Graphs of sine and cosine

Here are the rules of graphing sine and cosine graphs:

1. Sine/ Cosine ​​​​​​​

2. Y = a sin (bx + c) + d

3. A = amplitude

4. B = period : 2∏/b

5. (bx+c) = Start: bx+c = 0; End: bx+c = 2∏

6. D = vertical translation; d units

Matrices, and elementary row operations

Elementary row operations:

1.Interchange 2 equations

2. Multiply an equation by a non zero constant

3. Add a multiple of an equation to another equation. 

Trig Identities

The Path to of verifying a trig identity:

You must only use one side, you can not work with bothe sides

Try to; factor, add fractions, square a binomial, create a monomial denominator. 

Look to use fundamental identities, they work best. 

Last resort, try to convert everything to sines and cosines

Trr anything, even mistakes help to solve the problem in the end. 

Converting Polar Coordinates

Converting between Rectangular Equations, (equations that contain x and y) and Polar Equations (equations that utilize r and theta (u) instead). When converting from x and y to r and u, use these equations:

 

x = rcos(u)

y = rsin(u)

 

When converting from r and u to x and y, use these equations:

 

tan(u) = y/x

r^2 = x^2 + y^2 (r squared = x squared + y squared)

 

Partial Fractions

When dealing with partial fractions, it is very important to follow the steps below! If you follow the steps, it will make the partial fraction problems very easy. 

The first step is to multiply by the LCD (least common denominator)

Second you need to distribute

Once you do those two steps, combine the terms

Then factor out the variable

When you do that you need to equate 

Next it is time to solve. 

Most people believe that they are done there, but dont forget to write it as a partial fraction.

Cross Product

The cross product of 2 vectors is not a hard topic. 

1. Make a 3x3 matrix, first column should be the variables i, j, k. Second column should be the values of vector U. Third column should be the values of vector V. 

2. Then take the first 2 rows of the matrix and rewrite them next to it. 

3. Then you multiply the diagonal values going forward (see example below, it is the green portion), and add the values you get from that. 

4. Then you multiply the diagonal values going backward (see example below, it is the purple portion), and subtract these values. 

Limits in Calculus

Limits are the core to calculus. The limit do a function is essential in many topics and situations in calculus. Limits are the building blocks of calculus and here is a link to read more. 

http://www.khanacademy.org/math/differential-calculus/limits_topic

Tangent Lines

Today in Mathland we learned about Tangent Lines and how to find their slopes! Finding the slope is simple, but not easy. There is a formula you must memorize. It's f(x+h)-f(x) all over h. First, you. It's plug the values into your formula. Then you foil. Then you combine like terms. After that, you plug in 0 for your h. Then you just solve. There is an example below. This lesson isn't to tough, just keep track of your terms. 

Evaluating Limits

Today in Mathland we learned how to evaluate limits! It's not that tough of a topic, but it takes sometime to finish. The first step is to use direct substitution and plug in the value of the variable. You should get 0/0. Then you take your original equation, and from there, 2 things can happen. If you have a square root in your numerator, then you multiply by the conjugate. If not, then you factor. And the final step is to substitute the value of the variable, into the factored equation. 

Limits can be tricky, but they are essential for the next levels of math. 

Polar Coordinates

Today in Mathland we learned about polar coordinates. It's a way of graphing similar to rectangular coordinates. First you fix a point O that is called the pole, and construct from it a ray called the polar axis. Then each point P in the plane can be assigned coordinates in (r, θ). 

You must also know how to convert between rectangular and polar coordinates. The formulas are down below. 

Hyperbolas

Today in Mathland we learned all about hyperbolas! They have a pretty interesting conic section. The formula for a horizontal parabola is (x-h)^2/a^2 - (y-k)^2/b^2=1 and for a vertical hyperbola the formula is (y-k)^2/a^2 - (x-h)^2/b^2. As long as you stick to the formulas you can't go wrong. 

Ellipses

Today in Mathland we learned about the conic known as the ellipse. An ellipse is the set of all points (x,y) the sum of whose distances from two focus points is constant. The standard equation of an ellipse is 

Here is a graph of an ellipse. 

Ellipse can be challenging at times, but if you stick to the formula you can't go wrong. 

Parabolas

Today in Math Land we learned about Parabolas. Parabolas have a lot of rules and a lot going on. 

A parabola is a set of all points (x,y) that are equidistant from a fixed line, and a fixed point on the line.

The formula for a vertical parabola is: (x-h)^2=4p(y-k) and the directrix: y=k-p

The formula for a horizontal parabola is: (y-k)^2=4p(x-h) and the directrix:'x=h-p

Make sure you keep your eye on all the variables and make sure you don't make any silly mistakes. 

Lightning!

The chances of being struck by lighting are 1 in 3000. Yet, Lightning is one of the leading weather related causes of death. And don't think rubber shoes will help. Rubber shoes actually do nothing to protect from lightning. Here is a link to an article explaining the probability of getting struck by lightning: 

http://news.nationalgeographic.com/news/2004/06/0623_040623_lightningfacts.html

Measures of Central Tendency

Today we learned the !easures of Central Tendency. They are mean, median, and mode

Mean: is the average

Median: is the middle number, the set must be in order of least to greatest. 

Mode: the number that occurs most often. 

They are pretty simple to learn, but make sure you don't make any silly mistakes like forgetting to order your sets from least to greatest before searching for the median. 

Standard Deviation

Today in Mathland we learned about the measures of dispersion. Measures of dispersion will give us an idea of how much the numbers in the set differ from the mean of the set. The two measures of dispersion are called the variance of the set and the standard deviation of the set. Finding the standard deviation of a set and the variance of a set take a little effort. But as long as you are careful you can't go wrong. 

Well Ordering Principle

The well ordering principle states that every non empty subset of positive numbers contains a least element. So a subset of all positive natural numbers has a least element. Here is a link explaining more:

http://www.math.wustl.edu/~chi/310notesIV.pdf

Big Data

Big data sets are becoming increasingly more prevelant in today's world. We are constantly looking for new and better ways to analyze these data sets. New tactics and technology is being made to help improve this. By identifying patterns you can speed the process up. Here is a link to an article relating to this topic: http://www.sciencedaily.com/releases/2014/02/140218185128.htm

Sequences

Sequences are great but they can be very challenging at times. There are several formulas to learn and different types of sequences. There are geometric and arithmetic. There can also be infinite solutions. But, if you stick to the formulas and plug in the right numbers you can't go wrong. For arithmetic sequences. Here are the formulas:

Factorials

Today in Mathland we learned about sequences and factorial a. My favorite was the factorials! They are so neat. You take a number and multiply it with all the numbers that come before it. So if you have 4! you multiply 4x3x2x1. And you get 24. You can also divide and multiply factorials together. That's when it gets really fun. I can't wait for a quiz on factorials! 

Here is an example of dividing and multiplying factorials. 

Sum Formula

Today in Mathland we learned about the Sum Formula for arithmetic sequences. Now sequences can get very complicated pretty quick. But as long as you stick to the basics you should be fine. Make sure you keep track of your variable "n" and don't mix up Sn and An. So, stick to the basics, and they will work wonders for you!

Here is the formula:

The Best Website Ever!

Matropolis!

https://sites.google.com/site/matropoliscom/

Voice Thread

voicethread://voicethread.com/share/5446248

Educreations

We had to split our video up, but here are out examples.

Video 1

http://itunes.apple.com/us/app/educreations-interactive-whiteboard/id478617061?ls=1&mt=8

Video 2

http://itunes.apple.com/us/app/educreations-interactive-whiteboard/id478617061?ls=1&mt=8

Video 3

http://itunes.apple.com/us/app/educreations-interactive-whiteboard/id478617061?ls=1&mt=8

Cryptography!

What a wonder Cryptography. Today in Mathland we learned the Art of Cryptography. It is a simple idea, but an elaborate process. First you take a word or message, for example Cheese. Then you pick numbers that will symbolize each letter. Then, you create an encryption matrix, which just has to be a matrix that has an inverse. Next, you make 1x3 matrices out of your word or phrase. So, if we are using cheese, let's let 3 represent c, 8 represent h, 5 represent the e's, and 19 represent s. So you would take the matrix (3 8 5) and multiply it by the encryption matrix you chose, and your message is encrypted. To decode the message simply take the 1x3 matrices and multiply them by the inverse matrix of your encryption matrix. 

Inverse Matrices

Today in Mathland, we took a stroll down to Inverse Lane and learned how to find the inverse of a matrix. The key in this lesson is that not every Matrix has an inverse. We call those such Matrices singular. But, here is how you find the inverse.

You set a matrix with just 1's on the main diagonal next to your original matrix. Then you use Gauss Jordan to make your original matrix all ones on the main diagonal, but all the changes you do apply to both soft your matrices. And once your original matrix is solved, your inverse is the second matrix. Here is am example. 

This lesson wasn't to challenging. I hope my explanation was helpful, and will be a great guide for anyone who wishes to visit Inverse Lane. 

No Solution or Infinite Solutions

When given a system of equalities, there are 3 possibilities for the solution. No solution, infinite solutions, or exactly one solution. The key is knowing which of the 3 is correct for the system you're working with. 

You know your system has infinite solutions when you get a row of zeroes in your augmented matrix. For example in the picture the 3rd row is all zeroes. 

You know you have a system with no solutions, when you get an impossible equation such as all zeroes in a row except for the last number is not a zero. For example in the picture the second row has all zeroes besides the last number.

For further detail, here is the link to the website. 

http://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx

Mathematicians

New studies have shown that you must study many different types of mathematics to become truly good at it. These research could affect how Math is taught in school. It is also been found that you can not simply be good at math from birth. You can not just be born with math skills others do not posses. 

For more information Bolow is the link to the article:

http://www.sciencedaily.com/releases/2013/12/131216102844.htm

Matrix Vocabulary Prezi

http://prezi.com/3mudcdbdekml/matrix-a-rectangular-array-of-quantities-or-expressions-in/

Gaussian Jordan

Today in Mathland we ventured on down to down town Augmentia, and learned about solving systems using matrices. There are different ways to do so, but my favorite is using Gauss Jordan Elimination. It's just like Gaussian elimination, but you take it a step further. 

In Gaussian elimination as shown above, you must follow the 3 steps.

1. Write the Augmented Matrix

2. Use ERO's to rewrite in row echelon form

3. Write system of equations and use back substitution

The difference in Gaussian Jordan Elimination, is that you solve to get all 1's on your main diagonal, but also you must make all other spaces in the Augmented Matrix equal 0. Then you won't need to back substitute because your answer will be right there in front of you. Here are some examples. 

Augmented Matrices can seem elaborate at times, but following the steps of Gaussian or Gaussian Jordan Elimination will give you clear and concise guidelines to make it as easy as possible. 

Partial Fractions

In Mathland, there is a terrible place. Very few ever venture there and for good reason. It's the complicated route of Partial Fraction Pass. Now partial fractions can be very difficult. They can be very elaborate and may include long division, but if you follow these steps you can't go wrong. 

1. Multiply by the Least Common Denominator

2. Distribute

3. Bring the like terms together

4. Factor out the variable

5. Equate

6. Solve

7. Then rewrite your answer as a partial fraction 

Venturing into the realm of partial fractions can be challenging and precarious. You must make sure you are solid on your factoring and are carefully with every step. If you are a true Mathlete, you can handle it. 

The importance of Math is being overlooked

Finnish adolescents aren't motivated to do math. And their PISA results have dropped. In Finland they are afraid literacy is getting worse and it starts with the math problems they have. Kids aren't motivated to do scholastic work especially mathematics. This could be bad for the future. 

Here's the link to the article. 

http://www.sciencedaily.com/releases/2013/12/131219082752.htm

Linear Programming

Today in Mathland we took a trip to Programania and learned about Linear Programming. It's an interesting subject to learn about, but can be quite elaborate at times. If you follow these steps you'll never have a problem. 

1. Read the problem carefully

2. Write the constraints or inequalities

3. Graph the inequalities. Find the feasible region

4. Find the vertices of the feasible region

5. Write a function to find the minimum or maximum value

6. Plug the vertices into the function

7. Find the maximum or minimum

Today's trip was exhausting, but fulfilling. Now I know how to make the greatest profit. I can't wait to go back to Mathland tomorrow. 

Graphing Systems of Inequalities

Today in on adventure in Mathland, we took a little trip to Systemetropolis! We learned how to graph systems of inequalities. Graphing them can be quite extensive at times, but if you follow these rock solid steps, and you'll never be overwhelmed. 

1. Replace the inequality sign, and sketch graph of resulting equation.

2. Test one point in each of the regions formed by the graph in step 1. If the point satisfies the  inequality, shade the entire region to denote that every point in the region satisfies the inequality. 

3. A solution of a system of inequalities in X and Y is a point (x,y) that satisfies the inequality in the system. 

4. For a system of inequalities it is helpful to find the vertices of the solution region. 

My adventure today was a blast! Graphing is fun! I can't wait until I get to go back to Mathland tomorrow!

Solving Systems Using Matrices

In my exhilarating trips to Algebra City, Mathland, we learned 2 ways to solve systems. We learned how to solve by elimination and by substitution. But, there is another way. Here is a video on how to solve systems using matrices. This way can be just as useful as the others when solving systems. I remember my days back in 8th grade using matrices. In fact I had a test on solving systems using matrices. What a dreaded day in Algebra City that was. But, I studied hard and in the end did very well. It's a good skill to have. 

http://youtu.be/y28HRED15wo

Percentages Over the Years

The National Basketball Association(NBA) has a huge problem. The players free throw shooting percentage is getting worse and worse. Only 15 players shot better than 80% last year. That's an embarrassing amount and nearly an all time low. The art of making free throws is being ignored, the simple fundamentals have been overlooked, and free throwing shooting for players has become a real problem. The sport of basketball is in real trouble. 

Square Systems

This is the link to the video for my groups problem from class on Square Systems

https://www.edmodo.com/link?url=http%3A%2F%2Fwww.educreations.com%2Flesson%2Fview%2Fproblem-10%2F15536009%2F%3Fs%3Dpl0aeo%26amp%3Bref%3Dapp

Solving by Elimination

Today's adventure in Mathland, we took a trip to Algebra City and learned about solving systems by elimination. It's simple as long as you follow the 4 easy steps. 

1. Obtain coefficients that differ only in sign.

2. Add the equations to eliminate a variable. 

3. Back substitute to solve for the second equation. 

4. Check your solution. 

It's up to you to chose whether to solve by substitution or by elimination, but it can sometimes be easier to solve systems with equations that have coefficients by using elimination. 

Which way of solving systems do you find faster or easier?

Solve by Substitution

Today during my adventure in Mathland, I had a flashback to 8th grade and relearned how to solve by substitution. I followed the 4 simple steps. 

1. Isolate one variable in one equation.

2. Substitute the variable found in step 1 to the other equation. 

3. Solve.

4. Use back substitution to find the remaining variable. 

It's as easy as that. I hope my time in Mathland tomorrow will be equally as fun as today.