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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solutions to Hung-Hsi Wu's Teaching School Mathematics: Prealgebra/Algebra textbooks?
I am currently home-schooling my kids and picking up the math subjects that have long been neglected by me.
I have been looking for solutions to Hung-Hsi Wu's <Teaching School Mathematics: ...
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Best way to explain the thinking steps from x² = 9 to x=±3
I'm writing four young adult novels with math at the center. The first one is about complex numbers. I'd like to know what good teachers think of the implied pedagogy in the following conversation ...
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What skills do algebra teachers wish their students had mastered before taking algebra?
I am designing 10 hour weekly summer math learning opportunities for students taking algebra next school year. I would like to know from algebra teachers what skills they wish their students had ...
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Usage of cubic equation Vieta's formula
I explained to my kid Vieta's formula of quadratic equation and showed that it can be used to solve quadratic equation (from $x_1+x_2$ and $x_1x_2$ to get $x_1-x_2$) and cubic equation (Cardano's ...
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Math textbook for secondary school using Logo like language
What math textbooks for kids do you know that use Logo or similar languages with visual robots like Turtle (in "The Turtle Geometry") that demonstrate space motions, transformations of all ...
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Math textbook for secondary school using Logo like syntax
What math textbooks for kids do you know that use Logo or similar languages with visual robots like Turtle (in "The Turtle Geometry") that demonstrate space motions, transformations of all ...
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why or when do we use the general form of conic sections?
why write a conic in this form?
for any real A, B, C, D, E, and F, a conic may be expressed as
$$Ax^2 + Bxy +Cy^2+Dx+Ey+F=0$$
why wouldn't it be more natural to write the conic in standard form with ...
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When can students understand the intersection of two circles?
I'm interested in learning two transitions:
(1) When can students reason (intuitively, but accurately) to conclude that two circles in the plane
could intersect in $0$, $1$, or $2$ points, or are ...
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Advice and Remedial Algebra Resources for Students Committed to Calculus
I've got a student in my introductory calculus course. They're failing because they lack algebra skills. They understand the concepts just fine, and can articulate their understanding fine, but get ...
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Confusing variables with units in simple equations
I have noticed that beginning algebra (or pre-algebra) students can easily make the following type of error, confusing variables with units when being introduced to equations for proportional ...
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1st time math teacher 1st test feedback appreciated
I'm not a math teacher but I've stepped into the role to assist a small private school who lost their math teacher during the holidays (I'm a mechanical engineer by trade). The course is Algebra 1 (...
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What are common mistakes that students will make when solving "What's the original price" percentage problems?
Take this question for an example:
A smartphone is now $\$500$ after a $20\%$ discount. What was its original price?
Now, this is an example of a type of math problem that students face usually in ...
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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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