Questions tagged [precalculus]

Courses designed to prepare students for subsequent calculus courses

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Advice and Remedial Algebra Resources for Students Committed to Calculus

I've got a student in my introductory calculus course. They're failing because they lack algebra skills. They understand the concepts just fine, and can articulate their understanding fine, but get ...
10 votes
6 answers
1k views

Intuition for order of operations in compound transformations

This is a close cousin of the previous question asked here about transformations inside and outside a function and how they switch things around. I think some of the perspectives there will help here, ...
5 votes
4 answers
346 views

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

Back in 2018, I wrote a post about asymptotes of rational functions in which I addressed not only horizontal and slant/oblique asymptotes, but also the general case of "polynomial asymptotes.&...
3 votes
0 answers
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Examples of Financial Institutions that Compute Interest Atypically?

Are there examples of financial institutions that compound their interest more frequently than once-a-month? Are there examples of financial institutions that consider continually compounded interest ...
1 vote
1 answer
122 views

Chinese and Japanese most important high school textbooks

I would like to know the best high school math books from Japan and China. Can you suggest some books or free resources? I would like to compare the different approach betweeen China and Japan and ...
9 votes
10 answers
14k views

Requirements to learn calculus

I always was non math background student and programming is my hobby. I was attempting to program code instruction given here. Since I don't know calculus I'm stuck. I would like to know what are the ...
2 votes
1 answer
378 views

Sources on inequity in precalculus sequence

I'm trying to put together some thoughts on the importance of a strong college precalculus sequence (mainly I'm thinking College Algebra, where much of my experience is) for addressing socioeconomic ...
7 votes
3 answers
543 views

Why is isolating for $x$ taught before factoring?

I'm currently working on some precalculus packages for students who need review. For inspiration, I'm looking at some prealgebra books and I'm wondering why isolating for $x$ is taught before ...
19 votes
4 answers
4k views

Why do we teach linear algebra in precalculus classes?

When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't ...
0 votes
2 answers
272 views

Differentiation in integer solutions

What would you suggest as examples to demonstrate as applications of differentiation in finding integer solutions of an equation for advanced level students? Here you have one example which I have ...
5 votes
3 answers
3k views

What are some examples of great functions that are not too elementary (easy)?

I am teaching precalculus/basic calculus to my class (high schoolers of around 18 years of age), and I'm always searching for nice functions to plot and study (finding the domain, the function's sign, ...
30 votes
6 answers
3k views

f(x+h) in the difference quotient

When teaching students how to compute the difference quotient in a precalculus or calculus class, we need them to evaluate the expression $$\frac{f(x+h) - f(x)}{h}$$ for various simple functions, like ...
6 votes
3 answers
3k views

Are there examples of central symmetry, without axial symmetry, in nature?

Examples of axial symmetry abound, but I could not find an example of pure central symmetry (that is, without axial symmetry)! Do you know of any? A butterfly shows axial symmetry, what shows point/...
1 vote
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Math curricula\programs or any experience using "The Road to Reality" as the\a primary textbook

Primarily a reference request, collaborator search-tips requests, and question-improvement request (including improveent by deletion and re-posting to more appropriate stack, meta, wiki, etc). Rank ...
7 votes
7 answers
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Write $y=\sqrt{3+x}$ as the composite of two functions

For the question "Write $y=\sqrt{3+x}$ as the composite of two functions", what if a student gives the answer $f(x)=\sqrt{3+x}$ and $g(x)=x$? This answer would be technically correct but it ...
2 votes
1 answer
123 views

Do properties of exponents apply only to positive real numbers? [closed]

A common rule given in textbooks is: If $a$, $b$, and $x \in \mathbb{R}$, then $(x^a)^b=x^{ab}$. Suppose I write: $(-9)^{1/2}=(-9)^{2/4}=((-9)^2)^{1/4}=(81)^{1/4}=3$. But this contradicts the fact ...
4 votes
2 answers
310 views

Do horizontal asymptote rules require function to be fully simplified?

I am teaching high school precalculus and have a textbook that gives the following preamble to its rules for finding horizontal and slant asymptotes of rational functions: Suppose $f(x)=\frac{a(x)}{b(...
6 votes
3 answers
466 views

How can you elicit the $\log x = {\log} \cdot x$ error?

You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually $\log$ or a trig function, and you watch ...
8 votes
3 answers
407 views

Algebra/trig/precalculus review questions that elicit common student errors

This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the ...
0 votes
1 answer
336 views

Best PreCalculus Textbook

What is your favorite PreCalculus textbook for someone that needs to get their algebra skills up to snuff? Something comprehensive with some tricky problems. Stewart? Sullivan? Blitzer? Something ...
4 votes
4 answers
593 views

How to word this exercise about converting "English" into interval notation?

I am writing an exercise for a precalculus homework assignment that deals with the topic of interval notation. The point of the exercise is to convert open, closed, and half open intervals described ...
6 votes
2 answers
455 views

Is Trigonometry done differently in the US?

I'm Italian and I've watched some videos from Americans and noticed a weird thing. Let's talk about a linear trigonometric equation like this: $$\sin x+\cos x+\sqrt3=0.$$ I've seen Americans solving ...
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101 views

Is the AC Method of Factoring polynomials more popular and used by teachers than others methods of factoring polynomials?

This is an example of the AC Method: $ x^2 + 16x +63 $ (1) $x² + 7x$ (2) $9x + 63$ (1) $x(x + 7)$ (2) $9(x + 7)$ so we have: $x(x + 7)+ 9(x + 7)$ (1) with (2) The Result is: $ (x+9)(x+7) $ I have more ...
31 votes
10 answers
11k views

Why do we teach even and odd functions?

I've been either a student or an instructor in Precalculus or Calculus 1 at about 6 institutions now, and teaching the definition of even functions (where $f(-x) = f(x)$) and odd functions (where $f(-...
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1 answer
203 views

Matriculation exams like in Europe

I just looked at matriculation exams from Finland. They have both basic and advanced level exams. Most US high school seniors could not pass the basic exam. If each US state were to create its own ...
1 vote
1 answer
437 views

Honors Precalculus: what topics to cut?

We’re precalculus honors teachers. In this year of Covid and reduced instructional time, what topics can we cut (Demana textbook) that would not hurt our kids in either calc AB or BC?
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3 answers
634 views

When are students taught implicit and parametric representations of curves?

Do students learn implicit equations (such as $x^2+y^2-r^2 = 0$) and parametric equations (e.g., $x=a t^2,\;y= 2 a t$) in a first course in algebra, which in the US would be early high school, maybe ...
4 votes
4 answers
2k views

How are the basic trigonometric functions introduced to students?

The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is ...
2 votes
1 answer
90 views

Appropriate context for teaching derivative (undergraduate/graduate)

(Repost from MO, where the question will eventually be closed.) This question is related to lectures I have to make concerning differential calculus in one variable, but the students are quite ...
12 votes
6 answers
3k views

How do you explain concavity of a polynomial without any calculus?

How do you explain the concavity of a polynomial without any calculus? As the title suggests, I am struggling to explain when given a graph of a polynomial, how we determine when it is concave up or ...
8 votes
1 answer
191 views

How can I deal with the time pressure of teaching a short course?

I am an undergraduate applied math student. In about a month, I will be teaching two nine-hour math courses (one precalculus, one calculus) to a small group of motivated high school students. My broad ...
5 votes
2 answers
195 views

Is evaluating a Real Polynomial at a Complex Value a suitable task for Precalculus students?

In Korea, basically every teaching material for 10th grade math(about the level of precalculus) contains this kind of exercises in their first treatment of complex numbers: Evaluate $f(x)=4x^4-8x^3+...
14 votes
7 answers
987 views

How should I introduce the concept of a function to a precalculus student?

My brother has not taken a math class in $10-15$ years. He is studying for the GRE so I have been teaching him a chapter or two from my precalculus book. So far, he has learned (and excelled at) basic ...
4 votes
4 answers
2k views

Best Way to Learn Trigonometry

What are the best resources to learn trigometry? I recently decided to pursue a BS in mathematics at uni. I used to fail all my math classes with D's or F's until I started teaching myself, and so far ...
5 votes
2 answers
588 views

Which textbooks on College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE are written by world-class mathematicians?

For example, Trigonometry was written by Wolf-Prize winner Israel Gelfand, one of the top mathematicians in the 20th century. I am wondering if other world-class mathematicians have written textbooks ...
45 votes
21 answers
7k views

How to help new students accept function notation

I am struggling to help some of my new precalculus students accept function notation -- something new to them this term. I am looking for strategies to help them adopt this new notation. Their main ...
4 votes
3 answers
360 views

Is there a pre-calculus introduction to the formal definition of a limit?

To give an example of what I mean, I'll answer a similarly worded question: “is there a pre-calculus introduction to the derivative?” I would say yes, since there already are the ideas of a slopes of ...
4 votes
2 answers
424 views

Enlighten younger students about the concept of "procedural justice" in mathematics?

I am tutoring a 16-year-old student from my home country (in Asia) in, roughly speaking, precalculus. I would like to give him a feeling of procedural justice, so to speak, in modern mathematics, ...
4 votes
2 answers
653 views

Why "plug in numbers" when solving inequality?

Let me use this example, Solve $x^3-4x>0$ After factorization, we have $$(x+2)x(x-2)>0$$, in order to have product of several numbers positive, even(0,2,4,...) of them have to be negative ...
9 votes
3 answers
228 views

Physical devices for exploring calculus or pre-calculus

I saw this partial derivative machine yesterday, and it got me excited about other physical devices for exploring calculus concepts in a "lab" setting (e.g. make a prediction, collect data, etc.) Do ...
8 votes
4 answers
750 views

A more rigorous approach to Precalculus

I am a pure mathematics PhD student and graduate teaching assistant at a major state university. During the summers here, teaching assistants are typically appointed to teach an entire course, rather ...
10 votes
3 answers
424 views

The royal road to calculus

In the early 1900s Felix Klein lay out his vision for secondary mathematics curriculum. He wanted schools to teach calculus, so that universities would not be burdened by it. And at the core of the ...
4 votes
2 answers
284 views

Why are we even studying cyclotomic polynomials?

My students found an exercise about cyclotomic polynomials in the AOPS precalculus text. They asked me why this construction exists in the first place and what it's good for... I am looking to give ...
12 votes
3 answers
728 views

Examples (for beginners) of real functions which are not given by elementary formulae

Question: What are good examples of functions $f: \Bbb R \rightarrow \Bbb R$ (or $f: D \rightarrow \Bbb R$ with $D \subseteq \Bbb R$) which are not just given by "a formula" (or finitely many formulae ...
2 votes
3 answers
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Good (natural) motivational examples for quadratic equations

I am looking for good motivational examples of how quadratic equations can naturally arise in real life for someone starting high school. The high school book my child is using just jumps into ...
1 vote
1 answer
501 views

How can I make "complex" graphs that combine multiple functions with a software?

Til today I've been using geogebra to sketch functions for my students quizzes or homework. Sometimes I use the ones that I found searching in google, but this takes a lot of time specially because I ...
1 vote
0 answers
70 views

How to introduce trigonometric ratios (HS) through a cognitive model?

I'm teaching a precalculus course and also taking a class on how to teach mathematics constructing a specific cognitive model for different topics. So, I have this assignment to build a cognitive ...
6 votes
4 answers
1k views

Functions, Domains, and Ranges in Precalculus

Possibly related, though of a different flavour. Background In most of the precalculus texts with which I am familiar, readers/students are given a crash course in set theory, handed the definition ...
3 votes
2 answers
156 views

Should Measurement of Angles Using Degree (and perhaps Common Logarithm as well) be Avoided in Pre-Calculus?

People use degrees and radians to measure angles and though degree measurement is acceptable and is widely used in everyday life, it is not in the International System of Units and mathematically it ...
33 votes
8 answers
6k views

Math topics that reward going beyond cookbook methods

Students fresh out of high school are often under the impression that mathematics is a discipline based entirely in recognizing the type of problem and applying an algorithm or cookbook method. These ...