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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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23 votes
6 answers
8k views

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 ...
Thomas McLoughlin's user avatar
4 votes
1 answer
136 views

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 ...
athos's user avatar
  • 777
2 votes
0 answers
57 views

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 ...
paus's user avatar
  • 83
1 vote
1 answer
96 views

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 ...
paus's user avatar
  • 83
6 votes
6 answers
1k views

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 ...
Lenny's user avatar
  • 1,068
4 votes
0 answers
131 views

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 ...
Joseph O'Rourke's user avatar
7 votes
2 answers
126 views

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 ...
Mike Pierce's user avatar
  • 4,845
5 votes
8 answers
2k views

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 ...
3rdMoment's user avatar
  • 151
1 vote
0 answers
129 views

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 (...
Dan S's user avatar
  • 19
6 votes
3 answers
2k views

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 ...
CrSb0001's user avatar
  • 295
7 votes
7 answers
1k views

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 ...
dt688's user avatar
  • 173
11 votes
12 answers
3k views

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 ...
bobble's user avatar
  • 375
9 votes
3 answers
1k views

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 ...
Justin Skycak's user avatar
10 votes
6 answers
1k views

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, ...
Kevin P. Costello's user avatar
3 votes
0 answers
75 views

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 ...
Mike Pierce's user avatar
  • 4,845
11 votes
1 answer
1k views

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. \...
marco trevi's user avatar
2 votes
5 answers
986 views

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 ...
Janaka Rodrigo's user avatar
-1 votes
2 answers
162 views

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 ...
Abhishek's user avatar
4 votes
3 answers
282 views

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). ...
Maesumi's user avatar
  • 1,410
2 votes
2 answers
271 views

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 ...
Adam Rubinson's user avatar
0 votes
1 answer
271 views

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 ...
Janaka Rodrigo's user avatar
2 votes
1 answer
382 views

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 ...
TomKern's user avatar
  • 4,425
4 votes
2 answers
633 views

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. ...
muuzzmolz's user avatar
  • 171
1 vote
2 answers
120 views

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 ...
Clive Long's user avatar
2 votes
3 answers
170 views

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 ...
Janaka Rodrigo's user avatar
8 votes
5 answers
894 views

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 ...
Chris Cunningham's user avatar
30 votes
6 answers
3k views

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 ...
Chris Cunningham's user avatar
7 votes
4 answers
4k views

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 ...
Vivaan Daga's user avatar
7 votes
1 answer
232 views

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 ...
TomKern's user avatar
  • 4,425
3 votes
3 answers
247 views

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 ...
MichaelLink's user avatar
12 votes
10 answers
1k views

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 ...
doetoe's user avatar
  • 237
7 votes
7 answers
1k views

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 ...
Zuriel's user avatar
  • 4,275
0 votes
0 answers
190 views

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) ...
Humberto José Bortolossi's user avatar
0 votes
2 answers
194 views

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 ...
EXL's user avatar
  • 311
2 votes
2 answers
266 views

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 ...
Noob_Guy's user avatar
  • 121
6 votes
3 answers
472 views

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 ...
Mike Pierce's user avatar
  • 4,845
1 vote
2 answers
2k views

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 ...
Zuriel's user avatar
  • 4,275
18 votes
8 answers
2k views

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 ...
Chris Cunningham's user avatar
2 votes
0 answers
111 views

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 ...
Paul's user avatar
  • 121
3 votes
1 answer
760 views

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{\...
blakedylanmusic's user avatar
5 votes
2 answers
265 views

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 ...
pdmclean's user avatar
  • 967
3 votes
2 answers
404 views

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 ...
Wasp's user avatar
  • 187
0 votes
1 answer
105 views

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 ...
Bob's user avatar
  • 1
15 votes
5 answers
5k views

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 ...
James S. Cook's user avatar
3 votes
3 answers
261 views

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 ...
Prime Mover's user avatar
0 votes
0 answers
107 views

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 ...
rubengavidia0x's user avatar
3 votes
2 answers
266 views

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,...
Chaim's user avatar
  • 655
1 vote
3 answers
679 views

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 ...
Rick Does Math's user avatar
3 votes
3 answers
1k views

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 ...
Prime Mover's user avatar
2 votes
1 answer
198 views

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. ...
10understanding's user avatar

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