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.



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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