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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Special topics for introductory probability
I am helping to design a low-level college course whose purpose is to teach critical thinking, logic, finance and probability. I have been tasked with developing the probability section. I am ...
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Fixing wrong ideas about coefficients (e.g. subtract 3 from 3x to isolate x)
I tutor a student (9th grade, United States) who is in an algebra class. She consistently makes mistakes when dealing with coefficients.
The most common one is attempting to subtract away a ...
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Is there a standard convention for interpreting ambiguous absolute value expressions?
Consider the expression
$$|x + 2|x + 3|x + 4|.$$
One way to interpret this is that there are two products being added together:
$$|x+2|x \hspace{1cm} + \hspace{1cm} 3|x+4|$$
But you could also ...
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Intuition for order of operations in compound transformations
This is a close cousin of the previous question asked here about transformations inside and outside a function and how they switch things around. I think some of the perspectives there will help here, ...
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Examples of Financial Institutions that Compute Interest Atypically?
Are there examples of financial institutions that compound their interest more frequently than once-a-month? Are there examples of financial institutions that consider continually compounded interest ...
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Does there exist a (statistical) topology induced by students on the space of algebraic formulas? :)
It's kind of a serious question even if the title seems silly.
As math educators, we all know that students link together different algebraic expressions thinking that they mean the same thing, e.g.
\...
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Geometrical verifications for Algebraic formulae
What is the importance of using approaches related to Geometric Algebra in teaching,is it only useful when introducing Algebra to the students or can it be helpful to improve creative skills in ...
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I'm in dilemma while solving arithmetic problems [closed]
I'm competitive exam student learning Quantative aptitude what should i choose over solving more questions and skipping the one i can't solve or spending hours on one question till i solve it and then ...
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Looking for web app resources for symbolic Gaussian elimination
I am looking for a web app software that takes step-by-step directions from a student to perform the linear combination operation on a matrix with symbolic coefficients (as opposed to just numbers). ...
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Scepticism as the cornerstone for not making mistakes in arithmetic/algebra etc, especially for students who relentlessly make every possible error
As a maths tutor, some students I have tutored don't just make the odd mistake in arithmetic (including fractions) and algebra: they make every possible mistake and regularly.
My go-to approach for ...
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Limitations of applying the factor theorem
Here are three situations in which students might try to apply the factor theorem.
Proving that $x + 1$ is a factor of the polynomial $x^3 + x + 2$ can be done using the factor theorem by showing ...
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Sources on inequity in precalculus sequence
I'm trying to put together some thoughts on the importance of a strong college precalculus sequence (mainly I'm thinking College Algebra, where much of my experience is) for addressing socioeconomic ...
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What is Algebra 1 and 2 as it is in US highschool education?
I am a pre-university student who wants to help students with Algebra 1 and 2 in high school. I am curious to how the curriculum was built and what the goal of teaching both algebra 1 and 2 might be. ...
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Any online resources explaining why rearrangement of terms occurs in a particular order
Does anyone know of links to resources to explain why basic algebra rearrangement operations take place in a certain order?
A simple, seemingly absurd example, but not uncommon follows.
Say the ...
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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 $\iff$? Because that is what lets 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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