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38

One of the major themes of precalculus is what I call “connecting geometry to algebra”. Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the graph of $f$ is symmetric about the vertical axis is a great instance of this. This is just one more way to practice reinforcing function concepts, and the ...


24

A blanket problem I've observed over-and-over is that the deficits that scuttle calculus students are even more fundamental than what is discussed in (typical) pre-calculus courses. Specifically, kids, as well as adults returning to school, often cannot do middle-school (pre-?) algebra. Either they get stuck and confused, or immediately and frequently commit ...


24

Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples: The function that maps every non-empty list to its first element. The function that maps every finite set to its size. The function that maps color names to RGB triples. The function that maps days to sunrise times at a ...


23

You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun intended) we want to talk about the entire shape: we want to say that $f$ is symmetric, that $f$ is concave, that $f$ has an asymptote. We can't do that with $y$; ...


21

The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, restricted notion.


20

They are not usually well-prepared, but factoring is not a big issue. I would like students to be able to: Make meaning from graphs. [I want to get to introduce the connection between speed graphs and distance graphs, though, so I'd be happier if you focused on other graphs.] See what is algebraically sensible, and what's not. e.g. you can't cancel from ...


20

You should tell them these two main benefits: (1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable when dealing with long or complicated problems asking for a lot of information. We also shorten things like this all the time. For instance, instead of writing $...


18

Maybe draw this picture? Make it clear that the green hypothenuse is fixed in length, but the red altitude is growing and approaching that hypothenuse in length as the angle approaches $90^\circ$.          


18

My experience in teaching Precalculus is that you will be surprised about what you actually need to teach compared to what you thought you needed to teach. For example, if you ask students to compute and simplify the difference quotient for the function $f(x) = x^2$, even after several lessons and group work, you may be surprised that by far the most ...


17

One answer would be: any real-world word/application problems; insofar as one avoids making them in a repetitive template by "just changing the numbers". I find that most any textbook has an "Extensions" and/or "Real-World Applications" block of exercises in each section, after the repetitive symbol-manipulation exercises. But I would caution about pushing ...


16

The crucial thing the students need to realise is that the (e.g.) $x$ that turns up in the function definition is a bound variable. That's what allows it to be freely renamed or indeed omitted without changing the semantics. Unfortunately, education tends to completely obscure this facet by a) always using the same dumb variable names as if there were a ...


15

A linear function is not necessarily a first degree polynomial function: zero function is also linear. In France the terminology is more appropriate than the traditional English one: a linear function is a function of the form $f(x) = ax$, while a function of the form $f(x) = ax + b$ is called an affine function. So, strictly speaking, constant functions ...


15

Here you can see that knowing if the function is even or odd can help you when you are integrating over the interval $[-a, a]$. You can reduce really-hard-to-look-at integrals to zero just by knowing this. As an example, to calculate $E(Z)$ where $Z \sim N(0, 1)$, the standard normal distribution, you have: $\displaystyle E(z) = \frac{1}{\sqrt{2\pi}}\int_{-...


15

Because x and y are just variable names It happens that sometimes y=f(x), but other times z=f(x,y), w=f(x,y,z), or x=f(y) for that matter. All of these variable names are syntactically equivalent, and the mere existence of "x" and "y" in an equation does not necessarily connote that "x" is the independent variable and "y" the dependent. Thinking of the ...


14

Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry. Upon learning about flux in vector calculus, would this student be able to quickly see that the flux of the vector field ${\bf F}(x,y,z)= y^2{\bf j}$ through ...


12

Here's one I haven't seen mentioned so far: students coming into calculus are very uncomfortable with the idea that there might be more than one way to do something, and especially that they might have to make decisions, especially decisions that depend on context rather than just a rule. Two concrete examples of this phenomenon. 1) Simplifying. I see ...


12

By throwing in all of the $r$, $t$, and $d$, not to mention $x$, you're overly complicating things, especially if your brother is algebra-averse. There's really only one unknown, and you can call it $k$. It's the number of hours it would take Karen to paint $1$ house by herself. Everything else can be expressed in terms of $k$. Don't complicate things with $...


12

Besides applicability in topics like integration and Fourier analysis, it also connects algebra to calculus at least in the way that multiplication of even/odd functions behaves like addition even/odd numbers: Multiplying two even functions gives an even function. Multiplying two odd functions gives an even function, too. Multiplying an even and an odd ...


12

On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge. The practice of starting students out with trivial arithmetic proofs like proving 1+1=4/2 seems to be pretty common, but I'm very skeptical of it. ...


11

By their definition (using a right triangle), $\sin 90^\circ$ is undefined, since a triangle cannot have two right angles. You need the definition based on a circle ($\sin \theta = y/r$) to have a value for $\sin 90^\circ$.


11

While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no stipulation that $m \neq 0$. Thus, if you define linear to be a polynomial of degree 1 you are likely to be contradicting whatever textbook you are using. From a ...


11

When I teach people calculus, the big reasons they don't succeed tend to be problems with arithmetic and very basic algebra. For example, students won't know how to compute 2/(3/4), or they'll try to simplify $1/(x+y)$ to $1/x+1/y$. If you can handle this kind of stuff, then you're better prepared to learn calculus than most of my students. There is some ...


11

Even and odd parity are probably the simplest examples of function symmetries. In applied mathematics, the general observation of function symmetries allows to simplify calculations (as stated by others) and to produce more meaningful graphs. In physics, symmetrical parts of a function are sometimes associated to different physical phenomena. Two examples:...


11

One really good one is solving absolute value inequalities. Students can memorize the following pattern: When you have $|X| \geq Y$, the setup is $X \geq Y$ or $X \leq -Y$. Students with more comprehension will have the following more universal strategy: When you have an inequality, start by asking whether it is impossible, or always true, or ...


11

How about $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = y$ where $y$ is the unique solution to $y^5+x^2y+5=0$? This does not have an elementary formula, but students can understand that for any value of $x$ the function $g(y) = y^5+x^2y+5$ is increasing (and the limits at $\pm \infty$ are $\pm \infty$), so it must have a unique root for each $x$. You ...


11

TL;DR: A function is a verb. It's an action. Variables are nouns, objects. Verbs (functions) connect nouns semantically, i.e. how A (or x) relates to B (or y), how to get from here to there. Long version Some context: I learned maths from my father who was a physics / engineering guy at heart, so everything had to be 'tangible' or 'observable' to him. ...


10

My impression (as a former private tutor/current university employee and from my mind as a former student) is that mainly equation solving is a big problem for poor (sometimes also for advanced) students. This is somehow some task one has to perform to do subtask in calculus (e.g., a new thing is to calculate a minimum of a function. The students learn that ...


10

What those students need is a course that will help them see math differently. They need to learn to think mathematically. An excellent pre-calculus course can begin to do that. But at my college (for example), pre-calc has way too many topics. If I taught the content I'm supposed to, we could never dig in properly. I skip some topics, so that I can work ...


10

Mitchell, Great question. I think that precalculus is usually a wasteland for some random classical techniques that used to be required to study calculus. I think that you can take a completely different approach if you're interested and incorporate data and matrices in place of a bunch of algebraic manipulation. I just taught a precalculus class and ...


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