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.

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65
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, "...
67
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6answers
4k 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}=...
26
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14answers
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 ...
31
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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 ...
28
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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 ...
25
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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 ...
15
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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 $12$ ...
21
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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 ...
33
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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 ...
29
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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 ...
15
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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) ...
23
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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 ...
24
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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. ...
11
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3answers
599 views

Teaching algebra to visually impaired or blind students

I am currently writing an independent project investigating the teaching and learning of algebra for students with a visual impairment. I am struggling to find literature specifically about teaching ...
8
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3answers
2k views

Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?

For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the Abel–Ruffini theorem (there is no algebraic solution to general polynomial equations ...
7
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5answers
1k views

Constructing and sketching parabolas, conic sections and other curves

Whenever teaching or discussing parabolas, conic sections and other curves with my students, I always feel dissatisfied with the standard "find vertex, pick points, connect the dots" method to draw a ...
17
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4answers
890 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 ...
11
votes
3answers
500 views

Appropriate ways/sayings to discourage undergraduate students' overreliance on calculators

Main question: How do I, in a medium- to large-sized undergraduate class setting, appropriately and effectively discourage students from relying too heavily on calculators? There have been several ...
7
votes
2answers
282 views

Teaching logic through “high school algebra”?

I am going to be teaching a discrete math class in the fall. One of the major goals of the course is a solid understanding of the basics of logic: the precise meanings of "and", "or", "not", "implies"...
18
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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 ...
10
votes
3answers
316 views

How to teach a student algebra who misses too much previous knowledge?

I am now tutoring a student in Grade 9, who falls behind in math study. He lacks the basic understanding of operations and inverse operations, and have trouble dealing with negative numbers and ...
4
votes
4answers
398 views

Is This Trick Helpful?

I am no professional educator; I am a student myself! But apparently I come up with useful tricks that help my younger brother do better in maths. I just want to hear your feedback, is all. My ...
25
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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 ...
17
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5answers
597 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 ...
19
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4answers
647 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, ...
11
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6answers
1k views

Should Eisenstein’s criterion be taught in high-school?

Eisenstein’s criterion: If you have a polynomial with integer coefficients (and a non-zero constant term) and there’s some prime number $p$ such that $p$ goes into every coefficient but not ...
9
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4answers
474 views

Everyday Example Problems for Solving Linear and Quadratic Equations

I am going to teach some grade 9 students about solving linear and quadratic equations. I am looking for a question from every day life (of a teenager) or a puzzle which is hard to solve without using ...
5
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4answers
338 views

Extension activities in Algebra II

I'm in Algebra II this year, and I have to admit, it's kind of boring. The only new thing we've touched on so far this year is how to graph piecewise functions, and those are really easy to graph. ...
4
votes
5answers
3k views

Where do students learn to solve polynomial equations these days?

When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving ...
29
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6answers
797 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 ...
16
votes
3answers
393 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
7answers
931 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 ...
12
votes
7answers
704 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 ...
10
votes
1answer
397 views

The “rearranging” approach to teaching logarithms

Consider the following way to teach division: Division works this way: any product equation $xy = z$ can be rewritten as a quotient equation $x = \frac{z}{y}$. Just move the numbers in that way. ...
9
votes
2answers
1k views

What is the failure rate of students in Algebra 1?

Hello fellow educators, I've been hearing a lot recently that many students struggle with Algebra 1 in the US. The dropout rate is supposedly very high and some educators argue that Algebra 1 and ...
8
votes
1answer
158 views

Introducing the concept of variables to kids

Today I had a discussion on how to introduce the basic concept of variables in math using real life examples. We came up with ideas of using boxes containing matches, or M&Ms representing the ...
8
votes
4answers
755 views

Literature on development of algebraic thinking

I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8. We will do this through focus on the common core. I'm also putting together ...
11
votes
6answers
497 views

Pedagogical quandary with the definition of $i$

I'm not sure how the concept of $i$ is taught in other places, but in our district the curriculum defines $i = \sqrt{-1}$, which is how it has been traditionally taught (for a while now) and also how ...
8
votes
4answers
261 views

Shifting function graphs: writing vertical offset on the y-side?

Students tend to mix up signs when shifting function graphs around: consider $y=x^2$. To shift it one unit upwards ("increasing $y$"), you write $y=x^2+1$, to shift it to the right ("increasing $x$"), ...
7
votes
5answers
2k views

Best rote high school algebra textbook?

I am looking in too getting a textbook for my son who I will be teaching algebra soon (homeschool). I don't like math, never did and am not very good at it. Quite frankly, my son isn't good at it ...
7
votes
11answers
4k views

Why unlike terms cannot be simplified?

"Why $x^2+3xy^2+4xy+7x^2y$ can not be simplified? Why can these terms not be simplified?" I would like an explanation that is understandable by 8th-grade students. The only proof I know (based on ...
5
votes
2answers
357 views

A student's problem on solving simple trigonometric operations

In my class today as I was checking homework of trigonometric operations, I called one of the students and gave her one of the problems of book which was $\frac{1}{(1-\cot (\theta ))}+\frac{1}{(\cot (\...
2
votes
3answers
201 views

Why in the FOIL Method the terms are taken with their signs?

That was the most boring title I could choose but in all honesty, it is what the question is. Here is a reminder of the FOIL method that is used for multiplying two binomials. For example, to multiply ...
2
votes
4answers
203 views

Might it be helpful for students to have different symbols for subtraction (-) and negation ( _ )?

Might it be helpful for students to have two different symbols for subtraction (-) and negation ( _ )? Subtraction, after all is a binary operation (with 2 operands). Negation is a unary operation (...