Hot answers tagged

81 votes
Accepted

Issues with "equals", where does this come from and how do I combat it?

A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or ...
user avatar
  • 4,765
69 votes

How to respond to “solve this equation” in a basic algebra class

This is a really interesting question, because similar issues---the question of how demanding to be about formatting of answers---come up a lot, at all levels, and the answers aren't always ...
user avatar
45 votes
Accepted

How do I teach algebra?

As a personal tutor, I’ve been teaching algebra to kids from ages 8 to 16 for many years. Mostly I find myself in the position of picking up the pieces when the kids are failing and fearing more ...
user avatar
  • 1,214
40 votes

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

In a comment, the OP has suggested that he actually wants a practical example convincing students that the product of two negative numbers is positive. This is related to, but psychologically ...
user avatar
39 votes
Accepted

Rationale for not dividing both sides of an equation by $x$ (ex: $6x^2 = 12x$)

Just before dividing, you can reason "Either $x=0$ or I can divide by $x$." This creates two separate cases to be analyzed. This works for dividing by anything. You want to divide by $\sin(x)$? You ...
user avatar
  • 6,350
36 votes

Should I be teaching point-slope formula to high school algebra students?

As someone who teaches calculus to college students, I expect my students to have seen point-slope form. We just start using it (because it's the right way to talk about tangent lines and ...
user avatar
31 votes
Accepted

How to Teach Adults Elementary Concepts

But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it." Good! Students (at any age) should never be satisfied with "This is the way it's done,...
user avatar
  • 16.3k
29 votes
Accepted

Good way to explain why an absolute value in an equation does not automatically mean to make the other side +/-

One of the reasons your students are putting the $\pm$ on the 10 is probably because someone told them that when you remove a pair of absolute value signs that the $\pm$ goes on the number after the $=...
user avatar
29 votes

Is $a^0 = 1$ for a nonzero, real number $a$, a theorem or an axiom?

This is what really happens during education (not in the mind of a mathematician): The teacher introduces $4^9$ as just a way to abbreviate $4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\...
user avatar
  • 2,315
29 votes

Should I be teaching point-slope formula to high school algebra students?

Point slope form emphasizes the actual meaning of slope. Literally, $$ y - b = m(x -a) $$ Says "The change in the outputs ($y-b$) is equal to the slope ($m$) times the change in the inputs ($x-a$)"...
user avatar
29 votes

Why do we teach complex numbers?

We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has ...
user avatar
29 votes

What is the right feedback for incorrect cancellation?

My feedback might be more like this: When you are not 100% sure that a step works, try it with numbers. If x=2, does your expression keep the same value before and after your cancellation? Also, you ...
user avatar
  • 17.3k
27 votes
Accepted

Should word problems be reasonable?

Absolutely! In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is checking whether the answers make sense. This is an ...
user avatar
27 votes

How to respond to “solve this equation” in a basic algebra class

Ask your pedantic colleagues to reconcile their formatting expectations with the context "solve $F=m\cdot a$ for $a$". Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up ...
user avatar
  • 7,575
27 votes

How should a student's inefficient calculation be pointed out?

Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just ...
user avatar
26 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

$$3x^2 -14x - 5 = 0$$ Multiply through by A or here, 3 $$9x^2 -42x - 15 = 0$$ Now, use substitution, u=$3x$ (3X is the square root of this first term, and by using the u substitution, we now have an '...
user avatar
25 votes

How do I teach algebra?

For some students, the difficulty with solving $3x+5=14$ is even more basic than figuring out what operations to do in what order in order to reach the goal. Before getting to that, they need to know ...
user avatar
25 votes
Accepted

Should students get full credit for getting the correct answer (without work)?

Personally, I'd say "no." The students who don't write out the steps in their algebra classes appear in calculus thinking that they should be able to write down all the answers without any ...
user avatar
  • 19.1k
25 votes

Why do we teach that every line is a linear function?

The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, ...
user avatar
  • 16.3k
25 votes
Accepted

Is 'estimating' still considered a valuable skill?

Yes, of course it is. You've identified many of the important reasons in your question. In practice in quantitative fields, we are all estimating as a first-pass on whether problems are soluble all ...
user avatar
24 votes
Accepted

Why are $m$ and $b$ used in the slope-intercept equation of a line?

This is also borderline not-an-answer, but it might be a nice broadening of your students' worldview to know that the "$m$" and the "$b$" are not universally accepted. Showing them this map (even ...
user avatar
  • 19.1k
23 votes

How shall we teach math online?

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 ...
user avatar
20 votes
Accepted

How should a student's inefficient calculation be pointed out?

I like your second option the best: ...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part? ...
user avatar
  • 7,686
20 votes

A PEMDAS issue request for explanation

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 ...
user avatar
  • 17.3k
19 votes
Accepted

How to justify teaching students to rationalize denominators?

My thinking is that it is just so damn useful for students to be aware of these tricks. The examples/exercise should allow them to develop a sense of when and how it is helpful to simplify an ...
user avatar
19 votes

How does one explain that transformations 'inside' a function operate in the opposite direction than intuition suggests?

The way I help students understand this is to flip the idea on its head and say to them that replacing either variable with something always does the opposite to the graph, whether you replace the x ...
user avatar
19 votes

Rationale for not dividing both sides of an equation by $x$ (ex: $6x^2 = 12x$)

I think your student pointed out the key issue: "Well, obviously I didn't know that zero was an answer when I was doing the problem". That's exactly right: at the time the student was dividing, they ...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible