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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Geometrical approaches in algebra
Usually we describe proofs in algebra by algebraic means, I think it may be useful to introduce geometrical approaches to those proofs to improve creativity skills of students, what are the examples ...
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What should I call the "important" values of x?
When analyzing the functions
$f(x) = \sqrt{x-5}$
$g(x) = \frac{1}{x-5}$
$h(x) = 2^{x-5}$
we know that it is useful to think about what happens at $x = 5$.
For the function $f$, this logic will ...
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f(x+h) in the difference quotient
When teaching students how to compute the difference quotient in a precalculus or calculus class, we need them to evaluate the expression
$$\frac{f(x+h) - f(x)}{h}$$
for various simple functions, like ...
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What implication arrows, if any should I require in teaching?
Q: Solve $x+5=0$
A: $x+5=0\implies x=-5$.
This answer would be given full marks.
Isn’t it better to tell students to use $\equiv$ or $\iff$? Cause that is what let’s them say $-5$ is a solution to ...
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Research into how students read algebraic expressions
In answering another question What is the justification to teach the (redundant) use of parentheses in multiplications? I was left wondering what we actually know about students' progression in terms ...
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Relearning math after long COVID using AoPS or developmental math textbooks?
This is a little bit of a niche topic.
I've dealt with a pretty bad dose of long COVID that has caused some serious gaps in my mathematics (basically causing terrible arithmetic skills and a really ...
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Can it be defended that $\sqrt 4$ is both 2 and -2 (and likewise for general square roots)?
Over the past one or two years, at least two different teachers have told my children that $\sqrt 4$ is $2$ or $-2.$ I don't think this is useful, but if you want to define $\sqrt x$ as the set of ...
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Write $y=\sqrt{3+x}$ as the composite of two functions
For the question "Write $y=\sqrt{3+x}$ as the composite of two functions", what if a student gives the answer $f(x)=\sqrt{3+x}$ and $g(x)=x$? This answer would be technically correct but it ...
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Manipulative materials to teach functions
I am looking for manipulative materials to teach functions (the concepts including domain, image, etc.) and kind of function (affine, quadratic, exponential logarithmic, polynomial, trigonometric) ...
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Can 5 - 3a be described as five minus three times a?
The question asks students to describe the expressions in words. This is a student's answer:
"$(x + 4) \times 2$" is "$x$ plus four times two"
"$5 - 3a$" is "five ...
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Is coefficient same as constant?
I was studying about polynomials when I stumbled upon this video
https://www.youtube.com/watch?v=vBfdYuoc3x4&list=PLjS5lmipV2HJEaKfdeVSKdprfFxinzmNw&index=2
The video says that a monomial has ...
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How can you elicit the $\log x = {\log} \cdot x$ error?
You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually $\log$ or a trig function, and you watch ...
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Is there a College Algebra book that was written by a world-class mathematician? [duplicate]
I have taught College Algebra several times and will teach it again in the next semester.
College Algebra, according to the catalogue of my college, is described as follows:
This course provides ...
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What is the right feedback for incorrect cancellation?
Here are three "cancellations" seen during algebra simplification, two of which are invalid.
(1) $\frac{x + 6}{6} = \frac{x+6\hspace-1.2ex\diagup}{6\hspace-1.2ex\diagup} = x$
(2) $\frac{6x ...
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Measures to quantify complexity of algebra equation
Like the title says, I am looking for ways to measure the complexity of an algebra equation. For now, I am focused on linear equations, but I would think any metrics could be generalized for ...
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Why is my 8th grade Algebra 1 tutoring student learning mean absolute deviation and standard deviation?
I’m tutoring an 8th grade student in Algebra 1, and he showed me that their class learned how to find standard deviation and mean absolute deviation using the following formulas:
$SD=\sqrt{\...
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What's the best ratio for algebra tiles?
A good model for algebra are Algebra Tiles.
See: https://calculate.org.au/wp-content/uploads/sites/15/2019/03/lesson-sequence-for-algebra-tiles.pdf
In this presentation they recommend an aspect ratio ...
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Teaching Solving Linear Equations before teaching evaluating expressions
Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have ...
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What is Algebra 1 "Number and quantity; Interpreting Data"?
My sister received her scores for "End-of-Course Examination Program" test results. There is a section "Algebra 1" and underneath a table with rows:
Algebra
Functions
Numbers and ...
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Why the fear of polynomial long division?
Why do people think long division of polynomials is complicated ?
I heard this expressed recently and it seems like an odd sentiment. For me, synthetic division is complicated and totally adhoc ...
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How to train facility with numbers?
I have a student (age 14) who is valiantly striving to improve his algebra. He knows the "rules", and he more-or-less manages to apply them correctly, and hence is generally able to tackle ...
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Is the AC Method of Factoring polynomials more popular and used by teachers than others methods of factoring polynomials?
This is an example of the AC Method:
$ x^2 + 16x +63 $
(1) $x² + 7x$ (2) $9x + 63$
(1) $x(x + 7)$ (2) $9(x + 7)$
so we have:
$x(x + 7)+ 9(x + 7)$
(1) with (2) The Result is:
$ (x+9)(x+7) $
I have more ...
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How to create an equation of a given form with integer coefficients and variables
A math textbook gives the following problem:
What is the solution of 3(2x-1) – 2(3x+4) = 11x.
The solution is an integer.
Using Excel to choose some numbers at random and compute the remaining numbers,...
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Why is highschool math so unrigorous?
I am an highschool student (I'm starting soon the italian equivalent of 12th grade) and I have many problems with the way hs math education works. I don't understand why everything is explained only ...
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What is the expected fluency with fractions at UK key stage 3?
I have recently started some summer tuition for a student who has failed in mathematics at UK key stage 3 (age 14), and trying to plumb the depths of his knowledge. Having engaged him in some simple ...
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How can maths assistants at a college be extra helpful for professors?
I am a maths graduate student.
A little background: In the coming semester, I will be assisting a prof whom I admire and whom I also want to thank a lot, and I actually will have a lot of free time. ...
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A linear equation -- my approach
Here is an example of a lesson I did on linear equations where my objective was to show that they are equations of first degree. The reason I do it this way is because I tend to find that students ...
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Arithmetical Progression
I recently came across a very old Algebra textbook from the 1860s, and on the chapter discussing "arithmetical progression", it says there are "20 cases for arithmetical progression&...
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Why do we write $x$ instead of $1x$?
I am currently student teaching for an Integrated Math 1 class (which is similar to Algebra 1) that consists of 9th graders. I have been teaching my students how to solve linear systems using ...
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Geometric and Graphical perspective on completing the square
I just read an interesting article that helps to understand completing the square, and prove the quadratic equation from a geomterical perspective.
My question is how do I understand the graphical ...
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Multiplying the square roots of negative numbers before we calculate a result using $i$ [closed]
To evaluate $\sqrt{-1}$ $\times$ $\sqrt{-1}$ we cannot use
$\sqrt{A}$ $\times$ $\sqrt{B}$ = $\sqrt{AB}$ as the result would be 1.
I know (?) that we must first respect that the initial numbers ...
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Do my students know elementary algebra; do they just use online calculators or external help; and is this ok?
Background
I know the question in the title is very broad so I will try to explain it as succinctly as I can. Half my time is spent on as a researcher on didactics, while the other half is devoted to ...
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Looking for a rigorous middle school self-study math course
My son is in 5th grade (US) and since he is doing remote learning, we have been doing a lot of topics in pre-algebra just using worksheets. I'd like to start him on a formal middle school curriculum, ...
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What is the motivation for teaching Factoring by Grouping?
This seems like such a niche trick to teach students when factoring polynomials. Like, the polynomials I've seen textbooks ask students to factor by grouping seem so cherry picked that I can't imagine ...
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I'm in 8th grade and just finished Algebra 2. What math would I do for the next 4 years? [closed]
I'm in 8th grade and just finished Algebra 2. What math would I do for the next 2 years? In what order would math I would do in 9th, 10th, 11th, and 12th grade. Thanks!
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Why is isolating for $x$ taught before factoring?
I'm currently working on some precalculus packages for students who need review. For inspiration, I'm looking at some prealgebra books and I'm wondering why isolating for $x$ is taught before ...
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How do I teach my kid [closed]
I am struggling with teaching my 9th grade kid to solve math problems that are just outside of routine.
For e.g.,
An example problem given by math teacher at school.
x, y, z are in geometric ...
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Exercises for explaning homothety, homothetic center, similarity on line and plane, free vector and vector space
I need the collection of exercises for such topics as:
maps and transformations, composition of maps
homothety, rotation homothety, homothetic center
similarities of the line and the plane
free ...
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Practical case for solving with system of 2 equations
When I teach basic math I want to emphasize on it's power (algebraic part for starters) as a tool for solving certain problems you cannot solve with naked brain, so that one models a problem with ...
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When does thinking $(-8)^{1/3} = -2$ result in problems for an undergraduates?
In high school we learn that the cube root of $-8$ is $-2$. Much later some of us learn about the single valued natural logarithm of a complex number, and that $w^z = e^{z\cdot Lz(w)}$ when $w$ and $z$...
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What topics are considered to be part of pre-algebra?
I know pre-algebra is like a terminology thrown around to really basic stuff that are taught before high school algebra. Some stuff taught there are already considered as part of algebra in some ...
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How can I introduce the idea of eigenvectors and matrix decompositions to a general audience in an engaging manner?
So I'm doing a freelance writing job, writing a script for a YouTube video about eigenvectors/values. It took me a while to decide what the focus was going to be, but I finally settled on focusing on ...
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Courses equivalent to College Algebra in other countries?
In USA, there is a course called College Algebra and a course description may look like the following:
This course provides students an opportunity to gain algebraic knowledge needed in ...
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Is evaluating a Real Polynomial at a Complex Value a suitable task for Precalculus students?
In Korea, basically every teaching material for 10th grade math(about the level of precalculus) contains this kind of exercises in their first treatment of complex numbers:
Evaluate $f(x)=4x^4-8x^3+...
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How shall we teach math online?
Many universities, including mine, are now requiring we teach our courses online because corona. How shall we do this? Let’s brainstorm here.
Some challenges:
My school provides limited online ...
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How to motivate my ten year old math student
I work as a private math tutor.
I have a student, she is 10 years old. Her mother has asked me to provide assistance in preparation for the admission process to the eight-year high school.
My ...
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Should we stop teaching "interchange $x$ and $y$" when finding the inverse function?
In one textbook I use for College Algebra, the author teaches that one should interchange $x$ and $y$ when looking for inverse functions. For example, the inverse function of $$y=2x+2$$ is $$y=0.5x-1.$...
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How should I convince a student who proved $1=-1$
One of my high school students who has ZERO knowledge on complex numbers and the modulus function has showed me the following algebra:
$$(16)^{\frac{1}{2}}=(16)^{\frac{2}{4}}=((16)^2)^{\frac{1}{4}}=...
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Algebra/trig/precalculus review questions that elicit common student errors
This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the ...
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How to read chained equalities out loud?
I find that my community-college students are usually very hazy on the status and meaning of chained equality statements (or other relational statements). This seems like a really critical element of ...
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