Foundations of Algebra II

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Welcome to the new season of 12 and Beyond’s math-based series, It All Adds Up! As I embark on my sophomore year of high school, I’ve begun taking Algebra II. Already, in comparison to Geometry, I feel the course is much harder, and the pace is insane- just how I like it.

So far, we’ve jumped all over the place in terms of topics, since we essentially covered 2 1/2 chapters of algebra review in a week. Thus, I’m going to pick and choose a few interesting topics to discuss today: Absolute Value Equations and Interval Notation. 

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It All Adds Up: A Conclusion of Geometry

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It has been quite the year taking Geometry, but alas, with the new school year ahead, my lessons regarding such must come to an end. This will be the last It All Adds Up Geometry edition, and beginning with a monthly schedule this Fall, I’ll be moving on to Algebra II.

Thus, I must conclude this season with where we began: proofs. Proofs, as a refresher, are step-by-step ways to prove a mathematic hypothesis true. We’ve dealt with proofs at their basics, like in Algebraic Proofs | It All Adds Up, and then applied what we learned in geometry to proofs, like with definitions, triangles, and segment geometry.

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Finding Area of Polygons Using Trigonometry

New It All Adds Up LogoWe covered a lot in Geometry this year, and overall, I found trigonometry to be one of the easiest units. Once you’re familiar with the mechanics, it all comes down to calculation. However, its implementation can be a bit more complicated, like in finding the surface area of a 2D shape with more than four sides, but the process is actually pretty cool.

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Geometry of Circles

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For me, in Geometry this year, of all topics, circles were the one that I seemed to grasp most. Something about them just made sense, perhaps that every concept involving circles seems to tie back to a central point (pun intended).

Thinking about a circle on a simple level, a circle is made up of 360 degrees (which made understanding arcs a piece of cake), its circumference is 2πr, the area is πr^2, and every other aspect essentially builds upon these ideas. Given these, you can determine the values of interior angles built into the circle, find areas of inscribed/circumscribed shapes, and solve for almost any related missing aspect. We’ll try and get into a few of these today.

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It All Adds Up: Goals For The Future

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Welcome to the 50th edition of It All Adds Up, 12 and Beyond’s dedicated math series.

Over the past three years, It All Adds Up has been ever-changing, yet has remained at the heart of my mission here at 12 and Beyond. After all, the origins of 12 and Beyond find their roots in education.

In all this time, I’ve shared mini-lessons related to countless aspects of both Algebra and Geometry, and I have full intent to continue to do so. But as time passes, and I continue experimenting and improving these lessons, I’ve begun to plan out the near future of It All Adds Up.

The focus of It All Adds Up will always be to provide my readers with useful mathematic skills related to content I’m learning in school. And next year, I’ll be taking Algebra II, opening up many new opportunities for discovery.

As I bring my year of geometry to a close, reflecting upon my lessons, there are some really great things that we’ve accomplished, but also many ways in which I can improve. For one, most of my previous lessons have been very by-the-book unit lessons. For a high-schooler who can’t release posts every day, it became really difficult to give a comprehensive understanding of geometry as I remained focused on more specific topics as opposed to the bigger picture.

In the future, my lessons are going to be significantly more concept based. I’m not a math teacher. I’m a high school student. And the great thing about blogging about my education is that I begin to understand the topics I write about better. But by just feeding you the definitions and theorems found from each individual unit, there is little greater understanding apart from further review that results. Each of you has your own education to attend to- and the individual details of each unit are best fit for that.

By focusing on the broader picture, It All Adds Up can be significantly more beneficial both for me and for your understanding as a reader. It All Adds Up will continue to keep its core mission at heart, but will be presented in a new, fresh way.

I’ll be providing more details about the new year of It All Adds Up later this Summer before the school year begins. For now, It All Adds Up will continue as it has been and will wrap up Geometry over the next four editions that will release in July and August once 12 and Beyond returns in late June.

Thank you so much for your support over the past 50 editions, and I look forward to the bright future of It All Adds Up.

Gabe

VIEW THE ENTIRE COLLECTION OF IT ALL ADDS UP EDITIONS HERE.

Proving Triangle Congruence | It All Adds Up

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Today, I’m going to begin jumping ahead and covering several main portions of Geometry, especially as my in class learning is about 7 chapters ahead. As I’ve always said about proofs, there are ‘infinite’ methods that can be used to achieve a desired solution. What I find most useful is proving triangle congruence. Once you prove triangles congruent, you can prove so many different relationships, like congruent segments, angles, and even aspects like corresponding arcs when we begin talking about circles. No matter what, this is a tactic you’ll want to know.

Proving two triangles congruent is really quite simple, and there are five separate methods that you can use depending on the information you are given.

Let’s get started.

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Proving Segment Congruence | It All Adds Up

New It All Adds Up LogoFor necessary background information on this lesson, please click here.

Last time, we reviewed how to write a proof, and starting today, I’ll be sharing with you different purposes of writing proofs through basic geometry.

In Geometry class itself, we are about 3 chapters ahead of this, but given the importance of every postulate and theorem when writing proofs doesn’t allow for me to skip any content.

First off, let’s review what a postulate and a theorem are. You’ll see these terms many times over the coming months:

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Algebraic Proofs | It All Adds Up

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Hi there! Sorry for the delay on this post… High School has kept me very busy.

For the rest of the year, most of our Geometry units will center around writing proofs. A proof is a logical way of proving that a statement is true, and it consists of statements and reasons that logically follow each other. Today, I’m going to teach you about writing algebraic proofs, which involve solving for a variable. Most of the year, though, we will focus on writing geometric proofs, which involve lines, angles, etc. With this post, you’ll get a general idea for what a proof is. However, geometric proofs are something that really have to be taught in a class, or be taught over the course of several days. I can’t really cover that here, and that puts It All Adds Up at a dilemma.

Here’s the plan: in order for you to understand future editions, you will need to research geometric proofs on your own. Learn what they are, how they work, and how you form a proof (since there are so many possibilities for statements and reasons). For It All Adds Up, I’ll be going over the aspects that can be reasons or statements, along with some tips for writing proofs with them.

For today, though, we’ll start simple. We already know how to solve for a variable, such as in x + 68 = 20. Now, each step that we would use to solve that becomes a statement.

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Take the 4 step proof above. Each step, from top to bottom, shows how we solve for the value of x. We know that x does equal 38, but how would we prove it? For algebraic proofs, there are several properties that you’ll need to know.

Properties of Equality (For Algebraic Proofs)

Addition POE: When the same value is added to both sides of an equation

Subtraction POE: When the same value is subtracted from both sides of an equation

Division POE: When both sides of an equation are divided by the same value

Multiplication POE: When both sides of an equation are multiplied by the same value

Other Properties (For Algebraic Proofs)

Distributive Property: When a number outside of a parenthesis is distributed to each value within the parenthesis

Now, there are many other properties, postulates, and theorems to come, but for this week, I’ll be focusing on simply solving for a variable (for the purpose of getting you used to proofs) and thus, the properties stated above will be sufficient. I’ll be presenting you with numerous others in future editions.

So let’s fill in the proof.

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Here’s another aspect I want to focus on: the given. The given is the starting point of any logical process, and will always be the first step in the proof. It’s what you are, well, given. For example, we’d be given the first equation and then asked to solve for x. As for the other properties, you always want to ask yourself: How did I get from this step to the next step? Your answer will vary- it might be adding 72 to both sides, it might be dividing both sides by three.

Usually, when asked to write a proof, you’ll also be provided with a solution. (This isn’t the case with algebraic proofs, though). That will be your last step. For algebraic proofs, your last statement will be ____ = ____, but with geometric proofs, it can vary. You might have to prove two angles are supplementary, prove two segments are congruent, or two lines are parallel.

I’m learning right along with you- and believe me, it takes me a while to process everything too. But overall, this is a proof. And everything you’ll learn this year from here on out will build on this format. Once you know what a geometric proof can be, I’ll supply you with possible statements and reasons, as well as specific rules and patterns that apply. Do remember though: a proof can have many different solutions.

Thanks so much for reading. Have a great long weekend!

Gabe