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

Italladdsup

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

Italladdsup

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

Italladdsup

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

Italladdsup

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

It All Adds Up- Summer Review

Italladdsup

Hi! As it so happens, as of last week, I have presented information from every one of my math topics last year. The timing worked out extremely well, so until we begin a new unit this week in Algebra, I decided to do a review on some of the math topics I have discussed this past summer. Since answers were already published to these topics on their original problems, I won’t be releasing the answers to these- it’s up to you to figure them out!


Multi-Step Equations

Multi-Step equations are the center of several topics that I discussed, whether it be inequalities, finding the value of a variable, or finding the value of a shaded area. This problem is simple: just find the value of x.

24x +96.8 = 2x – 4.2

The answer will be a decimal!


Inequalities

Like I mentioned earlier, inequalities are very similar to multi-step equations, just with a < or > as opposed to a =. Can you solve this one? Pay attention to the direction of the <>!

-64.5 + 7x > -3x + 129


Graphing Inequalities

Which would the line be facing? Would the bubble be filled in or empty?

98 > -32


Angles

What do you call two angles that have a sum of 180 degrees?

90 degrees?


Volume of a Cylinder

What is the formula for finding the volume of a cylinder?

Find the volume of a cylinder that has a radius of 54 cm and a height of 6 cm.


Percentage

Find the percent for this word problem using the percent equation.

John got back his test, and he isn’t sure whether he got a passing grade. A passing grade counts as a 70%. His teacher told him that he got 64 out of 96 problems correct. Did he pass his test?


Ratios and Rates

How can you tell whether a statement is a ratio or a rate?

Is the statement 72 songs : hour a ratio or a rate?


Thanks so much for following It All Adds Up this summer! Next Sunday I’ll have new information from our first Algebra unit. Have a great week!

Gabe

It All Adds Up- August 21, 2016

Italladdsup

Hi there! I am back from vacation with a new installment of It All Adds Up! Today, I would like to discuss the topic of absolute value, and the idea that a number and it’s reciprocal are always the same distance away from zero on a number line.

Let’s take a standard number line, like this one:

NumberLine.jpg

We notice that are two copies of the numbers 1, 2, and 3, but one copy has a negative sign in front of it. For example, 3 and -3 are the same number, but are on different sides of the  number line. There is, of course, a reason for this. -3 and 3 are opposites, as they are on opposite sides of the number line. In mathematical terms, -3 is the reciprocal of 3, and 3 is the reciprocal of -3. We can tell that, while -3 and 3 are on opposite sides of the equation, they are both an equal distance away from zero.

Another idea that goes along with this is absolute value. For absolute value, we have to remember this: THE ABSOLUTE VALUE OF A NUMBER IS ALWAYS POSITIVE! 

If you were given a problem such as this, how would you solve it? Hint- || is the symbol for absolute value.

|-7|+4=y

This would be a very simple problem in the first place, and it makes little difference when the absolute value symbol is added in. For a number that is negative, we simply change it into it’s reciprocal on the number line, which will be positive. For a number that is positive, the number stays positive. After we figure out the absolute value of -7, we end up with this problem:

7+4=y

The problem now becomes very easy to solve. Do you understand the concept of absolute value? If so, try these problems. Check your answers on Saturday, August 27.

-21 + x = |-72|

|7| + |93| + |6.9| = y

|-7| – |-93| – |6.9| = y

Thanks for reading! Please be sure to check out my reading response to The Boys in the Boat later today!

Gabe

It All Adds Up- July 31, 2016

Italladdsup

Hi there! For this It All Adds Up math lesson, I wanted to discuss how to solve and graph inequalities. They are extremely similar to solving multi-step equations.

Let’s start with this inequality:

3a + 8 > 14 + a

We solve this as we would a normal multi-step equations. We subtract the 8 from both sides and subtract one a from each side. That leaves us with this inequality:

2a > 6

We then divide both sides by two to find out that a is greater than 3.

With inequalities, though, most are not that simple, especially when dealing with negative numbers. Let’s take the second inequality from above, and say that we changed it to this:

-2a > 6

We would solve this the same way, but this time dividing by negative two. But when multiplying or dividing by a negative number, we must then change the direction of the sign. The solution to that particular inequality would therefore be this:

a < -3

In a way, this makes sense, since we changed the 3 to a negative number.

This would work out the same way if the sign was a greater than or equal to sign, and the greater than sign would simply change directions. The only thing different about a greater/lesser than or equal to symbol is that it effects how we graph the inequality.

We graph inequalities using a line bar, such as this one:

Inequalities.jpg

We graph an inequality to find out which numbers could be a solution to the inequality. If the inequality is less than the variable, the line moves left. If the inequality is greater than the variable, the line moves right.

When we are using regular greater than or lesser than symbols, we use an open bubble. When we are using greater/lesser than or equal to symbols, we use a closed bubble.

Here is a solved inequality and its corresponding graph:

a > -3

Inequalities.jpg

This shows that any number beyond the bubble is a possible solution to finding the value of a. We would simply fill in the bubble if the inequality was using a greater than or equal to symbol, which means that -3 could also be the value of a.

Now that you know how to solve and graph inequalities, try solving and graphing this one, and check your answers on August 6th.

4a -5 ≥ -5a + 76

There will not be an It All Adds Up post next week as I will be on vacation, but when it returns on August 14, I will be discussing how to factor out an equation.

Thanks so much for reading!

Gabe

It All Adds Up- July 24, 2016

Italladdsup

Hi there! For this week’s math lesson, I wanted to discuss how to find the surface area of a rectangular prism. As with any algebraic problem, we always start with a formula, and for surface area of a rectangular prism, we use this one:

SA = 2lw + 2lh +2wh

We keep in mind that SA stands for Surface Area, l stands for length, h stands for height, and w stands for width. This formula may look complicated, but if you think about it, all you need to find is the length, width, and height and it is relatively easy to find the surface area.

While we are working with a rectangular prism, the idea is really to find the surface area of this figure:

Now that we know the formula, we can try it out on an actual problem, like this one:

SurfaceAreaDiagram.jpg

When we use the formula, we multiply length x width, length x height, and width x height and come up with a surface area of 1710 inches.

Now that you know how to find the surface area of a rectangular prism, test your skills with this problem. Check your answer on July 30, and come back for another question on August 1st.

SurfaceAreaDiagram.jpg

Thanks for reading!

Gabe