36
votes
Should I be teaching point-slope formula to high school algebra students?
As someone who teaches calculus to college students, I expect my students to have seen point-slope form. We just start using it (because it's the right way to talk about tangent lines and ...
- 11.5k
29
votes
Should I be teaching point-slope formula to high school algebra students?
Point slope form emphasizes the actual meaning of slope.
Literally,
$$
y - b = m(x -a)
$$
Says
"The change in the outputs ($y-b$) is equal to the slope ($m$) times the change in the inputs ($x-a$)"...
- 23.6k
27
votes
Accepted
What's the point of learning equivalence relations?
Perhaps emphasize to students the spirit of equivalence relations. They partitions sets into equivalence classes--cutting down the amount of cases necessary to prove something.
To illustrate this, ...
- 495
25
votes
Accepted
An introductory example for Taylor series (12th grade)
One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are ...
- 10.7k
22
votes
Dividing by zero
In third grade we taught division using repeated subtraction. To divide 6 by 2, subtract 2 until you get to 0. 6-2=4, 4-2=2, 2-2=0. It took 3 steps so 6÷2=3. This can also be shown on a number ...
- 7,939
21
votes
teach that $\frac10$ not defined properly
What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever ...
- 1,705
20
votes
Why do we care about multiple proofs of the same theorem?
I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances ...
- 369
19
votes
Should I be teaching point-slope formula to high school algebra students?
As someone else teaching calculus and higher math to college students, I use point slope form repeatedly:
In the/a definition of derivative, I use point-slope form. I have students think about $y_1-...
- 191
19
votes
Accepted
Replacement for the Pac-Man grid analogy
A possibility is to show your students Google's implementation of the game Snake. If you enter the term snake into Google's search engine, the there is a box at ...
- 7,433
19
votes
An introductory example for Taylor series (12th grade)
An excellent introductory example would be exponential function $\exp(x) = e^x$.
By definition, this is the function that is its own derivative, i.e. $\exp'(x) = \exp(x)$. That's all nice and swell ...
18
votes
Accepted
How to get students in a under-graduate linear algebra course interested in determinants?
I have found it motivates to explain the determinant as computing a volume.
One can work through and convince for $2 \times 2$ and $3 \times 3$ matrices,
and perhaps only hint at the
$n \times n$ ...
- 29k
18
votes
Are there direct practical applications of differentiating natural logarithms?
Have you thought about the fact that you’re asking this in the middle of a pandemic for which log plots are being used all over the place to visualize the growth of COVID cases?
At any rate,
$${d \...
- 5,157
16
votes
What's the point of learning equivalence relations?
In general, all applications are going to be more of the "here is what we have to check to make sure that our algorithm/theory/definitions work". We don't usually encounter practical ...
- 671
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?"...
- 299
14
votes
Accepted
Good way to explain fundamental theorem of arithmetic?
The way I have explained the fundamental theorem of arithmetic in the past is by establishing what it means to be prime (has exactly two positive divisors) and then having students construct factor ...
- 18.3k
14
votes
Are there direct practical applications of differentiating natural logarithms?
Whenever we measure a quantity on a log scale (such as Richter, decibels, musical pitch, or a log-plot axis), we are focusing attention on relative variation in that quantity. If $y = \ln x$, we have
$...
- 271
13
votes
How to teach the Pythagorean theorem in a satisfying way to high school students?
I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion.
I will try to come back and write a more elaborated ...
- 17.2k
13
votes
Does induction really avoid proving an infinite number of claims?
The "avoidance of proving an infinite number of claims" explanation for the need for induction has not yet resonated with me because there are obviously many universally quantified ...
- 1,404
12
votes
An introductory example for Taylor series (12th grade)
I feel it a bit strange that no one mentioned it, but a famous example is how Newton quickly estimated $\pi$ by the Taylor series.
Here is a quick sketch:
First note that the area of a quarter circle ...
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 ...
- 20.8k
11
votes
How to teach binary numbers to 5th graders?
When teaching binary to any age group I always start by taking a set of kitchen weights into the classroom. 2lb (32oz) 1lb (16oz) 8oz 4oz 2oz 1oz. Then make a table asking what weights yo1u would use ...
- 119
11
votes
Should I be teaching point-slope formula to high school algebra students?
Personally, I went many years not teaching the point-slope formula (college algebra and remedial algebra), and I just flipped on the issue this past weekend. Here's why: In making sure that I could ...
- 23.6k
11
votes
Should I be teaching point-slope formula to high school algebra students?
I am someone whose mind seems to work similar to your cooperating teacher. Why learn two forms of something when you can learn one and manipulate it? I made it through all of algebra 2 never ...
- 1,225
11
votes
Accepted
How can I motivate the formal definition of continuity?
Have a look at the paper written by Nunez et all:
EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION.
In essence, they argue that it is better to be causious if ...
11
votes
Accepted
Why should we study continuity?
Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to:
Computers and ...
- 2,552
11
votes
Good way to explain fundamental theorem of arithmetic?
To really understand why the integers $\mathbb{Z}$ have unique prime factorization, it helps to understand how unique prime factorization can fail in other settings. For example,
$$(2 + \sqrt{10}) \...
- 4,843
11
votes
Accepted
Why do we care about multiple proofs of the same theorem?
Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, ...
- 226
11
votes
What's the point of learning equivalence relations?
Equality vs. Identity
Alexei touched on this topic by mentioning hash tables, but I would like to spell it out more explicitly, because this is a critical and fundamental topic in software engineering,...
- 653
11
votes
Does induction really avoid proving an infinite number of claims?
My personal take on this, is that all the talk about "infinite this, and infinite that" is only mudding the waters. The emphasis should not be on wanting to prove $P(n)$ for all all $n$, but ...
- 1,078
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