37 votes
Accepted

To 17 year olds, how can I explain that two numbers with arbitrarily small difference are equal?

I'd recommend to play a little game with them. You choose a number $a.$ They choose a number $b$ that they think will satisfy $|a-b| < \epsilon \,\,\, \forall \epsilon > 0.$ You try to state a ...
Justin Skycak's user avatar
36 votes
Accepted

When writing log, do you indicate the base, even when 10?

And a computer scientist thinks that $\log=\log_2$. I am using $\log$ for the natural logarithm by default in all my courses though I clearly state that in the beginning of each course (I teach at the ...
fedja's user avatar
  • 3,831
36 votes
Accepted

Why not think of derivatives as fractions?

If you have a function $f(x,y)$ where $x=x(t)$ and $y=y(t)$ are themselves functions of a parameter $t,$ and you blindly cancel out differentials, then you can get to incorrect statements like $$\...
Justin Skycak's user avatar
35 votes
Accepted

Explaining Sigma-Notation

I've experienced positive results by first having students spend some time writing out sums in full (or using ellipsis notation if there are many terms). That way, it gets annoying to spend so much ...
Justin Skycak's user avatar
31 votes

Applications of High School Geometry

Every time I'm doing carpentry/DIY remodeling on my house, I use so much high school geometry that it makes me chuckle. I've had instances where I need to cut angles to make pieces parallel and used ...
Aeryk's user avatar
  • 8,025
23 votes

What benefit is there to obfuscate the geometry with algebra?

Tests seek to measure ability. Math ability, like most other forms of ability (including athletic ability), isn't solely dependent on one's ability to execute individual skills in isolation -- it also ...
Justin Skycak's user avatar
21 votes
Accepted

Can this be a better way of defining subsets?

In both formal and informal treatments of set theory, we need to specify which operations and relations are allowable and build from there. Usually we take sets and set membership as primitives. We ...
Steven Gubkin's user avatar
21 votes
Accepted

What benefit is there to obfuscate the geometry with algebra?

Is your ultimate goal really just to teach cofunctions? Or are you trying to teach cofunctions so that the students can apply them later? I am speaking as a student rather than an educator, but math, ...
PC Luddite's user avatar
18 votes

What benefit is there to obfuscate the geometry with algebra?

This multi-step question requires students to understand and apply multiple concepts or strategies to solve the problem. The goal of a standardized test is not to provide a correctly-sequenced list of ...
Steve's user avatar
  • 1,404
17 votes

How can we motivate that Newton's method is useful?

I'm going to respond from an applied math (or maybe CS) perspective: Part of the problem is that the functions that you look at in high school and the standard calculus sequence are unrealistically ...
Adam's user avatar
  • 5,382
16 votes

Why do we teach linear algebra in precalculus classes?

Vector algebra is a standard 3rd-semester calculus topic (e.g., see OpenStax Calculus 3, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard ...
Daniel R. Collins's user avatar
15 votes

When writing log, do you indicate the base, even when 10?

I'm in Germany (chemist, FWIW). I'm familiar with: $\log$: base is unknown/not needed (as in $\log (a) + \log (b) = \log (ab)$ natural logarithm: $\ln = \log_e$ base 10 logarithm: $\lg = \log_{10}$ ...
cbeleites unhappy with SX's user avatar
15 votes

How can we best motivate the study of polynomials to high-school students?

Using puzzles to attract attention: "Think of a number, subtract 7, multiply 3, add 30, divide by 3. Then subtract the original number. The result will always be 3. Why does this magic work?"...
Spai's user avatar
  • 299
15 votes

Should I choose Cox-based Jaynes' approach or Kolmogorov approach to base myself on to teach probabilities to high-school students?

I think you should (and likely will have to) use the assigned text and approach. It's incredibly unlikely you will just derive some new approach. That's not how high school teaching works. And on ...
Pro-pedagogy guest troll's user avatar
15 votes

Applications of High School Geometry

Here's my rant that I gave to my students: You are learning this right now because some of you will need this later in life. You may need this information for your daily job. You may need this ...
abestrange's user avatar
14 votes

How can we motivate that Newton's method is useful?

In video game animation, each second thousands of equations need to be solved with code. You can't tell a computer "look at a graph"! Your students need to think beyond the setting of ...
KCd's user avatar
  • 3,456
13 votes

Can this be a better way of defining subsets?

In addition to the very good answers that you've already received, it's probably worthwhile to also mention the following point: The alternative that you suggest might lead to a similar type of ...
Jochen Glueck's user avatar
13 votes

To 17 year olds, how can I explain that two numbers with arbitrarily small difference are equal?

If a student sees $=$ and $<$ somehow becoming equivalent here, they exhibit a common beginner mistake: They don't actually read the statement they are considering. Quantifiers are a particularly ...
Arno's user avatar
  • 871
12 votes

Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?

You could ask this series of questions; I don't know how "real world" it is but it does indicate the specific contributions of each individual term of the series. Suppose we have a car, and ...
Chris Cunningham's user avatar
12 votes

Why not think of derivatives as fractions?

There is a nice write up at the included link with an application to thermodynamics. I've included two of the images (subsequently transcribed) that give the core of the argument. Single variable ...
Adam's user avatar
  • 5,382
12 votes

Topological fun facts for high school students

Cutting a Möbius strip in half: or at one-third: I would bring a collection of ~20 scissors and tape dispensers, and have the students form Möbius strips (sharing equipment), predict what will ...
Joseph O'Rourke's user avatar
11 votes

Geometrical verifications for Algebraic formulae

In my experience, geometry can lead you into interesting questions that can be answered geometrically in special cases, and algebra can help you tighten up the rigor of your answers and generalize ...
Justin Skycak's user avatar
10 votes
Accepted

Responding to students' questions that aren't directly relevant to their exams

What would you suggest as the best way to deal with students questions which seems like not relevant to standards of exam comparing with past papers? First, commend the student. Second, either start ...
Peter Flom's user avatar
10 votes
Accepted

Special topics for introductory probability

A classic application of Bayes' Theorem is in medical testing, and the difference/conversion between "what is the probability I test positive, given I have the condition" vs. "what is ...
Kevin P. Costello's user avatar
10 votes

Do you think with the advent of Desmos/GeoGebra, the Moore Method is more accessible to high school?

It seems like there are several questions here. I'll try to untangle them and address each one individually. Do online graphing/visualization tools like Desmos and GeoGebra make projects more ...
Justin Skycak's user avatar
10 votes

How can we motivate that Newton's method is useful?

Mild challenge to the framing of the problem It seems that a by-hand Newton's method is being compared to a graphing program. That does not seem a fair comparison. Why not compare an equation solver ...
user15245's user avatar
  • 205
9 votes

Why do we teach linear algebra in precalculus classes?

The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled Functions ...
user52817's user avatar
  • 10.5k
9 votes

Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?

I think the one with useful interpretation is $$ \frac{1}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots = \exp(x) $$ You can check $\exp(x+y) = \exp(x)\exp(y)$, so it does behave like you would ...
Gerald Edgar's user avatar
  • 7,499
9 votes

Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?

A possible motivation is the solution to the so-called "hat check problem." There are $N$ people who throw their hats into a box and then randomly take one out. The probability that each ...
user52817's user avatar
  • 10.5k
9 votes

What benefit is there to obfuscate the geometry with algebra?

I have been on committees that write questions for standardized tests and placement tests. In this role, I have reviewed results of many trigonometry questions that were piloted and then revised for ...
user52817's user avatar
  • 10.5k

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