Dan Christensen
  • Member for 6 years, 1 month
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Why are induction proofs so challenging for students?
9 votes

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 ...

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Exponents with Negative Base; with or without Parentheses
7 votes

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 ...

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Should my 8th graders see a proof of the Pythagorean Theorem?
7 votes

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. ...

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Real-life exceptions to PEMDAS?
4 votes

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 ...

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What are the best places online where math educators can discuss their experience?
3 votes

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 ...

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What is a variable?
3 votes

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 ...

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How to resolve the new definition of subtraction and division seen in college algebra?
2 votes

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\...

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Teaching congruent triangles non-rigorously
2 votes

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 ...

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Inability to work with an arbitrary mathematical object
2 votes

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 ...

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Why do no students know to change the limits of integration when doing substitutions?
2 votes

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) \;\...

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Why do students have problems with showing that something is well-defined? How can this be improved?
2 votes

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 $N$ (...

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How to teach logical implication?
1 votes

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 ...

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Good, simple examples of induction?
1 votes

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{...

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Dividing by zero
1 votes

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, ...

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How to help new students accept function notation
1 votes

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 ...

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teach that $\frac10$ not defined properly
1 votes

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 $y\neq 0$ ...

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Can we explain to undergraduates how points make a line?
1 votes

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 ...

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How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
1 votes

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 $a=-x$ ...

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How should I convince a student who proved $1=-1$
0 votes

(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 ...

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Functions, Domains, and Ranges in Precalculus
0 votes

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 ...

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Defining vertical tangent lines
0 votes

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 ...

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Rigorously defining the concept of an angle for high school students
0 votes

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 ...

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How to explain the difference between the fraction a / b and the ratio a : b?
-1 votes

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 ...

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Simple, elegant ways to teach the idea of what functions are for the first time
-2 votes

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 ...

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What is abstraction and generalization ?
-2 votes

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 $P(x)$ ...

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