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 visualizing 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 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, might find very useful review and some new material here (approaches to visualization, symbolic algebra, …). But if they do not need review then 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:
|
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 mathematical structures 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 \(a^0\), \(b^{-3}\), \(x^\frac{1}{2}\).
A handy table for once we have understood them:
In class I give a lengthy explanation of why we introduce these messy exponents:
Write down the definition of powers, as we learned them in 6th grader or so, as a typographical definition (this is my own use of the word, losely based on Douglas Hofstadter’s “Typographical Number Theory”): you simply define \(3^4\) as “write down the number 3 four times, and put multiplication signs in between”:
\[3 \times 3 \times 3 \times 3\]Emphasize with lots of gesticulation that this only defines \(b^n\) if n is a positive integer!!!!
Then play around with those rules of adding exponents and show examples. (The students usually need to be reminded of this.)
Then show how you can subtract exponents and get the division of powers of the same base, but emphasize with great emphasis that this only works if the first exponent is strictly greater than the second one!!!! otherwise it doesn’t mean anything.
\[\frac{b^n}{b^k} = b^{n-k} \;\;\; \textrm{ONLY IF} \;\;\; n > k\]Now we imagine what would happen if k could be bigger than n, and we ask “if we extend the definition of subtracting exponents to allow a negative power, do we get a consistent arithmetic? Since we do, we decide to use the notation convention of
\[{b^{-k}} = \frac{1}{b^k}\]We note that it does not follow from the definition - it’s a notation convention.
We then do similar reasoning for \(a^0\), \(x^{1/2}\) and \(x^{m/n}\)
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 crucial to review what these look like: it is taught in schools, but somehow most students miss it and reach high school not knowing these expresions. You can use our glossary to remember what monomials and binomials are. Let us look at these expressions:
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 factoring
polynomials using those identities. To do so it’s useful to write
them flipping the left and right hand side of the = sign:
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.
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:
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. Reviewing fractions – what, seriously??
This is another topic where student preparation is uneven, and it is a good chance to show some tricks.
3.2.5.1. Simplifying fractions
Start with talking through simplifying these fractions:
\(\frac{36}{3}\)
\(\frac{128}{1024}\)
\(\frac{512}{384}\)
A useful tool to have handy is the “prime factorization”. You can simply do a web search for “prime factors 384” and you will get \(2^7 \times 3\)
If you are running linux you can go faster with the command line
program factor:
$ factor 512
2 2 2 2 2 2 2 2 2
$ factor 384
2 2 2 2 2 2 2 3
This tells us that:
These kinds of simplifications are very useful, and they become even more important when we multiply fractions.
Important mantra vis-a-vis fractions:
Note
Multiplying and dividing fractions is easy. Adding and subtracting fractions is annoying.
3.2.5.2. Multiplying (and dividing) fractions
Multiplying fractions involves just multiplying the top and bottom:
This is easy! Just what you would dream of. (With plus and minus our dream will be crushed…)
So you can do simple things like:
But sometimes the numbers get pretty big, and this is where you want to simplify before you multiply. Example:
Let us do together:
\(\frac{12}{5} \times \frac{20}{27}\)
\(\frac{1}{3} \times \frac{18}{19}\)
\(\frac{49}{3} \times \frac{12}{21}\)
Dividing fractions is just one step from multiplying them. You flip the second (which is called taking the reciprocal) and then multiply them:
Note that I will be trying to move students away from ever using the \(\div\) symbol, in favor of always using fractions. So you might end up with this kind of situation:
3.2.5.3. Adding (and subtracting) fractions
Adding (and subtracting) fractions is annoying. The famous “freshman’s dream” is incorrect:
and yet many kids write it down thinking it must be true, because life should occasionally give us something that simple.
Instead you have to do the procedure we learned way back of making a common denominator.
It’s worth the effort of doing a couple of examples of common denominators with the students, and then quickly moving on.
3.2.6. 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.
3.2.7. 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.
As we do each of these exercises we also plot them in our online graphing calculator.
Motivated students could also go on to the “completing the square” section, but in the working group I would recommend skipping it and moving on.
3.2.8. Some heavy emphasis on how functions are constraints
The crucial nexus
There is a crucial point that everyone has to get used to, so we will bring out our graphing calculator and keep talking about it until either everyone falls asleep, or everyone gets it: what does it mean to translate the function to a plot?
We will use the words satisfy and constrain quite a bit, and we will keep using them until people get comfortable.
The phrases we use to drive home how a function defines the set of \((x, y)\) points will be phrases like these:
“When I write \(y = 1.2 x - 1\), I am defining the collection of all points \((x, y)\) for which that equal sign is true”
“When I write \(y = 1.2 x - 1\), I am defining the collection of all points \((x, y)\) that satisfy that relationship.”
“When I write \(y = 1.2 x - 1\), I am defining the collection of all points \((x, y)\) that satisfy the constraint of that equation.
“When I write \(y = 1.2 x - 1\), I am restricting the possible \((x, y)\) quite dramatically. Before I wrote the function, the entire 2-dimensional plane was ‘fair game’, but now only a single line is part of that.”
“When I write \(y = f(x)\), I have reduced the dimensionality of the space from the 2-dimensional Euclidean plane to a 1-dimensional curve that lives in that plane.”
Let’s draw a lot of these with our online graphing calculator, and talk about dimensionality.
Finally, let us talk about what it means to satisfy two equations at the same time:
Put these two equations in your graphing program:
Then subtract a bit from the second one to have something like \(y = 2 x + 1 - 2\).
Here we talk about how having two constraints reduces the dimension of the space from 2 to 0 (a point has dimension 0; a discrete collection of points also has dimension 0).
Here we can gab a bit about a detective drama where the pool of suspects is cut dramatically when you talk about the color of their eyes, or if they are left-handed, …
We will talk more about the dimensions of spaces and subspaces when we solve more complex equations and systems of equations.
3.2.9. 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:
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.
Look at some simple exercises that can be solved either by inspection or with a small amount of calculation.
Remember the two often-used approaches: substitution and gaussian elimination.
With this in mind, let us do example 1 through TRY IT #5.