters are proportional to the
measures of corresponding sides.
Theorem 7-8
If two triangles are similar, then the
measures of corresponding altitude are
proportional to the measures of the
corresponding sides.
Theorem 7-9
If two triangles are similar, then the
measures of corresponding angle
bisectors of the triangles are
proportional to the measures of the
corresponding sides.
Theorem 7-10
If two triangles are similar, then the
measures of corresponding medians are
proportional to the measures of the
corresponding sides.
CCSS.Math.Practice.MP4:
Model with mathematics.
CCSS.Math.Practice.MP5:
Use appropriate tools strategically.

 

Systems of Inequalities Assessment A.REI.D

johnsonkatie Assessment of CCSS-Math, HS Algebra, PreCalculus

System of inequalities (CCSS.Math.Content.HSA.REI.D.12)

 

Alignment to Content Standards

Alignment: CCSS.Math.Content.HSA.REI.D.12

Alignment: CCSS.Math.Content.HSA.REI.D.10

 

Tasks

Find the system of inequalities for the graph below. Write the inequalities in slope-intercept form.

Then, determine if the following points are in the solution set of the system of inequalities:

(0,0), (1,1), (-1,1).

 

Commentary

The purpose of this assignment is to assess whether the students understand systems of linear inequalities and the solution set. In this task, the students will be working with two major skills. They will be creating equations for two inequalities in two variables for a system of equations from a graph. The students will also be determining if a point satisfies the solution set from the system of inequalities. The students will have to have a full understanding of the concept to be able to complete this assessment since the students are given the solution and must work backward to find the system that satisfies the solution.

 

Solution

The first step to completing this assignment, the students must first create inequality equations for each line. Since the assignment says the equations must be in slope-intercept form, the student can use the plotted points to calculate the slope and find the y-intercept for each then create the formula for the equation. They must then choose a point in the solution set, that is the shaded region, and plug that into their equations and choose the correct inequality symbol to create a true statement.

For the purple line, the slope is -3 and the y-intercept is 1; therefore, the equation for the line is y=-3x+1. Then by putting a point into the equation that is in the solution region and simplifying, we can find the correct inequality symbol. For this one I put in the origin point, (0,0) and simplified.

0 ? -3*0 + 1

0 ? 1, 0 ≤ 1

Therefore, the inequality should read y ≤ -3x + 1. The inequality has the equal to part since the line is a solid line and not a dashed line. If it was dashed then the symbol would only be less then, <.

For the blue line, the slope is 2 and the y-intercept is 4; hence, the equation for the line is y=2x+4. Similarly to the purple line, by putting in the origin, (0,0), which is a point in the solution shaded region, we can find which inequality symbol to use.

0 ? 2*0 + 4

0 ? 4, 0 ≤ 4

Thus, inequality is y ≤ 2x + 4. Again, since this line is solid the symbol contains the equal to part as well as the less then, <.

Finally, the student need to determine if the three points are in the systems solution set or not. The students can do this in two different ways. They can either look on the graph and see if the point is in the shaded region then it is in the solution set. Or the students can put the point into both equations and see if the statement is true for both inequalities. If the point creates a true statement for both, then the point is in the systems solution set. Therefore, (0,0) and (-1,1) are in the systems solution set and (1,1) is not.

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

johnsonkatie HS Functions, HS Modeling, ML Functions, ML Modeling, Technology Teaching Issues

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.

 

 

 

 

 

 

 

 

 

 

Pizza Price MG.A.3

johnsonkatie HS Geometry, Picture Problems

Image result for Dominos pizza

CCSS.Math.Content.HSG.MG.A.3
Apply geometric methods to solve design problems

For this picture, I would create a problem related to area of a circle and ratios. The students would be given the diameter of three different sized pizzas then the price of each of those pizza and the students have to determine what deal is the best, that is what has the lowest price per area of pizza ratio. At Domino’s Pizza they have three carry out deals, a Medium for $5.99, and Large for $7.99, and an Extra Large for $9.99. The Medium pizza has a 12 inch diameter, the Large has a 14 inch diameter, and the Extra Large has a 16 inch diameter. This problem relates to the CCSS because it is about using geometric areas and ratios to solve a problem. This picture and problem will get the students interested because I am sure all students like pizza and would love to find the best deal for when they want to buy pizza themselves. Since high school students do not have a lot of money, finding the best deal will definitely draw their attention. The students will work with the concept of ratios and how they can relate to areas.