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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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}=... 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, "... 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-... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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-... 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$xlike this. \begin{align*} 0 &= (x-2)^2-9 \\0 &= (x^2-4x+4)-9 \\0 &= x^2-4x-5 \\0 &= (x+1)(x-... 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 ... 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 ... 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. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 findV=\frac{1}{3}\pi ... 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 ... 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, ... 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, ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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?" ... 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 ... 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 ... 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$12$... 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 ... 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 ... 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 ... 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) ... 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 ... 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? 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 ... 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 ... 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 ... 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 ... 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 ...
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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 ...
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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 ...
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 ...