microsoft word - s1_syllabus.doc€¦ · web viewlarson & hostetler pre-calculus with limits...

Classroom Interactions

Title of Lesson: Down, Set, Interception!UFTeach Students’ Names: Zack Brenneman, Julie WalthallTeaching Date and Time: September 26, 2013. 1:00 – 2:45.Length of Lesson: 50 minutesCourse / Grade / Topic: Pre – Calc Honors / 11th & 12th / Rational Functions (Graphing)Source of the Lesson: Larson & Hostetler Pre-Calculus with Limits textbook

Embedding Strategies Based on Observations:Based on the readings and what happened in class, I am including the following teaching strategies with these students because… The goal of this lesson is for the students to learn and master graphing rational functions and specifically explore asymptotes. However based on observation, I am including the following strategies so that students will learn to put forth effort and become more active and engaged in learning. With these strategies, I am hoping that the students will absorb the material better from learning by doing and not just watching. One of my goals for the students includes taking responsibility for their own learning and being held accountable by peers and teachers.

Strategy Rationale for Choice Describe where in Lesson Plan this strategy would best fit

Group collaboration Exploring brand new concepts and graphs can be difficult and intimidating individually. So, allowing students to work in groups will build confidence and promote deeper thinking and effort.

During the exploration. They will be asked to graph types of functions they have never seen before. As a group however, I am confident they can discover strategies together to graph correctly.

Modeling and “Non-textbook” learning material

Students seem bored with traditional learning materials such as their textbook and note taking. Using an online simulation or graphing calculator will allow them to explore graphs in a new way with new tools.

During the engagement and exploration. The engagement will use a real life application (football) to present a graph of a function. The exploration will allow students to graph by hand and then check their work with the graphing calculator.

Questioning in Teaching Asking questions to the students and letting them provide the answers requires students to take an active approach to learning. This is crucial when learning new topics for the first time, like in this lesson.

During the exploration, the teacher will ask probing questions to each student. During the explanation, the teacher will have students come to the board and fill in some blanks from the worksheet. While doing so, the teacher will ask them questions to ensure the students learn the correct reasoning behind the solutions.

Common Core State Standards (CCSS) / Next Generation Sunshine State Standards (NGSSS):

Standards Number Benchmark Description Cognitive Complexity


MACC.912.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Level 2: Basic Application of Skills & Concepts

MACC.912.F-IF.3.7 Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available.

Level 2: Basic Application of Skills & Concepts

Concept Development:Information from the Textbook listed above in “Source of the Lesson”.Chapter 2: Section 6: Rational Functions.

A Rational Function can be written in the form f(x) = (N(x)/D(x)) where N(x) and D(x) are polynomials. D(x) is not the zero polynomial. The domain of a rational function will be all real numbers except the x values that make the denominator zero. When a value is determined to be excluded from the domain, we must look at the behavior of the function near that value. It will be noted that near this value, the function will approach different values from different directions, and thus it can be said that the function is not continuous. It will be observed when looking at the graph of a rational function; they will most likely have asymptotes. 1. The line x = a is a vertical asymptote of the graph of f if f(x) ∞ or f(x) -∞ as x a, either from the right or from the left.2. The line y = b is a horizontal asymptote of the graph of f if f(x) b as x ∞ or x -∞To determine the types of asymptotes of a Rational Function:Let f be the rational function be given by f(x) = (N(x)/D(x)) where N(x) and D(x) have no common factors.

1. The graph of f has a vertical asymptote at all the zeros of D(x)2. The graph of f has a horizontal asymptote determined by comparing the degrees of N(x) and D(x).

a) If the higher degree (power) is in the denominator, then the graph f has the line y = 0 (x axis) as a horizontal asymptote.b) If the degree (power) is the same for the numerator and denominator, the graph f has the line y = a/b (ratio of the leading coefficients of the highest degrees) as a horizontal asymptote. c) If the higher degree (power) is in the numerator, then the graph f has no horizontal asymptote.

Consider the rational function described in c) above, and the denominator has a degree of 1 or greater. If the degree of the numerator is exactly one more than the denominator, the graph of the function has a slant, or oblique, asymptote. This section is aimed at graphing rational functions and identifying asymptotes. As well as identifying previously learned concepts such as domain, zeros, intercepts, and continuousness.

Performance Objectives Students will be able to graph polynomial functions. Students will be able to find the domain of a polynomial function. Students will be able to decide whether a polynomial function is continuous or not continuous. Students will be able to identify x and y intercepts for polynomial functions.


Students will be able to identify any vertical or horizontal asymptotes for polynomial functions.

Materials List 27 Graphing Calculators (provided by Newberry High School) 27 copies of Exploration Worksheet 1, 27 copies of Exploration Worksheet 2, 27 copies of the Post

Assessment PowerPoint projection onto Smart Board

Advance Preparations In advance, the teacher will have a PowerPoint prepared with slides for the Engagement, Exploration,

Explanation, and Elaboration. In advance, the teacher will print out copies of Exploration worksheet 1, Exploration worksheet 2, and

Post Assessment for every student (27 copies). In advance, the teacher will make sure there are enough working, fully charged, calculators for every

student (27). Groups will be assigned by proximity (unless Master Teacher has overriding recommendations) Materials will be distributed only at the time of need, and will be passed out by the teacher and

observing teacher. There is no preparation in the PowerPoint other than projecting it onto the Smart Board. (No animations,

applications or links are necessary)

Safety I anticipate potential problems with the calculators. Students will be instructed to keep the calculators on their desks and use them for intended use only No other significant concerns


ENGAGEMENT Time: 3 MinutesWhat the Teacher Will Do What the Teacher Will Say

(include Probing Questions)Student Responses and Potential Misconceptions

[Teacher will have a PowerPoint set up before class and have the first slide ready]Switch to Slide 1 of the Power Point: Teacher will introduce himself/herself

Good afternoon everyone! My name is Miss Julie Walthall / Mr. Zack Brenneman and I’m from UF. We will be learning about Polynomial and Rational Functions today.

Hello Julie / Zack

Switch to Slide 2 of the Power Point and introduce the football engagement.

How many of you are Gator football fans? Any Seminoles out there? I sure hope not!

Me!

Teacher will present the engaging scenario and pose a question to be explored.

So here’s the scenario: it’s the rivalry football game Florida versus Florida state. The Gators are up 50-0, but the Seminoles are on offense and about to score (points to picture)! Luckily, the Gators intercept the attempted touchdown pass just in time and run it back. The interception happens right here (points) at the end of the curved black line. The player will run around on the left side headed towards the end zone on the path defined by the equation y = (10/x). Assuming he must follow y = (10/x), will he step out of bounds before reaching the end zone?

No! Yes! He looks out right now!

Teacher will transition to the exploration.

This is the question we will investigate today!

Great! I am curious as to if the Gators will score a touchdown before stepping out of bounds.

EXPLORATION Time: 25 MinutesWhat the Teacher Will Do What the Teacher Will Say

(include Probing Questions)Student Responses and

MisconceptionsTeachers will place students in groups of 4 or 5.

Before we start the activity, you all will be placed in to groups of 4 (maybe 1 or 2 groups of 5). (Teachers will place students in groups based on their seating location.)

Do we get to choose our groups?

Teacher Response: We will group you based on who you are sitting near.

Teachers will switch to Slide 3 of the Power Point and show students

Each of you will be receiving a worksheet to work on in a bit. This

I don’t have a worksheet.


the worksheet which they will be working on.

is an enlarged version of the worksheet.

Teacher Response: You all will receive a worksheet shortly.

Teachers will read instructions. Below are eight blank coordinate planes and four equations. There are two coordinate planes which correspond to one equation. Your goal is to plot the given equation on the aligned left coordinate plane. Do not plot or draw in the right coordinate plane. We will use the right coordinate plane in a bit.

What will we use the right coordinate plane for?

Teacher Response: It is a surprise. We will get to it later.

Teachers will pass out the Exploration worksheet #1 (which includes 4 given equations followed with 8 blank coordinate planes.)

I will be handing each of you a worksheet that pertains to our main activity. Do not start working on this yet. We will be doing number 1 together.

Misconceptions:

Students start on the worksheet anyway.

Teachers will give an example of how to complete the activity. They will do this by only going over the first question.

For the first question, I have the equation f(x) = 1/(x-3). Any ideas on how I could graph this equation?

[Start with x=-2, then find f(x). Afterwards, use x=-1 and find the following f(x) value. Then just repeat using x=0,1,2,…]

The equation f(x) = 1/x looks like “this”. Now all you need to do is move the graph over 3 units to the RIGHT since 3 is being subtracted from x]

Teachers will graph the first equation.

What I will do is choose a few x-coordinates and find their corresponding f(x) value. So for:X = -8, f(x) = - 1/8X = -2, f(x) = 1/5X = 3, f(x) = undefined.X = 4, f(x) = 1X = 7, f(x) = 1/4

Now I am just going to connect the dots and this is what I am expecting the function to look like.

What does it mean for f(x) to be undefined?

Teacher Response: f(x) being undefined implies there is no y value for the x value 3. This is because we can never divide by 0.

Teachers will ask students if they have any questions regarding this activity.

You will be doing a similar process for the remaining 3 functions. What questions do you have about this assignment?

[Are we supposed to guess as to what the graph reads on the left coordinate plane, or do you want us to actually plot.]

Teacher Response: Either is fine. You want to resemble what the graph will look like as best as you can.


Teachers will let students begin working on the assignment.

You have 10 minutes to complete this activity and you may work with your group mates. Go!

Misconceptions:

Students work with other students outside their group.

Teachers will ask probing questions while students are working on the activity.

What approach are you using to graph each equation?

[I am choosing a random x value and finding the corresponding f(x) value. Then I am just connecting the dots.]

Teachers will ask probing questions.

Why do you feel your graph represents the given equation?

[I know that every point I have graphed on the coordinate plane accurately portrays an ordered pair with the given equation. So I feel confident my graph represents the given equation correctly.]


(If students are completely lost and have no idea about how to start the activity): What if you plugged in the value 0,1,2,… for the x term. What corresponding y value would get?

I don’t see a y value. Do you?

Teacher Response: f(x) is another way to say the y value. In either case, it is the output of the equation. (x is the input)


(If a certain equation has many terms): What could I do to simplify the expression.

Factor and eliminate (hopefully)!

After 10 minutes have passed, teachers will gain the student’s attention and demonstrate the second part of the activity.

Going back to the first equation, I took an educated approach to determine what the graph would look like. At this point in time, we will be using the right coordinate plane. I will be using the TI – 80 *something* to see what the entire and correct graph looks like.

Note: (Teachers will plot this on the smart board and demonstrate how to use the calculator to plug in the equation).

Will we be plotting the graph as well? Does every student get their own calculator?

Teacher Response: Yes, you will be plotting the correct graph on the right coordinate plane for equations 2, 3, and 4. Also; every student will get their own calculator.

Teachers will explain the second part of the activity and pass out calculators to each student.

For this next part, each of you will be given a calculator. Next, you will place the given equation in the calculator and see what the graph looks like. Afterwards, you will draw the correct graph on the other grid provided. We have just completed the first graph, so once you receive your calculator, you may begin with the second graph.

Misconceptions:

Students think only 1 calculator is provided for every group.

How do I plot the graph again?

Teacher Response: Go to the calculator’s y= screen, type in the equation, and hit graph. To find the coordinates of the actual graph, hit


2nd and graph. That will give you every x and f(x)/y value.

After 5 minutes have passed, teaches will switch to Slide 4 of the Power Point, pass out Exploration Worksheet #2 and go over the next part of the activity.

Each of you will be given another worksheet. Do not start working on this until we have gone over the first question together.

Misconceptions:

Students start working on the worksheet without teachers going over instructions.

Teachers will read question 1 of the 2nd worksheet.

Let’s look at these questions and apply it to the first graph. What is the domain of this graph?

[(- ∞ , 3) U (3 , ∞) (Interval Notation.)]

[In words: From negative infinity to 3 (not included) united with 3 (not included) to infinity.]


Decide whether or not each graph has any holes, gaps, or is broken in to more than 1 line.

[The graph is broken up in to more than 1 line. So there is a gap.]


Find the coordinates where the graph crosses the x-axis. Find the coordinates where the graph crosses the y-axis.

Crosses the x – axis: NEVER (since y can never be 0)

Crosses the y – axis: (0 , - 1/3). (When x is 0, y is – 1/3).


Are there any imaginary lines where the lines of the graph do not touch/cross? If so, where are these imaginary lines located?

Yes! Imaginary lines take place when x = 3 and when y = 0.

Teachers will ask students if they have any questions.

So now that we have completed all the questions pertaining to the first function, you will answer the remaining questions for the other 3 functions. What questions do you have?

How much time do we have?

Teacher Response: You will have about 5 minutes to complete this.

Teachers will allow students to begin.

You will have 5 minutes to complete this second worksheet. You may begin.

Let’s do disssss!

After 5 minutes have passed, teachers will gain the students attention and begin the Explanation.

May I have everyone’s attention? We will be going over the second worksheet now.

Misconceptions:

Students continue to work on the activity.

EXPLANATION Time: 15 MinutesWhat the Teacher Will Do Teacher Directions and

Probing/Eliciting QuestionsStudent Responses and

MisconceptionsUsing the same Power Point slide as before (Slide 4), teachers will ask 4 students to come to the smart board and fill in data for the 2nd

I need 4 people (preferably one from each group) to come up to the smart board and fill in questions #1-4 for the 2nd graph. Who wants

[#1: Domain: (-∞,1) U (1,∞)#2: The graph is broken up in to more than 1 line. So there is a gap.#3: x-intercept: (1/3,0), y intercept


graph. [Displays Exploration Worksheet #2].

to share what they got? (0,1)#4: Imaginary lines exist! At x=1 and y=3.]

Teachers will ask the class if they agree with the data.

Do we agree with the data representing the 2nd graph? Who has something different?

Yes or No. Depends on Students answers.

Teachers will ask 4 different students to come to the smart board and fill in data for the 3rd graph.

I need 4 people (preferably one from each group) to come up to the smart board and fill in questions #1-4 for the 3rd graph. Who wants to share what they got?

[#1: Domain: (-∞,2) U (2,∞)#2: The graph is broken up in to more than 1 line. So there is a gap.#3: x-intercept: NONE… y-intercept: (0,-1/4)#4: Imaginary lines exist! At x=2 and y=0.]


Do we agree with the data representing the 3rd graph? Who has something different?


Teachers will ask 4 different students to come to the smart board and fill in data for the 4th graph.

I need 4 people (preferably one from each group) to come up to the smart board and fill in questions #1-4 for the 4th graph. Who wants to share what they got?

[#1: Domain: (-∞,-3) U (-3,3) U (3,∞)#2: The graph is broken up in to more than 1 line. So there is a gap.#3: x-intercept: (4,0) & (-2,0)… y-intercept: (0,- 8/9)#4: Asymptotes: x=-3 and y=1]


Do we agree with the data representing the 4th graph? Who has something different?


Teachers will Switch to Slide 5 of the Power Point and go over terminology and the definitions regarding the first question from the worksheet. [Domain]

If we define a function f to assign each element x in the set A exactly one element y in the set B, then the set A is the domain (or set of inputs) of the function f.

Misconceptions:

Students think that all x values are the domain, even if an element x has no output y.

Teachers will ask students if they know the term(s) associated with question 2.

Any ideas on what the 2nd question is asking. Who knows the term which tell us whether or not a graph has no breaks, holes, or gaps?

I have no idea.

[A Continuous function!]

Switch to Slide 6 of the Power Point. Teachers will go over terminology and the definitions regarding the first question from the worksheet. [Continuous/Non-continuous Graphs]

A polynomial function is continuous if the graph of a polynomial function has no breaks, holes or gaps.

Misconceptions:

Students may claim a function is continuous, not realizing that the function is not a polynomial function.


Any ideas on what the 3rd question is asking? Who knows the term(s) which tell us when the function crosses the x and y axis?

I have no idea.

[x and y intercepts!]

Switch to slide 7 of the Power The x-intercept of a line is the point Misconceptions:


Point. Teachers will go over terminology and the definitions regarding the first question from the worksheet. [x-intercepts/y-intercepts]

at which the line crosses the x-axis.

NOTE: Teachers should allow students to state what goes in the blanks:x-intercept = (x , 0)y-intercept = (0 , y)

Students think the x-intercept is when the line crosses the y axis.


Any ideas on what the 4th question is asking? Who knows the term(s) which tells us if a function has imaginary lines where the lines of the function do not touch or cross?

I have no idea.

[Vertical/Horizontal/Slant Asymptotes!]

Switch to slide 8 of the Power Point. Teachers will go over terminology and the definitions regarding the first question from the worksheet. [Vertical/ Horizontal Asymptotes.]

A vertical asymptote occurs when there is no input for a given output. (There is no x value for a given y value.)

A horizontal asymptote occurs when there is no output for a given input. (There is no y value for a given x value.)

Asymptote! I knew there was a word for it!

Misconceptions:

Students may mix up the order of the x and y values when finding the asymptote line.

Teachers will ask the stated question before switching to Slide 9 of the Power Point. Teachers will further discuss Vertical Asymptotes.

Who knows how to find the Vertical Asymptote when given an equation?

NOTE: Teachers will elaborate if students do not know.

[Vertical Asymptotes occur whenever the expression is undefined (that is where the denominator equals zero).]

Teachers will ask the stated question before switching to Slide 10 of the Power Point. Teachers will further discuss Horizontal Asymptotes.

Who knows how to find the Horizontal Asymptote when given an equation?

NOTE: Teachers will elaborate if students do not know.

[Horizontal Asymptotes occur in 3 different cases.

Case 1: If the degree of the numerator is GREATER than the degree of the denominator, no horizontal asymptote exists.

Case 2: If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the fraction created by the leading coefficients.

Case 3: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line y=0. ]

Teachers will ask the stated question before switching to Slide 11 of the Power Point. Teachers

There is one more type of asymptote. Anyone know what it is? (If students know), who knows

A slant/oblique asymptote!

If the degree of the numerator is


will further discuss Slant/Oblique Asymptotes.

how to find a slant asymptote?

NOTE: If students do not know the name of the other asymptote, teachers will explain Slant Asymptotes.

exactly one more than the degree of the denominator, the graph of the function has a slant asymptote.

Switch to Slide 2 of the Power Point. Teachers will refer back to the opening engagement question and ask a question regarding asymptotes.

Going back to our football scenario, let’s say the interceptor for the Gators ran the route given by the equation y=10/x If this was an infinitely long field, would he ever go out of bounds?

[No, he would never go out of bounds because there is a vertical asymptote at x=0. (And the side line is the line x=0).]

ELABORATION Time: 2 MinutesWhat the Teacher Will Do Probing/Eliciting Questions Student Responses and

MisconceptionsTeacher will ask students to take out a sheet of paper

Everyone please take out a sheet of paper and then turn your attention to the Smart Board.

Switch to Slide 12 of the Power Point. Teacher will explain the instructions for the elaboration activity.

For this activity, I would like you to look at the graph on the board and write down the corresponding function equation.

y = 2x 2 – 5x + 2 2x2 – x – 6

Teacher will walk around the room and ask probing questions

What do you know about this graph?

Its in two partsThe left part doesn’t touch the x or y axisIt has an asymptote [It has x and y intercepts][It has both a vertical (x=-3/2) and horizontal (y=1) asymptote][It has a hole at x=2][The domain is all real numbers except x=2]

What does that hole tell you about the equation?

The domain does not include that point[The domain is all real numbers except x=2][x=2 will make the equation undefined (will yield a zero in the denominator)]

Is this function continuous? Yes[No]

What does that tell you about the equation?

I don’t know[It will have a domain other than all real numbers]

I see you have ____ written. How do you know the equation will

(Circumstantial)


contain ____?EVALUATION Time: 5 Minutes

What the Teacher Will Do Assessment Student ResponsesTeacher will pass out evaluation worksheet

Good work today everyone! One last thing before you go! Please take the remainder of the class to complete this problem I’m passing out to you.

Do we have to?[Yes]

Teacher will read instructions aloud Please work alone.Please turn in before leaving class today. Please show all your work. Do your best to answer the questions that follow from the function equation.

What if I don’t finish?[Please do your best to finish and turn in what you have completed

Exploration Worksheet #1

Instructions: Below are eight blank coordinate planes and four equations. There are two coordinate

planes which correspond to one equation. Your goal is to plot the given function on the aligned left


coordinate plane. Do not plot or draw in the right coordinate plane. We will use the right coordinate

plane in a bit.


Exploration Worksheet #2

What is the domain of each graph?


1) __________________________ 3) ___________________________

2) __________________________ 4) ___________________________

Decide whether or not each graph has any holes, gaps, or is broken in to more than one line.

1) __________________________ 3) ____________________________

2) __________________________ 4) ____________________________

Find the coordinates where the graphs cross the x-axis and y-axis.

1) x-axis: _________________________ y-axis: ___________________________

2) x-axis: _________________________ y-axis: ___________________________

3) x-axis: _________________________ y-axis: ___________________________

4) x-axis: _________________________ y-axis: ___________________________

Are there any imaginary lines where the lines of the graph do not touch/cross? If so, where are these imaginary

lines located?

1) ______________________________ 3) _______________________________

2) ______________________________ 4) _______________________________

Exploration Worksheet #2 [ANSWER KEY]

What is the domain of each graph?


1) (-∞ , 3) U (3 , ∞) 3) (-∞ , 2) U (2 , ∞)

2) (-∞ , 1) U (1 , ∞) 4) (-∞ , -3) U (-3 , 3) U (3 , ∞)

Decide whether or not each graph has any holes, gaps, or is broken in to more than one line.

1) There is a gap; hence, NOT continuous. 3) There is a gap; hence, NOT continuous

2) There is a gap; hence, NOT continuous. 4) There is a gap; hence, NOT continuous

Find the coordinates where the graphs cross the x-axis and y-axis.

1) x-axis: NONE y-axis: (0 , - 1/3)

2) x-axis: (1/3 , 0) y-axis: (0 , 1)

3) x-axis: NONE y-axis: (0 , - 1/4)

4) x-axis: (4 , 0) & (-2 , 0) y-axis: (0 , 8/9)

Are there any imaginary lines where the lines of the graph do not touch/cross? If so, where are these imaginary

lines located?

1) Yes, imaginary lines exist. (x=3 and y=0) 3) Yes, imaginary lines exist. (x=2 and y=0)

2) Yes, imaginary lines exist. (x=1 and y=3) 4) Yes, imaginary lines exist. (x=-3 & 3 and y=1)

Rational Functions: Post Assessment Name:__________________________________

**Please turn in before leaving class today. **Please SHOW YOUR WORK


h(x) = -1 . x + 2

1. Graph the function:

2. Find the domain of the function: ___________________________________________________________

3. Decide if the function is continuous: _______________________________________________________

4. Identify ALL intercepts: _____________________________________________________________________

5. Identify any horizontal AND vertical asymptotes: ________________________________________Rational Functions: Post Assessment Name: KEY

NOTE (To Mrs. Weber): Questions #1 - #5 reflect the 5 performance objectives respectively.


h(x) = -1 . x + 2

1. Graph the function:

2. Find the domain of the function: all real numbers x except x = 2

3. Decide if the function is continuous: not continuous

4. Identify ALL intercepts: y intercept: (0, -1/2)

5. Identify any horizontal AND vertical asymptotes: horizontal: x = -2. Vertical y = 0.

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