Portland Community College | Portland, Oregon

The course I am running is considered a flipped classroom model.  Instead of lecturing during our class meeting,  the class meeting will be used to have you engage in the material, as well as other support activities, which may include a mini-lecture. 

The traditional lecture portion will take place outside of class by watching videos (10 to 15 minutes on average), attempting a question or two in MyOpenMath to make sure you understood the concepts,  then the following meeting we engage in an activity to allow you to ask questions or test what you have viewed in the lecture portion.  Then following that, the rest of the class is devoted to the activity we all know as homework.

This model has some advantages, if you, as a student, buy into this model (that means you must do your part). The advantage is getting real time support while attempting the homework.  Another advantage is that, in theory, you are engaged throughout the entire process.

In a traditional lecture classroom, if you have not prepared ahead of time by doing some reading, it is not uncommon to get lost during the lecture.  Now, some students have mentioned that in the traditional classroom, they get to ask questions while lecture is occurring.  While that is true, meaning, this should be occurring, my experience has been that most students, especially those that need to, don’t ask questions.  When I have asked why they do not ask questions, the typical response is, “I don’t even know what to ask.”

This response actually makes sense if the student has not read ahead of time.  It takes some thought, to formulate a question.  If a student is not practiced at asking questions, then the amount of time needed to gather your thoughts is not available in a traditional classroom setting.

The advantage of watching the lecture at home is that you get to stop watching when you loose focus.  You have the time to formulate questions, that you can then bring to class, or you can post questions in D2L.  Instead of relying on going to the tutoring center for help, there is a chance that the questions you have can be answered in class.  Being successful in college requires you to be engaged, be an active learner.  This model puts you in a position to be engaged immediately if you accept it.

The set up is as follows.

1. Watch lecture at home, answer questions in MyOpenMath; a pre-homework to make sure you are engaged as you watch the videos and examines the understanding. MyOpenmath is required and it is free.  Take notes on the videos, part of grade.
2. The pre-homework and video notes are due the following class period.  We start the class by answering questions on the concepts learned.  Do an activity to further cement the ideas or uncover ideas that still need further explanation.
3. The last hour to hour and half of class is dedicated to doing the homework (Medium to harder questions).
4. Finish the questions you were not able to complete at home.  You go home and watch the next lecture, and repeat the process.

This idea is nothing new and there have been reports on this approach.

• KGW report
• PBS news report
• PCC student testimonial

The prerequisite course is MTH 95 and this course requires that you are adequate at the topics covered in MTH 95. Some topics in MTH 95 we cover more in depth. 

This course focuses on the concept of functions and characteristics associated with functions.   Below is a listing of what we will cover and required prerequisite knowledge; you don't have to recall prerequisite knowledge perfectly, but you are expected to work on it if there are missing gaps to your knowledge.  We don't go over that prerequisite knowledge in depth if at all but there is mention of it as we engage in an example that requires it. 

Below is an outline of the course.

·         Define a relation and a function.

o   Using ordered pairs

o   Mapping

o   Graphical

·         Determine if an equation, a table, a set of ordered pairs, a graph defines a function.

·         Vertical line test.

·         Define function, f(x) = y

o   How to read the notation

o   How to write/communicate using function notation.

o   Evaluate an equation given a prompt using function notation.

o   Evaluate/Solve a function given a table of ordered pairs given a prompt in function notation.

o   Solve an equation (linear or quadratic).

o   Recognize and understand how to read the input and output value of a function.

o   Solve a basic radical

·         Memorize the Basic (no frills) Function Equations

o   Linear, Identity

o   Quadratic

o   Absolute Value

o   Cubic

o   Reciprocal, Rational, Reciprocal Squared

o   Square Root

o   Cube Root

·         Read ordered pairs from a graph.

·         Solve a linear equation.

·         Solve by factoring a quadratic equation: 4 = 3x2 -2x

·      Solve a Radical equation:  6=2x+1

·         What is the result of dividing by 0?  40 ?

·         What is the result of dividing zero by a non-zero number? 08 ?

·      What is the result of taking the square root of a negative number?-4 ?

·         Know that (x + 2)2 ≠ x2 + 4 and instead equals (x + 2)(x + 2) = x2 + 4x + 4.

·         Define what is the domain of a function.

·         Define what is the Range of a function.

·         Reading and writing interval notation to write the answer to a find the domain/range question.

·         Set builder notation to write the domain of a function

·         Determine the domain of a function:

o   From a graph

o   From an equation

§  Linear

§  Rational

§  Quadratic

§  Square Root

o   Know the domain and range of basic functions

·         Piecewise Functions

o   Given a situation (linear) create one.

o   Know how to read a piecewise function

o   Graph a piecewise function using linear, quadratic, basic functions.

·         Graphing linear equations

o   Using slope of a line to graph

o   Reading the slope and y-intercept from looking at a linear equation in slope-intercept form.

·         Graphing quadratic equations

o   Recall how to find the vertex of a quadratic.

o   Know how to use symmetry of a quadratic to help you graph

o   Know what a quadratic looks like.

·         Know the Graph of basic functions.

·         Solving inequalities

·      Know that -4= a non-real number, k0  is undefined where k is any real number not equaling zero, and 00  is undetermined.

·         Devine what the average rate of change of a function is and interpret the result of a situation.

o   Find the average rate of change given a function, table, graph.

·         Simplify the Difference Quotient

o   Given a linear or quadratic function

o   Know what the difference quotient indicates.

·         Define what is meant by a function is increasing or decreasing.

·         Define what is meant by a local maximum and local minimum.

·         Define what is meant by a absolute maximum and absolute minimum.

·         Apply increasing/decreasing, maximum an minimums to basic functions.

·         Know how to interpret the slope of a linear equation with units (in context).

·         What does a negative slope indicate?

·         What does a positive slope indicate?

·         Know how to multiply (x + 1)2 for example.

·         Reduce fractional expressions.

·         Factoring, adding like terms, algebra basics.

·         Reading a graph and finding ordered pairs from a graph.

·         Using interval notation

·         What is the algebra of functions????

·         Function notation used with algebra of functions

·         Given two, or more,  functions in table form, graph or equation form find:

o   (f + g)(x), (fg)(x), (f – g)(x), (f/g)(x)

·         What is a composition of functions?

·      Given two or more functions, in table, graph or equation form find (fog)(x)

·         In context interpret the meaning of  a composition or algebra of functions result.

·         Find the domain of a composite function

·         Simplifying an algebra expression by

o   adding like terms,

o   multiplying expressions

o   factoring

o   Simple complex fraction

o     Solve rational functions of the form 23x-7=2

·         Identify horizontal, f(x ± a), versus vertical shift, f(x) ± a,   given :

o   A table

o   An equation

o   A graph

o   A description

·         Identify a horizontal compression/stretch, f(ax), versus a vertical compression/stretch af(x).

·         Given an original function, f(x), determine what a transformed function:

o   Graphs to

o   Determine equation symbolically using function notation

o   Determine what the new table will look like.

·         Understand how a transformation works.

·         Applying more than one transformation to a:

o   Equation

o   Table

o   Graph

·         Given a transformation, describe in words what transformation was done to the original equation.

·         Even and odd functions

·         Function notation

·          

·         Basic Function Graphs

·      Simple factoring involving fractions: 13x- 1  = 13x-3

·         Know the graphs of your basic functions.

·         Graphing a basic linear, quadratic, cube, square root.

·         Know that a – h can be expressed as  a + - h and vice versa.

·         Know the difference between input and output using function notation symbols: f(x) = y, f(input) = output.

·         What is a factor

·         What is an inverse function?

·         Section 3.1 – page 193. One-to-One Functions. 

·         Apply the horizontal line test to a graph.

·         Domain and Range of an inverse function

o   Given a linear, radical, rational, find the inverse function.

o   Restricted domain

·      Notation for inverse functions: f-1x .

·      Composition of inverse functions, f(f-1x)

·         Symmetry between a function and its inverse.

·         Function notation

·         Solving a rational equation

·         Solving a linear equation

·         Solving a radical equation

·         Composition of functions

·         Graphing functions

·         What is an exponential function.

·         Domain and range of an exponential function.

·         Recognizing an exponential equation.

·         Define exponential growth and decay.

·         Find the equation of an exponential function:

o   Given ordered pairs.

o   A graph

o   A description of a situation.

·         Compound Interest Formula

o   Annual percentage rate (APR) versus nominal rate.

o   Base e and continuous compounding.

o   Continuous growth formula.

o   Continuous decay formula

·         Function notation

·         Transformations

·         Exponential Rules

o   Meaning of a negative exponent is most important

o   (ab)n = anbn, (am)n = amn , am(an) = am = n

·         Meaning of the y-intercept and how to find it.

·         Meaning of a one to one function

·         Know what a linear equation, quadratic, general polynomial looks like.

·      Finding the roots other than square roots: 327   ,

·         Recall rules of roots.

·               Identify essential characteristics needed to graph

·               Memorize the y-intercept and how to find it, what occurs when x = 1.

·               Growth versus decay and how to determine from an equation

·               Use transformations to graph.

·         Function notation

·         Transformations

·         Exponential Rules

o   Meaning of a negative exponent

·         Meaning of the y-intercept and how to find it.

·         Meaning of a one- to- one function

·         What is a logarithmic function?

o   The dirty secret of logarithmic functions

o   Connecting a logarithmic to a exponential function and vice versa.

o   Meaning of log bases.

·         Domain and range of a logarithmic function.

·         Properties of a logarithmic function.

·         Evaluating simple logs using exponential rules

·         Common log, what is it?

·         Evaluate logs on a calculator

·         Function notation

·         One to one functions

·         Inverse functions

·         Transformation of Functions

·         Exponential growth and decay

·         Basic graph of a logarithmic function

·         Transformation of logarithmic functions.

·         Transformations

·         Why log properties exist?

o   Identities

o   Connection to trigonometry

o   Transcendental functions

·         Rules of logs

o   Product and quotient rule of logs

o   Power rule of logs

o   Change of base

·         How we will use log rules

·         Practicing expanding and contracting log expressions

·         Function notation

·         Transformations

·         Exponential Rules

o   Meaning of a negative exponent is most important

o   (ab)n = anbn, (am)n = amn , am(an) = am = n

·         Meaning of the y-intercept and how to find it.

·         Meaning of a one to one function

·         Know what a linear equation, quadratic, general polynomial looks like.

·      Finding the roots other than square roots: 327  .

·      Root expressed as exponents: 34=413  , 56=615

·      Recall rules of roots: 3ab=3a3b  which is the same as rule of exponents, (ab)4=a4b4.

·         Use rule of logs equations to solve log equations

·         Use rule of log and exponents to solve exponential equations.

·         Exponential rules

·      Root expressed as exponents: 34=413  , 56=615

·      Recall rules of roots: 3ab=3a3b  which is the same as rule of exponents, (ab)4=a4b4.

·         Exponential growth and decay

o   Determine the Half-life

o   Determine the doubling or tripling time.

·         Use rule of log and exponents to solve exponential equations.

·         Newton’s Law of Cooling

·         Logistic Growth Model

·         Exponential rules

·         What is a power function?

·         What is a polynomial?

·         What is the degree of a polynomial?

·         What is the leading coefficient of a polynomial?

·         What is the zero of a polynomial?

·         What is the end behavior of a polynomial?

·         Turning points of a polynomial

·         Local and absolute maximums and minimums

·         Geometric situations that lead to a polynomial function.

·         What is the domain of a polynomial

·         Factoring  in general

·         Factoring trinomials

·         Factoring more advanced trinomials

·         Factoring by grouping for four term polynomials

·         Factoring by the difference of squares.

·         Multiplying polynomials.

·         Understand how to multiply expressions of the form (x + 3)3. 

·         Understand that (x + 3)2 does not equal x2 + 32

·         Reminder that x-1=1x

·         Local behavior of a polynomial around a zero

·         Use the knowledge gained in section 5.2 to graph a polynomial given an equation

·         Given a graph, find the equation of a polynomial.

·         Geometric situations that lead to a polynomial function.

·         Factoring  in general

·         Factoring trinomials

·         Factoring more advanced trinomials

·         Factoring by grouping for four term polynomials

·         Factoring by the difference of squares.

·         Multiplying polynomials.

·         Understand how to multiply expressions of the form (x + 3)3. 

·         Understand that (x + 3)2 does not equal x2 + 32

·         Define a rational function.

·         Arrow notation to indicate approach to a value from left or right.

·         Determine the zeroes and x-intercepts of a rational function.

·         Determine the vertical asymptotes of a rational function.

·         Determine the horizontal asymptotes of a rational function.

·         Local behavior of a rational function.

·         End behavior of a rational function.

·         Writing down the equation of a rational function given a graph.

·         Given an equation of a rational function draw a rough graph of the equation by finding all the appropriate characteristics: end behavior, zeroes, y-intercept, vertical asymptotes, … etc.

·         Physical situations involving rational functions.

·         Factoring  in general

·         Factoring trinomials

·         Factoring more advanced trinomials

·         Factoring by grouping for four term polynomials

·         Factoring by the difference of squares.

·         Multiplying polynomials.

·         Understand how to multiply expressions of the form (x + 3)3. 

·         Understand that (x + 3)2 does not equal x2 + 32

·         Reminder that x-1=1x

·         The linear equation of a vertical line is x = k, where k is a constant, and the line has an x-intercept of (k, 0).

·         The linear equation of a horizontal line is y = k, where k is a constant, and the y-intercept is (0, k).

·         Finding the inverse of a rational function

·         Find the inverse of a polynomial.

·         Find the inverse of a polynomial that is not one to one by restricting the domain.

·         Recall what are one to one functions.

·         Function notation

·         f-1(f(x)) = x

·         Concept of domain and range related to inverses.

·         The relationship between the graphs of inverse functions.

Desmos and an online graphing calculator is recommended to do some exploration of concepts.   It is a very intuitive in how you enter expressions just using your keyboard.