3Blue1Brown. I've sent a lot of remind messages lately about these videos, but this is absolutely one of my favorite YouTube channels. During the school closures he has been doing a series of videos called "Lockdown math", and it just so happens that a lot of the topics he's covered are ones that we've discussed this year. I really encourage you to watch one or two of these because they offer a great (and usually a bit different) perspective on these topics.
https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
This content finishes out the semester and finishes out the school year. You're amazing. You can do it. You've likely seen most of this before also.
Here is your weekly checklist.
There are a handful of topics in maths that are quintessential to Algebra. I would include in this category factoring, polynomials, and systems of equations. Being able to consistently solve systems of equations problems of different difficulty levels is pretty definitive proof that you have a solid understanding of how to manipulate mathematical statements algebraically. The main idea of systems of equations is introduced to you in your entry-level algebra course, and expanded upon basically until you get done with math. There are extremely complicated systems of equations that are present in subjects like differential equations, and in specific STEM disciplines like engineering (I am thinking ideas like stress and strain in structural analysis), and, perhaps unsurprisingly as the whole idea of math classes in high school is to develop the skills you may need later, the procedures for solving basic systems and far more complicated ones are largely the same. Here's a video introduction to systems of equations:
If you're in my precalculus class there is exactly 0% chance you've never done a system of equations problem before, but technically anyone can view the content on this website, so here's a quick rundown. When we are being asked to solve a system of equations problem, what we are trying to find are values for the variables that will satisfy all of the equations given to us. For example, if we have a linear system of equations with two variables, we are trying to find a value for x and a value for y that will satisfy both equations. In other words, if we plugged those x and y values into either equation, the resulting statement would be true. Graphically, this coordinate point would look like the intersection between the two lines. It is possible for a system of equations to have no solution (2 parallel lines will never intersect) or for their to be infinitely many solutions (two equations that represent the same line). As you scale up either the degree of the variables (x^2) or the number of variables (x, y, z), you get more complex interactions, but there are still going to be three primary ways that you can solve a system of equations problem: Graphing, Substitution, and Elimination. Graphing simply involves graphing each equation by whatever method you prefer and identifying their intersection point(s) as the solution. Substitution involves solving one of the equations for a certain variable, then substituting that in for that variable in another equation. This is actually a surprisingly useful strategy for solving systems of equations with four variables, effectively turning the problem into a 3-variable system. Gaussian Elimination (or simply just elimination), is when you manipulate one equation by multiplying it by a scalar (a number) so that if you were to add the equations together, variable will cancel out. Though the procedure is tedious, this is the most versatile method for solving systems of equations, particularly when you start to explore systems of many variables.
Here are some examples of fairly basic system of equations problems involving two variables, and one that involves 3-variables.
As is often the case in mathematics, we build our understanding of a skill by looking at simple examples and then apply that understanding to more complicated problems that usually manifest as word problems. I don't know if this is the best way to learn math, but in any case when solving word problems there are a couple of additional things to consider. It is important to define variables from the outset so that you know what result you've found after going through the algebra. It is also important to answer a word problem in a manner commensurate with the problem itself. For example if the problem is asking you how many eggs Laura should buy, you don't just say 4, you say, "Laura should buy 4 eggs."
Systems of equations with three variables add a layer of complexity to the procedures we develop for simpler two-variable systems, but no additional skills are needed. If using elimination, the idea is to use the three equations given to generate a two-variable system and then proceed to solve normally. Here is a good video from Khan Academy where Sal explains conceptually what a three variable system is describing and proceeds to solve one.
Lines are not the only things that intersect. There is a possibility of intersection(s) between any two curves on a coordinate plane. Although non-linear systems of equations can definitely get a little more hairy in terms of the algebra skills we need to recruit in order to solve them, the procedure, no matter the method we choose to solve, remains the same. Here are some example of non-linear systems of equations problems. There is a special challenge problem at the end of this video that you should definitely attempt.
If the first thing you think of when you hear the word Matrix is the movies or trilogy of movies about Neo becoming basically god and overcoming sentient robots, I just want to say that I am so extremely proud of you in this moment. If the first thing you think of is an array of data, then you should absolutely take a break right now and go watch all three Matrix movies. Once you've done that, you can begin to explore the world of matrices, which are simply rectangular arrays of things arranged in rows and columns. Usually in high-school level math we deal with matrices of numbers or expressions. Arranging information into matrices can be very useful for myriad applications, including engineering, computer graphics, business and sports analytics. Watch this video on some basics about matrices along with a very low-level look at some of the basic matrix operations.
I may not have convinced you that matrices are a necessary part of the precalculus (or Algebra) curriculum, but hopefully the next video will prove otherwise. Matrices are extremely versatile and there are a lot of properties of matrices and operations we can perform with them that are a bit beyond the scope of this abbreviated precalculus class, but the primary thing we'll be using them for is one of high importance. When we encounter very tedious tasks in math, there is an automatic incentive to develop ways to reduce the tedium. Such is the case with matrices and manipulating them into what is known as reduced row echelon form. Watch this video and be prepare to be shocked and amazed.
Something that is very important to note about augmented matrices is that they must be set up correctly. That is to say, if you have, for example, a system of equations with three variables, each equation needs to be ordered in the same way (for example: x + y + z = 1, where the x term, y term, z term, and answer all "line up"). Reduced Row Echelon Form refers to manipulating a matrix through a set of finite elementary row operations (things we can do to row of matrices) to get them into a form that allows us to solve for variables. There is more to it than this. Usually we end up with something called an identity matrix and the explanation behind why we can just multiply rows by a number is the same reason we can just multiply every term in an equation by the same number without technically changing what it represents, but for the scope of this class, just understand that the rref() function on your calculator is incredibly powerful.
So ends the new/required instruction for your precalculus class. I mentioned in an update letter recently that there are other precalculus topics that we were unable to get to this year for various reasons that have the potential to be very helpful to you as you progress to your next math class (if that class is Calculus or Algebra-based), and I will be providing materials that you can peruse on your own time if you choose to do so. For now, please complete this google forms activity by Friday 5/15 at 11:59PM so that I know you have an understanding of the material presented above.
In the letter I sent I also mentioned a final exam. You will not be asked to solve any math problems for this final exam. Please do not stress out about it even a little bit. You do not need to prepare for it. I will be asking you questions about your year in math and you can probably answer all of them without using a single number.