Proof Concepts: Definitions | It All Adds Up

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Today, we’ll discuss a concept found in proofs of all kinds. Definitions can be used to identify almost any aspect of geometry, and constitute an important step in solving a proof.

But first, some necessary background information:

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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

 

Writing Conjectures, Using Reasoning, & More | It All Adds Up

Screen Shot 2017-08-22 at 10.59.09 AMHappy Friday! Welcome to another edition of It All Adds Up! Once more, my Geometry class had gone further in content in two weeks than I can cover here. Below, you’ll find a bulleted recap of what we learned in Geometry this week. In red are the topics I will cover here, and in black are additional topics we’ve learned that you may want to research on your own:

  • Writing Conjectures using inductive and deductive reasoning
  • Writing the inverse, converse, and contrapositive of conditional statements
  • Structure of a conditional statement
  • Laws of Detachment and Syllogism
  • An introduction to everyday postulates and theorems that deal with geometry

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It All Adds Up Sample Problems: Radicals, Lines, Planes, & More

Hi there!

I meant to include this Quizlet full of practice problems in last week’s edition of It All Adds Up, but I left it out. Now, though, if you’d like to try out some extra problems involving radicals and vocabulary, click on the link below to access the accompanying study set to last week’s edition that I released today. In the future, accompanying Study Sets will release within the main post.

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Have a great long weekend!

Gabe

Radicals, Lines, Planes, & More | It All Adds Up

Screen Shot 2017-08-22 at 10.59.09 AMWelcome to the all-new edition of It All Adds Up! I’ve redesigned one of 12 and Beyond’s original series, and made it better and easier to understand!

For more details about what you can expect this Fall, check out the Global Interests page at the top of this post. In this post, I’ll be covering radicals, lines, planes, as well as providing additional opportunities for researchRead on for more!

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The New It All Adds Up | Announcement

ItAllAddsUp Banner 2.png Starting September 29:

This fall, 12 and Beyond’s original math series is back and re-vamped: with a new design, new theme, and new content, It All Adds Up is set to premiere biweekly with:

  • Recaps from geometry, based on that week’s explorations
  • Lessons based on each week’s geometry topic, but more focused and concise
  • Of course, I’ll continue creating graphics as needed to help with your learning!

*Please know- It All Adds Up (and 12 and Beyond as a whole) is not meant to be used as an educational resource.

Look out for the first post on September 29th!

Gabe

It All Adds Up: Polynomials | Part 2

ItalladdsupBEFORE YOU READ THIS POST, PLEASE MAKE SURE YOU HAVE READ PART ONE, WHICH YOU CAN ACCESS HERE.

Hi there! First off, in Part 1, I lied. I said that there would be only 2 parts to the unit on polynomials. Let’s change that to 6-8. 😬

Of course, since this is the last It All Adds Up before my June break, the remaining parts will be added throughout the summer.

When I refer to an exponent during this post, you will see this symbol (^) just before the exponent. For example, two to the second power would be 2^2.

Today, we’ll be focusing solely on multiplying polynomials. To access the lesson, please click CONTINUE READING.

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It All Adds Up: Polynomials | Part 1

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Hi there! I’m splitting this unit into two parts, over this episode and the next, as this unit is very long and gets complicated quick. Notice how the word polynomial has a prefix of poly. We’ll get into what that means later, but let’s start with what a monomial is.

A monomial can be thought of as just about any number. By definition, a monomial is a number, a variable, or a combination of the two. The following are all examples of monomials:

7

x

7x

A binomial is referring to an expression that contains two monomials, such as this:

7x + 5

Keep in mind that a binomial can’t have an like terms, like these:

7x + 9x

This is because we could combine these terms to make 16x, which would be just a binomial.

A trinomial is referring to an expression that contains three monomials, such as this:

7x^2 + 9x + 2

With a trinomial, we will always have at least one monomial with an exponent (in this post, I’ll prelude to an exponent with this symbol: ^). There could be two or three of them, of course, as long as the values of the exponents are different. Here are a few examples:

9x^2 + 6x^8 + 4

34x + 64x^2 -79

80,000x^1/2 + 98 + 23x

Don’t worry about the last one- you won’t need to know how to solve it yet.

Press CONTINUE READING for more of this post!

Continue reading “It All Adds Up: Polynomials | Part 1”