# Tag Info

30

I would say the standard implementations of the Rational Root Theorem (make a huge list for the sake of making the huge list) indeed feel like a complete waste of time. However, the theorem can sometimes combine nicely with technology tools. I'll provide an example of this. Consider the following algebra puzzle: Solve for $x$: $7x^3 -39x^2+52x+30 = 0$. If ...

16

I can think of two related reasons: The characterization via the decimal expansion might be perceived more strongly like a property of the number: "This number is irrational, because this number's decimal expansion does not terminate." The other one is rather a non-property and thus not perceived as a definition of some thing. The characterization via ...

12

I think an important learning outcome of this part of high school Algebra 2 (theory of polynomials) is developing mathematical capacity to juggle several disparate abstract tools all at once, similar to how in calculus, one uses first derivative test, second derivative test, concavity, asymptotes, intercepts, end behavior, etc., all at once, in some ...

11

Agreed...it's one of the less useful parts of high school algebra. But not because "you could use a computer"--you could say that about almost everything. And then we get some of the same people who push the "use a computer" who are surprised when their kids flounder because of lack of manipulational ability in calculus. ;) The reason ...

11

Let $x = \sqrt 2 + \sqrt 7$ prove that $x$ is irrational. \begin{align} x - \sqrt 2 &= \sqrt 7 \\ x^2 - 2\sqrt 2x + 2 &= 7 \\ x^2 -5 &= 2\sqrt 2x \\ x^4 -10x^2 +25 &= 8x^2 \\ x^4 -18x^2 + 25 &= 0 \end{align} So $\sqrt 2 + \sqrt 7$ is a root of $x^4 -18x^2 + 25$. According to RRT, if $x$ is rational, then $x = \pm 1$ or $x=... 10 Note that this definition of rational and irrational numbers is most commonly presented to high school students, who tend to have a strong and natural intuition of numbers in base$10$. At this stage, it's much easier to understand what it means for a number to be irrational in terms of its decimal expansion rather than the statement "a number which cannot ... 10 The RRT is not taught in isolation. It is taught as a collection of tools for (partially) factoring polynomials. It should be taught with Descartes' rule of signs and some form of polynomial division. It is direct preparation for understanding the proof of Eisenstein's criterion for irreducibility. The "shadow" of that application is via Gauss's ... 7 I would reluctantly agree that it's not a particularly powerful tool if you have electronics at your disposal. But I might double down and say that you should be teaching synthetic substitution as well so that students can factor cubic and quartic polynomials without their calculators. About a year ago, I went down a YouTube rabbit hole of watching videos ... 7 The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to probably does not know anything about Cauchy sequences or Dedekind cuts. So the complement of "rational" is at best only intuitively defined for them. 5 This is a hard question to answer formally, but an easy question from an educator's perspective: it is very unlikely that adding a pure number and a percentage is a good idea. While a percentage has the same units as a pure number, they're usually represent different types, so adding them is unlikely to give a meaningful quantity. Anyone who finds ... 4 Theorem: For every integer$m$, the polynomial$x^3 - mx^2 - (m+1)x - 1$is irreducible among polynomials with rational coefficients. Proof: This polynomial has degree$3$, so if it is a product of lower-degree polynomials, the decomposition is (linear)(quadratic) or (linear)(linear)(linear). Either way, there is a linear factor and thus a rational root ($...

4

The point is definitely called, in general, the radix point (as stated in a comment by @user52817). I'm not familiar with, nor succeeding at a search for, a general name for the representation method. I would be comfortable calling one an "$n$-ary representation", following the term decimal representation (similar to a comment by @mweiss).

3

Many good arguments have been presented. I would like to add that working with the decimal expansion does require much less understanding of what a real number is. The decimal expansion gives you rather touchable things to work with so that even without being able to see a "sequence of digits" as a number (which would allow arithmetic operations), pupils can ...

3

In the US, we have the controversial Common Core. A summary of it is available here. In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending ...

3

I really think this questions gets to a bigger issue. This is part of a general problem with our approach to math education. The point of high school math class, from any practical standpoint, is not to teach students things that they will constantly be using all the time (or even ever). Empirically this is verified in myriad examples, not just the rational ...

2

While this is not quite an app, I believe it fits within the spirit of the question: Desmos activities. https://teacher.desmos.com/collections/featured While some Desmos activities are very math-forward, others follow Dan Myers' "Three Acts" format, giving students an interesting scenario and asking them to solve it with math, and showing the ...

2

I have been searching the corect wording for a while in the same context, that is, implementing a function that represents the decimal expansion of $\frac{a}{b}$. But then, I landed in a word desert when I needed that function to handle base 2 to 10. I naturally came to google "n-imal" fractions and found that the term "n-imal" is used ...

2

It always makes me cringe when I hear someone present the definition of irrational as a number whose decimal expansion does not repeat. It does a huge disservice to the history of math, its beauty, and the notion of rational. A definition of irrational as a decimal number which doesn't repeat lacks beauty in that the word irrational bears no relation to ...

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