19

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea. Connections to other mathematics The notation with AB, CA and BC might be something the students have used or will use in less analytical ...


19

I think the basic answer is that there are all sorts of things that we could write, but don't. We usually leave off things that are redundant, but we can add them back in when convenient. For instance, instead of $1x$ we could also write $1\times 1\times 1\times x$. Or we could write $x + 0$ or even $0x^3 + 0x^2 + x + 0$. We could also write $0yz + 0y^2 + ...


17

My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a university level mathematics course. $\newcommand{\u}{\mathbf{u}}$ $\newcommand{\v}{\mathbf{v}}$ $\newcommand{\w}{\mathbf{w}}$ $\newcommand{\R}{\mathbb{R}}$ Let ...


16

What's wrong with this?: $$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ with $\frac{dt}{dt}=1$. It's what you would get if you had $f(x,y,z)$, except that $z=t$. You do have to be a bit careful, though. You want to be clear that $f(x,t,t)=xt^2$ ...


14

I think you're correct, strictly speaking, but overreacting. I just checked two calculus books, Rogawski and Strang. Rogawski defines the cross product as the "symbolic" determinant (his words) in question but then defines exactly what he means by this determinant in the same line, so I don't see a problem there. Strang mentions the determinant ...


13

Sure, give up on the mixed-object determinant formula. Instead use tensor arithmetic. Let $\epsilon_{ijk}$ be the completely antisymmetric symbol. Also, while we're at it, let's give up on quaternionic notation for unit-vectors and instead use $\hat{x},\hat{y},\hat{z}$ or $\hat{x_i}$ for $i=1,2,3$ then you have an elegant formula: $$ \vec{A} \times \vec{B} = ...


11

Do you ever introduce the Jacobian matrix, or the derivative of a function at a point as a linear map? This clarifies everything. If not, and you must restrict yourself to "traditional multivariable calculus" notation, you could write the chain rule as $(f \circ \gamma)'(t) = \nabla f|_{\gamma(t)} \cdot \gamma'(t)$ In your example $f(x,y,t) = xyt$ ...


10

I guess you are aware of it, but since you don't say so explicitly: the chain rule on the top of your post does not apply to your example, since that $f$ is not a function of $x,y$ but of $x,y,t$. Also, as written, you are confounding the modern concept of "function" $f:\mathbb{R}^3\to\mathbb{R}$ with the original notion of "function of". ...


10

In Olympiad geometry, $a$, $b$, $c$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing Pythagoras' theorem states that $a^2+b^2=c^2$. or Pythagoras' theorem states that $...


9

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a picture instead For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...


7

Absolute time is an example of what is called an affine space; a mathematical structure where you can consider the difference of two points (which in this case would be duration), and that second space is linear. The affine space doesn't have a linear structure itself in the most abstract case, but it gets an interesting structure of its own when you think ...


7

this makes no sense But it does. A determinant exists for any $n\times n$ matrix for which the Leibniz formula$$\det A:=\sum_{\sigma\in S_n}\epsilon_\sigma\prod_{i=1}^nA_{i\sigma_i}$$is fully antisymmetric. This is why, for example, quaternionic matrices lack determinants: their elements don't commute. But if we apply it to the case at hand, we don't have ...


7

I had a lecturer who thought the same as you, so suggested we just learnt the definition of the cross product (which he derived along the same lines as James S Cook's answer). Nobody liked his lectures because he pitched them way to high for our first-year brains. The point is, it's a very useful mnemonic device, and when you teach it make that clear, but ...


6

You are not being pedantic. The name of the process that slices is $g$, and the result of slicing $x$ is $g(x)$. On the other hand, the textbook presentation seems to be for students who are just encountering function notation. In the language of the Dubinsky school of constructivism, students at this stage are not ready to distinguish between $g$ and $g(x)$....


6

The only one of these that looks objectionable to me is the one that calls the hypotenuse $h$, since in a triangle the letter $h$ usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse).


6

Clearly, this student did not recall how to add $x+2x$ and $-y+y$, but he did correctly add $10+5=15$ (presumably because the $10$ and the $15$ were "visible"). After discussing with my cooperating educator, she suggested that I rewrite the original system as $$\begin{cases} \color{red}{1}x-\color{red}{1}y=10 \\ 2x+\color{red}{1}y=5 \end{cases}$$ ...


6

I would like to comment on the question: For the purpose of teaching, should we normalize writing 1x instead of x? not for attempting an answer but to ask why one should consider it a (not so) reasonable idea. I think that behind such question there is the idea that if I write $1x$ rather than $x$ then the algorithm for sum would be clearer. I think what's ...


6

To explain why any particular ordered pair is in $S \circ R$, you can just show that it satisfies the definition, which says that $(a,c)$ is in $S \circ R$ if there exists $b$ such that $(a,b) \in R$ and $(b,c) \in S$. To show this is true, you can just give an example of such $b$ and observe that $(a,b) \in R$ and $(b,c) \in S$. The two things here are ...


6

I'd avoid introducing a new notation for such a limited scope: you won't probably use it elsewhere during the class, and it wouldn't be used elsewhere in the literature. I have three suggestions. The first is a simple variation of Trevor Wilson's answer, which mirrors the first relation sign to keep the bridging elements together: $$R\ni(1,3) \And (3,2) \in ...


5

I taught elementary school. I think the clearest way to write this for elementary school children is: 11 hours 40 min - 15 min = 11 hours 25 min You could then conclude from the above that 11:25 is 15 minutes earlier than 11:40. You could also regroup as necessary. 11 hours 15 min - 40 min = ? This should be regrouped as 10 hours 75 min - 40 min = 10 hours ...


5

I read the left-hand side as 11 minutes and 40 seconds minus 15 minutes. (Analogous to 11.4 - 15). So I expected the right-hand side to be negative. I would not write this, especially as a teacher. I would prefer $11:40-00:15=11:25$ hours, with the understanding that the units refer to the leftmost grouping


5

To clear up confusion from this, when I teach I make a massive emphasis of the fact that a function takes in an ordered list of inputs (and not something like 'variables'). Then I use notation like $D_1f$ to denote the partial derivative of $f$ with respect to its first input, $D_2f$ the second input etc. I go on about how each of these is a new function ...


5

There is, as others have said, nothing wrong with $$\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ Note that a partial derivative like $\frac{\partial f}{\partial t}$ is really to be understood as the derivative of $f$ with respect to its third variable,...


4

I agree with the OP: the determinant is only a mnemonic device, but (a) students may not know how to evaluate a determinant, so it's useless as a mnemonic, and (b) it is not a "legal" determinant and so mathematicallly misleading. I would rather start with $|a \times b| = |a| |b| \sin \theta$ as a contrast to $a \cdot b = |a| |b| \cos \theta$, and ...


4

I would argue that there is an issue with the diagram as labeled regardless of whether you are or are not being pedantic. For example: Consider the case where $f$ and $f(x)$ can be used interchangeably, as is done in many US high school courses. In this situation, the labeling on the slicer and fryer are fine, because they denote the functions in question....


3

Whatever you choose, make sure to follow some basic rules: Clearly state the preconditions and make sure they are understood (right triangle in your case). Clearly state the meaning of the symbols (e.g. which symbols stand for the sides adjacent to the right angle, and which for the third one). Use the symbols consistently (don't make e.g. the same symbol &...


3

I really wish people would stop teaching the Pythagorean Theorem as $a^2 + b^2 = c^2$, for the following reason: Give your students the diagram below, and ask them to solve for $c$. At least 1/3 of a typical high school class will write $a^2 + b^2 = c^2$ and report back to you that $c = 5$. The problem is that the equation $a^2 + b^2 = c^2$ is so ...


3

It is important to become comfortable with disambiguation. For example, in listening we must use context to distinguish between the homophones “reed” and “read” as in “I will read a book tonight.” In reading, we must use context to know when to pronounce “read” with a long e sound as in “reed” or with a short e sound as in “red.” The brain is an amazing ...


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