Questions tagged [precalculus]
Courses designed to prepare students for subsequent calculus courses
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Hard PreCalculus Problems
My AP PreCalc students are ready to tackle problems a bit harder than the usual fare in textbooks and AP Calc practice books. Are there any books, materials, or websites which have harder PreCalc ...
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Generating function example
I'm about to introduce the Generating Function concept to a couple of kids. The plan is just to roughly follow Herbert Wilf's Generatingfunctionology's first 12 pages, until Fibonacci numbers and Ch 1....
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Why do some (pre-) calculus text allow $r<0$ in polar coordinates?
Form this question, I was surprised to learn that it is common for calculus textbooks in the US to allow $r<0$ when discussing polar coordinates. This answer by Dan Fox summarizes some mathematical ...
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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 ...
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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, ...
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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.&...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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/...
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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 ...
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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 ...
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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 ...
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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(...
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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, ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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+...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $f^{-1}(x)$...
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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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