Cool interactive, web-based, activities,
for learning trigonometry (Chapters 6 and 7)
Learn and have fun at the same time!
I searched youtube for trig lessons. Most seemed slow and dull. I did not find an interesting lecture, and most of the lectures are advertising teasers to get you to buy their whole course. The activities I recommend (linked above) are the best I have found on the web. I recommend you read your text and try to follow it closely.
I strongly recommend you enter two versions of the Law of Cosines as programs into your calculator. The programs are very short and given here:
Here is a TI-83 calculator program to do the Law of Cosines (SAS version to solve for the opposite side). Program 1:
Follow each line here with ENTER. Comments you do not type are in black.
Go to NEW
Enter the name, letter by letter, say, LAWCOS (the blinking "A" means Alphabetic mode which refers to the letters in green on your keyboard. Otherwise, to enter "D" hit "ALPHA (a green key) D")
sqrt(A2 + B2 - 2*A*B*cos(C)) -> D
[In the above, "sqrt" means the square root key, and
"->" is the arrow or "STOre" key.]
In the above program A and B are adjacent sides and C is the included angle. D is the opposite side (which we would call lower-case "c" if we could).
1) Hit "PRGM"
2) go to "NEW" and hit "ENTER"
3) Type in a name, say, "LAWCOS" and ENTER
4) Find the "Prompt" command by hitting "PRGM" [again], arrow over to "I/O" (input/output), arrow down to "Prompt" and hit ENTER.
5) Type in "A,B,C" [Use the alpha key to use the alphabet. There is a comma key]
6) Type in the formula (above)
7) "Display" the result (D) by typing PRGM, I/O and selecting "Disp" and alpha D
8) Quit and run it on a problem where you know the answer, to check it. (Reproduce one of the examples in the text.)
Program 2: To use the Law of Cosines to solve for an angle, use this program (short!):
cos-1((C2-A2-B2)/(-2*A*B)) -> E
of this Course. Read this!
The world is
changing rapidly. Everyday things like iPads, smartphones, apps,
Facebook, Twitter, and massive multi-player on-line games didn't even
exist 10 years ago. What you need to learn about mathematics is
changing rapidly too. Most of the math you learned in high school can
be done by machines. Most of what students used to learn in a typical
precalculus course or even a calculus course can be done by machines
(for free!) So, if you want to learn something with real value, you
need to think about mathematics differently.
This course requires you to change your thinking about math.
This course is different. It is intended to teach you things that machines can not do for you. It is designed to teach you what you need to know about mathematics at the precalculus level that will make you capable and valuable. Software (like wolframalpha.com -- try it!) can compute answers to symbolic algebra and calculus problems, if you know how to communicate with it. So, learning to communicate mathematics to machines is one step. But the real world does not pose problems in symbols, so another important step is to learn how to take real-world problems and convert them to mathematical notation (The term "indirect" in Section 1.1 refers to this.)
In the past math courses emphasized
1) computational skills (because, not many years ago, computations were done by hand) and
But now if you want to solve an equation, there is software that can "do" that type of math problem. Now, everyone knows you can look up facts on the web. If you can pose a math question well, you can probably find an app or web page to answer it, if you can read and write mathematics well enough. Nevertheless, there is still a lot of math to learn.You need to learn how to communicate with your software, interpret graphs, write and read mathematics to understand how to formulate questions and interpret answers. Also, you need to have the right question come to mind, which is a non-trivial skill. We want you to have the prior knowledge required to think of the right questions and techniques.
In summary, if all you know are the facts and computational skills that math classes previously emphasized (in your school!), you have little value to anyone. The best person at multiplying three-digit numbers can be replaced by a $5 calculator. The best person at solving symbolic algebra problems can be replaced by free software on a website! We want to add value to you, not teach you valueless skills.
That is why this course is different--even from what it used to be a few years ago. The world is changing and what we teach and what you need to learn are changing too.
1) This course encourages you to use machines to do problems that are already given in symbols,
2) but, you will not get a lot of credit for being able to do symbolic problems like you learned to do in high school.
3) The intention is to help you develop essential concepts, abilities, and interpretation skills required to apply math to the real world and do things machines won't do for you.
4) You will learn to have the right things "come to mind". (It is one level of learning to know an answer when prompted, another to have it come to mind without prompting.)
5) Exams will focus on concepts and skills you need to develop, not on solving problems that machines can solve for free.
6) The point is to add value to you (knowing how to solve for x in problems posed in symbols is no longer a valuable skill -- sorry!)
7) Reading and writing math forces you to focus on essential, valuable, mathematical concepts.
8) Expect exams to be quite different from math exams you have taken before (even different from previous exams given in this class!) If you have read your text closely, written symbolic math daily on homework, and assimilated the concepts covered in the text, you will do well. However, if all you can do is compute numerical answers to symbolic problems, you will not do well (because the value you added to yourself would be close to zero.)
Take a look at this brilliant artwork created from words on this page. How much do you suppose it cost to have this complex work created?
It cost nothing. It was free on a web site! I just copied and dropped in the text of this page and it arranged the words with size corresponding to word frequency. This (previously) $1000 artwork was free! The point is, machines can do a lot of what we used to train people to do (and pay them well for--but not anymore!). This course is aware of that and will seem different to you because it teaches useful and valuable skills you were not taught in high school.
job is to
1.1. Learn what it means for a problem to be indirect and to become comfortable with working indirectly, that is, writing about operations you don't actually do.
1.2. Learn how order is expressed in written mathematics and on your personal calculator so well that you can rapidly evaluate complicated expressions correctly. This requires learning how to insert parentheses that are not in the usual written mathematics. Practice until you can get five problems from B2-B16 correct in a row.
1.3. Learn how functional notation is used to express sequences of operations. Learn to distinguish the function from the notation used to express it, and how to apply that function to expressions other than "x". (This is also discussed in Section 2.2.)
1.4. Learn how to read definitions and theorems (They express mathematical methods). Learn how to write mathematical methods in symbolic notation. (This is a course-long project, begun in Section 1.1 and treated as the focus of Section 1.4. Reading and writing are essential to word problems, so learning to read and write symbolically is critical.)
1.5. Learn to read graphs (that is, extract information they contain). Learn how to graph with your graphing calculator and obtain a "representative" graph. Learn to select and modify windows, and use the calculator's capabilities to determine key points on graphs. (This is continued in Section 2.1.)
1.6. Recall the usual algebraic ways to solve equations. Learn to rapidly classify equations according to which way to solve them, chosen from the "four ways to solve an equation." (This is a section on making good algebraic decisions.)
2.1. Learn how to choose windows that make graphs "representative" or have a particular look. Learn how changes in the window will change the appearance of graphs. (This section continues Section 1.5)
2.2. Learn how notation expresses functional composition. Learn how given graphs are affected by composition with addition, subtraction, multiplication, division, and attaching a negative sign.
2.3. Learn that solutions to "f(x) = c" may not be unique if the function is not "one-to-one." Learn how to recognize when a function is not one-to-one and how to deal with the complications that occur when your calculator has an inverse function but the original function is not one-to-one.
Read each section. Do not skip the harder parts. In fact, when the going gets rough you need to slow down and read it several times until it makes sense. If it remains unclear, ask!
text: Precalculus, 6th edition,
(The 5th or 4th editions will serve just fine, but correct their typos).
Required graphing-calculator capability: Calculators play a large role, and you must have access to graphing-calculator functionality, but this semester you do not need to buy a stand-alone calculator if you have an iPad or Smartphone or laptop with an equivalent calculator app or software.(The iPad app "Graphing Calculator HD" will serve. I have had the app "Algeo calculator" highly recommended and it looks good and is free. Probably many other app would work fine too. If you get an app, you must take the time to learn to use it!)
You must satisfy our special prerequisite to stay in Math M-151. Have you satisfied it? (Many incoming students who imagine they have actually have not. Check it!)
Course Policies (which includes sections about these and other topics):
calculus) in High
School, so I have satisfied the prerequisite, right?" No! You
must test into the course.
What you took in high school
is does not
count. What you know
counts. Here are the rules
(you must satisfy the prerequisite!), Work, Calculators, Exams and Grading, Course Goals
In this course, calculators
learning tool, not just a calculating tool. Calculators help in
two main ways. By making lower-level work less time-consuming, we can Calculators.
In this course you are supposed to
develop essential algebraic concepts. Graphing calculators or graphing
apps can help and are required. We will use a stand-alone TI calculator
in class, however, you may use any technology which is more or less
equivalent, including smart apps or any internet-based graphing
program. If you already own an iPad, why pay $100 for a calculator when
you effectively already have one?
In this course, calculators are a learning tool, not just a calculating tool. Calculators help in two main ways. By making lower-level work less time-consuming, we can 1) Concentrate attention on essential points, and
2) Increase the rate at which students gather experience with the subject. Other important information that you will want to know. For example, copies of previous exams are available on reserve in the Library. They are also on-line here.
We have free
tutoring! The Math Learning Center (1-110 Wilson) has free
tutoring 8:30am -9:00 pm M-Th and 8:30-2:00
Fridays. Click here
for more about its hours and when you can find a Precalculus instructor
are common-hour exams given at 6:00 pm. The dates are on our calendar.
Mark your personal calendar with these dates and times.
Be there! If you have an unavoidable academic conflict, or a disability, see here.
If you are taking other common-hour exam
may have a conflict. Look up their exam times
see. If you have an academic conflict, you may be able
to resolve it by signing up (with Dr. Esty in 2-238 Wilson Hall) for
our alternative exam time (probably 4:45 the same day for common-hour
exams). However, you must sign up well in advance. Signing up
the last day is not an option.
We use calculators a great deal. Instructors will use the TI-83 or TI-84, but you may use other models or iPad or SmartPhone apps. Learn to use technology. If you use a TI calculator, one program you will need many times is given next.
Activities. Chapter 1
Program your calculator with the Quadratic Formula. Here is a simple four-line program for the TI-83 or 84. Here it is:Prompt A, B, C
If something goes wrong, don't worry. Just QUIT (= 2nd QUIT in yellow) and resume from where you were by hitting PRGM and, this time, EDIT (instead of NEW).
[Now "quit" and try it out on an example where you know the answer. For example, to run it, hitPRGM
Try to solve x2 - 8x + 15 = 0. Did you get 5, 3? If not, check your keystrokes.] If you want to do a second example, you need not begin over, just hit ENTER and it will ask you for the next value of "A".]
You can quit here. The rest gives some interesting links, relevant to education, but not required for Math 151.
Ourselves to Death"
is a prophetic book that was written in 1985. Here is a cartoon that
illustrates its preface. http://www.recombinantrecords.net/docs/2009-05-Amusing-Ourselves-to-Death.html
The book itself is extremely interesting. It is amazing the something written then could still be so relevant (even more relevant) now.
Brain Rules: 12 Principles for
Surviving and Thriving at Work, Home, and School, by John
"The brain is an amazing thing. Most of us have no idea what’s really going on inside our heads.
"How do we learn? What exactly do sleep and stress do to our brains? Why is multi-tasking a myth? Why is it so easy to forget—and so important to repeat new knowledge?
"Brain Rules is about what we know for sure, and what we might do about it."
Here is the fascinating site. Learn about how to learn. Pay attention to the "12 rules".
fatty food appears
to take an almost immediate toll on both short-term memory and exercise
performance, according to new research on rats and people. 'We
expected to see changes, but maybe not so dramatic and not in such a
short space of time,' said Andrew Murray, the study’s lead author.’’
a link to a
commercial site on learning (believe it!), and a link to its page
on the role of
sleep in learning.
Practice beats talent
when talent doesn't practice
-- unknown author
This site has information about
Calculator programs (like the Quadratic Formula) we use frequently.