Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

5
votes
4answers
348 views

Real-life exceptions to PEMDAS?

What are some real-life exceptions to the PEMDAS rule? I am looking for examples from "real" mathematical language --- conventions that are held in modern mathematics practice, such as those appearing ...
1
vote
1answer
87 views

Explicit Cross Method

When factoring quadratic expressions $ax^2+bx+c$ it is common to the guess and check factors (AKA the cross method). This would involve factoring $a$ and $c$ and considering particular combinations ...
2
votes
3answers
186 views

Why in the FOIL Method the terms are taken with their signs?

That was the most boring title I could choose but in all honesty, it is what the question is. Here is a reminder of the FOIL method that is used for multiplying two binomials. For example, to multiply ...
5
votes
3answers
211 views

Complex numbers and encourage justification

In remedial algebra, we learn that the graph of $y=(\sqrt x)^2$ is only in the first quadrant. We know this is the correct graph for the equation. This is because we know $y=x$ and $x \ge 0$. However,...
0
votes
1answer
129 views

Integrated math curriculum in different countries

One of the selling points of re-hashed American 1990s high school math programs is that they are "integrated", that is, combine algebra, geometry, statistics, trigonometry just like the European ...
2
votes
3answers
167 views

How to explain to pupils that “$\frac n{100}$ OF $a$” is equivalent to “$a$ TIMES $\frac{n}{100}$”?

How to explain to pupils that "$\frac n{100}$ OF $a$" is equivalent to "$\frac{n}{100}\times a$"? There is some difficulty in explaining that the first sentence, containing "OF" (which could suggest ...
2
votes
1answer
95 views

Classical references on equation solving

If I'm not in error, old style algebra books ( before 1945) concentrated on equation solving, and modern ones concentrate more on functions and their graphs ( as a preparation to calculus). Are there ...
4
votes
6answers
3k views

Where do students learn to solve polynomial equations these days?

When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving ...
1
vote
1answer
108 views

Could this visual explanation of horizontal shift be helpful ? …( if not beautiful…)

With the image below I try to explain in which way substituting (x-a) ( with a> 0) for x in the expression defining a function results in a shift to the right, although " intuition" tells us it ...
5
votes
0answers
256 views

What books properly address the properties of $a^b$?

Many students think $\sqrt{a} \sqrt{b}=\sqrt{a\ b}$ $\sqrt{a^2}=a$ $\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}$ but none of the above are true when (a) and (b) are negative. To avoid such problems, ...
2
votes
0answers
129 views

According to Nathan Jacobson, what is Intermediate Algebra and Advanced Algebra?

Nathan Jacobson's Basic Algebra I, II covers many topics in Algebra that is probably even beyond many pure mathematics full professor's scope of knowledge, unless the professor is specialised in ...
8
votes
3answers
2k views

Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?

For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the Abel–Ruffini theorem (there is no algebraic solution to general polynomial equations ...
4
votes
0answers
210 views

“Indicated Arithmetic” or “Delayed Evaluation”

In the recent past, I've come across a pedagogical strategy for teaching/learning algebra that is sometimes called "Indicated Arithmetic" or "Delayed Evaluation". However, I've been unable to find any ...
5
votes
0answers
202 views

Teaching methods to make a weak student good at math? (particularly student from social science background)

I am currently teaching a high-school student, 1st grade Social Science. He is weak in mathematics. My initial strategy was to explain basic concept but with high repetitions, so that he can have a ...
19
votes
2answers
2k views

Algebra 2 textbooks that incorrectly claim that all solutions of polynomial equations can be found

Over the years I have occasionally encountered a number of Algebra 2 textbooks that make an incorrect (or at very least extremely misleading) claim along the lines that "all solutions of a polynomial ...
4
votes
4answers
395 views

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 ...
2
votes
3answers
317 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 ...
9
votes
5answers
459 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 ...
1
vote
3answers
309 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
votes
2answers
2k 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 $y = f(x) \pm C$,...
6
votes
2answers
239 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"...
4
votes
2answers
179 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 ...
3
votes
4answers
171 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&&...
7
votes
3answers
251 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 ...
5
votes
3answers
346 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 $7$." $-\...
3
votes
6answers
207 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 ...
10
votes
2answers
277 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 ...
5
votes
2answers
249 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 $- ...
4
votes
2answers
108 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
votes
4answers
203 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 ...
8
votes
4answers
254 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 $x$"), ...
8
votes
3answers
175 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 ...
18
votes
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
votes
10answers
8k views

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-...
10
votes
3answers
313 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
votes
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 $x = -18/12$. ...
2
votes
4answers
201 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 (...
6
votes
3answers
488 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! ...
4
votes
2answers
268 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
votes
3answers
158 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: ...
5
votes
4answers
324 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. ...
14
votes
3answers
477 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 ...
8
votes
3answers
246 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 ...
-1
votes
3answers
264 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
votes
9answers
366 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 ...
-1
votes
2answers
179 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 ...
12
votes
6answers
463 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 ...
11
votes
2answers
249 views

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 ...
5
votes
3answers
167 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 ...
7
votes
2answers
147 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,...