# Tag Info

32

The specific identity $$\tag{A} \tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x}$$ as such is probably not often encountered, but simplifications akin to $$\tag{B} \tfrac{1}{1-t} + \tfrac{1}{1 + t} = \tfrac{2}{1 - t^{2}}$$ occur frequently. For example, integration via partial fractions ...

15

Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year. So you can imagine that "How much do you really need to know to see the bigger picture in calculus?" has been on my mind lately. Here's where my thoughts have fleshed out in regards to trig so far. Periodic behavior is widespread ...

12

If you are a private tutor, hired by an undergrad or adult student, or hired by the parents of a student in 6-12 (middle school/high-school), then I'd suggest that when you meet with a "client" as a potential tutor, that you develop a contract with the student and/or parents to make clear your expectations: what is the minimum level of participation/effort ...

11

This is not an answer, but an assertion that what you are experiencing is not something new. Here are some quotes from a 1993 article of a Russian (actually, native Estonian) math prof, who moved to the U.S. in the early 1990s, so the problem is at least thirty years old. Some say that the commoditization of universities started from the Reagan times. This ...

8

Well, there is a recent and excellent book about this question: Why Students Resist Learning: A Practical Model for Understanding and Helping Students by Anton O. Tolman, Janine Kremling and Anton O. Toman. The authors say resistance in learning may be a joint consequence of several factors, including resistance from teachers and schools. Here is a ...

8

This is how most students perceive math tests. Whether it's fair or not, this is the perception and it is the normal response to a broken math education system. Imagine you are a teenager and your driving test for your full license is in a week. The Department of Motor Vehicles is massively understaffed, so if you fail, you can't book another test for a ...

7

From a historical perspective, knowing these identities used to be somewhat more important than now. Prior to the invention of logarithms, people who needed to do lots of sophisticated calculations [esp. astronomers] resorted to a technique know as 'prostapharesis'. This involved combining certain trigonometric identities to produce equations (e.g) having ...

6

There are many calculus textbooks that use no trig. They may be called "Calculus for Business" or "For Biology" or "For Social Science". random example: Of course math, physical science, and engineering, definitely use parts of calculus connected with trig functions. I would have thought that Business would be interested in cyclic phenomena, but what ...

6

I teach in the U.S. at a community college. Although I prefer distributing without drawing an area box when I'm doing math myself, I often show the box in class to help students see how things work. When we use an area model to help students visualize the workings of the distributive property, it makes much more sense for many students. In fact, the box ...

6

I run into this issue frequently. As a high school in-house math tutor, students visit to show me their quiz/test scores and ask about their work. The FOIL method is fine, if it works for the student. For those who are prone to making mistakes, I show them the Box method (call it what you will, that just my name for it). The benefit, if any, to this method ...

5

To demonstrate the distributive law, we often use an area model. Then $(2+4)\times(5+7+9)$ can be visually decomposed into the sum of 6 subproducts. Rather than actually drawing a grid, you might eventually start just labeling the edges and the products, without really caring about the relative sizes of the pieces. Finally, this table method'' could ...

5

I gave an earlier answer [to a rather different question] in which I pointed to the Regents Exam Archives. One approach that you could take would be to look over a few tests from the 1990s as compared to now to see whether you sense disconnects. Of course, the Regents are rather geographically specific; so, I am not sure how well this will apply to your ...

4

To keep my answer specific, I'd probably say the biggest three factors in any personalized teacher would be (1) flexibility in the material (2) compatibility with students (3) inventiveness in the online classroom. For 1) since the teaching will be personalized, the teacher would need to know the material well enough to interact with spontaneous questions, ...

4

The most profound shift that the Common Core introduced is scattering statistics into the standards across most of the grade levels. For instance, in my state (New York), high school freshmen are expected to know how to calculate linear regressions from a two-column table and discuss the degree of correlation between the two factors. Also, upperclassmen ...

3

The core of concepts of calculus: that is Differentiation and Integration can be defined rigorously and intuitively without any reference to trigonometry. To be functionally able to use these tools well also requires a very strong command over algebra which other posters have commented. Trigonometry is in some sense a “nice to know” since it lets you apply ...

3

Trigonometric substitutions are useful for solving many integrals in closed form and learning how to solve integrals is a major part of most calculus courses. Often more than half of university-level "Calculus II" is concerned with integration techniques. Without trigonometric identities, it may not be obvious how to solve $\int \left( \frac{1}{1- \sin x} + \... 3 Algebraic skills to do manipulations in general are important. And this one is not that hard. Good practice. Get dirty and do it. There's some applications when you get to trig subs of quadratic radicals and the like. 2 My advice is to accept 3/4 of a loaf. Tailor your instruction to cover exactly this: "solution to problems that are known to be on their school's exams, and prefer step-by-step instructions free from any context/theory/mathematical properties." This is really still useful content to cover and better than nothing. Also, I wouldn't kill yourself in terms ... 2 I've notices that rigid motion transformations, rotations, translations, and reflections, is getting more attention now in geometry. Ive seen that topics from John Tukey's "Understanding Robust and Exploratory Data Analysis" have made it into the classroom. I saw one class where they were teaching voting theory, which I had no idea how to do. 1 IMO knowing basic identities such as$\sin^2x + \cos^2x=1$should be as fundamental as knowing$12\times 12 = 144$(without needing to reach for a calculator). That is not to say that students should know all the identities by rote, but they should at least recognise the identities. Categorising "prove trig identities such as$\frac{1}{1−\sin x}+\frac{1}{1+\...

1

Once the student figures out the process - start from one side of what is to be proven, use known formulas to get the other side, done (or start from both sides or do some other variation) - the rest is getting intuition about which formula to use in the process and when. I guess this intuition might be useful if they need to evaluate tricky integrals ...

1

3blue1brown makes the best math videos in existence in my opinion, and he uses a Python library called manim.

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