As already touched on here, perhaps proof by induction should not be the first real method of proof that students learn. (The two-column proofs of geometry common in North American schools don't ...

You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the ...

I think the usual visual proof of PT uses something like: (From Math Is Fun website) Make cardboard cutouts of the outer and inner squares and the 4 identical triangles. Fit them together as shown. ...

I would add the following to Paracosmiste's reply: Numerical negation (e.g. the '$-$' sign in in the expression $-x^2$) usually has an order of precedence that is less than that of exponentiation and ...

For an absolute free-for-all with no restrictions on extended discussions (unlike here) or anything else for that matter, you could try sci.math at Google Groups. WARNING: It is currently infested ...

Suggestion: Instead of using single letters for variables, you might consider using meaningful words or phrases. Instead of s=d/t, for example, write speed = distance / time. Computer programmers ...

The problem stems from effectively defining the division operator twice. You should begin by defining division on $R$ as usual: For all $x, y, z \in R$ where $y\neq 0$, we have $x/y=z$ iff $x=y\...

You might consider defining a pair of triangles to be congruent based on 3 pairs of equal sides. That would take care of SSS. Make some puzzle worksheets asking students to pick out the congruent ...

Believing that they can infer that something is true for all real numbers based on a few examples points to lack of understanding of basic methods of proof, specifically how we go about making ...

Following on Mike's example, you might insist on explicitly specifying the variables being integrated over in the limits as well as the substitution used: $\int\limits_{x=0}^{x=2} 2x\cos(x^2) \;\...

Here is an interesting but simple example of something that is not well defined: Let $f$ be a binary function (resembling exponentiation or repeated multiplication) on the set of natural numbers $N$ (...

If you will eventually be teaching the basic methods of proof (conditional proof, proof by contradiction, etc.) in your course for math majors, you might consider starting with the truth tables for ...

Prove that, for all $x$ in $\mathbb{N}$, we have $x+1 \neq x$. (Requires use of proof by contradiction.) Use only the following properties of addition on $\mathbb{N}$: Addition is closed on $\mathbb{...

You might explain, at any level, that for any real number $x$, we have $0x=0$. So the equation $0x=0$ has no unique solution. And $0x=1$, for example, has no solution whatsoever. For these reasons, ...

You could start with two tables of values for x (the input variable) and y (the output value) in both. To start, each should represent a permutation on say, the set {1, 2, 3, 4, 5}, but don't use the ...

If the students have sufficient algebraic skills, point out that division is defined in terms of multiplication as follows: For any real numbers $x$, $y$ and $z$, we define $x/y = z$ iff $y\neq 0$ ...

They want to know how a bunch of zero length points make a line of length 1? It may help to think of the Euclidean plane as a set, each element of it being a point in the plane. Then any geometric ...

Proof that $-(-x)=x$: By definition, for all $a\in R$, we have $-a\in R$ and $a+(-a)=0$. Apply this definition for $a=x$ where $x\in R$ to obtain $x+(-x)=0$. Now, apply this definition for $a=-x$ ...

(My new and hopefully improved answer) Should we require that $(a^m)^n=a^{mn}$ only when $a \gt 0$ ? That might "solve" the problem in some sense, but we do have legitimate cases with negative ...

Let $f(x) := \sqrt{x+3}$. What are the the domain and range $f$? You are right to be concerned about such questions. Technically, you cannot define a function without specifying its domain and ...

Suppose function $f$ is continuous at $a$. Then $f$ has a vertical tangent line at $a$ iff $$\displaystyle\lim_{h\to0}\frac{h}{f(a+h)-f(a)}=0$$ i.e. the inverse of the slope of the tangent there will ...

Just a suggestion for the purposes of discussion... Let A, B, C be points in the Euclidean plane. Then (A,B,C) is said to be an angle with vertex B if and only if B=/=A and B=/=C. Let A, B, C, D ...

I'm not sure how you would translate the following into the language of 10-year-olds, but the teacher should first understand this much: The variables x and y are respectively in the ratio a:b means ...

Suggestion: Introduce calculator functions. Very concrete. Use function notation to record the results. Examples sin(1) = 0.017 (rounded) sin(2) = 0.035 log(1) = 0 log(2) = 0.301 Then introduce ...

I'm not sure what you mean by abstraction, but you can make a universal generalizations from a specific case in one of two ways: Direct proof: Start by assuming for that sake of argument that $P(x)$ ...