Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.

Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!

Linear Functions

GeoGebraBook: Exploring Linear Functions, which contains:

Interactive Linear Function Graph: Slope-Intercept Form

Interactive Linear Function Graph: Point-Slope Form Continue reading Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Absolute Value: Notation, Expressions, Equations

What Does Absolute Value Mean?

The term “Absolute Value” refers to the magnitude of a quantity without regard to sign. In other words, its distance from zero expressed as a positive number.

The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. These bars mean: evaluate what is inside and, if the final result (once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars; if the final result inside the bars is zero or positive, you may drop the bars without making any changes:

\lvert ~1-4~ \rvert\\*~\\*=~\lvert ~-3~ \rvert\\*~\\*=~3

Another example is:

\lvert ~4-1~ \rvert\\*~\\*=~\lvert ~3~ \rvert\\*~\\*=~3

Note that absolute value signs do not instruct you to make “all” quantities inside them positive. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.

\lvert ~1-4~ \rvert~\ne~\lvert ~1+4~\rvert~~\text{ Do not make this mistake!}

Absolute Value expressions that contain variables

Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either Continue reading Absolute Value: Notation, Expressions, Equations

Combining or Collecting Like Terms

The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.

Quantities With Units

Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?

Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?

The quantities of apples may be combined because addition or subtraction only work with  Continue reading Combining or Collecting Like Terms

Simplifying Fractions

Three concepts help explain the process of simplifying fractions:

  1. Multiplying a quantity by 1 has no effect
  2. A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
    \dfrac{17a^2b}{17a^2b}~~=~~1~~~~a\ne 0,~~b\ne 0
  3. A product of two fractions can be rewritten as a fraction of two products (and vice versa)
    \dfrac{a}{b} \cdot \dfrac{c}{d}~~=~~\dfrac{ac}{bd}\\*~\\*\dfrac{ac}{bd}~~=~~\dfrac{a}{b} \cdot \dfrac{c}{d}

To simplify a fraction:

  • Rewrite both numerator and denominator as products of factors (if they are not already factored)
  • Examine both numerator and denominator to see if they share any factors
  • If they do share factors, use concept (3) above to move the shared factors into a separate fraction
  • That separate fraction should now have a numerator that is exactly the same as its denominator, which by concept (2) above means that it must equal 1, therefore by concept (1) above we can drop it from the expression

Consider the following fraction… can it be simplified? Continue reading Simplifying Fractions

Negative Differences

Algebra is a set of rules that allow us to change the appearance of an expression without changing the quantitative relationship that it represents. Sometimes the changes in appearance are greater than expected, causing us to doubt whether two expressions really do represent the same quantitative relationship.  The ways in which negative differences can be rewritten seem to surprise people until they become accustomed to them.

Consider a difference that is being subtracted:


If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign in front of them first.  Leaving the parentheses in place while Continue reading Negative Differences

Negative Fractions

Where should I put the negative sign when I am writing a fraction like negative two thirds?  As long as you write only one negative sign, it does not matter where you put it.

Subtraction is the same thing as the addition of a negative. The negative of a number is created by multiplying the number by negative one. These rules apply to fractions as well, so:


Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to Continue reading Negative Fractions

Geometric Sequences and Geometric Series

Geometric Sequences / Progressions

Pick a number, any number, and write it down.  For example:


Now pick a second number, any number (I’ll choose 3), multiply it by the first number, then write their product down to the right of the first number:


Now, continue multiplying the number you chose (3 in my example) by the previous product and writing the result down… over, and over, and over:

5,~15,~45,~135,~405,~1,215, ...

By following this process, you have created a “Geometric Sequence” or “Geometric Progression”, a sequence of numbers in which the ratio of every two successive terms is the same.

Vocabulary and Notation

In the example above, 5 is the first term (also called the starting term) of the sequence or progression. To refer to the first term of a sequence in a generic way that applies to any sequence, mathematicians use the notation


This notation is  Continue reading Geometric Sequences and Geometric Series

Arithmetic Sequences and Arithmetic Series

Arithmetic Sequences

First, a terminology aside: the terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic progression”. From here on in this post, the term “sequence” will be used… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create an arithmetic sequence (also known as an arithmetic progression).

Pick a number, any number, and write it down.  For example:


Now pick a second number, any number (I’ll choose 3), add it to the first number, then write their sum down to the right of the first number:


Now, continue adding the second number to the sum and writing the result down… over, and over, and over:

5,~8,~11,~14,~17,~20,~23,~26,~29, ...

By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:


Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation


This notation is Continue reading Arithmetic Sequences and Arithmetic Series

Piecewise Functions and Relations

While many relationships in our world can be described using a single mathematical function or relation, there are also many that require either more or less than what one equation describes.  The behavior being described might start at a specific time, or its nature changes at one or more points in time. Two examples of such situations could be:

Acceleration up to a speed limit
Free fall then controlled descent

In the graph on the left, note that the blue line starts at the origin. It does not appear to the left of the origin at all. Furthermore, when x = 3 the blue line stops and the green line begins – but with a different slope.

In the graph on the right, note that the blue curve starts at x = 0.  It does not appear of the left of the vertical axis at all.  And when x = 3 the blue parabola turns into a green line with a very different slope. And the green line stops at x = 5.5, just as it reaches the horizontal axis.

These graphs do not seem to follow all the rules you were taught for graphing lines or parabolas. Instead of being defined over all Real values of x, they start and stop at specific values. The graphs also show (in this case) two very different functions, but in a way that makes them look as though they are meant to represent a single, more complex function.  Both of these graphs are Continue reading Piecewise Functions and Relations

Linear Systems: Why Does Linear Combination Work (Graphically)?

A system of linear equations consists of multiple linear equations.  The solution to a linear system, if one exists, is usually the point that all of the equations have in common. Occasionally, the solution will be a set of points.

There are four commonly used tools for solving linear systems: graphing, substitution, linear combination, and matrices. Each has its own advantages and disadvantages in various situations, however I often wondered about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical perspective.

Consider the linear system:


which, when graphed, looks like: Continue reading Linear Systems: Why Does Linear Combination Work (Graphically)?

Where’s the mistake?

I have started a separate blog devoted to helping students learn to find mistakes in worked problems (their own, or someone else’s). If this is of interest, check it out:

7/17/11 Update: There can be great value in work that contains mistakes. Learning to catch your own mistakes is a critical life skill, as is learning to review other people’s work while seeking to understand it fully (the best way to do this is by looking for mistakes).

Along these lines, I came across an interesting blog posting by Kelly O’Shea.  She came up with the idea of insisting that each group who is presenting try to sneak a mistake in their work past their peers.  Brilliant!

Algebra Intro 1: Numbers and Variables

This post begins a series intended to help introduce or re-introduce some of the core concepts of Algebra. It is often very helpful to re-visit these concepts with students who may have memorized their way through previous math courses without slowing down to contemplate some of the concepts behind Algebra.


Numbers are used in the English language as both nouns and adjectives. However, the only physical instance of a “number” in our world is a symbol or group of symbols such as “23”. Most of the time, we seem to use numbers as adjectives: “Look at the two trees.” Continue reading Algebra Intro 1: Numbers and Variables

Algebra Intro 2: Addition

Mathematical thinking probably started with addition. Someone may have combined two piles of bricks, and wondered how many were in the single large pile. Addition is the mathematical term that describes “joining quantities together”.

Properties of Addition

Looking at the patterns that can occur when quantities are joined together, you might have noticed that it does not matter whether 2 bricks are added to a pile of 3, or 3 bricks are added to a pile of 2… either way, we still end up with a pile of 5. So, the order in which we add two quantities does not change the result. This probably makes intuitive sense to you when you visualize the situation above and the final pile that results. Continue reading Algebra Intro 2: Addition

Algebra Intro 3: Subtraction

Once addition has been explored a bit, it leads pretty naturally to a new question: if there are three bricks in a pile, how many bricks do I need to add to it so that there will be five in the pile?

Our addition problems were all phrased using a pattern like

number + number = what?

and the question above rearranges it a bit:

number + what? = number
what? + number = number

This question is usually asked as “What is the difference between Continue reading Algebra Intro 3: Subtraction

Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

Negative Numbers

Negative numbers are something new and interesting to think about. What do they mean?

They arose from changing the order in which we subtracted two numbers. While we usually think of a “difference” by starting our thought process with the larger number, when we fail to do that and try to Continue reading Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites