Questions tagged [precalculus]

Courses designed to prepare students for subsequent calculus courses

Filter by
Sorted by
Tagged with
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 ...
user avatar
  • 7,713
32 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 ...
user avatar
  • 19.1k
30 votes
10 answers
10k 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(-...
user avatar
30 votes
6 answers
1k views

What holds your students back in Calculus?

I teach Precalculus to high school kids, and I know a lot of you all teach Calculus. What are some issues that your students have in Calculus classes that you wish had been addressed in a ...
user avatar
17 votes
3 answers
934 views

What deficiencies are present in Precalculus curricula that causes so many students to fail Calculus I?

At our university we now require one semester of Pre-calculus instead of one semester of Algebra and one semester of Trigonometry before you take Calculus I (for those who do not test into Cal I). ...
user avatar
  • 1,178
16 votes
7 answers
1k views

Why do we teach that every line is a linear function?

Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read: "A linear function is a function defined by ...
user avatar
  • 1,736
15 votes
5 answers
2k views

For purposes of teaching, should constant functions be considered "linear functions"?

I can see arguments both for and against classifying constant functions as linear functions. Against: "Linear function" means "first-degree polynomial function", and constant functions are not first-...
user avatar
  • 16.4k
15 votes
7 answers
902 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 ...
user avatar
  • 709
13 votes
3 answers
613 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 ...
user avatar
12 votes
6 answers
2k 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 ...
user avatar
  • 231
12 votes
4 answers
970 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 ...
user avatar
  • 299
10 votes
5 answers
264 views

Teaching about absolute values not centered at $0$

One of my peers is studying for the GRE and ran into the following problem: When is $|x-4|$ equal to $4-x$? I tried to attempt teaching this by developing the intuition for why $$|x| = \begin{...
user avatar
10 votes
3 answers
389 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 ...
user avatar
  • 201
9 votes
2 answers
343 views

How to explain the range to students?

I mostly tutor students ranging from beginning algebra to calculus level. I think the explaining the range as "the set of all possible outputs" would not really cut it for someone struggling with math ...
user avatar
  • 709
9 votes
3 answers
204 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 ...
user avatar
  • 7,713
8 votes
9 answers
12k 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 ...
user avatar
  • 183
8 votes
5 answers
407 views

How to convince the students of grade 8 that $\sin 90^\circ =1$? ( calculator not allowed )

In general the secondary student may not ask why is $\sin 90^\circ = 1$ because they can see the answer in the graph of the sine wave. However the students in grade 8 are not familiar with the graph ...
user avatar
8 votes
4 answers
625 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 ...
user avatar
  • 191
8 votes
3 answers
349 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 ...
user avatar
  • 5,877
8 votes
2 answers
674 views

Vector problems in high school precalculus

I was helping my kid study for a precalculus examination and looking at her old tests from the year, and came across a question about vectors. Below is the typed up version of the question and her ...
user avatar
8 votes
2 answers
195 views

Can a constructivist learning model be applied to online lower division math courses?

I'm using the word constructivist as it is used in this paper, not in the sense used in logic. The abstract should be sufficient to understand at least roughly what the model is. The important part, ...
user avatar
8 votes
2 answers
274 views

Is there a good notation for "ratio" comparable to the use of $\Delta$ for "difference"?

It is standard to use the symbol $\Delta$ to indicate a change in a quantity between two points on a curve, two rows on a table, and so forth. For linear functions, we write slope = $\Delta y / \...
user avatar
  • 16.4k
8 votes
1 answer
182 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 ...
user avatar
  • 81
7 votes
3 answers
287 views

Explaining the domain of a function to students?

I mostly tutor community college students ranging from beginning algebra to calculus level. There are several ways in which I explain the domain: "The domain of a function is all the values you can ...
user avatar
  • 709
7 votes
0 answers
1k views

Teaching a Pre-Calculus Course using Basic Mathematics by Serge Lang

I am considering using Basic Mathematics by Serge Lang as the primary text for my High School Pre-Calculus course. My students have all spent a year working through the first six books of Euclid and ...
user avatar
  • 299
6 votes
3 answers
824 views

Is there a more intuitive way to solve combined rates of work problems?

I am helping my brother study for the GRE and we have come across some problems like this in my old precalculus textbook: 1) Karen and Betty have been hired to pain a house. Working together, they ...
user avatar
  • 709
6 votes
3 answers
397 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 ...
user avatar
  • 4,474
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 ...
user avatar
6 votes
2 answers
377 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 ...
user avatar
  • 61
6 votes
3 answers
394 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
6 votes
6 answers
988 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 ...
user avatar
  • 281
6 votes
2 answers
360 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 ...
user avatar
  • 161
6 votes
4 answers
315 views

Are convergent and divergent sequences a prerequesite for calculus or are they a part of it?

Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: http://www.stitz-zeager.com/szprecalculus07042013.pdf However, it does not cover ...
user avatar
  • 143
5 votes
3 answers
389 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 $f^{-1}(x)$...
user avatar
  • 7,713
5 votes
2 answers
468 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 ...
user avatar
  • 4,025
5 votes
1 answer
122 views

Suitable Curriculum for Algebra through Pre-Calc students in One Class

I run a high school outreach program at UCLA called DMAP: Diversifying Mathematics And Physics (webpage: http://dmap.pbworks.com), which focuses on exposing groups underrepresented in advanced maths ...
user avatar
  • 153
5 votes
1 answer
134 views

Consolidating three descriptions of a parabola in precalculus

I want to present these three descriptions of a parabolic curve to my precalculus class: The graph of a quadratic function $f(x) = ax^2+bx+c$. Given a line called the directrix and a point called the ...
user avatar
  • 4,474
5 votes
2 answers
172 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+...
user avatar
4 votes
4 answers
569 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 ...
user avatar
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 ...
user avatar
  • 1,077
4 votes
4 answers
1k 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 ...
user avatar
  • 496
4 votes
2 answers
570 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 ...
user avatar
4 votes
2 answers
253 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 ...
user avatar
  • 313
4 votes
2 answers
260 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(...
user avatar
  • 41
4 votes
3 answers
280 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 ...
user avatar
  • 331
3 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, ...
user avatar
3 votes
2 answers
403 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, ...
user avatar
  • 147
3 votes
2 answers
148 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 ...
user avatar
  • 4,025
3 votes
1 answer
291 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 ...
user avatar
3 votes
1 answer
321 views

I want to learn math from the beginning [closed]

I finished high school 2 years ago and now I'm stuck in a university in Turkey. I am interested in learning precalculus, discrete mathematics, physics and chemistry. Question: I need to learn math ...
user avatar
  • 31