Questions tagged [algebra]

Algebra is the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.

Filter by
Sorted by
Tagged with
67
votes
6answers
5k views

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

An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=...
66
votes
11answers
7k views

Whence the “everything is linear” phenomenon, and what can we do about it?

$$ \color{red}{(a+b)^2 = a^2+b^2}$$ $$ \color{red}{\sqrt{x^4+y^4} = x^2+y^2} $$ $$ \color{red}{e^{t^2+C} = e^{t^2}+e^C}$$ I've observed this phenomenon -- wherein, implicitly, students say, "...
42
votes
4answers
4k views

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

I asked this question once on math.se, but don't follow the link unless you want to risk biasing your own response: https://math.stackexchange.com/questions/444696/how-to-respond-to-solve-this-...
33
votes
13answers
14k views

Why do we teach complex numbers?

In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex ...
31
votes
3answers
2k views

How to cure students from the idea that root and squaring are identity operators?

I tutor high school algebra and I’ve noticed that a lot of my students don’t seem to understand what they’re doing when they “convert” between different ways of writing numbers involving perfect ...
29
votes
6answers
830 views

When $-x$ is positive

This recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative ...
29
votes
5answers
2k views

What fraction of the population is incapable of learning algebra?

In the comment thread of this academia.SE question, the following generated some strong reactions: My very different (community-college) perspective is that the math discipline will end up as a ...
28
votes
4answers
3k views

Open-Source Math Textbooks

It seems to me that an open-source model could work quite well for textbooks, with issues being raised by the users of the book and different forks of the project being created for different ...
27
votes
15answers
3k views

How do I teach algebra?

I find that soon I'll be working with high school students that are struggling with math. In particular, we'll be talking a lot about algebra and some basic trigonometry. The latter I have experience ...
27
votes
13answers
7k views

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

I'm student teaching this semester, and so far I'm loving it! Our next section in the book teaches point-slope formula, and my cooperating teacher (a 24-year veteran teacher) is convinced that point-...
27
votes
10answers
8k views

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

One time I watched a student solve the equation $0 = (x-2)^2-9$ for $x$ like this. $$\begin{align*} 0 &= (x-2)^2-9 \\0 &= (x^2-4x+4)-9 \\0 &= x^2-4x-5 \\0 &= (x+1)(x-...
25
votes
18answers
2k views

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

Actually, there is no algebraic problem to show that $-(-x) = x$. This proof can be build upon the concept of the addition of the opposite like this: $- x + x = - x + [- ( - x) ]$, and thus by ...
25
votes
9answers
7k views

How to justify teaching students to rationalize denominators?

I'm teaching an "intermediate algebra" college course ($\approx$ junior high school or beginning high school algebra) and we have a bunch of problems on rationalizing denominators. How do I motivate ...
24
votes
5answers
2k views

Should word problems be reasonable?

I've recently run across a series of problems that didn't reflect reality. For example - An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH. ...
23
votes
4answers
2k views

How to Teach Adults Elementary Concepts

I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school ...
23
votes
5answers
663 views

Do all high school students need the same 3-year sequence of math courses?

I continue to be troubled by the amount of symbolic manipulation in a typical Algebra 2 course. Once a student has completed Algebra 1 and Geometry, shouldn't there be another option for them if a ...
22
votes
7answers
3k views

When should we first teach variables in school math? And how?

From a pedagogical point of view, when is the "right" moment to introduce letters and variables to school children? Let's say we taught arithmetic, the four operations, did computation exercises, or ...
22
votes
2answers
2k views

Is this just a mistake or a more serious misconception?

One of my main research areas is algebraic thinking at different levels. Yet, from time to time, I observe something that I cannot relate to anything else that I know. This is the story of one of ...
21
votes
11answers
13k views

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

Consider a real function $f(x)$ and imagine its graph in the plane. Then the graph of $f(x+2)$ is simply the graph of $f$ shifted to the left 2 units while the graph of $f(x-2)$ is that of $f$ shifted ...
19
votes
9answers
3k views

Is 'estimating' still considered a valuable skill?

I was with a 2nd year high school class, preparing for our (US) state's standardized test. I asked the class how they would solve this, and they flipped through the sheets to find $$V=\frac{1}{3}\pi ...
19
votes
5answers
810 views

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

I was helping out in a learning support class today and we were working through some absolute value problems when something like $|x + 4| - 5 = 10$ came up and both students I was working with split ...
19
votes
4answers
656 views

How to help motivate math when tutoring low level algebra (High school)

I was tutoring a student today and we were doing basic factoring of quadratics and expanding terms like $(x+2)(x+5)$. Now he ended up being able to do this by the end of our 2 and a half hour session, ...
19
votes
4answers
681 views

Difference in meaning of 'algebra'

The other day, in a conversation with colleagues, I realised that the word 'algebra' means different things to us. To me, it brings to mind the study of algebraic structures: vector spaces, groups, ...
19
votes
4answers
2k views

Is algebra really the gatekeeper to higher math, or is it multiplicative reasoning?

The National Mathematics Advisory Panel final report states that algebra is the gateway to higher math, to a college degree, and higher earnings from employment. It also states that success in algebra ...
19
votes
2answers
2k views

Algebra 2 textbooks that incorrectly claim that all solutions of polynomial equations can be found

Over the years I have occasionally encountered a number of Algebra 2 textbooks that make an incorrect (or at very least extremely misleading) claim along the lines that "all solutions of a polynomial ...
19
votes
3answers
971 views

Group theory for high schoolers, want the opinion of other educators

So I am going to be teaching the basics of group theory to high schoolers in a few weeks, and I want to hear what the Stack Exchange network has to say on the matter. What are the applications and ...
18
votes
10answers
2k views

How to Write Steps of Solving Equations?

This is a common way to write the steps during solving equations: But in GeoGebra the steps are shown this way (the highlighted part): I'm going to use GeoGebra to teach equations. Is it OK to let ...
17
votes
4answers
329 views

Are teaching about finding the missing member(s) of the sequences really appropriate?

I notice that in current mathematics education they always have sections teaching about finding the missing member(s) of the sequences e.g. in this way: $1,2,4,8,16$ , the next term is what? Someone ...
17
votes
4answers
637 views

Mindless use of “antisimplifications” such as $1/x+1/y=(x+y)/xy$ and $1/\sqrt{2}=\sqrt{2}/2$

I recently gave an exam problem that roughly 2/3 of the class made much more difficult by using the identity $1/x+1/y=(x+y)/xy$. Their work would have been much simpler if they hadn't done this. It ...
17
votes
5answers
622 views

What is a variable?

There are two kinds of answers I'm looking for: What do students think a variable is? What do YOU, the teacher, think a variable is? I'm also interested in why you think a variable is what you think ...
17
votes
4answers
910 views

Are fractions hard because they are like algebra?

It occurs to me that to really understand the ways that people work with fractions on paper requires a good grasp of the ideas that numbers have multiple representations and that expressions can be ...
16
votes
6answers
553 views

How to teach students when they can and can't cancel factors in a fraction?

I mainly tutor adults in college algebra classes or lower. Sometimes an expression like $\dfrac {x+5}{5}$ will come up, and the students will say: "We can cancel out the $5$'s and get $x+1$, right?" ...
16
votes
3answers
451 views

Good lessons/activities for one-day subs

In my school district, and I'm sure most others, every teacher needs to have a set of "emergency lesson plans", in case they are sick or need to be out for a day, so that the substitute can have ...
16
votes
3answers
402 views

Evidence for or against the claim that some students are “algebra people” and others are “geometry people”

Where I live and work, there is a widely-accepted and often-repeated claim that there are two kinds of students: "algebra people" and "geometry people". This claim sometimes gets expressed in ...
15
votes
10answers
2k views

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

so we are working through various methods of factoring quadratic equations and the students seem comfortable factoring basic quadratics such as: $$x^2 - 7x + 12 = 0$$ by finding the factors of $12$ ...
15
votes
7answers
938 views

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

Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read: "A linear function is a function defined by ...
15
votes
3answers
498 views

Explaining to students why $m$ and $b$ are used in the slope-intercept equation of a line

The slope-intercept form of the equation of a line is often presented in textbooks as $$y = mx + b\,,$$ where $m$ is the slope of the line and $b$ is the $y$-intercept. How did $m$ and $b$ become ...
15
votes
4answers
626 views

Student converted $\sqrt{x^2}$ and ended up with just $x$ instead of $|x|$

I asked a student I tutor what $\sqrt{x^2}$ was so I could show him why the solution is $|x|$ instead of just $x$. He ended up changing the problem to $(x^2)^{1/2}$ and then multiplied the exponents ...
15
votes
4answers
1k views

Remedial students struggle with factoring $x^2+bx+c$ and $ax^2+bx+c$

Remedial students have seen quadratics before but, perhaps they don't elicit positive memories. The textbook (designed for people taking the course for the first time, not for remedial students) ...
15
votes
1answer
4k views

What basic algebra skills and techniques are most important for calculus students to know?

In my experience, algebra is one of the biggest stumbling blocks to calculus students. For instance, sign errors are common, and exponent laws (and log laws!) cause a lot of headaches. Many courses ...
14
votes
11answers
2k views

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

For the students of grade 9: Is $a^0 = 1$ for a non zero real a, considered a theorem or an axiom?
14
votes
12answers
997 views

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

Pre-algebra If the student is taking this branch of mathematics, they are expected to show their work because they're expected to solve specific problems in a certain way. Ex, when they're solving ...
13
votes
4answers
756 views

How to visually demonstrate basic algebra

I'm a private tutor working with a 7th grader who is struggling with solving equations. Given a simple equation, he is able to solve it using a formulaic procedure, but it is very obvious that he has ...
13
votes
2answers
1k views

A simple explanation or derivation of Cramer's rule, suitable for secondary Algebra 2?

I am tutoring a high school student in Algebra 2. Her class has just covered systems of linear equations, mainly via the "substitution" and "elimination" methods. Her textbook concludes that unit ...
13
votes
1answer
1k views

High-school level algebra textbooks for gifted students

Note. I asked the question below on Math Stack Exchange (link), but didn't get a really satisfactory answer there, so I'm posting it here too. I am looking for high school algebra/mathematics ...
12
votes
7answers
712 views

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

this came up in class yesterday and I feel like my explanation could have been more clear/rigorous. The students were given the task of finding the zeros of the following equation $$6x^2 = 12x$$ and ...
12
votes
7answers
1k views

The sum - product problem

I have long been a fan of all of the different methods for factoring quadratics, yet I hardly ever use them in my classroom. The first task they are confronted with, in factoring trinomials, is to ...
12
votes
4answers
2k views

Why are university algebra courses often harder than the corresponding high school courses?

Someone I know recently took an online intermediate algebra course to prepare for college algebra. Thus course had 70+ sections, each with 10-30 poblems, beginning with set-builder notation and going ...
12
votes
3answers
284 views

Best way to find out what math topics a person struggles with when tutoring

I will soon tutor someone I know in math. Because of the bad results on tests, combined with the feedback from the teacher, he has a very bad sense of achievement. Which in turn, has given him a low ...
12
votes
6answers
511 views

Teaching Completing the Square

I'm trying to teach some secondary school students on how to complete the square. The goal is to rewrite: $$y = ax^2 + bx + c \ \ \Rightarrow \ \ y = a(x-h)^2 + k$$ The first thing I did was to ask ...