42 votes

Why do we teach even and odd functions?

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 ...
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26 votes

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

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,...
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  • 13.4k
25 votes

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

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, ...
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  • 16.1k
24 votes

How to help new students accept function notation

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 ...
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  • 241
23 votes

How to help new students accept function notation

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. ...
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  • 570
23 votes
Accepted

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

Here is a proposed definition: A function $f$ is said to be concave up on an interval $[a,b]$ if for all $x,y \in [a,b]$ with $x<y$, the line $L$ connecting $(x,f(x))$ and $(y,f(y))$ satisfies $L(t)...
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22 votes
Accepted

What holds your students back in Calculus?

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 ...
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  • 17k
22 votes

How to help new students accept function notation

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 ...
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  • 496
20 votes
Accepted

Math topics that reward going beyond cookbook methods

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 "...
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19 votes

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

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 $90^\circ$. &...
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18 votes

Why do we teach even and odd functions?

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 ...
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18 votes
Accepted

A more rigorous approach to Precalculus

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 ...
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  • 19.1k
17 votes

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

I'm personally a fan of simple examples: $x e^x$ (has nice critical point, point of inflection) $e^{-x^2}$ (with appropriate rescaling, the normal distribution from statistics) $\frac{x}{x^2+1}$ (a ...
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  • 1,919
16 votes

Why do we teach even and odd functions?

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 ...
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  • 2,922
16 votes

How to help new students accept function notation

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 ...
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15 votes

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

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 ...
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  • 271
14 votes

What holds your students back in Calculus?

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 ...
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14 votes

Why do we teach even and odd functions?

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....
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  • 7,575
14 votes

How to help new students accept function notation

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, ...
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13 votes

How to help new students accept function notation

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), ...
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  • 131
12 votes

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

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 ...
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  • 17k
12 votes

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

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 ...
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  • 1,426
12 votes
Accepted

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

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 $...
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  • 663
12 votes

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

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 ...
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12 votes

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

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 ...
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11 votes

What holds your students back in Calculus?

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) ...
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  • 9,096
11 votes

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

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 ...
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  • 17k
11 votes

What holds your students back in Calculus?

When I teach calculus, there are two preparedness issues that frequently surface: Understanding functions: I don't mind if my students can't write down the definition of a function (though ideally ...
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  • 1,102

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