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

Continue reading “Writing Conjectures, Using Reasoning, & More | It All Adds Up”

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!

Continue reading “Radicals, Lines, Planes, & More | It All Adds Up”

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.

Continue reading “It All Adds Up: Polynomials | Part 2”

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”

It All Adds Up: Arithmetic & Geometric Sequences

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In algebra, you can describe the constant increase or decrease of numbers in a pattern in two ways: as an arithmetic sequence, and as a geometric sequence.

Arithmetic

In an arithmetic sequence, there is a pattern of numbers that are either increasing or decreasing through addition or subtraction by a common difference. Here’s an example of one:

56, 54, 52, 50, 48

As you can see, the numbers decrease through subtraction by a common difference of 2. The common difference is the value that an arithmetic sequence increases or decreases by.

Get it? If you don’t, here is another example, this time with addition:

4, 8, 12, 16, 20

This particular sequence increases with a common difference of 4.

Press “Continue Reading” for geometric sequences and nth roots! (You’ll see!)

Continue reading “It All Adds Up: Arithmetic & Geometric Sequences”

It All Adds Up- Equation Forms

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Gosh, where to start?

Welcome back to It All Adds Up! I’ve got quite a lot to cover… but i’m already busy enough with math! I am taking my very first midterm EVER- in fact, I have another It All Adds Up post scheduled for that date (which will most certainly be changed).

I wasn’t sure where to go with this episode, so I thought I would go simple but complicated. Hopefully, you already know about writing an equation in slope-intercept form, such as this example:

y=mx+b

y value = slope * x value + y-intercept

Did I ever discuss how to find a y-intercept? Or an X-INTERCEPT? Probably not. What if i told you that there are also other ways to write an equation like this?

Let’s stick with slope-intercept form for a bit. We know how to find slope, and we know that the x value is how many times the slope will be multiplied by. For example, if ice cream cones cost $3.23 each, the mx would be y=3.23x. If he wanted to buy 2 ice cream cones, we would substitue in “2” for x and then we would be able to find the value of y. Algebra!

What about that ‘b’? That stands for the y-intercept. A y-intercept is the point on the y-axis where the line touches. This would be best represented by a rental company that charges a base charge in addition to a cost per __. The base charge is basically the amount with an x value of zero. Keep in mind that in order for an equation to be proportional, it must have a y-intercept of zero (starts at the origin [0,0]).

How do we find a y-intercept? All we have to do it substitute zero in for x in the equation and then solve for y. An x-intercept is the same thing, except that it is the value of x when y is zero, or when the line crosses the x-axis. An x-intercept can be found by substituting in zero for y and solving from there.

I’ve got plenty more to discuss, so click “Read More”!

Continue reading “It All Adds Up- Equation Forms”

It All Adds Up: Absolute Value Equations

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Wow! It feels like it has been forever since I last wrote an It All Adds Up post! Luckily, we aren’t too far into the school year, but I’ve got lots of material to cover.

Today, I wanted to expand the concept of Absolute Value. I believe I discussed this in 7th grade, but the principle you always want to remember is: ABSOLUTE VALUE IS ALWAYS POSITIVE! Even if you had |-7| (The absolute value of -7), the answer would be 7. If we had |7|, the answer would still be 7.

However, I want to bring this concept one step further: absolute value equations. What if you had an absolute value inside of an equation? An absolute value with a variable inside of it? Sounds confusing. It is, at first. There is plenty to cover on this topic.

Let’s take this equation:

|x+3| = 7

So how do we solve this? We have to create two separate equations. First, we remove the absolute value sign (for now) and keep the side opposite to the absolute value positive on one side and the same for the second equation but negative. These would be the two equations:

x+3 = 7        x+3 = -7

And so we solve both of them like regular one-step equations and we get the two possible solutions that could equal x. HOWEVER: sometimes we have what is called an extraneous solution. Remember, first, that an absolute value cannot be negative. An extraneous solution is a solution that does not satisfy the original equation.

On another note, we can also have an equation that has no solution. Take this one:

|9x| = -81

We would break both down into the two equations, and we would get the answers: -9 and 9. But how can it be either? An absolute value can not equal a negative number! This would be considered ‘no solution’.

What is we also had a number on the outside of the absolute value as well? Let’s take this equation:

4|4x|=48

Well, we treat the absolute value bars as if they were attached to the number (which they are). So, before we can begin working with the absolute value, we have to perform the normal operation when we would normally be isolating the variable (in this case, we want to isolate the absolute value) For the equation above, we would first divide both sides by 4 and then begin working with |4x|.

What if… we had an absolute value on BOTH sides of the equation? We would first remove the absolute value bars, then make the right side negative for one equation and positive for the second equation. However, for this one, we have to make the ENTIRE right side negative. However, ‘negative’ doesn’t really describe it right: we have to take the inverse of every integer on that side. For example, if we had the equation |5-x| = |-8+x|, our two equations would be these:

5-x = -8 +x       5-x = 8-x

The entire side is not negative: it is just the inverse of the original one (the inverse of -8 is 8). We would solve both equations like the ones with absolute value on one side- but we have to isolate the variable by combing like terms on both sides. Be sure to ALWAYS check for extraneous solutions!

What if there was only a variable inside the absolute value? We solve it the same way! We make sure that the variable is positive, and if there is no number outside of the absolute value, the variable is already isolated!

What if… we dealt with absolute value in inequalities? We would solve them exactly the same way. Just remember: when we divide or multiply both sides by a negative, we must flip the direction of the greater than/lesser than symbol.

Sorry about the great number of ‘what if’s in this post- but there are so many different possible ways to write and solve an absolute value equation- and its best that you know most of them!

Finally, here is this week’s math problem for you to solve. When you have finished, check your answers here.

Solve this absolute value equation:

5|7-x| =  |5x + 5|

Hint: when dividing both sides by a number, divide the numbers inside of the absolute value equation, but keep the bars until you begin solving for x.

Thanks for reading! See you in two weeks for another It All Adds Up post!

Gabe