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20

The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous. If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than ...


17

For the project, you would need to define what are "classical results in mathematics". I suspect that different people would disagree on the classicalness of various results. Furthermore, it is doubtful if everyone should learn the same classical results - mathematics is a big field. You can, of course, cut it down by a sufficiently strong definition of "...


14

If the expression were, say, $48\div 4\times 12$, there would not be much disagreement (multiplication and division are performed from left to right). But an expression such as $48\div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying. If one interprets the parentheses ...


13

This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to "know" a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a car usefully? Must we be able to prove the ...


11

At the moment, I can answer bullet point two: Are there any high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree polynomial equations, and that in higher degrees that are no general methods at all? Yes, you can find this on p. 267 of CME Project's (2009) Algebra 2 text. ...


11

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something. Unless they specialize in mathematics at the college level, they do not learn any more. Why not? Because we have computers now, so most people do not need to solve polynomial ...


9

The examples you give aren't exceptions. The parentheses aren't needed because there is no other way to interpret the expressions. In applications in engineering and the physical sciences, variables have units associated with them, and the units often disambiguate the expression without the need for parentheses. For example, if $\omega$ has units of radians ...


8

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the ...


8

$2x$ is an expression, a phrase. Compare it to "two ducks". This is neither true nor false. It doesn't have a 'truth value'. $2x = 4$ is an equation, a statement. Compare it to "two ducks have four legs". This is true (edit: for the ducks, but not necessarily for the $x$). The meaning of the word "ducks" has not changed. The grammar of what is with that ...


7

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division and synthetic division in Algebra 2. This is also when students will learn the remainder theorem, Descartes’ Rule, and how to identify the only possible rational ...


7

Your last para was very reasonable. (I was going to give a mean sarcastic answer, but can't now.) We can crowdsource this: Frank Ayres, First Year College Math (algebra 1 to precalc; Schaum's Outline) 1958 but still in print: Only has geometric mean of 2 objects, but does spend quite a lot of time on geometric progressions. And also discusses getting ...


7

I am with you on this one. I feel like concatenation (implied multiplication) is of higher precedence than explicit division. For me $8:2x$ means "8 divided by 2 x'es" - $8:(2x)$ not $4x$. Replacing $x$ with $(2+2)$ shouldn't change anything. But the formal answer is that it's undefined. There is no C for concatenation in PEMDAS. For me it should be PECMDAS....


7

Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority? The conventional order-of-operations in textbook math (or any math) simply don't prioritize ...


6

That's a good question. In addition to PEMDAS there's grouped symbols; they occur under a radical ($\sqrt{~}$, $\sqrt[n]{~}$), in the numerator and denominator of a fraction and in exponents. These grouped symbols are treated as if they are between parenthesis. That means that the expressions $a^{\text{expression}}$, $\sqrt[n]{\text{expression}}$ and $\dfrac{...


6

I'm going to answer with something of a polemical frame challenge: FOIL is evil, and probably shouldn't even be taught. Okay... that's a bit extreme. How's this: FOIL is a mnemonic that is, in my opinion, not all that useful, and should not be taught. In my own experience teaching college algebra and precalculus courses, students come to rely on FOIL ...


6

It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to do so. In my thinking, the modern perspective of Cardano's formula is an algebraic perspective, not analytic, after Galois. Polynomials of degree 2, 3, 4 are ...


6

The problem underlying the discussion in the question can be summarized as that it is necessary to choose a branch cut to define a complex logarithm (or arctangent). It is a mistake and pedagogically a bad practice to allow negative values of $r$. It is a mistake because pairs $(r, \theta)$ with $r$ possibly negative have no right to be called coordinates. ...


5

Some students that I've encountered, especially those who have struggled with solving linear equations in the past, have taken well to what is sometimes called "backtracking" [in CME Project: Algebra 1] or "the arrow method" [in LINCT Transition Course curriculum (note: this is the program I work for)]. The illustration below may be enough for a master ...


5

It just depend on what do you mean by "knowing very well". To me "knowing very well" means: if you give me 1 hour I can tell you the basic ingredients that go in the proof (and the reason why the result is relevant) and if you give me 1 full day I can sketch a reasonably detailed proof. This means: I know how to fit the result in an area of math; I know ...


5

My disagreement lies in the dismissal of the parentheses, as my explanation would be that $8÷2(x+x)$ would, as a first step, simplify to $8÷2(2x)$ and then $8÷4x$ It seems to me that you are: ignoring the Left To Right rule, not making the implied multiplication between $2$ and $(2+2)$


5

I run into this issue frequently. As a high school in-house math tutor, students visit to show me their quiz/test scores and ask about their work. The FOIL method is fine, if it works for the student. For those who are prone to making mistakes, I show them the Box method (call it what you will, that just my name for it). The benefit, if any, to this method ...


5

I teach in the U.S. at a community college. Although I prefer distributing without drawing an area box when I'm doing math myself, I often show the box in class to help students see how things work. When we use an area model to help students visualize the workings of the distributive property, it makes much more sense for many students. In fact, the box ...


4

The earliest mathematical insight I remember from childhood is that the word "of" almost always means "times." Half of a dozen $= \frac12 \cdot 12 = 6$ Three-fourths of a mile $= \frac34 \cdot 5280$ feet $= 3960$ feet. I'll take 6 of those thousand-count boxes $= 6 \cdot 1000 = 6000$. I remember feeling like I had secret knowledge that no one ...


4

You will find textbooks for all of these at artofproblemsolving.com. They formed in order to help students like you. I have used their number theory book, while tutoring a young student (who was great at math), and I loved it. === Edited (9/28/19) to add: Another good resource is Henri Picciotto's mathed.page. You'll find free pdf's there for at least two ...


4

To demonstrate the distributive law, we often use an area model. Then $(2+4)\times(5+7+9)$ can be visually decomposed into the sum of 6 subproducts. Rather than actually drawing a grid, you might eventually start just labeling the edges and the products, without really caring about the relative sizes of the pieces. Finally, this ``table method'' could ...


3

For factoring a quadratic, when "A" = 1, it's relatively easy. Answer the question, "What 2 numbers multiply to C but add to B?" This process may be guess and check, but when A is 1, there's little need to spend much time on this. When one factor has a negative integer, the rule doesn't change, so much as the student needs to be mindful of where the minus ...


3

The issue isn't unique to factoring or the FOIL process. The roots of $(ax^2 + bx + c)$ can be found via the quadratic equation. $$x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}$$ Now, this prompts the issue; $(2x^2 - 3x - 4)$ What is the value for a,b,c that we will plug into the quadratic equation? Isn't c = -4 ? And we need to accept that the minus ...


3

Simple answer: Including the signs/negative numbers is algorithmic. Working out the sign afterwards requires thinking. Imagine writing computer code to multiply out the brackets. You would do it by keeping the minus signs with the relevant terms. (Pedagogically: keeping the signs with the magnitude will extend to bigger brackets in a more natural way than ...


3

The word "of" means $\times$. For example "half of $a$" means $\frac12\times a$. Note that the product of $a$ by a number isn't necessarily less than $a$ ; it'll be less than $a$ iff the number is less than $1$ ; in that case we can write the number as $\frac{m}{n}$ ($m<n$) so that $\frac{m}{n}\times a$ means to divide $a$ into $n$ equal parts and take $m$...


3

I don't see any good way to talk about this without explicitly addressing the issue of domains and the fact that the same square root symbol is used for two different functions depending on context. In the first context, the square root symbol refers to the single-valued, nonnegative real square root. In the second context, the square root symbol refers to ...


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