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New York State academy for teaching and learning

Learning Experience

Informational Form-Final: 01 May 2012

Name: Anthony J. MussariLocation: Buffalo, New YorkE-Mail: [email protected]

Current Teaching Position: Student TeacherGrade Level: Tenth and Eleventh (All Boys)Name of School: St. Joseph's Collegiate InstituteCooperating Teacher: Thomas QuagilianaSchool District: Independent College-Prep School Address: 845 Kenmore Ave Buffalo, NY 14223School Phone: 716.874.4024

Title of Learning Experience: Graphing Logarithmic Functions with Transformations. Grade Level: Ten and Eleven (Mixed Grade Level) Common Core Learning Standard: Mathematics Domain/Level: Functions. Building Functions. (F.BF.3)

Standard F-BF.3.Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Peer Review Focus Question: Besides graphing how can students demonstrate clear conceptual understandings of the effects on graphs due to transformations?

Learning ContextPurpose and Rational

The ability to translate a given equation into an image expedites the process of finding a solution to a given problem. Moreover, when learners see a graphic image and receive some key facts about the graph, with an understanding of transformations, one is able to derive the equation for the image. This process of translation and transformation is a necessity in the informational age. Overall, graphs allow a learner to solve mathematical equations, physical problems, as well as make logical inferences with an image. Lastly, graphing is an essential part on the New York State Algebra II and Trigonometry Regents Exam, to prevent disservice to the learner as well as to insure proper preparation for the exam graphing with transformations ought to be addressed.

Enduring Understanding:

Graphing with transformations can generate images on a host of parent functions. Graphs allow data to be represented visually and opportunities to find and compare

trends, to make logical inferences.

Essential Question:

What will happen to the graphs of y i=f (x ) when a constant k i is introduced?

Guiding Questions:

What are the characteristics the parent function y i=f (x )? How does one read a graph? Where does the constant k i translate the image? When is it appropriate to use transformations?

Congruency TableCourse Name and Grade Level Domain/Level NYS Common Core Learning

Standards

Anthony J. Mussari 01 May 2012 pg. 2

Algebra II Trig. Grades 10 and 11 F.BF.3 Mathematics (Page 59).Standard: F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Instructional Task Learning Objective Student Work Assessment Tool

Using lecture, modeling, and examples, found within the Guided Note Packet ,

state the definition of logarithm, and the three unique parts of a logarithmic equation. Moreover, detail how to represent a function f(x) in both

logarithmic and exponential forms

The learner will be able to define the three unique

parts of a logarithmic equation

In Addition, The learner will be able to solve problems by using

function notation and changed forms.

Specific guided examples, found within

the guided notes, of how to solve problems

by using function notation and how to evaluate logarithmic equations in changed

form.

Since the learners are expected to copy, store, and maintain the

guided notes, the rational assessment to implement is to

collect each of the learners note books at the end of each quarter

and give a grade for completion. The Learner’s

notebooks must list and identify the effects on graph of f(x)=lo by replacing f(x) with f(x) + k, k∙f(x), f(k∙x),

and f(x + k) for specific values of k. (both positive and negative);

and how to find the value of k given the graphs. In addition, how

to experiment with cases and illustrate an explanation of the

effects on the graph using technology.

Using lecture, modeling, and examples, found within the guided notes illustrate

the effects, on the graph of

f ( x+k1 )+k2=logb ( x+k1 )+k2

For the specific values k1=0 , and

k 2=0.(Introduce this function as the

Parent Function)

The learner will be able to list the characteristics of the Parent Function when f(x) is replaced

with the composition of f(x) + k and f(x + k) for specific values k1=0 ,

and k 2=0.

A graph off ( x )=logb x is

sketched in the guided note packet with the

given specific values of ki

A list is then made asserting the six main characteristics of the

effects of specific values of ki using f ( x )=logb x.

Using the In Class Worksheet illustrate the effects, on the graph of f(x) by

replacing f(x) with f(x) + k, k∙f(x), f(k∙x), and f(x + k) for specific values of k

such that k i ≠ 0.

With aid the learner will be able to identify the

effects on graphs of f(x) by replacing f(x) with f(x) + k, k∙f(x), f(k∙x),

and f(x + k) for specific values of k.

Assist, if need be, the learners in the

completion of the In Class Worksheet

Assign the Take-Home Worksheet for homework.

Independently, at home, the learner will be able to

identify the effects on graphs of f(x) by

replacing f(x) with f(x) + k, k∙f(x), f(k∙x), and

f(x + k) for specific values of k.

Complete theTake-Home Worksheet

The Take-Home Worksheet is collected the following class and given a grade of either

Complete/SatisfactoryComplete/UnsatisfactoryIncomplete/Satisfactory

Incomplete/Unsatisfactory

Assign Logarithmic Quiz # 3.8

Independently, in class, the learner will be asked to identify the effects on

graphs of f(x) by replacing f(x) with f(x) + k, k∙f(x), f(k∙x), and

f(x + k) for specific values of k, without Aid.

Complete the Logarithmic Quiz # 3.8

The Logarithmic Rubric will be used to assess the students ability to identify the effects

on graphs of f(x) by replacing f(x) with f(x) + k, k∙f(x), f(k∙x), and f(x + k) for specific

values of k.

Class Background


This learning experience is designed to address two class periods of Algebra II and Trigonometry honors students, of mixed grade levels (tenth and eleventh) at Saint Joseph's Collegiate Institute, in Buffalo New York. The three sections of students, totaling sixty-five learners, share the same set of classroom rules. There are no students with special needs and therefore no need for modifications environment, or instruction type. Moreover, this Learning Experience is developed with modifications in place which, are specifically identified, rationalized, and supported in the Modification Table (pg.17), to insure every learner is successful.

Overview of Prior, Current, and Post KnowledgePrior to Implementation:

The effects on a familiar graph of f (x) by replacing f (x) with:f ( x )+k ,kf (x), f (k x), and f (x+k ) for specific values of k .

How to identify the parent function given y i=f (x ). How to graph a function given y i=f (x ) using a table of values.

During Implementation:

The characteristics of the logarithmic function y i= f ( x )=logb x. How to construct a table of values. How to graph a function.

After Implementation:

The effects on y i=f ( x )=logb x when replacing f (x) with:f ( x )+k ,kf (x), f (k x), and f (x+k ) for specific values of k .

How to construct a transformations table of values. How to graph the function y i=f ( x )=logb x after all transformations have been completed.


VocabularyFunction A rule or relationship that assigns to each number xin the function's domain a

unique number f ( x ) .

Graph A visual model of a mathematical relationship.

Image The resulting point or set of points under a given transformation; in any function f , the image of x is the functional value f ( x )corresponding to .

Domain The input value of a function, or independent variable.

Range The output value of a function, or dependent variable.

Notation

Symbol Definitionf ( x ) The value of the function when evaluated at x.

k The mathematical notation for a constant it can be either positive or negative.* ).

¿ Less-than> Greater-than⟶ Implies, Then

*N.B. c is not used as it stands for eitherCircumference, C, or in the Theory of General

Relativity, c, is used for the speed of light in Einstein’s world famous equation, E=mc2.


Assessment PlanDiagnostic:

As this is the first time the logarithmic functions are introduced the diagnostic assessment is informal where general questions about the logarithmic function* are asked by the instructor and orally responded to by the learners. An oral diagnostic takes place to mitigate the influence over poor grades, and prevents setting the learners up for failure.

* The questions for the oral diagnostic, (answers provided in the procedure section).

Has anybody heard of a Logarithmic Function before? Does anybody remember what the Exponential Function looks like? Which line do you reflect a function over to find its inverse image? If an Exponential Function is the inverse to the Logarithmic Function what should the

image of a Logarithmic Function look like?

Formative:

The students will actively participate during the lecture by responding to individual issues** when asked. Moreover, the learner completes the guided note packet during the lecture. Please note the learner's completed notebooks are collected at the end of the quarter and receive a grade. The questions to be utilized during the guided not lecture are:

**The questions for the formative diagnostic, (answers provided in the procedure section).

What is the difference between a logarithm and an exponent? What is the difference between Exponential and Logarithmic form? How does one find the inverse of a function algebraically? What base do common logarithms have? What base do natural logarithms have? Why do we change from Logarithmic form into Exponential form when graphing

Logarithmic Functions? What is the difference between a function of “x” and a function of “y”? If the base of a Logarithm is greater than one, does the function increase or decrease? If the base of a Logarithm is greater than zero but is also less than one, does the function

increase or decrease? What point is a characteristic or featured in a Logarithmic Function? What is the domain of a Logarithmic Function? What is the Range of a Logarithmic Function? Does a Logarithmic Function have an Asymptote?


Summative:

The students are given a homework assignment, which specifically addresses the topic of graphing logarithmic functions with transformations. The assignment is collected at the beginning of the next class after the opportunity for questions and corrections. The assignment is given either a complete (reasonable effort, both questions answered) or incomplete (less-than reasonable effort, only one or no questions answered) grade. After the home work is turned in issue the formal assessment (Quiz # 3.8 ) allow for tem minutes to pass(fifteen minutes max.), collect and grad against The Logarithmic Rubric.

Rubric AlignmentStandard: F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Distinction Level Description Alignment to Indicator

DistinguishedOr

Point Level (4)

A correct graph is drawn and all

calculations are correct

The leaner finds ( Solves ) and identifies ( Sketches ) all of the effects on the graph by replacing f(x) with k1f(x+k 2) + k3,for specific values of k i (both positive and negative;, by producing the correct equation for the image of a given

function under the effects of k i.

ProficientOr

Point Level (3)

A graphing error is made or one

mathematical error

The leaner f inds ( Solves ), all the effects on the values of the graph by replacing f(x) with k1f(x+k 2) + k3, for

specific values of k i (both positive and negative); but is unable to identify ( Sketch ) any of the effects of k i on the

image of f ( x )=k1f(x+k 2) + k3.

Developing +Or

Point Level (2)

More than one mathematical error is

made

The leaner f inds ( Solves ) some of the effects on the values of the graph by replacing f(x) with k1f(x+k 2) + k3

for specific values of k i (both positive and negative); and is unable identify ( Sketch ) any of the effects of k i on the

image of f ( x )=k1f(x+k 2) + k3.

DevelopingOr

Point Level (1)

Completely incorrect, irrelevant, incoherent, or a correct response

with obviously incorrect procedure

The leaner is unable to f ind ( Solve ) any of the effects on the values of the graph by replacing f(x) with

k1f(x+k 2) + k3, for specific values of k i (both positive and negative); and is unable identify ( Sketch ) any of the effects of k i on the image of f ( x )=k1f(x+k 2) + k3.


The Logarithmic Rubric:Name:________________________________________

Value Specific Criteria4 A correct graph is drawn and all calculations are correct3 A graphing error is made or one mathematical error2 More than one mathematical error is made1 Completely incorrect, irrelevant, incoherent, or a correct response with obviously

incorrect procedure

The grade is calculated by dividing the value earned by four. The Quotient is then multiplied by 100 to establish the learners received percentage. No rounding is necessary.

Total Earned:_______

Grade:_____________

Comments:

Homework counts for forty percent of the learner's overall quarter grade. Therefore, it is to the learner's benefit to establish a proficient or distinguished grade for all the homework.


Student WorkClassifications and Distribution

Classifications:

Distinguished Classification requires the learner to earn a rubric score of four. Proficient Classification requires the learner to earn a rubric score of three. Developing Classification requires the learner to earn a rubric score of one or two.

Distributions

Pre Assessment:

Classifications Number of students Percentage of studentsDistinguished 0 00.00 %

Proficient 3 04.60%Developing 62 95.40 %

Post Assessment:

Classifications Number of students Percentage of studentsDistinguished 43 66.15 %

Proficient 13 20.00 %Developing 9 13.85 %

Graphic Representation of the Pre and Post assessment Data:


Developing

Proficient

Distinguished

0 10 20 30 40 50 60 70

Number of Students

Clas

sific

ation

s

Post

Pre

ConclusionPre –Assessment Data (

xr ¿Post-Assessment

Data (xs ¿Descriptive Statistics

xr ≈ 1.6308 xs ≈ 3.4923

Null (H 0)∧¿Alternative(H a)Hypotheses:

{H 0: xr=xs(instruction isineffective)H a : xr ≠ xs( instructionis effective )

sr ≈ 0.5747 ss ≈ 0.8125

Using A Two Tail Testy, With a Level of Significance 𝛼=0.01or with 99% certainty we find our critical

region bound by → Zα=± 2.575

nr=65 ns=65Zx=

(( xr−xs )−( μr−μs ))

√ sr

1+

ss

ns

¿

*In this Calculation we must assume that ( μr−μs )=0 because of the null hypotheses,xr=x s.

Zx=( (1.6308−3.4923 )−(0 ) )

√ 0.57471

+ 0.812565

≈−3.1904Results:

Since Zx<Zα there is significant evidence to reject the null hypotheses and conclude, instruction is

effective.

Using the graphic representation compiled from the raw data one finds that instruction appears to be effective. In addition when viewing the collected data one sees about 86% of the learners demonstrated their understanding at either a distinguished or proficient level, this also alludes to the same conclusion, as the graphic data, or that instruction appears to be effective.

However, the learners still at the developing level require further direct instruction to mitigate the magnitude of conceptual misunderstanding. Help is urged and available for the learner to seek either before or after school based on whichever is more convenient for the learner and / or their guardian to arrange.


ProcedureCurricular Placement

This year we have covered the transformations on lines, parabolas, exponential functions, and now we are moving to logarithmic functions

Logarithmic and Exponential Functions Unit Time LineSection 1 2 3 4 5 6

TopicExponents, rational and

negative

Radical Equations

and Equations

with Exponents

Exponential Functions

Logarithmic Functions

Section forLearning

Experience

Logarithmic Rules

Applications of

Exponential Functions

Class Time Required 2.5 Days 2.0 Days 3.0 Days 2.0 Days 1.0 Day 2.5 Days

Diagnostic Assessment:

The questions and answers for the oral diagnostic,

Has anybody heard of a Logarithmic Function before? Does anybody remember what the Exponential Function looks like?

See y=bx on graph below.

Which line do you reflect a function over to find its inverse image? The line y=x .

If an Exponential Function is the inverse to the Logarithmic Function what should the image of a Logarithmic Function look like?

See y=logb x on graph at right.


-10 -5 0 5 10 15 20

-10

-5

0

5

10

15

20

y=logb x

y=x

Graphing Key

Anticipatory Set:

Using a graphing calculator graph and identify the change on y=x2 under the given transformation. for

y1=x2+2(shift up two units) y2=x2−2 (shift down two units)

y3=( x+2 )2(shift to the left two units) y4=( x−2 )2

(shift to the right two units)

y5=12

x2 (closer to x-axis) y6=52

x2 (closer to y-axis)


Input:

The input for the lesson will be a lecture on logarithmic functions and the effects of transformations on logarithmic graphs.

What is the difference between a logarithm and an exponent? A logarithm is an exponent

What is the difference between Exponential and Logarithmic form?

x=by⏟

Exponentia l

∧ y=logb x⏟

Logarithmic

How does one find the inverse of a function algebraically? Switch the x and y in a given equation and then solve it for y.

What base do common logarithms have? Common logs have a base of 10.

What base do natural logarithms have? Natural logs have a base of e.

Why do we change from Logarithmic form into Exponential form when graphing Logarithmic Functions?

It is more efficient to represent x as a function of y, and then let the values for y vary.

What is the difference between a function of “x” and a function of “y”? With a function of “x” one lets the x values vary. With a function of “y” one lets

the y values vary. If the base of a Logarithm is greater than one, does the function increase or decrease?

Logarithms with a base greater than one increase. If the base of a Logarithm is greater than zero but is also less than one, does the function

increase or decrease? Logarithms with a base greater than zero but less than one decrease.

What point is a characteristic or featured in a Logarithmic Function? The point (1,0)

What is the domain of a Logarithmic Function? { x | x ∈ (0,∞) }

What is the Range of a Logarithmic Function? { y = f(x) |y ∈ (-∞, ∞) } or all real numbers, or ℝ.

Does a Logarithmic Function have an Asymptote? Yes, the parent function will never cross the line x = 0, or the y-axis.


Model:

Since the input for the lesson is a lecture on logarithmic functions and the effects of transformations on logarithmic graphs the Guided note packet issued at the beginning of the unit will be utilized for the modeling the conceptual aspects of the lesson, through interrogatives and examples.

Example 1, to write 2=log525 in exponential form.

Remember that y= logb x → by=x . So given example 1

2=log525 → {52=25 } Example 2, to write 9

12=3 in logarithmic form.

Remember that b y=x → y=logb x . So given example 2

912=3→ {1

2=log93}

Example 3, to Solve log12 x=2 for x . Remember that y= logb x→ by=x . So we can rewrite 3 as

log12 x=2→ 122=x→ { x=144 }

Example 4, to Solve 12=log x 16 for x .

Remember that y= logb x → by=x . So we can rewrite 4 as

12=log x 16 →x

12=16→( x

12 )2= (16 )2→ { x=256 }

Example 5, to evaluate log √3 9.

Remember that y=logb x→ by=x. First we rewrite 5 as

y=log√39→√3y=9→ 3y2 =32 → y

2=2 → { y=4 }

Lastly Graph the three examples at the end of the logarithmic function section in the guided note packet. Be sure to allude to the fact that one must first change the given function from logarithmic form to exponential form, then pick an arbitrary set of values for y and not x then solve the new function of “y” for x. In general the set A = {-3, -2, -1, 0, 1, 2, 3} is sufficient for the values of y. After the values for y have been determined, substitute them in to the function of y and then solve each equation for the corresponding value of x. After all the values for y have been substituted in make a table of values and plot the points. (See page 2 of appendix A6.).


Guided Practice:

A note sheet, with two examples, is issued to the learners at the end of the lecture. This worksheet provides an opportunity for the learners further their own conceptual understanding of how to graph logarithmic functions using transformations by following along and completing the given task at hand with support and guidance. (Please note a detailed key for the note sheet is provided in appendix A9.).

Independent Practice:

A worksheet with two examples is assigned for homework. Remember the assignment is given either a complete (reasonable effort, both questions answered) or incomplete (less-than reasonable effort, only one or no questions answered) grade. (See appendix A10. for a completed worksheet).

Closure:

An informal discussion about the geometric translations due to algebraic transformations. Also a handout will be given which states the effects of transformations.

Formative Assessment:

The learner is given a ten minute quiz, Quiz #3.8 approximately on the day the homework is corrected and collected.


ResourcesMaterials:

Guided note packet Guided practice work sheet Home worksheet Transformations closure sheet Quiz 3.8

Supplies:

Pen or pencil Straight edge Guided not packet in notebook Worksheets

Technology:

Calculators


Time Required:Planning:

For this learning experience planning takes three to five hours and includes all the time needed for copying and organizing the guided note packets, the in class and take home assignments, the in class and take home assignments’ keys, as well as the key for the quiz. The total time for the copying takes about two hours. As for the remaining one to three hours, ample time ought to be given by the instructor to internalize the diagnostic and formative assessments, and to establish self-proficiency with the material covered in the lesson.

Implementation:

This learning experience requires the learners to participate in a lecture, complete a guided practice assignment, receive and complete an independent practice worksheet, and finally take a quiz the following day. To accomplish the task of implementation, two class periods (or about ninety minutes) is allotted to complete the learning experience.

Assessment:

As stated earlier the assessment takes about ten minutes of class time for the students to complete (allow for fifteen minutes as an absolute maximum if need be). The grading time for the Homework and Quiz ought to be about three to five minutes per student.


ReflectionThis learning experience is executed during the exponential and logarithmic functions unit and addresses the effects of transformations for logarithmic functions. Moreover, to accomplish proficiency in the experience, one must first learn how to change form, for logarithmic to exponential form, and the algebras for both logarithmic and exponential functions. I feel this task is accomplished if the learner participates in the lecture and follows along in their notes. However, since not all of the students are classified as distinguished, further refinement of this lesson ought to be considered. To bolster the struggling students’ conceptual understanding extracurricular help is made available. In addition, the developing students are urged to seek help during their lunch period by me, the department chair, and the assistant principal.

This learning experience was peer reviewed at Daemen College, on 20 Mar 2012. The reviewers, Christopher, Shelly, Jessica, Mary, Karissa, Meghan, Sarah, and Dr. Arnold, provided may useful comments, both warm and cool, that I would like to formally acknowledge. The comments provided enough support to insure this learning experience is of a fine grain and high quality. Thank you all so much for your insight, time, and thought.


Appendices

A1. Guided Note Packet (blank)

A2. Take-Home Worksheet (blank)

A3. Logarithmic Rubric

A4. Classroom Rules

A5. Floor Plan

A6. Guided Note Packet (complete)

A7. In-Class Worksheet (blank)

A8. Closure Handout

A9. In-Class Worksheet (complete)

A10. Take-Home Worksheet (complete)

A11. Peer-Review Comments


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