Randy’s current tutoring subjects are listed at the left. You
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The ACT Mathematics Test is a 60-question, 60-minute test designed to measure the mathematical skills students have typically acquired in courses taken by the end of 11th grade. The test consists of multiple-choice questions that require you to use reasoning skills to solve practical problems in mathematics. You need knowledge of basic formulas and computational skills to answer the problems.
So, what can I do for you? Well, if you’re in need of tutoring, there are areas that you’re probably not sure of, or maybe your test taking abilities need to be improved, or both. Covering the weak spots or gaps, as teachers like to call them, in your knowledge is just a matter of identifying those gaps and thinking of new ways to approach them. Weak test taking skills? Well, that’s an area I know a lot about, unfortunately, from personal experience. While in college, I was the worst test taker, and my grades suffered miserably. That is, until one of my favorite professors taught me the trick to taking tests. Want to know what it is?
Not now! That’s what I’ll teach you!
My favorite math subject! It has the most puzzles to it. Think about it. All those letters in the equation are unknown factors in the puzzle. Each step in solving the equation is like finding a clue to solving the puzzle. (Ok, maybe you don’t get as excited as I do, but it can be fun if you think of it as puzzles instead of problems!)
Algebraic reasoning actually starts way, way back. Remember when your teachers in grade school would phrase the question this way, “What number plus 3 is equal to 4?” Well that’s algebra! The equation is obvious now, X + 3 = 4. Solve for X.
This goes on year after year until a teacher finally writes X + 3 = 4 on the board and asks, “What number does X stand for?” Wow, you’ve just been introduced to algebra! That usually happens in 5th or, for sure, by 6th grade.
Remember the equation for a line: y = mx + b? Well, that’s algebra! Try to relax; you already know a lot about the subject before you go to the first class.
Algebra is about finding the unknown or it is about putting real life problems into equations and then solving them. Algebra is a branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale. What is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra develops your critical thinking, specifically logic, patterns, problem solving, deductive and inductive reasoning.
Algebra can include:
Real numbers, complex numbers, matrices, vectors
Algebra is my favorite math subject! This course is a continuation of Algebra 1. Algebra is the course that springboards you into Calculus. Mathematics is the language of Physics, the rules that govern the universe. In Physics you will learn to do what is called “Deriving your own formula.” You’ll use Calculus to piece together an equation, or system of equations, that describes whatever your “system” is, whatever piece of the universe you’re looking at. This could be the arc of a ball in the air or the orbit of a planet or a complex molecular chain. Whatever it is, you begin with Calculus.
You then manipulate these equations, simplifying them until you get down to Algebra, Trig and Geometry; from there, you crunch the arithmetic to get a final answer. All of Math builds upon itself.
Arithmetic is the beginning, Calculus is the end.
But, even if you don’t go on to Calculus, Algebra is used in many disciplines. As a machinist, I had to be very good at Algebra, Trig and Geometry.
Algebra 2 is designed to build on algebraic and geometric concepts from Algebra 1. It develops advanced algebraic skills such as systems of equations, advanced polynomials, imaginary and complex numbers, quadratics, and includes the study of trigonometric functions. It also introduces matrices and their properties.
The contents of this course are important for students’ success on both the ACT and college mathematics entrance exams.
I received my high school diploma from Permian High School in Odessa, Texas (go Panthers!) and went on to get my BS in Aerospace Engineering from UT Austin (Hook’em Horns!). I was the foreman of a machine shop when I was 16 (lot’s of math, lot’s of mechanical aptitude!). I also obtained my private pilot’s license (again, lot’s of math!) at the age of 20. After graduating from UT, I took the Air Force Pilot’s exam and aced it. I was one of a very few selected from Texas that year to go into the pilot program. Unfortunately, during my years as a martial arts instructor (yeah, I taught martial arts too) I received one too many injuries and did not pass the physical, so I wound up as an Aerospace Engineer. My strength lies in math and science but, I was also a strong student in other areas like reading and writing (avid reader and writer, even wrote a book once).
Your dream of going into the military is an honorable one. Let me help you achieve your dream.
If you’re not sure what the ASVAB test battery covers read on.
The Armed Services Vocational Aptitude Battery (ASVAB) is a multiple choice test, administered by the United States Military Entrance Processing Command, used to determine qualification for enlistment in the United States armed forces. It is often offered to American high school students when they are in the 10th, 11th and 12th grade, though anyone eligible for enlistment may take it. There has never been a requirement that a test-taker with a qualifying score enlist in the military, and the test may simply determine personal aptitude at a particular career.
The ASVAB currently contains nine sections. The duration of each test varies from as low as ten minutes up to 36 minutes for Arithmetic Reasoning; the entire ASVAB is three hours long.
The ASVAB Subtests: Paper Version
If you take the ASVAB at a Military Entrance Test (MET) site, you’ll likely take the paper-and-pencil version of the test. The following outlines the subtests on the paper version of the ASVAB, including information on content, the number of questions, and time limits.
General Science (GS) has 25 questions with an 11 minute time limit. It covers general principles of biology and physical sciences.
Arithmetic Reasoning (AR): 30 Questions, 36 Minutes. Simple word problems that require simple calculations.
Mathematics Knowledge (MK): 25 questions, 24 minutes. High school math, including algebra and geometry.
Word Knowledge (WK): 35 questions, 11 minutes. Correct meaning of a word.
Paragraph Comprehension (PC): 15 questions, 13 minutes. Questions based on several paragraphs a few hundred words in length that you read.
Electronics Information (EI): 20 questions, 9 minutes. Electrical principles, basic electronic circuitry, and electronic terminology.
Auto & Shop Information (AS): 25 questions, 11 minutes. Knowledge of automobiles, shop terminology, and tool use.
Mechanical Comprehension (MC): 25 questions, 19 minutes. Basic mechanical and physical principles.
Assembling Objects (AO): 25 questions, 15 minutes. Spatial orientation.
Navy applicants also complete a Coding Speed (CS) test.
Basic math is well, just arithmetic. But just because it’s only arithmetic doesn’t mean it’s easy. Can you do your multiplication tables in your head? That’s 1x1 through 12x12 by memory. Most kids coming into 6th grade can’t. The Texas TEKs (Texas Essential Knowledge and Skills), requirements that teachers must go by in the instruction of their students, says that a student must have mastery and an “instantaneous recall” of multiplication facts by the end of 5th grade!
Why is it so important to know these facts? After all, I can just crunch it on a calculator, right? They’re taught to memorize those facts so they won’t be slowed down punching every arithmetic problem into a calculator. Later, especially in algebra and calculus, you don’t have time to use the calculator for the trivial stuff like arithmetic. You need to be able to add, subtract and at least multiply numbers in your head. Notice I left division out. You should be able to divide and get estimated answers in your head but by the time you get to higher level math and need an answer to several decimal places of accuracy, a calculator comes in handy.
So, why don’t we know them? It’s the memorization part that kills us. Yes, even me. My students knew that I did not have all my multiplication facts memorized. They also knew that I could calculate those problems in my head, very quickly. There are some mental tricks to this that I taught my students. Soon the counting on the fingers stopped as they learned the clues to the puzzle!
Basic math, or arithmetic, is math before it becomes too abstract. Math from pre-school through to the 6th or 7th grades would qualify to be basic math.
Topics covered include:
Fractions, Multiplication, Pre-Algebra, Word Problems, Addition, Subtraction, Multiplication and Division
I received my high school diploma from Permian High School in Odessa, Texas and went on to get my BS in Aerospace Engineering from UT Austin. I was the foreman of a machine shop when I was 16 (lot’s of math!). I also obtained my private pilot’s license (again, lot’s of math!) at the age of 20. My strength lies in math and science but, I was also a strong student in other areas like reading and writing (avid reader and writer, even wrote a book once).
I did all this through hard work and perseverance. Getting your GED takes a lot of hard work and perseverance as well. Let me help you.
If you’re not sure what the GED test battery covers read on.
General Educational Development (or GED) certifies that you have American high school-level academic skills. The GED is the equivalent to a high school diploma. If you’re going for your GED I commend you. One in every seven Americans with high school credentials received the GED test credential, as well as one in 20 college students so, you’re in good company.
The GED comprises five tests: Language Arts: Writing, Language Arts: Reading, Social Studies, Science, and Mathematics.
Language Arts: Writing
The Writing test portion is divided into two parts. The first covers sentence structure, organization, usage, and mechanics. You’ll read text from business, informational, and instructional publications and then correct or improve the text. 50 items - 75 minutes.
This part of the Writing test requires you to write an essay on an assigned topic in 45 minutes. A passing essay must have well-focused main points, clear organization, specific development of ideas, and demonstrate your control of sentence structure, punctuation, grammar, word choice, and spelling. Assigned topics are always an opinion or perspective that does not require special knowledge.
Language Arts: Reading
This 65-minute, 40-question test examines your ability to read and understand texts similar to those encountered in high-school English classrooms. The test has five fiction and two nonfiction passages, each about 300–400 words long. The fiction passages include portions of a play, a poem, and three pieces of prose. The nonfiction passages may come from letters, biographies, and magazine articles. Each passage is followed by questions that assess reading comprehension, as well as your ability to analyze and apply the information given to other situations.
This test covers American and world history, civics and government, economics, and geography; 70 minutes - 50 questions.
In the social studies test, you read short passages and answer multiple-choice questions. Some passages come from the Declaration of Independence and U.S. Supreme Court decisions.
Questions involving economics as well as civics and government rely heavily on practical documents, such as tax forms, voter-registration forms, and workplace and personal budgets.
This test covers life, earth, space, and physical science. It measures your skill in understanding, interpreting, and applying science concepts to visual and written text from academic and workplace contexts. You should expect to see tables, graphs, charts, and diagrams, as well as complete sentences. 50 multiple choice questions – 80 minutes.
This 90-minute, 50-question test has two equally weighted parts, the first of which allows you to use calculators, while the second forbids their use. You must use the calculators issued at the testing center, no other.
Forty of the 50 questions are multiple-choice; the other 10 require that you record answers on either a numerical or coordinate-plane grid. Both portions of the test have questions of both types. The test booklet offers a page of common formulas as well as directions for using the calculator.
The test focuses on four main mathematical disciplines: Number operations and number sense; Measurement and geometry; Data analysis, probability, and statistics; Algebra, functions, and patterns.
Passing the GED tests:
Possible scores on an individual test within the GED battery, like those on an individual section of the SAT, range from a minimum of 200 to a maximum of 800. A score of 800 on an individual test puts you in the top 1% of graduating high school seniors.
Geometry can be very challenging. (Sorry about that) The description of the course below uses the phrase “spatial sense and geometric reasoning.” Huh? What it really means is: can you see objects and shapes in three dimensions in your head? Can you project the movement and consequently the change in the shape of the object as it moves through space? The arc of a ball flying through the air, the angle the eight ball will go after the cue ball makes contact? This is all geometry and the better you get at geometry the more you develop your sense of “spatial sense.”
Why is it so hard? Even though kids play video games that mimic 3-D space most can’t visualize very well in 3-D. Even adults can struggle with this.
In the classroom, we start in 2-D space with simple polygons and move toward 3-D space. (A cube for example) Moving from the 2-D concept of area to the 3-D of volume to the 4-D of motion.
This can get quite exciting, when after a day of learning all about the dimensions, area and volume of a Frisbee, and then talking about and drawing diagrams of the angular momentum imparted on said Frisbee, I would then throw the Frisbee into a mass of expectant kids. Havoc ensued, much was learned and fun was had by all.
And now a very dry description of the course of geometry and subjects covered:
Geometry is the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. In geometry, you will explore spatial sense and geometric reasoning. Geometry is found everywhere: in art, architecture, engineering, robotics, land surveys, astronomy, sculptures, space, nature, sports, machines, cars and much more. Geometry is in every part of the curriculum K-12 and through to college. Since most educational jurisdictions use a spiraling curriculum, the concepts are re-visited throughout the grades advancing in level of difficulty. In the early years, students identify shapes and solids, use problem solving skills, deductive reasoning, understand transformations, symmetry and use spatial reasoning. Throughout high school there is a focus on analyzing properties of two and three dimensional shapes, reasoning about geometric relationships and using the coordinate system.
Some Common Terms in Geometry:
Lines and Segments, Shapes and Solids, Triangles and Angles, Coordinate Grids, Radians, Conic Sections, Circumference, Polygons, Trigonometry
Linear algebra is the branch of mathematics concerning finite or countable infinite dimensional vector spaces, as well as linear mappings between such spaces.
Remember doing a system of equations with several unknowns? That’s what you do in Linear Algebra but you’re going to use matrices and vectors to manipulate your system. These kinds of manipulations (transformations) make it much easier to solve your system of equations.
A linear transformation is a formal mathematical way of presenting the idea of proportionality or that something "is proportional to".
What’s that mean? Well, if a machine can make 5 widgets in 1 hour I can expect it to be able to make 10 widgets in 2 hours. That is proportionality and the new part (in Linear Algebra) is that the transformation is linear.
But, what about my running ability? If I can run a 12 minute mile should I expect to run 5 miles in 60 minutes? Not me! I tucker out pretty quick! So, if you plot time per mile, you get a curve. In Linear Algebra this is called a non linear transformation.
Beginning to get the idea? A lot of Linear Algebra is stuff you’ve done before i.e., finding the solution to a system of equations, but learning new ways to do it i.e., matrices.
I am an Aerospace engineer. We use a lot of math. Linear Algebra is just one example. But what good is it? What does it do in the real world? Aren’t there computer programs that can solve a system of equations?
Yes, there are programs that can do almost anything. But who programmed the programs? People really good at math and who understand its concepts like the back of their hand. In the real world hardly anything is linear. Systems get so complicated, especially in physics, that it is almost impossible to just “see” if something is linear or not. Linear Algebra teaches you how to determine if your system is linear and if the changes made to the system (the transformations) change this linear system into a non-linear system.
You already know how much easier it is to work with a straight line (linear equation) than curves (non-linear), so if you can figure out how to transform your linear system and keep it linear, well, linear is easier.
You can even get approximate results for a non-linear system with Linear Algebra. This is great for determining if you’re going in the right direction with a multi-million dollar project before you get too far down the road with the design.
Here are a few of the topics (and fancy names) used in a Linear Algebra Course:
Eigen value - One of a set of special scalars associated with a linear system of equations that describes that system's fundamental modes.
Eigenvector - One of a special set of vectors associated with a linear system of equations.
Inner Product - A synonym for dot product.
Linear Transformation - A function from one vector space to another.
Matrix - A concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix is an extremely important concept in linear algebra.
Matrix Multiplication - The process of multiplying two matrices (each of which represents a linear transformation), which forms a new matrix corresponding to the matrix representation of the two transformations' composition.
I have used spreadsheets as an engineer and teacher for 25 years. If you’re old enough you may recall a little program called Lotus 1-2-3. Yep, I was the first engineer in my group to start using that thing, oh those many years ago.
I eventually moved over to Excel when it first came out and have been hooked on it ever since. It’s a wonderful tool for data analysis and the graphs make it very easy to see all that data in a visual and colorful format.
Basically, Excel is an electronic spreadsheet program that is used to manipulate numbers. Its widespread use makes it easy to exchange files with almost anyone, or to display information in printed form. Because it is part of the Microsoft Office Suite, it can easily exchange data with that suite's database, word processing and presentation program.
Set the calculations and formulas between variables only once. These numbers change automatically when you change a value, making it easy to try what-if scenarios.
You can sort and filter data for easy analysis. Conditional formatting (different colors) allows you to highlight information that meets certain criteria.
Cut through the incomprehensibility of too many figures by displaying them as a colorful chart. You have full control over the color and appearance of bar graphs, pie charts and line charts.
Need to automate complicated entries and procedures? Use the built in Visual Basic for Applications program language to show dialog boxes, forms and buttons that process data with one click.
Excel is a really cool program, an essential one if you work in a field that’s collects, collates and analyzes a lot of numerical data.
Pre-algebra is designed to prepare a student for Algebra 1. Wow, that helps doesn’t it? What is it really? It’s a refresher course. Everything you’ve learned up to now will be gone over again, one last time, before you take that step into what I consider to be the beginning of real mathematics, Algebra. Everything that is covered, you should know by now, if not, this is your chance to hone those skills needed for the next step!
Some of the topics covered:
Integers, Fractions, Fundamentals of Graphing, Ratio/ Proportions/ Percents, Statistics and Probability, Using Formulas, Geometry, Solving One and Two step equations/ inequalities
Uh oh! There’s that dreaded word: Calculus! Yep, Calculus is where the rubber meets the road. If you’re going into any field that is remotely technical and riddled with geek-speak, such as Aerospace engineering, you need Calculus. Sorry, no way around it.
So what is Calculus? Calculus is the mathematics of change. Calculus deals with areas and volumes, rates of change, the orbits of planets, and infinite sequences. Ok, what’s that mean? Well, it’s the math you need when the fourth dimension, time, is brought into the equation. Not only time, but when an object changes how fast it moves through space and time, or when the area or volume of an object is not uniform but irregular in shape. And, what in the world is an infinite sequence? Well, did you know that an equation can “blow up?” Yes it can! In Calculus an equation can have many solutions, not just one; in fact, it can have an infinite, never ending, number of solutions, all equally valid. When you learn Calculus, you will find out how to tell when an equation will traipse off into infinity land and how to coral it. These are called the Limits to an equation. That’s kind of a nice thing to know before you program the equation into a computer and tell it to solve the thing. Your computer will quickly loose its proverbial mind.
But, Mr. A., this is Pre-calculus. Oh yeah, I forgot. Got excited for a moment there talking about Calculus!
Pre-calculus is a check-up course to see if you’re ready for the real thing. It consists of those subjects, skills, and insights needed to understand calculus. It includes arithmetic, algebra, coordinate geometry, trigonometry, and, most of all, functions (just an equation). A pre-calculus course builds on all of your previous mathematical knowledge and experience. It gives you a chance to hone your skills.
Calculus is so important the powers that be put in one last course just to make doubly sure you know all you need to know before making that leap into Calculus. In the past, I have described mathematics as a language, the language of physics, how everything in the universe works. That’s one of the reasons so many folks have trouble with math. Learning a new language is hard for us normal folks. Everyday mathematics, up through trig, geometry and algebra, is like learning Spanish. I took a year of Spanish, still don’t know it and thought it was hard. But, I’ve been told it’s a relativity easy language to learn. Yeah, right! But then, you get to Calculus, a most beautiful language indeed. But, it’s more like learning Latin! When you learn Calculus, you are speaking the language of mathematicians, physicists, and engineers.
Sound scary? Trust me, if you really know all that other stuff, algebra, trig, etc., you can handle Calculus. Believe me, if I could get through it, so can you!
Teachers and engineers take a lot of tests over the years. I was terrible at tests until a friendly college professor taught me a few tricks. My test scores improved dramatically. If not for him I would have flunked out of Aerospace Engineering.
As you are well aware, students take a lot of tests, too! As a teacher I have taught many students these tricks. If you have gaps, let’s identify them together and I’ll show you a different way to think about them to clarify your thinking. If you’re nervous or your mind just goes blank, well, we can do something about that as well.
And now, a little about the PSAT:
Over three million high school students take the Preliminary SAT, or National Merit Scholarship Qualifying Test (PSAT or NMSQT) each year. Like the SAT, the PSAT is designed to measure the ability to understand and process elements of reading, writing, and mathematics. Many students use this exam to get feedback on their strengths and weaknesses in preparation for the SAT. In addition, students may qualify for the prestigious National Merit Scholarship program based on their PSAT scores.
Test Format -The PSAT consists of five sections:
Two 25-minute Critical reading sections
Two 25-minute Math sections
One 30-minute Writing Skills section
The whole test requires 2 hours and 10 minutes. The two Critical Reading sections and the two Math sections can appear in any order among the first four sections. Writing Skills is always the final section of the test
The Math sections of the PSAT measure the ability to reason quantitatively, solve mathematical problems, and interpret data presented in graphical form. These sections focus on four areas of mathematics that are typically covered in the first three years of American high school education:
Algebra and Functions (but not 3rd year level math that may appear on the new SAT)
Geometry and Measurement
Data Analysis, Statistics and Probability
There are two 25-minute Math sections on the PSAT. One consists of 20 multiple-choice questions; the other consists of 8 multiple-choice questions and 10 grid-in questions. Either section can be the first Math section to appear on the test. Grid-in questions require test-takers to enter answers into a special grid on their score sheets. There is no penalty for incorrect answers on grid-in questions.
I have had years of experience preparing for and taking tests. In college, homework was not counted for a grade, only test scores. To be certified as a teacher, you have to be a good test taker. We take lots of tests. As a teacher, I taught my students how to prepare and take tests to maximize their performance.
But, at one time, I was not a good test taker. It wasn’t until a professor in college taught me a few tricks that I started to do well. If you’re nervous or your mind just goes blank come test time, you’re in trouble.
Allow me to show you how to overcome your fears and anxieties. Or if you just need those gaps in understanding filled, I can help you identify those and find new ways of thinking about them that will help you to understand.
If you’re unfamiliar with the SAT read on:
The SAT is a standardized test that colleges use to evaluate applicants. Over two million students take the SAT every year, and it is used by nearly every college in America for evaluating a student’s college preparedness. It is designed to measure a student’s ability to understand and process elements in three subjects: reading, writing, and math. SAT scores are calculated based on a student’s performance relative to other test-takers.
The math sections measure a student’s ability to reason quantitatively, solve mathematical problems, and interpret data presented in graphical form. These sections focus on four areas of mathematics that are typically covered in the first three years of American high school education: Arithmetic, Algebra and Functions, Geometry, and Data Analysis. The Algebra section was recently expanded to include basic College Algebra. To test these skills, the SAT employs two different question types:
Multiple-Choice and Grid-Ins
The multiple-choice questions carry a .25-point penalty for incorrect answers. The grid-in questions carry no penalty for wrong answers, because the likelihood of guessing the correct answer is negligible.
The format of the three sections is:
25 minutes: 20 Multiple-Choice questions
25 minutes: 8 Multiple-Choice questions followed by 10 Grid-ins
20 minutes: 16 Multiple-Choice questions
Getting a BS in Aerospace Engineering required that I hone my study skills to a fine edge. During my college years I worked full time as a machinist, taught martial arts at night and even found the time, but only rarely, to practice my flying and do a little skydiving.
As a certified teacher in McKinney ISD, I also taught study skills to my students.
I would be honored to help you or your child improve your skills.
As a teacher in McKinney ISD I administered the Math TAKs test every year.
If you or your child has gaps in knowledge, I can identify them. If you just "don't do well on tests" I can help with that as well, as I was a poor test taker myself until a professor friend of mine showed me a few tricks.
You've probably heard that TAKs has been replaced by the STAAR. If not, a little dry info follows:
STAAR replaces TAKs:
The new assessment test, called STAAR and pronounced star, was first used beginning in the 2011-2012 school year. Students in the graduating Class of 2015, will be the first students who must meet the end-of-course testing requirements, as well as pass their classes, in order to earn a diploma. The new tests will be significantly more rigorous (or so it is promised by the powers that be) than previous tests and will measure a child’s performance, as well as academic growth. The grade 3-8 STAAR tests in reading and mathematics, by law, must be linked from grade to grade to performance expectations for the English III and Algebra II end-of-course assessments. The last TAKS-based school accountability ratings were issued in 2011.
As an Aerospace engineer I designed a lot of widgets. Guess what was used most? (Well, maybe the most) Yep! Trig!
“SOH-CAH-TOA!” Learn it, memorize it, love it. You will use that mnemonic forever if you have any notions of doing anything remotely techie in the future.
What do I mean by “remotely techie?” Ok, how about, astronomy, navigation (on the oceans, in aircraft, and in space), music theory (What? Music theory? Yep!), acoustics, optics, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), chemistry, number theory, cryptology, seismology, meteorology, oceanography, many physical sciences, land surveying, architecture, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography and game development.
But, what is Trig Mr. A? Remember the Pythagorean Theorem? (a^2 + b^2 = c^2) That formula is great as long as you know the length of 2 sides of your triangle. What if you only know 1 side and 1 angle? Voila! Trig!
Trig is going to take the Pythagorean Theorem and change it a little using, sine, cosine, tangent and other functions you’re probably already familiar with from using your calculator. But, Trig does much more than that. It will show you how to explore the unknowns of a circle and other shapes.
Trig is one of the most useful and important of all the math subjects.