Blog Post 2

Part a:

1. 1. http://money.cnn.com/2010/02/02/news/companies/napster_music_industry/
2. 2. Functions have exactly one output per input. They can be shown as a table, a table, a graph, or in words. A function also has pass the vertical line test where it can only touch a vertical line once. 
3. 3. This graph is a function because there is an output (dollars in millions) for every input (each year).
4. 4. This function shows how each year the sales in the music industry has been falling since 1999. This article is slightly outdated, although CD sales have continued to fall since then. For every year (x value) the graph shows the sales in millions of dollars for that year (y value).
5. 5. This graph is not a linear function because there is no ARC since the number by which the sales fall each year is not constant. The graph even showed a spike in sales around the mid 2000s showing that there is no constant slope. 
6. 6. N/A  
7. 7. The function is not linear because there is no average rate of change in the graph. While the article states that the average rate at which the sales have dropped over 10 years is 8%, the average rate of change is not constant from year to yearIf the function is not linear, explain in detail how you know it is not linear (be sure to refer to the average rate of change).
8. 8. This function is not a mathematical model because the x inputs ( year) do not directly correlate to the y inputs (sales). That is to say that the y inputs could be any number and it would not make a difference.

Part b:
1.You could use the vertical line test to determine whether something is not a function. However, because functions can only have one output per input any table/graph that has more than one y value for an x value supplied is not a function.
2. http://blog.thecurrent.org/2014/02/40-years-of-album-sales-data-in-one-handy-chart/
3. This is a bar graph that shows music sales over the past 40 years (from 1973-2013) based on format (ie: CD, LP, Tape, 8 track, MP3). 
4. This is not a function because if it we're either put into a line graph or a table, we would see that for every x value (input/year) there would be several y values (output/sales in millions of $) as this graph focuses on several music formats rise and decline of sales. For instance, we see earlier in the graph that sales for cassettes were MUCH higher then than CD, and in the 90's you can see that reverse. 

Blog Post #3 (The King's Chessboard)

Mathematics in a Story

Jeffrey Williams

I 
 '"The King's Chessboard" by David Birch, is a story about an Indian king who wants to reward a wise man for his years of service.  However, the wise man refuses to accept any kind of reward. But, king commands  the wise man to choose a reward. While thinking of a reward, the wise man notices the king's 64-tile chessboard asks to receive a grain of rice for the each tile on the chessboard starting with one grain of rice on the first day and amount of rice he receives doubling the previous days amount for the next 64 days.  The king, a bit little perplexed, grants him his reward after being to proud to admit that he need help calculating the amount of rice this reward would add up to. The next day, a royal servant delivers one grain of rice to the wise man as per the agreement.  The next day, the amount of rice the wise man receives doubles to 2 grains, the third day the amount of rice doubles to 4 grains, and so on and so forth.  As the amount of rice being sent the wise man rapidly increases, the staff in charge of the royal granary begin to take notice and notify the king about how much rice this is costing him.  Growing more concerned over the amount of rice being being sent to the wise man, he has the royal mathematicians calculate how much rice this will cost him by the 64th day.  The number is staggering; realizing that he could pay the wise man the amount of rice that he owes him, The king summons the wise man to the royal palace and commands the wise man to tell him what would satisfy the wise man.  The wise man tells the king that he was satisfied serving his king and did not want a reward. The wise man point outs that it was the king who was not satisfied and insisted that the wise man pick a reward. The wise man then concedes that he picked the reward he did to teach the king a lesson about being too prideful. 

"The King's Chessboard" explains exponential functions by demonstrating how the number of rice increases if the amount is doubled everyday for 64 days.  At first, the wise man receives just 1 grain of rice.  The next day, he receives 2 grains of rice.  The day after that, he receives 4 grains of rice. And on the fifth day, he receives 8 rains of rice, and the number keeps on doubling.  This demonstrates the real world application of the exponential function 2^n. 

 

Literature is an effective way to learn mathematical concepts because its a good way to show readers  the application of mathematical concepts in a real world context.  For example, the use of an exponential function in "The King's Chessboard" shows readers the affects of doubling.  In the case of the book, one grain of rice is doubled for 64 days which quickly adds up.  Readers can then take the concept exponential functions and apply it to other parts of their lives such as personal finance. Another reason literature is such an effective way to learn math is it keeps the reader engaged with the material by conveying it inside a story.  This keeps the  reader is entertained and at the same time incentivized to learn the math in order to help understand the story.

Part b:

1.     After completing your blog entry, thoughtfully and critically comment (praise and/or critique) on the posts of members in your blog group.   

Lemonade for Sale by Stuart Murphy - Blog Post #3

1. Lemonade for Sale by Stuart J. Murphy features the story of Elm Street’s kids, Mathew, Meg, Danny, Sherry and their pet, Peter, and their brilliant solution to earn the needed money to cover for the fixes of their dwindling clubhouse’s repairing. They figure that if they sell 30 to 40 lemonade cups for $0.25 each during a hot summer days they will earn sufficient money.

The kids work together to keep track of the lemonade they sell each day for an entire week using a bar graph. On Monday they sold 30 cups. On Tuesday they sold 40 cups. On Wednesday they sold 56 cups of lemonade. However, on Thursday, after begging and Peter’s squawking of “Lemonade for Sale!”, they only sold 24 cups. As they begin to worry that something might be wrong with the lemonades they put so much hard work into, they notice that a new neighbor, Jed, was juggling down the street, taking everybody's attention away from the lemonades and into Jed. The kids finally decide to work together on Friday; Jed juggled next to the lemonade stand while the rest of the children sold lemonade. They sold over 100 lemonades making it hard to keep track on a graph, and they made even more money than they originally planned which was therefore enough to repair their clubhouse, of which Jed was now a part of.

2. Lemonade for Sale creatively shows easy and realistic ways of graphing obtained results in a real life and very common situation. The kids keep track of their lemonade sales in a bar graph. On the first day, the amount of cups sold was 30. On the following two days, the amount of lemonade cups sold increased; by 10 cups on the second day and 16 cups on the third day. Thus far the sales were higher by day. On the fourth day, the amount sold dropped by 26, with only 24 lemonade cups sold, meaning that the sales lowered that day to even less than the cups sold on the first day. And on the last day, the sales increased to the maximum. Therefore, the graph had ups and downs, which represented the increases and decreases. The book also vaguely touches on estimating numbers, such as when graphing number 56, Sheri chooses to estimate that it will be somewhere near the middle of number 50s and 60s, inclining a little more towards 60. 

Since the amount of cups sold hugely varied each day, there is not a constant expression that represents the pattern.

3. Literature is an effective way to teach/ learn a mathematical concept as it allows readers to identify and understand the importance of the meaning and usage of mathematical concepts in real life. Particularly in children’s books, the use of visual creativity to complement the words make it seem less intimidating to children, easier to relate to, more meaningful and at the same time it helps them experience and understand math in different ways. (i.e. through drawings, pictures, words and graphs).

Blog Post 2

Part A

1. http://www.theatlantic.com/business/archive/2014/01/the-growth-of-college-grads-in-dead-end-jobs-in-2-graphs/283137/
2. A function is defined as when an input number has exactly one output number, making the input a function of the output. Another way of finding a function is seeing if it passes the vertical line test, checking to see that every y value only corresponds to one x value.
3. (see first graph)
4. This graph looks at the function of underemployment in recent college graduates over time (from 1990 to 2012)
5. This graph cannot be a linear function, because to be so, the graph would have to have a constant rate of change, and the line clearly has differing slopes.
6. N/A
7. Because this graph is not a linear function, it has no one average rate of change simply because the rate of change shifts at different intervals
8. This graph is a mathematical model because it exemplifies the correlation between the year and the percent of recent college graduates' unemployment rates.

Part B

1. A graph does NOT qualify as a function when an input number has more than one output number, or has the potential of having more than one output number (ie. the input is NOT a function of the output)
2. http://www.huffingtonpost.com/jared-bernstein/two-scatterplots-regardin_b_3912735.html
(referencing the first graph in the article)
3. In this article, they plot the change in labor force participation against job growth by state.
4. If you were to give this graph the vertical line test, it would not pass.

What’s Your Function?

Math Blog post #2

Ana Maria Lopez

Part A

1.          http://www.worldpopulationstatistics.com/china-population-2013/

2.         A relationship is a function if there is exactly one output per input

3.          In the article, there is a relationship between years and the historic population in China. In this case since each output has only one input it is a function.

4.         In this case, the meaning of this relationship is that (X) is the year and (Y) is the number of people or the population (in million) in the specific year (x). It means that there is exactly one number of people or population per year.

5.         In this case, the function is a not linear because the rate of change is not constant at every interval.

6.         It is not linear

7.          The function is not linear because the rate of change is not constant at every interval.

Proof:

 a)     ( 1960, 667.1) and ( 1965, 715.2)

715.2 – 667.1     =  48.1    = 9.62 ( R.O.C)

1965- 1960                   5

b)     ( 1980,961.2) and (1985, 1051)

1051 – 961.2   =  89.8    = 17.96

1985 – 1980           5

This is proof that the rates of change among the intervals are not constant since between 1960 and 1965 the R.O.C is 9.62 and between 1980 and 1985 the R.O.C is 17.96

8.         This is not a mathematical model because the output values do not depend on the input values. Meaning that as the years (x) change, the output values or population (Y) does not necessarily changes. This is because as years go by, population might increase of might decrease. In summary, there is no correlation between the year (x) and the amount of people in china.

Part B

1.          A relationship is not a function when there is not exactly one output per input.

2.         The chart I found represents the Average student grades on Math Exam #1

Average Student grades on Math Exam #1
Student	A	B	C	D	E	F	G	H	I
Grades	90	88	72	90	98	81	88	96	81

3.          In this chart, students had his/her own grade but there are repeated grades amongst them. For example, Student A has a 90 as well as student D.

4.         In this case, the relationship is not a function because there is more than 1 output per input, meaning that the outputs repeat themselves among the inputs. For example, both students B and G got an 88% in his/her exam.