Proposed Intermediate Algebra Course:
>> >1. Solving Equations in one variable
>> >2. factoring polynomials
>> >3. solving simultaneous equations
>> >4. graphing linear and quadratic equations
>> >5. Basic Definitions of: "term", "coefficient", etc.
>> >6. graphing "equalities" and "inequalities"
>> >7. working with fractional exponents.
>> >8. translating English sentences into algebraic expressions
>
>> IS THE ORDER LISTED ABOVE THE ORDER OF PRESENTATION?=20
>
>There was no attempt to list the above in any particular order.
>
>IT SEEMS DISJOINTED.
>
>I would agree that the list is disjointed; do you think these
>topics, are candidates for Algebra I High School instruction??
>
>> ALSO, WHY AREN'T YOU USING FUNCTIONS AS THE UNDERLYING STRUCTURE THAT
>> CONNECTS ALL THE TOPICS?
>
>This is where I need some help; your definition of a function (or
>anyone's definition of a "function") would be helpful here.
>
>My initial reaction to a function is that you've got a collection
>of two sets of numbers; you perform some math calculation on any number
>of the first set to get some member belonging to the second set.
>The name of the first set is "X" ; it contains all the "x's"; the
>name of the second set is "Y"; it contains all the calculated
>values, the y's. Is this the definition of function or is there
>a more meaningful defintion of function?
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Below is a rather wordy response to Jim regarding his question about the
function approach in intermediate algebra. I was attending a conference
when I wrote this and didn't have much else to do. Investigations,
explorations, projects, and collaborative learning are not addressed.
Just an opinion,
Ed Laughbaum
________________________________________________________
On teaching intermediate algebra from a function approach.
First, we should look at the traditional approach, which might be called
the equation approach. Most traditional intermediate algebra text=92s
(therefore, courses) start with a little background on number properties
and maybe a little simplification of expressions. Then they move directly
to solving equations, usually linear equations in one variable. From here,
they will do more symbol manipulation work needed to solve more equations.
The solving of equations drives the symbol manipulation in the rest of the
text/course. For example, quadratic equations can=92t be solved until
students learn how to factor. Once factoring is taught and rational
expressions manipulated, then solving equations containing rational
expressions can be solved. The equation-solving approach requires that the
manipulation of expressions with exponents be done before students can
solve simple exponential equations. This process is repeated for radical
expressions and radical equations. Usually there is a chapter on functions
near the middle or near the end of the text -- for the purpose of
introducing function notation, domain and range, and function arithmetic
operations. And depending on what the author has convinced the publisher as
being a good idea, there may be a variety of other related material at the
end.=20
The idea of equations has been so ingrained in math education that many
(most?) of us talk about graphing =93equations=94 as opposed to graphing
functions. So, the traditional approach is driven by the desire to teach
students to solve equations such as linear, quadratic, rational, radical,
exponential, and logarithmic plus solve systems of equations. What is
needed to solve these? Symbol manipulation and algorithms are needed and
taught.
As reform started to take place in intermediate algebra within the last few
years, traditional authors who wanted to placate the reform curriculum (as
recommended by the AMATYC Crossroads) moved the function chapter near the
beginning of their texts. Some even added a few calculator exercises in
some exercise sets. But the textbooks were still driven by the solving of
equations because function concepts were not addressed anywhere in the text
except in the function chapter. A few new authors have moved from the
=93equation approach=94 mold and have written intermediate algebra textbooks
that take another step toward the full function approach.
So, what is the function approach to intermediate algebra? What are the
benefits? How does it satisfy the equation solving needs of the service
departments and traditionalists? Why does it demand the use of hand-held
technology?=20
Suppose the course starts with an analysis of data pairs. Related data
pairs coming from real-world data such as found in almanacs, world wide web
sites, life situations, magazines, newspapers, or collected live in front
of the class from a Texas Instruments CBL or CBR. All you may do in the
first week of class is make the connections between the numeric
representation of the functions and the graphic representations. You may
follow this with transforming these numerically represented functions to
their symbolic representation. (I don=92t necessarily mean to imply that we
should use the calculator regression models to make the conversions, but
some people do this.) So, on a very intuitive level, you are using
functions from day one. You have shown that function data can be
represented numerically, graphically, and symbolically. Students are
expected to be able to convert from one form to another under simple
situations, and they know that certain kinds of symbols generate specific
kinds of shapes. They know that these shapes are the same as found in the
data studied. At this time, calculators are used to convert data to graphs
and back. The list features of the TI-83 can be used to help teach the
conversion to symbolic form.
The next topic may revolve around exploring connections between symbolic
function parameters and function behaviors. For example, in the function dx
+ e, students may be asked to figure out that d controls the direction of
the graphical representation of the function and it tells them how fast
(the rate) the graph is rising or falling, and whether it rises or falls.
Similar explorations can be done for other types of functions such as d(x +
e)^2 + f, d abs(x + e) + f, and d sqr(x + e) + f. So, a daily topic might
start with data pairs that can be represented symbolically by a quadratic
expression. Once you have convinced students that quadratics show up in
various real-world data relationships, they are ready to start analyzing
the parameter-behavior connection. Function behaviors are: when is the data
or graph (function) increasing/decreasing, when does the data or graph
(function) have a maximum/minimum, what is the maximum/minimum, when is the
data (graph) positive/negative/zero, what is a reasonable domain for the
data (function), and how fast is the data (graph) increasing/decreasing in
value? At this time students are learning how to use the calculator and
they are discovering interesting behaviors about the data relationships.
(For example, zeros now have a real-world meaning other than just a value
for x that makes the function zero.) They are analyzing function behavior
that is needed later when they will be solving equations or problem
solving. All of this activity revolves around real-world data that also
happens to be a form of a function.=20
At this time students may be ready for a more formal definition of a
function, the associated notion, and operations with functions. They have
seen the symbols and know they can represent data relationships from the
real world. The arithmetic operations with functions can now give rise to
the traditional +, -, x, \, and factoring of polynomials. Maybe four days
of class time is sufficient maybe not. Did you ever wonder why traditional
texts never include arithmetic operations on functions like absolute value
functions? Or why have we never studied sums of any function and a square
root function? Well, not much happens when you do these operations if you
only study symbol manipulation. So why bother? But did you know that there
are many situations in the real world that can be modeled by sums of
absolute value functions? You can change the domain of any function
(without changing its behavior) by simply adding a square root function to
it. So instead of saying =93Just ignore this part of the function.=94, stude=
nts
can actually eliminate any non-needed piece of a function in a modeling
problem situation.
If you haven=92t stopped reading by now, you may be wondering where are the
=93traditional=94 topics found in intermediate algebra? We have already used=
3
weeks of the quarter. They have all been used to develop intuitive
understandings of the representations of functions and how they are
connected to the real world. Well, each unit in the remaining intermediate
algebra course might look like the following:
Linear Functions (first example)
1 Linear relationships in the real world
2 Slope
3 Slope-intercept form
4 Point-slope form
5 Parallel and perpendicular lines
6 Solving linear equations
7 Solving linear inequalities
8 Applications of the linear function
1 was introduced in the first 3 weeks and students already have an idea of
the parameter-behavior connection. Now is the time for a formalization of
nomenclature.=20
2 has been thoroughly developed in the name of rate of change. It was one
of the behaviors analyzed in the first 3 weeks. Students already know what
it means and that it is the change in the function value per change in the
function variable. Or more than likely, they know it as how fast a ball was
traveling, or a rate of change of position, or as a rate of change of
volume or weight. Just add the word slope to their vocabulary.=20
3 has been developed in the intro because of the investigations on the
parameter-behavior connection. They already know the coefficient of x is
the rate of change and the constant is the initial condition.
4 is new.
5 is new and can easily be included with topic 4.
6 and 7 have already been covered thoroughly when students were finding
when a function is zero, positive, or negative. They have used the
numerical method for solving equations and inequalities (finding when a
function is zero, positive, and negative). They have used the trace method.
You may now want to add the algebraic method, refine the zeros method (let
them use the zero finder), and develop the intersection method to add to
their repertoire of methods. All of these methods REQUIRE that students
know the behavior of the function in order to avoid problems related to
solving the equation.
8 is a simple step in level of rigor because of all the work done before in
using the idea of developing all mathematics in the context of a problem.
Application problems become simpler to solve.
Exponential functions (second example)
1 Exponential relationships in the real world
2 Behaviors of the exponential function
3 Simplification of exponential expressions
4 Solving exponential equations
5 Applications of the exponential function
1 - students already have an idea of the parameter-behavior connection, but
a new behavior is introduced for the first time here - an asymptote. Now is
the time for a formalization of terminology and moving to a higher level of
sophistication.=20
2 provides opportunity to further develop a method for creating symbolic
form from known exponential data. All the behavior concepts were developed
in the first 3 weeks.
3 Properties of exponents can be confirmed and/or developed by using
investigations of exponential functions. Now add some traditional symbol
manipulations.
4 has been done. The numeric, trace, zeros, and intersection methods are
the SAME for ANY equation. Just add the algebraic method. Except remember
that function/technology-based methods will be able to be used when
algebraic methods won=92t work. You can now solve exponential equations that
use to require logarithms, just use the function/technology-based methods.=
=20
5 is a simple step in level of rigor because of all the work done before in
using the idea of developing all mathematics in the context of a problem.
Application problems become simpler to solve.=20
Here might be an intermediate algebra course using the function approach.
As you will see, the topics may not be a whole lot different - just the
approach.
NUMBERS
Properties of Numbers, Equality, and Inequality
Behavior of Data
Describing Sets of Numbers Using Interval Notation
Big and Little Numbers
INTRODUCTION TO BEHAVIOR OF FUNCTIONS
Symbolic Representations of Data Relationships
Introduction to the Analysis of Behavior of Functions
Functions Represented Graphically
An Introduction to the Analysis of the Linear Function=20
An Introduction to the Analysis of the Quadratic Function=20
An Introduction to the Analysis of the Absolute Value Function=20
An Introduction to the Analysis of the Square Root Function=20
FUNCTIONS: NOTATION AND OPERATIONS
Definition of a Function, Again
Addition and Subtraction of Polynomial Functions
Multiplication of Polynomial Functions
Factoring: Common Factors, Grouping, and Difference of Squares
Factoring the Trinomial
Function Operations from a Graphical Perspective
ADVANCED ANALYSIS OF THE LINEAR FUNCTION
Rate of Change, Initial Conditions, and the Zero, Slope and Intercepts of
the Linear Function
Slope-Intercept Method of Graphing
Point-Slope Form
The Linear Function as a Mathematical Model
EQUATIONS AND INEQUALITIES
Solving Equations Containing the Linear Function
Solving Inequalities Containing the Linear Function
Solving Inequalities and Equations Containing the Absolute Value Function
Formulas and Direct Variation
INTRODUCTION TO THE ANALYSIS OF THE EXPONENTIAL FUNCTION
The Exponential Function (might introduce geometric transformations here)
Simplifying Exponential Functions
Equations and Inequalities Containing the Exponential Function
The Exponential Function as a Mathematical Model
INTRODUCTION TO THE ANALYSIS OF THE RATIONAL FUNCTION
Rational Functions (continue to reinforce geometric transformation ideas=
here)
The Fundamental Property of Rational Functions
Multiplication and Division of Rational Functions
Addition and Subtraction of Rational Functions and Simplification of
Complex Functions
Solving Equations and Inequalities Containing the Rational Function and
Inverse Variation
ADVANCED ANALYSIS OF THE SQUARE ROOT FUNCTION
The Square Root Function(continue to reinforce geometric transformation
ideas here)
Properties of Irrational Expressions
Operations with Irrational Expressions
Fractional Exponents
Solving Equations Containing the Square Root Function
The Square Root Function as a Mathematical Model
ADVANCED ANALYSIS OF THE QUADRATIC FUNCTION
The Quadratic Function (continue to reinforce geometric transformation
ideas here)
Solving Quadratic Equations
The Quadratic Function as a Mathematical Model
INTRODUCTION TO ANALYTICAL GEOMETRY
The Distance and Midpoint Formulas
Triangles
Parallelograms
Circles
INTRODUCTION TO TRIGONOMETRY
Conversions Between Degrees and Radians
Trigonometric Definitions
Solving Right Triangles
Trigonometric Functions as Models
SYSTEMS OF EQUATIONS AND INEQUALITIES
Introduction to Systems of Equations
Solving Systems Graphically
Solving Systems by the Addition and Substitution Methods
Solving Systems Using Cramer's Rule and Matrices
Systems of Equations as a Mathematical Model
INTRODUCTION TO THE ANALYSIS OF THE LOGARITHMIC FUNCTION
The Logarithmic Function
Properties of Logarithms
Solving Logarithmic and Exponential Equations
The Logarithmic Function as a Mathematical Model
An advantage of the function approach to teaching intermediate algebra is
that data can be used to MOTIVATE student=92s interest in mathematics.=20
An advantage of the function approach to teaching intermediate algebra is
that functions can be used to MOTIVATE the study of traditional symbol
manipulations and equation-solving algorithms.
An advantage of the function approach using hand-held technology is that we
can now teach a multiplicity of methods for solving problems, solving
equations and inequalities, factoring, checking work, =85=20
An advantage of the function approach is that students see that functions
are useful BEFORE they ever do those symbol manipulations that they could
never figure out why we were teaching.=20
An advantage of the function approach is that many (most) of the
traditional properties/laws can be developed/confirmed using graphs or
numeric representations of functions.
An advantage of the function approach is that students now can see why some
equations don=92t have real solutions.
Another advantage is that our service fields still get students that can
solve equations but now by several methods. (Unfortunately, they may not
be ready for what these students know.)
An advantage to the function approach is that if intermediate algebra is a
terminal course, students have some mathematics to take with them in their
work lives. (modeling and the data-symbolic connection)
Students now have a =93reasonable=94 method for checking their traditional
symbol manipulation work.
Students can now FACTOR polynomials by using the zeros of the related
functions and the zero-factor connection.
An advantage of using the function approach is that students can now solve
more complicated equations and inequalities.
An advantage of using the function approach is that new topics can be
included.
An advantage to the function approach in intermediate algebra is that
students can =93visualize=94 mathematics.
An advantage of using the function approach is that students bring a solid
number sense to the course and it is capitalized upon through the work on
functions in numeric form.
An advantage of using the function approach is that math can also be
treated as an experimental science by using calculator technology with
explorations and projects.
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