# 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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### How should one approach the concept of “plus or minus”, such as in the numerator of the quadratic formula?

The numerator is structured like: $$(-b)\pm\sqrt{b^2- 4ac}.$$ Is it confusing or acceptable to distinguish between the following two things? An idiom; and What is or seems to be a compositionally ...
318 views

### Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?

In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial $$f(x)=a_nx^n+\cdots a_1x+a_0$$ with $a_n\neq 0$ is defined to be $n$. But I ...
202 views

### Why not write “or” inequalities as $a>x>b$? [closed]

This seems like a stupid question . I just don’t understand why the algebra textbooks I see don’t really address this with students. I boy that I am tutoring brought it up and I was slightly ...
128 views

### Symmetry in polar functions - how to explain

In the precalculus curriculum I am teaching (using Stewart's book Precalculus: Mathematics for Calculus, 7th ed.), we do a bit of polar graphing, which includes discussion of symmetry on polar graphs. ...
557 views

### “Table” method for expanding brackets vs “each term in the first bracket gets multiplied by each term in the second bracket”

Hi I've just discovered mathseducators stackexchange. As a maths tutor in the UK, I am irritated that some of my students - particularly GCSE and sometimes below - use the table method for expanding ...
69 views

### Examples for environmental topics in the context of terms or linear inequalities

I want to emphasize the aspect of environmental education in my math class. Now I'm reasoning whether to do that with linear inequalities or terms with two variables - these are our next topics. The ...
430 views

### Does anyone use the cubic formula these days?

I am writing a story for young people about the history of the development of the cubic formula and complex numbers, partly because it has so much drama and partly because it's amusing that complex ...
157 views

### High school maths textbook for talented students

I am looking for a math textbook. I'm 15 and I'd like to complete algebra 2 geometry and perhaps something about probability/ number theory or trigonometry would be nice too. Later I wanna do ...
2k views

### A PEMDAS issue request for explanation

This question made the rounds recently - $8÷2(2+2)=?$ Now, I glanced at this, answered "1" and then saw the full article printed in the New York Times, The Math Equation That Tried to Stump the ...
191 views

### When (and why) did geometric means of more than two numbers exit the secondary curriculum?

In contemporary US secondary mathematics textbooks, geometric means occasionally make a brief appearance. For example: In Geometry, students learn that when an altitude is dropped to the hypotenuse ...
423 views

### Different Kinds of Variables

Students sometimes ask whether the $x$ in the expression $$2x$$ the same kind of thing as the $x$ in the equation $$2x = 4.$$ In the expression $2x, \;x$ can be any real value. However, in the ...
550 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 ...
103 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 ...
219 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 ...
227 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,...
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### 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 ...
187 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 ...
100 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 ...
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 ...
116 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 ...
274 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, ...
157 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 ...
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### 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 ...
229 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 ...
216 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 ...
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### 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 ...
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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 ...
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### 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 ...
705 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 ...
590 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 ...
3k views

### Good real-life examples of transformations of function graphs

I am a graduate student teaching college algebra at a larger state school, and currently I'm covering transformations of graphs of function, i.e.: Given the graph of a function $y =f(x)$, what do ...
357 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"...
198 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 ...
178 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&&...
279 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 ...
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### 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 ...
223 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 ...
289 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 $x$"), ...
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### 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 ...
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 ...
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-...
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### 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 ...
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### 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 $x = -18/12$. ...
238 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 (...
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! ...