Bill Dubuque
  • Member for 7 years, 10 months
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  • Shoulders of Giants
Remedial students struggle with factoring $x^2+bx+c$ and $ax^2+bx+c$
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13 votes

The AC-method is often presented as an unmotivated rule-based algorithm, e.g. here. If that is the way that you learned it, then I can understand why you hate it. Below I explain the (little-known) ...

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How to correct visualization of mathematical expressions?
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9 votes

The problem is due to imprecise specification of the intended result. Here's a more precise way. $\text{Recall that the }{\bf commutative\ law}\ \color{#c00}X+ \color{#0a0}Y = Y + X\ \text{ is true ...

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'Low-algebra' examples of induction
9 votes

Tiling problems might meet your constraints. A nice simple example is Golomb's Theorem that a chessboard of side $2^n$ with any square omitted can be tiled by trominoes ("L" shapes of 3 squares). In ...

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Algebraic Solving and Uniqueness Proofs
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9 votes

Both are valid ways to prove the uniqueness of solutions, i.e. that the set $\,S\,$ of solutions contains at most one element. Indeed, one easily proves that the following are equivalent for any set $\...

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Greatest common divisor applications
8 votes

Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like ...

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Examples of proofs that use a cycle of implications to prove equivalence
8 votes

Proving equivalent the below characterizations of squarefree naturals is quite elementary employing cyclic inferences (assuming one knows the existence and uniqueness of prime factorizations). ...

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How to show $(x - a)$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$ (without division)
5 votes

One way to avoid (explicit) division is to make a change of change variables $\, X = x-a\,$ which reduces it to the following simpler special case $$ X\mid P(X) \iff \color{#c00}{P(0)} = 0$$ ...

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Beyond cubic polynomials: Applications?
4 votes

Quartics and higher degree polynomials frequently arise when intersecting lower degree curves, e,g, the intersection of two conics, or a line and a torus. Such curve intersections often occur in ...

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Intuition for the mean for elementary school kids
3 votes

[Note: I interpreted the question as how to explain the relationship between the two definitions] Let's first consider the underlying algebra before turning to real-world models. As an example, ...

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How to explain the flipping of division by a fraction?
3 votes

Below are a various methods of presenting fraction inversion based on various innate symmetries. Unlike some other methods, these ideas are more to the heart of the matter, so they generalize more ...

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Simple Number Theory Task
3 votes

A nice third problem arises by continuing from the second problem to show the power of parity arithmetic for solving problems in integer arithmetic. For example, we can use it to show that large ...

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Experiences with Venn diagrams as didactic tool for factors, GCD, LCM?
3 votes

Perhaps worth emphasis is that the use of prime factorizations (either directly or implicitly via (multiset) Venn diagrams or exponent vectors) may actually obfusctate the essence of the matter. For ...

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

I won't try to give a specific explanation here because I think that depends highly on the level of knowledge of the student. Rather, I'd like to emphasize some general points that I think are helpful ...

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Simple question on radical expressions
3 votes

There is no standard form for such expressions. However, some forms may prove more convenient for certain applications, e.g. transforming the radicands to be squarefree helps to ascertain ...

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Unique steps leading to a non-unique answer
2 votes

Often students wrongly believe that integer factorization is unique because there is a deterministic algorithm for factorization: simply continue to pull out the least prime factor (found by trial ...

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Small 'new things' to confront talented high-schoolers with
2 votes

Telescopy (or telescopic induction) is a nice example. It is a simple, prototypical example of induction that will provide strong intuition for later studies. Telescopic cancellation is usually ...

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Harnessing misuse of equals sign
1 votes

Update. It appears that the intent of this answer was not clear to some readers, so I will elaborate. The most common way to fix the student's computations is simply to break out the subexpressions ...

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Undergraduate Math Seminar topic
1 votes

I recommend choosing an elementary proof that includes both elements of surprise and beauty - one that might spark students to further study mathematics. Along these lines I've had much success ...

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What could be good non-mathematical analogies to explain the difference between the words theorem, proposition, lemma and corollaries?
1 votes

If I had to explain common usage to a layperson then I might use an analogy with travelling to beautiful places, as follows. A theorem is like a beautiful place that you would highlight to anyone ...

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How to arrive at infinitude of primes proof?
0 votes

First, we find an integer universally coprime to any prime $\,p.\,$ A moments thought yields the solution $\color{#c00}1$. Next, we generalize to larger solutions. Since the coprimality of $\,n\,$ to ...

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