3. Visualizing algebra: motivation and review of prerequisites

3.1. Motivation and plan

Purpose:

The goal for this working group on visualizing algebra is to build:

Agility and wide experience in visualizaing and solving equations.

In-sequence:

This fits in the broad sequence of skills that you might call “practical math for research”.

Computer use:

We will use plotting programs and plotting web sites to draw functions and data. But:

Programming:

Almost no programming. Some explorations can include canned python programs which you can execute without knowing more than a smattering of the language.

Examples from:

Social science, news and current events, science, sports, …

Auxiliary textbooks:

OpenSTAX textbook on Algebra and Trigonometry [Abramson, 2021].

Other interesting books to accompany this material:

“No Bullshit Guide to Math and Physics” by Ivan Savov [Savov, 2014].

Who should join?

This does not fit in school curriculum boundaries, but if I had to narrow it: the material is aimed at students who are taking the US courses Algebra 1, Geometry, and Algebra 2. It supplements and makes real what students will learn in those courses, and might give them the feeling of excitement that the courses sometimes do not provide.

Users taking more advanced courses, like the US pre-caluculus and calculus, will find some new material here (approaches to visualization, symbolic algebra, …), but most of it will be old-hat and maybe not the best use of their time.

Those more advanced students might be interested in the other math working group we have: “Math for Research”, discussed at this link: Math for research: motivation and review of prerequisites.

Are there other things we get out of this?

A “working group” is very different from a classroom setting: you can ask conceptual questions, advanced questions, super-basic questions, questions that revisit the things you learned way back, and have a discussion with the instructor and other students. Semi-formal conversations with an experienced mathematician can be a new and useful experience.

Now let us start by reviewing some of the prerequisite materials, before we start with straight lines and then move on to polynomials.

Information on signing up for the next working group should be in the chapter Logistics for specific courses.

3.2. Review of prerequisites

In this section we will look at some basic material which we have seen before, but probably not in the context of producing visualizations and other practical results that we want to see.

We plan to make quick work of these topics, but we will still treat them thoughtfully. For each topic we will:

look at the definition, explanation, and exercises in the textbook do some visualization and application to an area of interest

Note

This might be the most important chapter in the course: it turns out that there is a lot of diversity in what people have studied so far, and everyone appreciates the review. Another important part of the review is that we discuss the reasons for formulas like \(a^0\), \(x^-3\), and \(y^\frac{1}{2}\), including putting it in the context of how we had have had to introduce new types throughout our life (naturals, integers, rationals, reals, complex, …) Some students really resonate with that approach.

3.2.1. The “why” of messy exponents

This will be an interactive discussion of why we have expressions like \(a^0\), \(b^{-3}\), \(x^\frac{1}{2}\).

3.2.2. A nostalgic romp through coordinates and plotting

Quickly read the Linear Equations in One Variable chapter in [Abramson, 2021].

Exercises: 2, 4, 6, 8, 10 in Review Exercises

3.2.3. What are functions?

https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-introduction-to-functions

It is useful to point out that functions have a very general meaning, even though we will mostly work with functions of real numbers. So start by reading:

Quickly read the Functions and Function Notation chapter in the OpenStax algebra book [Abramson, 2021].

and work the examples and “try it” exercises given in the rubrics from “EXAMPLE 5” all the way to “TRY IT #6”.

3.2.4. Special powers of binomials

It is worth reviewing what these look like: it is taught in schools, but somehow many students miss it. You can use our glossary to remember what monomials and binomials are. Let us look at these expressions:

\[\begin{split}(a + b)^2 = a^2 + 2ab + b^2 \\ (a - b)^2 = a^2 - 2ab + b^2 \\ (a + b)(a-b) = a^2 - b^2\end{split}\]

There are many other such basic identities, but these are the ones I always look at.

We will do two things with this: the first is to actually work them out and see why they are correct. Everyone has their way of doing this - I put my two index fingers down and scan each expression until I have multiplied everything out.

The second thing we will do is to get used to :term:`factoring` polynomials using those identities. To do so it’s useful to write them flipping the left and right hand side of the = sign:

\[\begin{split}a^2 + 2ab + b^2 = (a + b)^2 \\ a^2 - 2ab + b^2 = (a - b)^2 \\ a^2 - b^2 = (a + b)(a-b)\end{split}\]

One approach to factoring polynomials is a “visual inspection”, where your brain does pattern-matching on things that “look like a perfect square”. Here are some examples: we will look at them together and discuss how we found the factors.

\[\begin{split}x^2 - 4 = (x + 2) (x - 2) \\ x^2 - 3 = (x + \sqrt{3}) (x - \sqrt{3}) \\ 25 x^2 + 20 x + 4 = (5x + 2) (5x + 2) = (5x + 2)^2 \\ 49 x^2 - 14 x + 1 = (7x - 1) (7x - 1) = (7x - 1)^2\end{split}\]

The first one is the simple recognition that you have a difference of perfect squares. The second one shows that the technique is useful even if the number is not a perfect square.

The third involves recognizing that both the coefficient of \(x^2\) (\(5^2\)) and the constant term 4 (\(2^2\)) are perfect squares, and the coefficient of \(x\) looks like the “double product” \(2 \times 5 \times 2 = 20\).

The fourth again involves recognizing that both the coefficient of \(x^2\) (\(7^2\)) and the constant term 1 (\(1^2\)) are perfect squares, and the coefficient of \(x\) looks like the “negative double product” \(2 \times 7 \times (-1) = -14\).

More discussion, examples, and exercises on factoring polynomials is in the chapter Factoring Polynomials in the OpenStax algebra book [Abramson, 2021], which also discusses more elaborate factoring techniques than this simple visual inspection.

I recommend becoming quite comfortable with factoring polynomials: it comes up a lot in the life of a researcher. The “visual inspection” approach should is one you should always know, while the more algorithmic technique shown in the book is one that you should understand and be ready to look up.

Finally: try factoring these same polynomials using SymPy! You can put these expressions into the SymPy Gamma web calculator:

factor(25* x**2 + 20 *x + 4)
factor(x**2 - 3)
factor(49*x**2 - 14*x + 1)

As a final note on the \((a + b)^2\) expansion, it’s good to remember the classic “Pascal triangle”:

0             1
1            1 1
2           1 2 1
3          1 3 3 1
4         1 4 6 4 1
5       1 5 10 10 5 1

which allows us to automatically write down, with no effort:

\[\begin{split}(a + b)^0 & = 1 \\ (a + b)^1 & = a + b \\ (a + b)^2 & = a^2 + 2ab + b^2 \\ (a + b)^3 & = a^3 + 3a^2b + 3ab^2 + b^3 \\ (a + b)^4 & = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \\ (a + b)^5 & = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 +5ab^4 + b^5\end{split}\]

One fun thing to note is that the sum of the exponents of a and b in each term is always the power of the binomial we are expanding.

For example, \((a+b)^4\) has terms like \(6a^2b^2\) and \(4ab^3\), where 2+2 and 1+3 are both 4.

This relates slightly to ideas in physics: if a and b had physical dimensions (like length or mass) you would get inconsistent expressions if the exponents did not add up to the same amount – you cannot add an area to a volume, for example!

3.2.5. Getting comfortable with visualizing functions

We have two angles here: one is to become comfortable with what various functions look like. The other is to look at how the intersection (simultaneous solution) of functions in higher dimensions gives insight into the solutions to equations.

The functions most of our students will have seen are linear functions and quadratic functions, so we will visualize a few of these.

In geogebra or desmos enter the following straight line functions of x, and discuss the slope and intercepts:

(1/2) x + 3
x - 2
-2 x + 4
-x - 1

Now insert the following quadratic functions:

x^2
x^2 - 4
-x^2
-x^2 - 4
-x^2 - x
-x^2 - x - 1
-x^2 - x + 4

and discuss issues like how many roots there are, and whether some of those roots are complex.

More on this topic, which will not be review for most people, will be in Section 5.

3.2.6. Quadratic equations

See the OpenStax chapter on Quadratic equations

This will give us several examples and “TRY IT” exercises that we can do by inspection. For example, we can do example 1, TRY IT 1, example 2, TRY IT 2, example 3, TRY IT 3.

Motivated students could also go on to the “completing the square” section, but most of the time I would recommend moving on.

3.2.7. Two equations with two variables

This is probably still “review” for most students, so we take a brief look at systems of two linear equations with two unknowns.

We will look at the OpenStax chapter Systems of Linear Equations: Two Variables

Our approach here will be to:

1. Graph some pairs of equations like those in Figure 2 in the OpenStax book and discuss visually when solutions exist and when they do not. We will then make similar graphs in geogebra or desmos for each system we look at.

2. Look at some simple exercises that can be solved either by inspection or with a small amount of calculation.

3. Remember the two often-used approaches: substitution and gaussian elimination.

With this in mind, let us do example 1 through TRY IT #5.