kitchen table math, the sequel

From Dolciani's Algebra and Trigonometry, Chapter 13-2 Circular Functions, p. 555:

sin s = y
cos s = x
tan s = sin s/cos s if cos s ≠ 0
cot s = cos s/sin s if sin s ≠ 0
sec s =1/cos s if cos s ≠ 0
csc s = 1/sin s if sin s ≠ 0

s is the length of an arc of a circle.

This may be asking too much, but I need help.

I have never seen sine, cosine, tangent, etc. applied to an arc. I've only learned sine and cosine in relation to angles in a right triangle.

I've looked back through chapter 12, but I don't see a section that explains this. I'm sure it's there, but it's not obviously there, and I'm in a hurry, sad to say.

Is there a short way anyone can explain to me how we get from sine, cosine, angles, and SOHCAHTOA to sine, cosine, and arcs?

Is there a website that has a succinct and lucid explanation? 

And is there a book you like for self-teaching trigonometry and algebra 2? (Do we know what book(s) homeschoolers use?)

I need a royal road to circular functions.

I remember being very discouraged (in the old traditional math days, no less) trying to understand mixture problems because the book we used approached it using tables and grids. When the problem changed a little bit, I couldn't figure out which numbers went into what boxes. I finally learned to approach the problems using governing equations and defining variables.

That understanding didn't come from solving one or two problems. I had to work at it. There were so many times when I thought I understood what I was doing only to feel completely lost when I tackled the homework set. That's when the real lightbulb goes on. Look at any proper math text book and you will see homework sets that give you all sorts of problem variations of the material in the section.

I also want to make a case for speed in helping understanding too. As you move along to more complex math, you need this speed or else you will be completely bogged down. In high school, I got really good at "seeing" right triangles in word problems, even if the triangles weren't explicitly drawn. I was very fast at finding any side or angle given "enough" information. I could state that a length was something like d*cos(theta) just by looking at it. I didn't have to draw a picture and stew over which leg is for sine and which leg is for cosine.

The mechanical monkey paradigm leads to all sorts of wrong conclusions. It also conveniently fits in with their predisposition to equate mastery with rote learning and drill and kill. When they talk of balance, they really don't mean it. They still think it's just for convenience rather than understanding.

This position might seem reasonable when it comes to the basic algorithms of arithmetic, but it falls completely apart as you head into algebra.

Reading this post makes me want to go do, right this minute, two things that cannot be done at the same time:

• fire up ALEKS and finish the geometry course I was taking before my mom fell last summer
• finally write my post on just exactly how much money Response to Intervention (pdf file) is going to cost us once RTI gets going in public schools with a) lousy curricula and b) no focus whatsoever on deliberate practice (pdf file) & mastery
  

Maybe I should spent 10 minutes de-cluttering my desk before I do either of those.

On our next to last day in St. Romain en Viennois we went to the local Intermarché, where I purchased a Mini Kit Incassable in the school supplies section.





contents:
ruler
protractor
right triangle 45-45-90
right triangle 30-60-90

I'm pretty sure it is not possible to buy trigonometry paraphernalia in U.S. supermarkets.

I also scored a Travaux Pratiques for writing proofs and a Vocabulary Coach for memorizing Spanish & French vocabulary. Vocabulary Coach is very cool.

The Intermarché is a big-box Target-type store, like Meijer's in the Midwest.





When I get back to Springfield, IL in a couple of weeks, I'll see if Meijer's has right triangles idéal pour la trousse.

I'm guessing no.

Vicky S sent me notice of a workbook Stephen Wilson has posted on his web site: SAT/ACT Math and Beyond: Problems Book by Qishen Huang.

The book is listed here. I've just ordered the solution manual, which Dr. Huang says is highly detailed (460 pages for the manual, 131 for the workbook). That's critical for those of us teaching ourselves, and not easy to find.

Dr. Huang estimates that 20% of Chinese high school graduates can work 90% of these problems, which he says are not as difficult as those on China's SAT equivalent.

And...on the subject of workbooks, I've emailed Myrtle, who is using the NEM Workbooks (New Elementary Mathematics Syllabus D 1 and New Elementary Mathematics Syllabus D 2).

Meanwhile, I have done no math at all this summer, because I am busy reading C's massive Summer Assignment list, all 2549 pages of it. As to that, please know that you are in the presence of a woman who has read every last word of Guns, Germs, & Steel. There are few amongst us who can say the same.

from a comment left by Le Galoisien:

But really. If they for example, require you to know logarithms, the general attitude is like, "You probably forgot how to do these. Here's a refresher." and you learn the concept over again ... because it is true even the top students in the school forget them because the syllabus is structured in such a way that it is hard to exercise them all year.

I remember the seniors would cry, "awww, you mean we had to remember those?" when there was a rare AP problem that required us to know a trig identity we had learned a year before. And the teachers would respond, "of course. You didn't learn them for nothing," implying it was our fault (but begrudgingly teaching them to us again). But somehow, even though it was partially our fault for not revising the concepts we had learned over the years (even when we had been assigned no work that dealt with them after we finished the unit) I often wonder if it is someone else's fault as well.

I mean, imagine all the time that has to be used reteaching concepts, and generally just in time for the examinations, before we put them in the closet again.

If we reinforced them all along, I wonder if students would save so much time with progress so much quicker that doing linear algebra in your senior year would be no big deal.

That's the story around here, only worse.

Learn percent, forget percent.

Learn percent again, forget percent again.

Repeat, repeat.

Meet with math chair; math chair says class had no business flunking latest test because "they saw that material last year."


inputs, not outputs

I'm realizing, again, how deadly the inputs model is. When school quality is defined by class size, per pupil spending, and number of Masters degrees held by teaching staff (pdf file), there isn't much incentive to design curriculum & instruction that ensures students will actually remember what they've "learned."

In my next life I plan to live on a planet where schools and curriculum designers focus on:

a) how to get content and conceptual understanding into students' long-term memory

b) how to keep it there


Here's Stanley Ocken addressing the National Mathematics Advisory Panel:

My second suggestion is that you investigate and make recommendations regarding common sense issues of pedagogy. It's important to think about the sequence of tasks and knowledge that lead to success in algebra, but it is critical and possibly easier to find out why so many entering college students seem to have forgotten the algebra they learned in school. You could begin by stripping away the obfuscating rhetoric of blind rote and drill and kill. Then you might examine the proposition that repetition and practice, properly implemented, are essential to success in mathematics, just as repetition and practice, properly implemented, are
essential to success in music, sports and the study of foreign languages. You could conclude by identifying prior indicators of successful college math students.

Before they got to college, did they experience rigorous and frequent in-class assessments? Were they required, for example, to master the multiplication facts by the end of third or fourth grade, or were their programs grounded in the principle that it doesn't matter if children master the material this year, since they are going to relearn and re-relearn the same elementary material in later grades? In other words, please investigate the role of basic interventions that clarify the scheduling and rigor of learning goals, these may be more effective and easier to implement than complex manipulations of curriculum and pedagogy.

Here’s my third and final suggestion. Enunciate the importance of a coherent K to 16 mathematics curriculum, one grounded in the principle that K to 12 math instruction must permit and encourage students to prepare for the rigors of calculus. To bring that principle to life, we'll need to see fundamental changes in the dynamics of K to 12 curriculum design.
source:
National Mathematics Advisory Panel (pdf file)
Thursday September 14, 2006

I'm fairly certain the only group of people who measure the success of their instruction by student achievement, which means, among other things, student remembering, are the behaviorists. I may be wrong about that, but I suspect not.

worksheets: arithmetic through calculus

yowza!

Now I can spend the rest of the night downloading stuff onto my desktop instead of revising my chapter.

two birds, one stone

Does anyone have algebra 2 & trigonometry recommendations?

Any thoughts on Dad's Trigonometry?

Or on College Mathematics?

Another question.

Foerster's Algebra 1 is fantastic. (I think someone from ktm-1 sent that book to me - right?? ... yup, I thought so) When I was struggling to teach myself & Christopher function notation in 24 hours, Foerster's was the book for the job. I Xeroxed pages for my neighbor to use, too. Wonderful.

It has rave reviews on Amazon, and Mathematically Correct likes it, too, as does Greta F.

Here's a review from one of his students:


A former student of Mr. Foerster's, November 2, 2001 Until I met Mr. Foerster, I thought I desperately hated math. I scored well in it, yet I just hated the whole subject.

Mr. Foerster is truly an inspiring man; the whole high school was in awe of him. His courses were reputed to be extremely tough. But the hallway gossip was soon dispelled. Although Algebra isn't always "easy", I was quite surprised and delighted to discover that Mr. Foerster's classes - and especially his textbooks - were extremely user-friendly! Mr. Foerster writes clearly, and is able to address Algebra from the beginning, rather than talking several levels over students' heads. His kindness, humor, and gentle personality show through in the books. Wow! Math is fun after all!

I am now homeschooling my three kids, and Foerster's books are the texts of choice in this family.

So... is his second book, Algebra and Trigonometry, also great?

Does anyone know?

Last but not least, I notice he has a calculus text, too, which apparently is used in AP calculus classes.... and I spy a notation from teacher-2-teacher in my textbook file: "a great reform book."

hmmmm....

Sample chapter & material from instructor's manual here.


Prentice Hall Classics (math & social studies)
Prentice Hall Math Classics

Is this site good?

Dave's short course in trigonometry.

I worked an "overlapping triangle" problem in Saxon Algebra 2 today that threw me for a loop.

I finally got the correct answer, but I don't understand the solution in the solution manual.

(Image from Understanding sine at Homeschoolmath.net. I can't add labels to the illustration, unfortunately.)



Look at the bottom side of the left-most triangle, the one with two overlapping triangles.

Assume that the red segment measures 6 cm, the green segment 4 cm.

I've been taught that you would find the scale factor using this equation:

6 x SF = 10

However, the solution manual shows:

6 x SF = 4

I started checking various right triangle problems to see whether you can find a correct scale factor this way....and I'm stumped.

Just based in the triangles I've looked at, the 6 x SF = 4 formula for the bottom red & green segments also holds true for the corresponding 6 and H3 segments of the hypotenuse:

6 x SF = H3

Obviously it does not hold true for the ratio between the two vertical sides labeled 2.6 and 3.9.

What's going on?

++++++++

oh wow!

The homeschool.net page explains it!

hmmm . . .

I think Saxon blew it here. This was too big a leap for me inside a problem set.

Of course, Saxon isn't supposed to be a self-teaching book.

It may be time for me to take a class.

An actual class with an actual teacher.

Though I have to say, attempting to teach myself math I've never seen before is kind of cool.

My dad told me some relative of his taught himself calculus out of a book.

I like that idea.

+++++++

Someone needs to write a sci-fi novel about homeschoolers preserving knowledge for the future.

Which reminds me, I'm still worried about the solution manual for Moise and Downs.

Once it's gone, then what?