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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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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 ...
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3answers
289 views

Harnessing misuse of equals sign

Students often misuse the equals sign to indicate "I've done this operation" rather than the proper use indicating numerical equivalence. Eg. Tax is paid using the rule: \$3 572 plus 32.5c per \$1 ...
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5answers
326 views

Motivation for polynomial long division

In the U.S. students in grades $\{9,10,11\}$ often learn long division of two polynomials, e.g.: $$ \frac{x^4 + 6x^2 + 2}{x^2 + 5} = x^2 + 1 - \frac{3}{x^2 + 5} \;. $$ I believe it is fair to say that ...
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3answers
254 views

Good (natural) motivational examples for quadratic equations

I am looking for good motivational examples of how quadratic equations can naturally arise in real life for someone starting high school. The high school book my child is using just jumps into ...
9
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2answers
611 views

Good real-life examples of transformations of function graphs

I am a gradudate student teaching college algebra at a larger state school and transformations of graphs of function, i.e.: given the graph of a function $y =f(x)$, what do the graphs $y = f(x) \pm C$,...
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2answers
200 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"...
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2answers
156 views

Why bother completing the square to find the minimum/maximum of a quadratic function?

Given a question like Find the coordinates of the minimum point on the curve $y=3x^2+2x+9$. students are often taught to solve this by completing the square. The class I am currently teaching ...
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4answers
164 views

Method of Showing Algebraic Work

I have seen two different methods of showing algebraic work when solving equations. I show both of them below for the same simple math problem: \begin{alignat}{8} x+3 &\;=&\; 5 \qquad&&...
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3answers
227 views

How to resolve the new definition of subtraction and division seen in college algebra?

Here's the foundational thing that irritates me the most when teaching college algebra. Up through the secondary level, I think that instructors and students are trained to understand subtraction and ...
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3answers
332 views

How to Teach Middle School Students to Read Square Roots?

This exact quote from my standard American Algebra 1 textbook states when first introducing rational square roots: $\sqrt{49} = 7$ is read "The positive square root of $49$ equals $7$." $-\...
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6answers
175 views

Multidisciplinary problem

I am looking for ideas for an activity for high school students, which involves plane geometry and another field, such as algebra, series, etc... For example, in junior high there is a nice activity ...
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0answers
178 views

implication vs equivalence when solving equations

I remember we were taught in high school (Eastern Europe) the difference between implication ($\Rightarrow$) and equivalence ($\Leftrightarrow$) and were instructed, when solving equations to be ...
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2answers
238 views

How to solve $a x = b$?

I'm teaching algebra to lower ability grade 11 students. I've tried to give them fair grounding in algebraic manipulation. I'm trying to explain how to solve a linear equation like $2 x +1 =3$ (or $- ...
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2answers
95 views

Make a matrix algebra course (1st university year) more “project-based”

Among other courses, I'm teaching a (basic) matrix algebra course for 1st year university students (they are studying Economics, and the cursus leads them to management, finance, or econometrics in ...
10
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4answers
196 views

Algebra best practices for students

One thing I notice frequently is that students don't have 'best practices' for doing algebra. Let me given an example: If students are trying to differentiate, say, $f(x) = (x^2 + x)^2$, they will ...
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4answers
226 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 $x$"), ...
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3answers
161 views

Resources on solving systems of polynomial equations in multivariable calculus setting

Whenever I teach multivariable calculus I find students really struggle with both finding critical points and the method of Lagrange multipliers. I think that the reason is the same: solving systems ...
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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 ...
27
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10answers
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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-...
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3answers
277 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 ...
7
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2answers
139 views

What can we learn as we reduce the fraction at the end of the quadratic formula process? [closed]

In the final throes of the quadratic formula, you reduce a fraction. Consider the following two examples. $y = 6x^2 + 11x + 3$; the quadratic formula reveals the roots $x = -4/12$ or $x = -18/12$. ...
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4answers
174 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 (...
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3answers
484 views

Stating identity is not the same as knowing value

A discussion with a frustrated 10th grade student sent me here. I had provided two linear function expressions, $f(x)=2x+2$ and $g(x)=-\frac{1}{2}x-2$, now find the intersection of the two lines! ...
3
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2answers
160 views

How to explain Chinese remainder theorem?

I want to explain Chinese remainder theorem to master level computer science students. There are two versions of CRT one is number theoretic and second requires the definition of ideals, groups etc. ...
4
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3answers
153 views

Are there any good exercises on point-slope form of a linear equation by itself?

Let's say we're running a basic algebra course and we're committed to showing proofs of everything we reasonably can. The development of equations of lines seems most straightforward in this sequence: ...
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4answers
241 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. ...
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3answers
445 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 ...
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3answers
235 views

Is there a resource that formally develops the topics of elementary algebra?

Let's say you're a university lecturer who regularly teaches remedial and/or college algebra courses. A standard textbook for such a course usually starts out with a series of facts about real number ...
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3answers
259 views

As it is appears here, how many constants do you see in $2x^3 + y^4 = \sqrt 5$?

My original phrasing was "how many constants are there in...". I am trying to determine if this is more clear. This is not intended for an exam, just a basic question to get intro-level algebra ...
10
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9answers
344 views

Simple examples that violate group axioms

In a course for non-math-majors at a liberal arts college, I would like to give a few lectures and activities about groups and symmetry. I think it's straightforward to explain the group axioms and ...
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2answers
168 views

What is “subtract 4 from 3 times X” and why?

I saw this phrase/sentence in a worksheet, "subtract 4 from 3 times X" The question asks the students to write the statement using numbers and symbols. I think the correct answer is 3X - 4, but ...
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6answers
401 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 ...
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2answers
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Can number theory help me create equations with nice solutions?

I'm teaching a remedial algebra class, and I recently put a radical equation on a quiz. At this point, the students had only solved polynomial equations by factoring, so the equation had to turn out ...
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3answers
158 views

Online course for Algebra II

I'm going into highschool. Thanks to an acceleration program at my district, I've already taken Algebra I and Geometry. I am very much a math and physics nerd (take a look at my activity over on MSE ...
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2answers
145 views

How can I help an eighth grader learn to write mathematically?

I'm tutoring an eighth grader in math, and while he is particularly bright he is not very good at taking the time to show his work and write things down. He likes to try to solve a problem in his head,...
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1answer
203 views

What is the name of this discipline in mathematics education?

I am struggling with my students who can think only in concrete terms, they can compute with concrete numbers but are not able to think in terms of e.g. functions on natural numbers and come up with ...
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0answers
92 views

Advanced algebra circa 1969 [closed]

For almost 50 years, I've been looking for a textbook I had as a 12th grader in Abe Lincoln HS (Brooklyn) in 1969; a book that opened my eyes up to math- which resulted in me going into actuarial work....
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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 ...
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3answers
313 views

Apply the inverse operation on both sides, or know the inverse function?

My old question here was about logarithms, but as I teach more and more (precalculus) algebra, I've generalized the question a bit in my mind. Should students do the same thing to both sides, or ...
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1answer
211 views

how to prove my 6th grade son knows algebra 1?

My son is taking pre-algebra in 6th grade. He mastered algebra 1 in detail more than what school can teach. How to prove to school that he mastered Algebra 1? He wants to start Algebra 2 in 7th grade. ...
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1answer
149 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 ...
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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-...
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2answers
257 views

Looking for a “Stable” College Algebra Textbook

I have taught College Algebra several times and will teach it again in the next semester. College Algebra, according to the the catalogue of my college, is described as follows: This course ...
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4answers
515 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 ...
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3answers
262 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 ...
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1answer
92 views

Using Substitution in place of the balance model

How does substitution work as an alternative to the balance model in introducing solving equations? My biggest worry is that, lacking a concrete representation, is too abstract for middle schoolers. ...
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1answer
230 views

Good Textbook for combined Beginning and Intermediate Algebra Course

We'd like to create a course that covers beginning and intermediate algebra in one course (8 hours a week), getting students ready to succeed in precalc and calculus. I am at a community college. I ...
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3answers
262 views

Is there a more intuitive way to solve combined rates of work problems?

I am helping my brother study for the GRE and we have come across some problems like this in my old precalculus textbook: 1) Karen and Betty have been hired to pain a house. Working together, they ...
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3answers
155 views

Mathematical Task with Various Solutions

I need to come up with a mathematical task for middle school (9th grade), which involves either algebra, functions, probability or statistics (anything but geometry actually). My problem is, that the ...
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6answers
525 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?" ...