Tag: teaching functions

Posted in Algebra

What is an algebraic function?

Posted on by

An algebraic function is a function created by applying the operation of addition, subtraction, multiplication, division, and extracting the nth root. Let me give an example. Suppose you have the function f and g where f is a linear function and g is a constant function.  Let f(x)=x and g(x) = -3. We can create another linear function h by multiplying f and g that is h(x) = -3x. We can also create another linear function l where l = f – g, that is l(x) = x-3.

What about quadratic functions? A quadratic function (with real roots) is a product of two linear functions. So we can make a quadratic function n by multiplying f and l for example. That is, n(x) = f(x) x l(x) = x(x-3). And cubic function? A cubic function is a product of three linear functions or of a quadratic function and a linear function. And quartic function? Well, you must have figured it by now. This process of creating function by multiplying linear functions produces a family of functions called polynomial functions so called because its algebraic representation is a polynomial.

functions
Polynomial Function Family

What kind of function is produced when you divide a function by a function in x? Using the function defined earlier, what is g÷f?  g÷l? f÷l? Getting the quotient of two polynomial functions give us a new family of functions: p(x) = -3/x; q(x) = -3/(x-3); and, r(x) = x/(x-3). These expressions defining the functions will not simplify to polynomial expressions so they do not belong to the family of polynomial functions. They belong to what is called the family of rational functions so called because they are defined by rational expressions.

You can also raise a function to a fractional power, that is get the nth root of the function. For example we can have t(x)= x^0.5. That is t(x)=sqrt of x. I don’t know what this family of function is called. Maybe we can call then nth root functions.

These three families — polynomial functions, rational functions, and nth root functions, all belong to the family of algebraic functions. Functions that are not algebraic functions are called transcendental functions.

You may also want to read ideas for teaching functions.

Posted in Algebra

How to teach the inverse function

Posted on by

In What  is an inverse function? I proposed a way of teaching this concept starting with its graphical representations using GeoGebra applets. Al-Zboun Lilliana in our Linkedin group shares her idea for introducing the inverse function. She says that the most difficult part in teaching this concept is to make it make sense to students and not so much in making the students understand its definition or teaching them the process of finding the inverse function of a given function(by a graph or by a formula)  or to “verify algebraically” that the functions are inverses.

Here’s her proposed teaching sequence starting with examples that students can relate to in Levels 1 and 2. I would suggest inserting the activities I described in What is an inverse function? before Level 3 which introduces the algebraic solution.

Examples SET(1)-Level 1:
1. If we need to call someone we are asking for her/his name on the list of our phone contacts …
2. If someone of our contacts is calling us “our phone shows who is calling” This is the job of an inverse function: “finding the name corresponding to the number”

Examples SET(2)- Level 2:

1. If George makes $100/day. We know how to answer questions such as “After 7 days, how much money has he made?” We use the function W(t)=100t
But suppose I want to ask the reverse question:
2. “If George has made $700, how many hours has he worked?” The students know the answer : Time : t(W)=W/100. Given any amount of money, divide it by 100 to find how many days he has worked.
This is the job of an inverse function. It gives the same relationship, but reverses the dependent and independent variables.

Examples SET(3)- Level 3:

In this set the teacher includes examples to show simplifying solutions of mathematical questions

Example (1): Solve log (3 x – 2) = 3
• Since logarithmic and exponential functions are inverses of each other, we can write the following.

a = log (b) if and only b = 10^a
• Use the above property of logarithmic and exponential functions to rewrite the given equation as follows.
3x – 2 = 10^3
• Solve for x to obtain.

3x = 1002
x= 1002÷3=334

Example (2): Find the Range of the function ( or any RATIONAL function) :
F(x)= (3x+1)/(3 -x) or [y=(3x+1)/(3 -x)]
• Since the RANGE of a one to one function is the DOMAIN of its inverse. Let us first show that function f given above is a one to one function.
• Hence the given function is a one to one. let us find its inverse.

• Interchange x and y and solve for y.
x =(3y+1)/(3 -y)
And find y = (3x-1)/(3+x)
The inverse g(x) of function f(x) is given by.

g(x) = (3x-1)/(3+x)
• The domain of g(x) is R except x = -3. Hence THE RANGE of f(x) is R/{-3}.

Posted in Algebra

History of algebra as framework for teaching it?

Posted on by

In many history texts, algebra is considered to have three stages in its historical development:

  1. The rhetorical stage –  the stage where are all statements and arguments are made in words and sentences
  2. The syncopated stage – the stage where some abbreviations are used when dealing with algebraic expressions.
  3. The symbolic stage – the stage where there is total symbolization – all numbers, operations, relationships are expressed through a set of easily recognized symbols, and manipulations on the symbols take place according to well-understood rules.

These stages  are reflected in some textbooks and in our own lesson. For example in in pattern-searching activities that we ask our students to express the patterns and relationships observed using words initially. From the students’ statements we can highlight the key words (the quantities and the mathematical relationships) which we shall later ask the students to represent sometimes in diagrams first and then in symbols. I have used this technique many times and it does seem to work. But I have also seen lessons which goes the other way around, starting from the symbolic stage!

Apart from the three stages, another way of looking at algebra is as proposed by Victor Katz in his paper Stages in the History of Algebra and some Implications for Teaching. Katz argued that besides these three stages of expressing algebraic ideas, there are four conceptual stages that have happened along side of these changes in expressions. These conceptual stages are

  1. The geometric stage, where most of the concepts of algebra are geometric;
  2. The static equation-solving stage, where the goal is to find numbers satisfying certain relationships;
  3. The dynamic function stage, where motion seems to be an underlying idea; and finally
  4. The abstract stage, where structure is the goal.

Katz made it clear that naturally, neither these stages nor the earlier three are disjoint from one another and that there is always some overlap. These four stages are of course about the evolution of algebra but I think it can also be used as framework for designing instruction. For example in Visual representations of the difference of two squares, I started with geometric representations. Using the stages as framework, the next lesson should be about giving numerical value to the area so that students can generate values for x and y. Depending on your topic you can stretch the lesson to teach about functional relationship between x and y and then focus on the structure of the expression of the difference of two squares.

I always like teaching algebra using geometry as context so geometric stage should be first indeed. But I think Katz stages 2 and 3 can be switched depending on the topic. The abstraction part of course should always be last.

You may want to read Should historical evolution of math concepts inform teaching? In that post I cited some studies that supports the approach of taking into consideration the evolution of the concept in designing instruction.

For your reading leisure – Unknown Quantity: A Real and Imaginary History of Algebra.

For serious reading Classical Algebra: Its Nature, Origins, and Uses and of course Victor Katz book History of Mathematics: Brief Version.

 

Posted in Algebra, Graphs and Functions, High school mathematics

What is an inverse function?

Posted on by

In mathematics, the inverse function is a function that undoes another function. For example,  given the function f(x) = 2x. If you input a into the function f, the output is 2a. The inverse function of  f(x) is the function g(x) such that if you input 2a into g(x) its output is a. Now what is g(x) equal to? How does its graph look like? Is the inverse of a function also a function? These are the basic questions students need to answer about inverse function.

How to teach the inverse function
Functions and their inverses

The idea of inverse function can be taught deductively by starting with its definition then asking students to find the equation of the inverse function by switching the x and y in the original function then expressing the equation in the form y = f(x). This is an approach I will not do of course as I always like my students to discover things for themselves and see and express relationships in all three representations: numerical (ordered pairs or table of values), geometrical (graphs) and symbolic (equation) representations.

In teaching the inverse function it is important for students to realize that not all function have an inverse that is also a function, that the graph of the inverse of a function is a reflection along the line y = x, and that the inverse function does not necessarily belong to the same family as the given function.

The concept of inverse function is usually taught to introduce the logarithmic function as inverse of exponential function. Important ideas about inverse function such as those I mentioned are not usually given much attention. Perhaps teachers are too excited to do the logarithmic functions.

I suggest the following sequence for teaching inverse. I’m sure many teachers and textbooks also do it this way. What I may just be pointing out is the reason behind the sequence. I also developed three worksheets using GeoGebra. The worksheet is interactive so that students will be able to make sense of inverse of function on their own.

Start with linear function. Its inverse is also a function and it’s easy for students to figure out that all they need to do is to switch the x‘s and y‘s then solve for y to find the equation. You may need to see the inverse of linear function activity so you can make sense of what I am saying.

The next activity should now involve a quadratic function. The purpose of this activity is to create cognitive conflict as it’s inverse is not a function. The domain needs to be restricted in order to get an inverse that is also a function. Depending on your class, the algebraic part (finding equation of the inverse) can be done later but it’s important for the students at this point to see the graph of the inverse of a quadratic to convince them that indeed it is not a function. Click the link to open the activity inverse of quadratic functions.

The third activity will be the inverse of exponential function. By this time students will be more careful in assuming that the inverse of a function is always a function. Except this time it is! It is also one-to-one just like linear, but it’s equation in y belong to a new family of function – the logarithmic function. Click the link for the activity on inverse of  exponential functions.

Teaching principles

There are at least three math teaching principles illustrated in the suggested lesson sequencing for teaching the inverse function and introducing logarithmic function.

  1. Connecting with previously learned concepts. Start with something that students can already do but in a different context. In the above examples they are already familiar with linear function and they already know how to find its equation.
  2. Creating cognitive conflict. The purpose is to challenge possible assumptions and expose possible misconceptions.
  3. Making connections. Mathematics is only understood and hence powerful when there is a rich and strong connections among related concepts, representations, and procedures.

You may find the Precalculus: Functions and Graphs a good reference.