Happy Monday! Our Math Monday blog assignment for today is to explain what steps we will take in order to graph a line on a coordinate plane after we know its equation. We find this equation, as I explained in the previous Math Monday blog entry, by plugging variables such as the slope and the y-intercept into the algebraic formula y=mx+b.

Welcome back to Math Mondays! Our blog prompt for today is to name some questions we could ask once we know the equation of a line. This ties into our Math curriculum for Chapter 7, which is what we are studying at the moment.

Lines can be translated to equations by using the formula y=mx+b. The formula uses variables as substitutes for vocabulary terms. "Y" and "X" are output and input, respectively. "M" stands for the slope of a line, which can be found by dividing a line's rise (change in y values) by its run (change in x values). "B" stands for the "y" intercept, which is found anywhere on the "y" axis and 0 on the x-axis.

I was a little confused by the blog assignment. What did he mean by "questions we could ask"? I've figured that a few sample questions you might ask after finding what a line's equation is would be "what are the rise and the run?" When you divide them together, you get "M". Our long-term substitute in math explained to us that the slope is usually represented as a ratio or a fraction. Fractions are parts of numbers designed to show division. We get this ratio by finding the change in "y" values from the y-intercept to our "y" coordinate and dividing it by the change in "x" values from the y-intercept to our "x" coordinate. This results in the slope. So, we could ask ourselves what the rise and run, respectively, are by counting the number of changes on our plotlines from our "y"-intercept to our coordinate.

This sounds really confusing, I know. But another question we could ask is "How can the slope be expressed differently?" If a slope is 0 - meaning that there are no changes from "x" to "y" coordinates, we say that the line is horizontal, or flat. There are a lot of terms for the slope. We also call the rise/run division problem the "rate of change" because it tells us how much change occurred when we moved in the coordinate plane. We can find out what the coordinates would be for any equation with the same slope but different variables by making a table and writing down what our input (x), rule (y=mx+b), and output (y) would be. This way it is easy to get a bigger and wider picture of what the slope of the function is. 

Our blog assignment for Math Monday today required research. Our assignment was to find out the cost of a 12-pack of cans of Mountain Dew and the cost of a 2-liter bottle of the same soda. Then we had to explain why which one was the better deal and apply principles about decimals we learned in Math class.

What I found out was that the 12-pack of soda cost about $1.99 and that the 2-liter bottle cost around $1.50. Keep in mind that prices of certain items go up and down frequently due to sales, tax, and whatnot. These were based on average and what Google could tell me since I haven't purchased Mountain Dew in a while.

This is where our math standards come in. Each can of soda in a 12-pack is about 12 oz whereas a 2-liter bottle is about 33.8 oz. We need to divide to find something of a unit rate and then compare our results. You'd find that the 12-pack accumulates to about 144 ounces of soda for almost 2 dollars and the bottle was about 34 ounces for $1.50. The first proportion simplifies to 72 ounces for 1 dollar and and what would be 41.8 or so ounces for 2 dollars -- with a unit rate of just over 20 ounces for 1 dollar, it's clear that the 2-liter bottle is the better buy.

Happy New Year! Today is our first day back from winter vacation, and it is also the first day of the spring semester! As such, our Math Monday blog assignment for today was to reflect on a concept we learned during the first semester (quarters 1 and 2) that we remember most, and give evidence that suggests that we remembered it clearly and properly.

One of the mathematical concepts that I remember learning about most clearly was probably the Pythagorean theorem in quarter 2. My dad is a math teacher and I remember going into his class when I was in the 3rd grade. He was teaching his students about the Pythagorean theorem and I copied it down as a joke. At the time all I had written looked like a scribbled, unintelligible mess, but now I know what the Pythagorean theorem is and especially how to relate it to real-world situations.

At first I was confused about how exponents tied in to the number formula and how it had anything to do with triangles and geometry. I was always told that triangles had 180 degrees and anything that had to do with finding lengths of triangles required that knowledge. Now I know that that is incorrect -- the Pythagorean theorem is used to find the lengths of the sides, not the measures of the angles, in a triangle. Learning about the Pythagorean theorem taught me how to think outside the box as far as triangles go. The Pythagorean theorem is an excellent example of how exponents and square bases can be handy in real life. 

As I mentioned above, the Pythagorean theorem is a good example of how math lessons relate ro real life because until this year, I never saw use for math. Although I don't think it's the most important thing we need to be necessarily set in the real world, a lot of geometrical concepts are useful in adult daily life. This is why I'm excited to learn more about geometry. This learning of the Pythagorean theorem and reflecting about how it relates to geometry and exponents calls back to my sixth grade year, when my teacher explained in essence to her class how the California educational standards work -- concept upon concept until all of our knowledge ties into itself. I like the way we did this during semester one at Computech.

Happy Monday! Today is the last Monday of the quarter and the last Monday of the semester. Our blog assignment for today is to name a math topic we've struggled with in the past and explain how and when we overcame it and finally began to understand it.

In years past, I've had a hard time with multiplying numbers with multiple place values. I got over it in the fifth grade, but for awhile it was seemingly impossible for me to do. 

As I stated above, it started to click for me in the fifth grade when we were doing multiplication and place value, at the beginning of the year as always. It seemed so hard and no matter who I asked -- my friends or the teacher -- it wouldn't make any sense. I'd been taught to multiply numbers with zero as one of the place values, but not with actual digits in more than one. There were places in which I didn't know whether to add or multiply. My teacher said that she'd never had a student who wasn't able to do this at the beginning of the year. I was so embarrassed that I could never learn how to use what I called the "old-fashioned" method.

I looked in the math book. The instructions all seemed like nonsense until I looked at the problem I was doing. It told me to refer back to Example 1 in the lesson article. I did, and there it was! At first the instructions looked like jumbled gray matter to me, but then I saw the step-by-step instructions and it became easy. This helped me learn to take better notes and pay more attention to the content in my textbook.

Nowadays, I use both of these methods (multiple-place multiplication and textbook referral) all the time. I've found that the math textbook often contains valuable information and easy tricks that your teacher might not explain. I've also found that the textbooks we use now are more reliable in information than the company who made my elementary school math books were. My math motto is: "always look back in the textbook". Often there will be footnotes that name a place, usually in the back of the book, where extra practice and review are available. I've also found test prep inside and outside of class helpful at the end of difficult units or chapters.

If you're reading this, I hope you have a good winter vacation and enjoy my hiatus. I've really enjoyed using this blog to write about my new experiences this half of the year. (:

Our Math Monday blog assignment was to write about the Pythagorean theorem, explain how it is used in math, and apply it to two real-life situations; one funny, one serious. The Pythagorean theorem is named after a mathematician named Pythagoras in Ancient Greece.  Pythagoras theorized that in a scalene right triangle, the two smaller measurements when squared would add up to the largest measurement squared. 

The Pythagorean theorem can be used in everyday situations, and is used by architects and engineers constantly. One example of a serious situation would be in the case of an architect who is building a modernist building in the shape of a scalene triangle. If the measurement of the shortest wall is 60 feet and the largest wall is 80 feet, what is the wall in the middle's length? We can find this by using exponents and algebra. 60^2 + x^2 = 80^2. 3600 + x^2 = 6400. The missing measurement is 2800. The measurement we need is the square root of 2800.

Our Math Monday blog assignment for today was to explain what square roots are, where we think their name comes from, and what an alternate name for them could be. 

In short, square roots are numbers that are equal factors of a large number. These large numbers are called perfect squares when their square root is a whole. Every number can be a square root and every number has a square root, but not every number is a perfect square. We use the power of 2 ("squared") in finding the areas of rectangular surfaces.

The term "perfect square" probably comes from the fact that when multiplied by the same number, the number is perfectly even in relation to these factors. The factor that creates the perfect square is a root, so it is called a "square root". This seems really confusing at first, but once it is shown to you hands-on, you'll probably start to get it a bit more. Square roots are really just a fancier way of multiplying a number by itself, the same way that multiplication is just a fancier way of repeatedly adding the same number. 

A different term for square roots could be "corners", since squares have four equal corners, or bases, to make their sides equal. Another name might be "area bases", since as mentioned above, perfect squares and the power of 2 are used to find the area of surfaces.

Square roots are not just something you have to learn in school, though. Mathematicians, architects, engineers, and scientists all use exponents and squares. Exponents and square roots might even come in handy in classes besides math. Like formulae, exponential form and square roots make long, tiring processes easier and math a little more understandable. 

Our Math Monday blog assignment for today was to explain why numbers with negative exponents come out as decimals and not negative numbers. This concept seems pretty hard to grasp at first, but it's easy to understand once you get the hang of it. 

In short, negative powers result in decimal numbers because the powers are not positive. Seems confusing, I know. When a base has a positive power, the number multiplies itself by however many the exponent calls for. However, when an exponent is negative, the number multiplies downward instead of upward, so we get a fraction, which can be written easily as a decimal. To simplify these exponents, carry them out as if they were positive powers and place the result in a fraction under 1. The negative sign only signifies that it will be a fraction. This is why 2^-5 is 1/32 rather than -32. -32 in exponential form would actually be (-2)^5. This can be figured out with a similar process. 

This concept ties in with our current math unit on exponents because we're expanding what knowledge we already had of exponents and learning new concepts and principles used in exponents. It is also "layered" with our knowledge of positive and negative integers as well as our knowledge of fractions, decimals, and how they are interrelated. 

Our Math Monday blog assignment for today was to think and write about exponents. We were to write about what they are, what they do, and how they can be applied to real-world situations.

An exponent, in short, is an easy way to abbreviate a multiplication sentence that uses the same number over and over again. The number of times the number is repeated is called a "power". The number being multiplied is called the "base". For example, 8 to the third power is actually just 8 x 8 x 8.

Exponents can not always be used. They're simply a fancier way to write repetitive multiplication equations. But in the real world, they can be useful when that situation is met. For example, if a buyer buys 3 packages containing 3 apples 3 times over the course of a week, he has bought 3 to the third power apples, or 27 apples, in a week. Exponents are usually hard and confusing, especially when they are applied to real life.  Sometimes powers and bases are mixed up and confused.

You can use the Associative Property of Multiplication when solving exponents. For example, 5 to the third power or 5 x 5 x 5 is solved as (5 x 5) x 5, or 25 x 5. The answer is 125. 

Exponents are used a lot in geometry to find the area and volumes of surfaces and objects, too.

As you can probably see, exponents are pretty easy to solve once you know how!

Our Math Monday blog assignment for today was to click on the link to a math game we were given, play a few rounds, and then reflect on it. The game was called "Diffy" and the actual game objective was to find the differences between certain numbers -- certain kinds of numbers, too, depending on how and if we adjusted the game. Once a "frame" of numbers was completed, we progressed to a smaller one until the level was complete. 

The game seemed really simple at first, but as the levels progressed, more mental math was required to solve it. It was, as expected, easiest in the whole number stages as well as the integer stages, with some frames being much harder than others. The most challenging number frames were probably the ones that dealt with negative integers. Those ones needed the most effort and mental math. I was successful in figuring all of them out using the formulas and skills I've been taught so far this year. For example, when finding the difference between -16 and 11, one can simply leave out the negative sign on the -16 and add 16 + 11 in your head. Your answer is 27, which is the difference between their absolute values, but there's a negative sign on the 16 and it's said that "the darkness rules". The final answer is -27. -16 - 11 = 27.

One of the most unseen challenging things was the fact that we could not refer to any sort of number line or other reference point. All of the work was to be done in our heads and tricks and formulas were to be used. I think it was a nice way to tie in all of our reflective math skills and knowledge of rational numbers including wholes, fractions, and integers.