# Tag Info

## Hot answers tagged algebra

22

Instructor at University of Washington here - we were one of the early closures, so I feel like we're starting to get the hang of it. Here's what I'm using: Zoom: Zoom is similar to Skype, with better support for many-participant calling and additional features. It has a built-in whiteboard you can write or type on, mechanics for allowing students to "raise ...

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 ...

16

Already put two comments but ideas keep coming to me so I'll just package them here. Keep in mind I'm not an educator, I'm just trying to think of practical solutions to the problem as a whole. Another answer recommends YouTube to upload source material but I feel this might be inadequate interaction. Other alternatives: Discord Recently in response to ...

14

The case for WeBWorK tl;dr - use WebWork as an easy way of giving students as much practice solving problems as they can handle. Learn by doing lots of problems with a tight feedback loop. It engages the game-playing, obsessive nature in us. Main use case - Homework engine The two biggest features of a VLE/LMS are the presentation of materials and ...

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

I gave a presentation to my department about this today. Like you, as I see in a comment, I am also at a CUNY math department. I haven't done all-online classes before, but I've used Blackboard heavily for ~20 years and have had a hybrid (partly online) class for the last two years. I have access to Blackboard Collaborate. My only cameras at home are ...

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. ...

9

Should we impose that $(a^m)^n=a^{mn}$ only when $a \gt 0$? Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule $(a^m)^n=a^{mn}$ cannot be true with negative bases and rational exponents, and this is a great achievement. Try to explain that he/she proved that If the said rule is true for negative ...

9

I like You Tube for posting videos. Once you get started it's pretty simple. There are various levels of privacy possible which you can read about. If your school has a convenient way to post videos and you have broadband (we're talking about 1-1.5 Gb files here, do NOT use HD resolution or worse yet the 4k resolution...). Given all that, basically the thing ...

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

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 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}... 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 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 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 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 I asked the What are the Laws of Rational Exponents? question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. While it wasn't the top-voted, my answer there represents my best thinking about the status of this issue in classes at that level. As I wrote: Regarding the ... 6 Well, since I have also been obliged to teach from home due to CoVID-19 these days, I will describe here a possible solution to your problem. As a fast and cheap solution - I have been granted no access to any platform, unfortunately - I use the following: Skype, as a platform to communicate with my students and conduct the major part of the lesson. ... 6 In some sense this question is waaaay too broad, but it is attracting a useful collection of hints, and it's super topical for thousands of college math instructors (likely to be followed by primary/secondary ones), so here are a few things which I don't see mentioned yet, collated from the far too much time spent on this subject today. Web/Doc cam. There ... 6 I am surprised that a school would effectively say "go figure it out". You ask for "brainstorming"... here are my thoughts.. You haven't quite defines your goal, although I did hear, loud and clear, you'd like 'free' or close to it. Still, there are a number of outcomes. Live video - I believe there are many options, but in general, a multiuser ... 6 In the UK, what you describe lies somewhere in between GCSE and A-Level Maths. Both of these exams are intended for secondary school students. It is unusual for these topics to be taught at the university level, and if they are taught, it is often not by the Maths department. For example, the Economics department might have such a course for their first ... 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

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....

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