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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 ...


16

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

I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable definitions: $k[x]$ is the ring of formal expressions of the form $\sum_{j=0}^{\infty} p_j x^j$ with $p_j \in k$ and we require that $p_j$ is $0$ for $j$ sufficiently ...


12

The convention is that "$b<x<a$" means "$b$ is less than $x$ and $x$ is less than $a$." What you suggest is only wrong in that it goes against the shorthand that everyone (?) has already agreed on.


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 ...


10

$2x$ 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 ...


10

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 ...


9

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 ...


9

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. ...


8

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 ...


8

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

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{...


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

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 ...


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 ...


7

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 ...


7

As a different view, I would say that this is "wrong" in the sense that we usually expect transitivity with (many of) our relations. E.g. if I write $a=b=c$ then usually we would say $a=c$ as well. There are certainly counterexamples for more general relations, such as if $aRb$ means $a$ and $b$ share a hobby, so that $aRbRc$ wouldn't necessarily mean $...


7

Here's why you should take great care when considering $\pm$ as an operator. It's not unusual to see a sentence of the form We deduce that $A=\pm B$ and hence that $C=D\pm E$. This isn't simply saying that both ($A$ is either $B$ or $-B$) and (either $C=D-E$ or $C=D+E$). When two $\pm$'s appear in the same sentence it is implied that they are both to be ...


7

I made a handout with 10 true false questions, titled Algebra Temptations. I've put it on google drive here. I have students work in groups to decide which are true. I think it makes a good activity, though there is the risk mentioned in another answer of reinforcing the wrong idea.


7

Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work. So please don't accept this answer. Anyway, the educational answer is to see that student is using the fact that $1^2 = (-1)^2$ in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand ...


6

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 ...


6

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 ...


6

This: "On the other hand, without the proof the definition of the degree of polynomials is not even logically established." is not quite right. What is needed to establish the definition is the fact that the degree is well-defined, but this fact is stated (implicitly) by stating the definition. The expectation that all facts be proven is out of place in a ...


6

When solving a quadratic equation, $$ax^2+ bx+ c = 0$$ we use shorthand for the two solutions, to include both $$x_1 = \frac {-b +\sqrt{b^2-4ac}}{2a}$$ and $$x_2 = \frac{-b - \sqrt{b^2 -4ac}}{2a}$$ Hence, the shorthand, $$x_i = \frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$ I.e., the solutions to $ax^2+bx+c = 0$ are given by $$x\in \left\{ \frac {-b +\sqrt{b^2-4ac}...


5

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 ...


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

Here's another computer-graphics example, which may or may not count as "nowadays". When I was in college 25 years ago I started a project to write a ray-trace renderer, which I then continued to expand after college. Ray-tracing is an elegant model that just recently has gained the hardware support to make it feasible in real time, e.g.: https://developer....


5

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 ...


5

When an inexperienced student sees $a=b=c$, I'd assume that both $a=b$ and $b=c$ are clear but the transitivity that yields $a=c$ might not be obvious. That's why I'd focus on this hidden equality when reading it out loud, by not just reading the equation (your first option), but rather saying "a, b, and c are all equal" or "a, b, and c are the same [number/...


4

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 ...


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