Be the Professor

Lecture Outline:

- Ask class room to recall steep hills, or mountains, or slopes, that they've had experiences with & then just pick a number they think that represents the steepness of their hill
- Then explain slope
"The slope of a line is a number that represents the degree of steepness of the line."
- After explaining that, have students make a graph and draw lines on the graph with specified different degrees of steepness
"On your graph, start your pencil at the origin, (0,0) and plot a point here. Then, move your pencil to the right one unit and up k units. The k units should be your slope. So for example, if the mountain or the hill you picked had a slope of 7, move your pencil up 7 units. Then, connect the points and draw a line. Here, you have a line with a slope of k, and k>0. Now, make a line with a slope of 10, 5, and 15."
- After that, have students create a new graph & explain
"So what happens if on my new graph, I plot a point at the origin, (0,0) and then I move over three units to the right? And then I move three units up? If I connect these points and draw a line, what is my slope? For a line that isn't as easy as the first, you can use a formula:

rise
---------

run

This formal basically is asking, 'how much did you rise from the origin point, and how far did you run?' In the case of the line we just created, we rose three units up from the origin, and we ran three units to the right, so that would make our formula

3

-----

3

Which equals one. The slope of this line is one. Now, on the same graph, start at the origin and move to the left four points. Plot a point there. And then move six points to the left and go down five. What would our formula be here? Well, we didn't rise here, we fell. But that's okay, it just means we have to figure out how much it fell. So since the first point's location is (-4, 0) and the second point's location is (2, -5) we can use this formula on it

yfinal - yoriginal

-----------------------

xfinal - xoriginal

This formula is the same as the rise over run formula because points along the x axis determine how much you are running, because running is something you do horizontally, while points along the y axis determine how much you are rising, because rising is something you do vertically. So in the case of the line we just made, the point (0, -4) would be our original and (2, -5) would be our final.So let's plug that in and see what we get

-5 - (-4)

----------

2 - 0

=

-9

-----

2

So here, our slope is negative 4.5, or -9/2. And now you know how to find the slope, or steepness in lines on a graph."

Blog 4

Here is a math lesson on the rate of change!
https://www.youtube.com/watch?v=J871gF4cqB0&feature=youtu.be

Enjoy!

"Be The Professor"-- Anita Tjahyadi

**Today I will be talking about Domain and Range.**
The set of all the starting points is called "the domain" and the set of all the ending points is called "the range." The domain is what you start with; the range is what you end up with. The domain is the x's; the range is the y's.
As we all know, Domain is the set of all input values (also known as x-value) and Range if the set of all output values (also known as y- value). 
                      -There are 5 types of numbers that can Identify either domain or range.
                                  - NATURAL NUMBERS ---> |x|
                                               |x| = {1,2,3,4,5...}
                                  - INTEGERS --- >  Z
                                               Z= {-3,-2,-1,0,1,2,3...}
                                  -RATIONAL NUMBERS ---->  Q
                                               Q = { (-2/3), (-1/2), 0...)
                                  -IRRATIONAL NUMBERS ---> P
                                                P= {e, π... }    
                                  - REAL NUMBERS ----> R
                                                R= {all types of numbers}

To put these numbers into use, We have three different ways to notate-- Roster, Set-builder,& Interval notation.
                      
                        - Roster Notation  
                                  -Listing each element of a set inside a pair of {}.
                                  - Example 1: The set of all the letters in the name RAMONES
                                                   D= {R, A, M, O, N, E, S}          
                                  - Example 2:

                                                    D= {2, 4, 6, 8, 10, 12, 14}
                                                    R= {1, 6, 11, 16, 21, 26, 31}  
                         - Set-Builder Notation
                                   - Lists the rules that determines whether an object is an object of the set,
                                      rather than the actual elements.
                                   - Example 1: The set of all women such that the woman has won a Grammy.
                                                     D= {woman| the woman has won a Grammy}
                                   - Example 2: y= √(x-4)                                                         
                                                              x-4 ≥ 0                  D= { xϵ Q| x ≥ 4}    ( ϵ -- > "element of" )
                                                               x ≥ 4                     R= { yϵ Q| y≥ 0}  
                          - Interval Notation
                                   - A method of writing down a set of numbers. Usually, this 
                                     is used to describe a certain span or group of spans of numbers 
                                     along an axis, such as an x-axis. However, this notation can be
                                     used to describe any group of numbers. 
                               For example, consider the set of numbers that are all greater than 5. 
                                   - Example 1: y= √(x-4)
                                                   D= [4, ∞ ) 
                                                   R= (0, ∞)  


                                             

Blog Post #4

Rate of Change

A rate of change is a rate that describes how one quantity changes in relation to another quantity. When determining the rate of change we use the rate of change formula, which is:

                                                        y₂ – y₁

Rate of Change      =              ----------

     x₂ – x₁

In order to use this formula, you must have numbers or points to work with. Typically you use the formula by plugging in numbers from a graph or table.

Lets look at an example of how we would use the rate of change formula to find the slope of a line. The diagram below shows how this works.

  

Example)

Lets say that you want to find the rate of change, or in this case the slope, of a linear line:

                                          

The first step would be to pick two different

points on the graph. For this example lets use

(1,1) and (2,2). We take these points and plug

them into the rate of change formula as so:

Rate of change =     2-1

                             2-1

Then we solve it:

                                    2-1   =  1   =   1

                                 2-1       1

The rate of change, or slope, of the line is 1

Keep in mind the numbers wont always work out so perfectly. The rate of change can be a fraction/decimal, a negative number, or there could be no rate of change at all (in a case like that we would have a horizontal line).

*Also, one thing to note about the above example is that we only tested this with two sets of points. We don’t need to test anymore simply because we know it is a linear line, and any points that we plug in will give us the same rate. However in other situations you will need to test multiple points in order to confirm that they all give you the same point.

Now lets look at an example of finding the rate of change based off of a table:

Say you have just driven from Los Angeles to New York, and you want to know what your average rate of speed was.

The table below shows the time spent driving in hours (x), and the distance traveled in miles (y).

Time in hours (x)	Distance in miles (y)
12	400
24	800
36	1200
48	1600
60	2000

To figure out the your average rate of speed we will need to plug at least two numbers into the formula.

y₂ – y₁      

_{---------    =}        800 – 400    =     400      =    33.3 

x₂ – x₁                 24 – 12              12

We get an average rate of change of 33.3, however we need to plug in two more point in order to confirm that this truly is the average rate of speed.

1600 – 400              =    1200        =        33.3

   48 – 12                         36

Since we used multiple points and got the same rate for both, we have confirmed that the average rate of speed is in fact 33.3 mph. 

Blog Post 4

Concavity

This lesson will teach students what concavity is and what it looks like on a graph. Also, I will go over how the graph of a line can be increasing and decreasing at the same time that it is concave up or down.

­­

 The graph of a linear function is a straight line because the average rate of change is constant. But, not all graphs of functions are straight lines, they can be bent upwards or downwards. This upwards and downwards is called concavity.

• ·      A graph that is bent upward is concave up.
• ·      A graph that is bent downwards is concave down.

The graph of a line can also be increasing or decreasing at the same time that it is concave up or down.

• ·      A graph is increasing when the line is rising from left to right.
• ·      A graph is decreasing when then line is falling from left to right.

Note: If a function has a constant rate of change, its graph is a straight line and it is neither concave up nor concave down. BUT, it can still be increasing or decreasing depending if it is rising or falling from left to right. 

EXAMPLES OF CONCAVITY UP/DOWN AND INCREASIND/DECREASING

Concave Up / Increasing

Concave Up / Decreasing

Concave Down / Decreasing

Concave Down / Increasing

Photos taken from:

http://www.biology.arizona.edu/biomath/tutorials/functions/Properties.html

Ashley Maddox Lesson Plan

Hello everyone!

Please view this link to learn about compound interest formulas and choosing the right bank!
http://prezi.com/xifly1eryykh/?utm_campaign=share&utm_medium=copy&rc=ex0share
]


-Ashley Maddox

Blog #4: Be the Professor (Compound Interest)

_____________________________________________________________________________

Hello Class!

Prezi Presentation on Compound Interest! Please click the follow link: 

http://prezi.com/otzzczqsae4x/?utm_campaign=share&utm_medium=copy

ENJOY! 

-Professor AJ Ledesma 

Blog Post 3

1. For my blog post I decided to choose the book Bigger, Better, Best! By Stuart J. Murphy.  This picture book is about three siblings who argue about who has the nicest things.  Their family moves into a new home and each child got a new room which starts a new argument: Who has the bigger, better, and/or best room?  In conclusion no one won the argument, but they were able to determine who had the largest room by finding the area of each room.

2. Like mentioned before the picture book uses area to see which child who has the largest room.  Each room is square so  the way they found the area is the children multiplied the number of sheets on their wall(sheets where placed on the floor for the perimeter and on the walls) by the size of the room. Once they found the area they were able to determine who had the biggest.

3. I think that the book Bigger, Better, Best! does a great job at describing how to find the area and would be a great tool to be used in class. Not only does it allow the people to learn a mathematical concept but you get to learn it in a fun way! You learn and you don't even realize it because you are enjoying a book!

Blog #4

The concept that I will be explaining in this post is the concept of  interest.

Interest is arguably one important concepts that we will learn in math because people will use this concept almost everyday for the rest of their lives. Interest is used by banks, credit card companies, in bonds, mortgages and so on. If you don't understand interest it can be incredibly easy to be ripped off, taken advantage of and become financially unstable.

There are two types of interest: Compound interest and Continuously compounded interest.

The best way to remember compound interest is that the interest is added only to the principle amount

The formula for compound interest is A= P(1+r/n)^nt.
 Where A=Amount
P= Principle
R= Rate
N= Frequency
T= Time

an example of compound interest would be that a bank will give 2% simple interest every quarter for 5 years. How much will you make if you deposit $5,000?

The answer is A=5000(1+.02/4)^4x5 = $5,524.48. This means you will be paid $524.48 by the bank to deposit $5,000!

You divide by 4 in this situation because there are 4 periods in the year in which the bank will give you interest. You also multiply these 4 periods by 5 because over 5 years you will be given interest 20 times!

This is very easy to remember because it essentially just the interest rate divided by the number of times it is factored in a year raised to that same number multiplied by the time period of the problem.


The next type of interest is continuously compounded interest. To find this all you need is to use the formula A=Pe^rt

Where A=Amount
e=A mathematical concept
R=Rate
T= Time

An example of this would be a credit card company charges 5% interest compounded continuously. You buy a pair of shoes for $50 If you don't pay off your bill for 3 years, how much money in interest will you be charged?

A=50e^.05x3 = $58.09. This means you will be charged $8.09 in interest so those shoes won't actually cost you $50!

It is easy to remember this formula because all you have to remember is that continuously compounded interest is equal to the word PERT, which is the formula for continuously compound interest.

Blog Post #4: Rate of Change

Good morning class, My name is Professor Moran and today I am going to teach you the mathematical concept of Rate of Change. Now, let us begin.


Before we can discuss Rate of Change, we must first consider what "Rate" in itself really is. Rate is the value of one amount/the value of another amount.

The definition of Rate of Change is as follows: The speed at which a variable changes over a specific period of time.

Now, rate of change can be used  to solve and explain various real life scenarios. Most commonly, rate of change is used when discussing momentum, and it generally is depicted mathematically as a ratio between a change in one variable relative to a corresponding change in another.

Rate of Change in essence, is a rate that describes how one quantity changes in relation to another quantity.

Lets make an example on a graph to illustrate this concept.

By looking at this graph even before considering Rate of Change, we can deduce that the x-axis represents time, in the unit of seconds and that the y-axis represents elevation, in the unit of feet. 

This graph represents the Rate of Change of a helicopter's elevation over the first 20 seconds of flight from initial takeoff. 

At "O" seconds, the helicopter is stationary on the ground. 

After five seconds, the graph tells us that the Helicopter has risen to an elevation of 50ft. 

At 10 seconds, the graph tells us that the Helicopter has risen to an elevation of 100 ft....And so on.

Now, here is where we see Rate of Change come into play. As we said in the beginning of the lesson, Rate of Change is the speed at which a variable changes over a specific period of time. In this example on the graph above, the Rate of Change is consistently changing by the same amount each interval on the graph. At "0 seconds" we see the helicopter is at "0ft." Then at "5 seconds" we see by the graph that the helicopter is at "50 feet." And so on. 

Now, one might ask, is calculating ROC always this easy?! Or, is there a shortcut to calculating ROC when the numbers provided aren't as pleasant? 

YES!!!

There is one simple equation that allows mathematicians to skip analyzing graphs and find the ROC quickly and easily.

The ROC equation is quite simple. In words, simply take any "Y" value on the graph (besides the original) and subtract it by the "Y" value that comes right before your original "Y" value. Then, divide this number by any "X" value subtracted by the "X" that comes right before your original "X" value it. 

Why does this equation work you might ask??

The ROC formula works because it simply determines the vertical change of a graph divided by the horizontal change of a graph. 

All together, the ROC formula looks a little like this when all things are considered. 

                               Vertical Change   = Y2-Y1  = Rate of Change

                             Horizontal Change    X2-X1

This concludes today's lesson, class! See you all tomorrow!