I have completed a GeoGebraBook of nine Unit Circle Symmetry applets, which you can use by clicking here. There are three applets per type of symmetry on the unit circle, one focusing on the unit circle only, and the other two linking unit circle properties to patterns in the graphs of the sine and cosine functions.
When two angle expressions, such as and , exhibit symmetry on the unit circle, mastery of unit circle symmetries and reference angles often allow function arguments to be simplified. Mastery of symmetries and reference angles will also be very handy when expanding inverse trigonometric function results to describe all possible answers to a problem.
Suggestions for improvements to these applets, or additional applets, are always welcome via comments on this post.
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:
Another example is:
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.
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
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
Three concepts help explain the process of simplifying fractions:
- Multiplying a quantity by 1 has no effect
- A fraction whose numerator is exactly the same as its denominator is equal to 1 (unless the denominator equals zero)
- A product of two fractions can be rewritten as a fraction of two products (and vice versa)
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
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
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:
So, placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to Continue reading Negative Fractions
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:
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
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:
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
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
VEX Robots can be more competitive when they have addressed several drive motor control challenges:
- Stopping a motor completely when the joystick is released. Joysticks often do not output a value of “zero” when released, which can cause motors to continue turning slowly instead of stopping.
- Starting to move gradually, not suddenly, after being stopped. When a robot is carrying game objects more than 12 inches or so above the playing field, a sudden start can cause the robot to tip over.
- Having motor speeds be less sensitive to small joystick movements at slow speeds. Divers seeking to position the robot precisely during competition need “finer” control over slow motor speeds than fast motor speeds.
These challenges can be solved using one or more “if” statements in the code controlling the robot, however using a single polynomial function can often solve all of these challenges in one step. A graph can help illustrate the challenges and their solution:
Continue reading Polynomials and VEX Drive Motor Control
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)?
Do you ask questions in class at least once per week? For many students, the answer is probably “no”. Reasons for such an answer may include one or more of:
– I don’t want to let my peers or the teacher know I don’t understand something
– I am uncertain about what to ask… I just don’t get what the teacher is talking about
– I don’t wish to appear to be the teacher’s “pet”
– I am not being called on when I raise my hand
– Someone else asked a question first, and the teacher needed to move on
– The teacher has not answered my past questions – they just said “see me after class”
A number of small preparatory steps may help get your questions answered in class, particularly if your class is a large one. The need for such steps will vary greatly from one school to another, or one teacher to another, but they will not hurt your efforts to master the subject even if they are not necessary to get your questions answered during class time:
- Ask questions of your teacher outside of class time on several occasions early in the semester, particularly if you have never had this teacher before. This will help you get to know your teacher a little better, and will also help your teacher:
– associate your name with your face
– gain some Continue reading Successfully Asking Questions In Class
As a parent, I look for two categories of attributes when choosing a school for my child:
– Ones which benefit my child directly
– Ones which benefit my child indirectly, by helping others (teachers, parents) do their jobs more effectively
Schools that satisfy more of the attributes in both categories are likely to have happier parents and more successful students.
The Administration and Teachers Should Help My Child
- Being aware of history. Before the start of each school year, my child’s current teacher(s) should have reviewed all of
– last years’ teacher comments for my child
– my child’s transcript (all courses, all years at the school)
- Helping my child to both pursue existing Continue reading What A Parent Wants From A School
When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and co-domain.
The “domain” of a function or relation is:
- the set of all values for which it can be evaluated
- the set of allowable “input” values
- the values along the horizontal axis for which a point can be plotted along the vertical axis
For example, the following functions can be evaluated for any value of “x”:
therefore their domains will be “the set of all real numbers”.
The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:
The values for which a function or relation cannot be Continue reading Domain, Range, and Co-domain of a Function
Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are the same value):
Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
– roots, to undo exponents
– logarithms, to undo bases
Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many versions of the “root” function: one for each exponent value to be undone.
The symbol for a root is , and is referred to as a “radical“. It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a grouping symbol (like parentheses) to Continue reading Roots and Rational Exponents: a summary