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 ...
Mike Pierce's user avatar
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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, ...
Kevin P. Costello's user avatar
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.&...
Justin Skycak's user avatar
3 votes
0 answers
73 views

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 ...
Mike Pierce's user avatar
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1 vote
1 answer
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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 ...
ryuk's user avatar
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2 votes
1 answer
379 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 ...
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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 ...
Joel Croteau's user avatar
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 ...
Janaka Rodrigo's user avatar
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 ...
Chris Cunningham's user avatar
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/...
Maesumi's user avatar
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1 vote
0 answers
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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 ...
Ayer AGG'TDd'E-A's user avatar
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 ...
Zuriel's user avatar
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2 votes
1 answer
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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 ...
Chad's user avatar
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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(...
Chad's user avatar
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5 votes
3 answers
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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, ...
marco trevi's user avatar
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 ...
Mike Pierce's user avatar
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0 votes
1 answer
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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 ...
MichaelLink's user avatar
6 votes
2 answers
456 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 ...
Alfonso's user avatar
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0 votes
0 answers
102 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 ...
rubengavidia0x's user avatar
0 votes
1 answer
204 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 ...
user50896's user avatar
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?
Jojo18's user avatar
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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 ...
user avatar
1 vote
3 answers
635 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 ...
Joseph O'Rourke's user avatar
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 ...
ACL's user avatar
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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 ...
Nicholas Roberts's user avatar
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 ...
2132123's user avatar
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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+...
Hyobin Lee's user avatar
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 ...
Jon Bannon's user avatar
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5 votes
2 answers
589 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 ...
Zuriel's user avatar
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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 ...
Sat's user avatar
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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 ...
Nick C's user avatar
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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 ...
Lex_i's user avatar
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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, ...
Yes's user avatar
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9 votes
3 answers
231 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 ...
Nick C's user avatar
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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 ...
Sprinkle's user avatar
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4 votes
2 answers
286 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 ...
Mason's user avatar
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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 ...
Manya's user avatar
  • 201
12 votes
3 answers
729 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 ...
Torsten Schoeneberg's user avatar
2 votes
3 answers
3k views

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 ...
John Smith's user avatar
1 vote
1 answer
504 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 ...
Grouper's user avatar
  • 279
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 ...
Grouper's user avatar
  • 279
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 ...
Zuriel's user avatar
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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 ...
Chris Cunningham's user avatar
3 votes
1 answer
395 views

Deriving Jerk Equations without using Calculus

I am thinking about the links between SUVAT equations (constant acceleration), and equations for motion when higher-order measurements are constant (for example, when jerk is constant, or snap is ...
Jezza Judge's user avatar
6 votes
6 answers
1k views

Ideas for a 2 weeks project focused in polynomial functions

Right now I’m teaching precalculus in high school and I want to propose a project to my students about polynomial functions. They already know enough about quadratic functions and we study variation ...
Grouper's user avatar
  • 279
5 votes
3 answers
504 views

When discussing inverse functions, how can our notation and methods reinforce student understanding?

Yesterday in my precalculus class, I decided to teach students how to find the formula for an inverse function in a new way (to me). In this curriculum, they have already used the notation $f^{-1}(x)$...
Nick C's user avatar
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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(-...
Nick Matteo's user avatar
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 ...
Xander Henderson's user avatar
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6 votes
2 answers
410 views

How can I improve my problem solving/critical thinking skills and learn higher math?

I'm a rising sophomore in high school. So far, I've taken Algebra One, Two, and Geometry in school. I want to learn higher math such as precalculus/trigonometry, calculus, linear algebra, and more, so ...
Ansh.23's user avatar
  • 161
12 votes
4 answers
1k views

The Order in Pre-Calculus Textbooks

Every Pre-Calculus I have examined starts with functions in general, then polynomial and rational functions, followed by exponential and logarithmic functions and Trigonometry, and ending with ...
Mitchell's user avatar
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