Blog Post #3: Slopes

1. The book Slopes, by Michael Dahl, creatively talks about slopes and ramps for children. Dahl uses colorful drawings of situations kids are familiar with that exemplify different kinds of slopes. Some of his examples include slides, conveyor belts in factories, and stairs. He examines slopes as an element in getting one thing (a person or an object) from one spot to another--I personally think this is a good way of easily getting the idea of slope across to children. With the slide, for example, a child goes from a given height to the ground. This and other of Dahl's examples is perfect for children because it's not only something that kids learn/hear about or do but it's also something very simple and easy for children to look at and understand from one drawing.

2. Dahl's method of explaining slope is very clever because it takes things kids are already familiar with and shows these real world examples of what slope is. Dahl uses conveyor belts as an example at one point in the book, talking about how boxes in a factory go from the second floor to the first floor on a rolling conveyor belt. Although children might not see factory workers and conveyor belts on a day-to-day basis, they are still able to clearly understand this idea from a drawing. While there are no given number calculations, Dahl clearly explains how the upper levels of a building are above the ground floor, and how the connecting line (conveyor belt) between those two heights is a slope. It's a very simplistic way of explaining what the slope of a line is.

3. Literature is an effective way to teach kids about math because, even when it seems to be blatantly obvious to adults that it's about math, it isn't always obvious to kids. This is good because it means that children grow up understanding basic mathematic ideas and how they exist in the real world without seeing it as a yucky, difficult subject they're supposed to have trouble with in school. It takes out the whole stigmatization that elementary school math teachers seem to deal with about how children don't like learning math. Overall, learning a mathematical concept through literature is not only fun and seemingly easier, but also more relevant because it teaches children real-world examples of what they'll learn in a math classroom as they get older.

The Ants Go Marching Kirsten Jay

1. “The Ants Go Marching” is an example of a constant rate of change. The ants first march one by one, then two by two, three by three, four by four, increasing by one until they are marching ten by ten. Several ants engage in other activities besides marching, but most stick to marching throughout the book.

2. There is a constant rate of change, which can be proven by subtracting y2 by y1 and dividing this by (x2-x1). In this case, the first rate of change would be 2-1/2-1, which is 1. To ensure that the rate of change is indeed constant, this needs to be repeated with all of the values.

(3-2)/(3-2)= 1

(4-3)/(4-3)= 1

(5-4)/(5-4)= 1

(6-5)/(6-5)= 1

(7-6)/(7-6)= 1

(8-7)/(8-7)= 1

(9-8)/(9-8)= 1

(10-9)/(10-9)= 1

As shown in the above calculations, the rate of change is constant throughout.

3. I think that literature is a good way to teach and learn a mathematical concept because it effectively works both sides of the brain: the reading word side and the analytical number-driven side. Science has shown that when both sides are involved in learning, more neuropath ways between the two sides are formed, which improves learning and memory of the subject learned. This is why dancers, who have to equally use both sides of their body, are better at math than most because of the neuropath ways that working both sides has formed.

Blog Post 3

Once you have selected a book, read the book and write a synopsis (summary) of the book.  Be sure to include the title of the book and the author and give a summary of the proper sequencing in the story.

How Many Seeds in a Pumpkin is written by Margaret McNamara and illustrated by G. Brian Karas and tells a story about a boy named Charlie, who is the smallest kid in his class. One day Charlie walks into his class and finds three pumpkins on his teacher’s, Mr. Tiffin’s, desk. There is one large pumpkin, one medium pumpkin, and one small pumpkin. Mr. Tiffin asks the class “How many seeds in a pumpkin?” Different students estimate how many seeds each pumpkin has; they guess that the largest pumpkin has the most and the smallest pumpkin has the least amount of seeds. Mr. Tiffin suggests that they open up the pumpkins to count the seeds. The next day at school Mr. Tiffin opens up the pumpkins and the kids break into groups to pull all the seeds out of each pumpkin. Mr. Tiffin asks the kids to think about how they should count all the seeds, and ultimately the class decides to count them by twos, fives, and tens. “The Twos Club” will count all the seeds from the large pumpkin by twos, “The Fives Club” will count all the seeds from the medium pumpkin by fives, and lastly, Charlie will count the seeds from the small pumpkin by tens. “The Twos Club” had 170 pairs of seeds, “The Fives Club” had 63 groups of five seeds with one left over, and lastly Charlie finds that the smallest pumpkin had 35 groups of 10 seeds. The kids use multiplication to find the total number of seeds for each pumpkin, so 2 x 170 = 340 seeds, 5 x 63 = 316, and 10 x 35 = 350. This means that the smallest pumpkin had the most number of seeds; all the students were surprised, especially Charlie. Mr. Tiffin reminds the children that “you can never tell how many seeds are in a pumpkin until you open it up” and asks them to make observations about each pumpkin. They observe that the smallest pumpkin is the darkest orange and has the most lines on the outside. Mr. Tiffin tells them that each of those observations is an indicator as to how old the pumpkin is, and the older the pumpkin, the more seeds it will have. At the end of the day Charlie concludes that, “small things can have a lot going on inside them.” The End.

After reading and summarizing the book, explain in detail the mathematical concept presented in the book. 

How Many Seeds in a Pumpkin uses a handful of mathematical concepts: making estimations, adding, multiplying, skip counting etc. However, the one that most applies to what we’ve been doing in class is repeated sums and repeated products. Repeated sums are the same as a product, which is evident in the way that the students chose to count the seeds: 2 times x would be 2x, or 5 times x would be 5x, and ten times x would be 10x. We used this same concept in class to draw the distinction between linear functions (y = mx+b) and exponential functions (f(x) = ab^x). In a linear function, m is added an x number of times: m+m+m+m…, which is the way the students were adding the seeds: 10+10+10+10…+10 over and over 35 times (=350). This is different from an exponential function because an exponential function would have 10x10x10x10… with 10 being multiplied times itself 35 times (=1E35). This story underscores that the concept of addition is the foundation of multiplication, and it relates to our class because we have also made the distinction that multiplication is at the foundation of exponents.


Explain in your own words why you believe literature is an effective way to teach/learn a mathematical concept.

I borrowed this book from my little cousin, who is now in her first year of school in kindergarten. She got the book for her birthday from one of her relatives that thought it would be a good introduction to math. I’d imagine learning math for the first time could probably be a little boring at times and daunting for a young child. I am 21 now and I still find it to be daunting and boring, but I thought this book was a fun way to learn some of the basic concepts without realizing they’re listening to a math lesson. It is also a fun activity one could repeat with their child at home around Halloween, so that carving a pumpkin could also help them practice their math skills. Sometimes it helps to learn math in a different context than a textbook, the story book was definitely less intimidated and allows children to develop the concepts by applying them to a real life experience they have every year at Halloween.

The King's Chessboard Rachel Cohen

The King's Chessboard
By David Birch

1. Long ago in India, there lived a king and a wise man. The king wanted to reward the wiseman. The two fought about a reward because the wise man felt he didn't deserve or want one. Finally, the wise man noticed the king's chessboard. He said to the king, "Tomorrow, for the first square of your chessboard give me one grain of rice, the next day, for the next square two grains of rice, and the next day four grains," and so on. And so for every square of the chessboard, the amount of grains is doubled from the previous day. 

The king was curious how much rice it would amount to in total because there were 64 squares on the chessboard. He didn't say anything, however, so as not to make himself look like a simpleton. Then the king granted the wise man's request. The king sent a servant dressed in the best manner possible to deliver the first grain. The wise man simply thanked him and placed the grain on his chessboard. The king heard this, and placed a single grain on his chessboard. This went on and on. The wise man would place a single grain of rice on his board and give the rest away.

The counting of grains became too much for the king's weigher, so he began to simply weigh the rice, beginning with one ounce. The weigher grew concerned with the amounts of rice being given to the wise man and tried to warn the king's advisor, but it backfired. 

A few days later, the advisor saw four grainery workers carry sacks of rice and asked where they were taking it. One of them responded that it was going to the wise man who then distributed it to the poor and the needy. The advisor felt that the weigher had made a mistake, but it was explained to him how the amounts grew and grew, doubling each day. 

After a few days, the king saw sixteen wagons carrying over a ton of rice each and heard people cheering. He demanded to know where the rice was going. The advisor explained to him that the rice was for the wise man and that despite it seeming impossible, there had been no mistake, the math was all correct. 

The king then asked for his mathematicians so as do determine how many pounds of rice he had promised the wise man. At seeing the toal number the king became angry. 549, 755, 830, 887 tons of rice were to be given to the wise man. 

The queen then insisted that the king ask the wise man to be released of the agreement. The king ignored her, but the next day, the thirty-second day, he summoned the wise man. The king told him that there wasn't enough rice in India to fulfill the promise. the wise man agreed that there wasn't even enough in the whole world. 

The king demanded another way to satisfy the wise man, but the wise man responded that he had always been satisfied. The wise man argued that it is the king who must be satisfied, not the wise man. The king said he was satisfied and that he understood that the wise man had taught him an important lesson. 

The wise man served the king many more times, but never again was he asked about how he could be repaid. The king kept the chessboard for the rest of his life, to remind him "how easy it is for pride to make a fool of anyone, even a king."  

2. The King's Chessboard explains exponential growth in the form of a folk tale, using grains of rice to show how rapidly exponential growth occurs. 1 grain on the first day, 2 on the second, 4 on the third, 8 on the fourth, 16 on the fifth, 32 on the sixth. On the eleventh day, the switched to ounces, which after five days grew to pounds. On the fifteenth day,16 pounds. On the twentieth day 1512 pounds. On the twenty-fifth day 16,384 pounds. And on and on increasing exponentially until what would have been the sixty-fourth day. This exponential growth can be expressed as 2^n, where n is greater than or equal to zero. This fable is an effective and engaging way of explaining a somewhat complicated mathematical concept in simple terms. 

3. Literature is a good way of explaining mathematical concepts because it literally spells out what is happening in the mathematical concepts in everyday terms. By providing a background story or anecdote, the concept is given a context and real life expression, allowing for it to be more easily digested. 

Math Fables by Greg Tang-- Blog Post 3, Matthias Ng

Math Fables, Lessons That Count by Greg Tang and illustrated by Heather Cahoon. Tang’s book doesn’t have a continuous plot; instead, he uses each number to express a new story or virtue that would be deemed admirable. He uses animals to get his point across and to hook readers in by keeping them interested. Tang goes in chronological order as he goes from one to ten, using a new animal and life lesson for each number. For the number one, Tang starts with a spider waiting for a “dinner guest” while preaching the value of patience. Throughout the book, Tang uses Cahoon’s illustrations to supplement his lessons by showing the value of traits such as diligence, courage, perseverance, diligence and friendship. Other examples used combine puns with specific lessons. For example, Number nine is titled antics, while ants are used to tell the story and to promote the value of cooperation. Furthermore, for the number 8, crabs are used to illustrate the value of courage. Tang always starts off with the next number in sequential order and from there; he will tell a story by using that number. In the eighth section, there are eight crabs and one crab needs help in the water. Then, one brave crab goes into the water to help save the other crab. Eventually, all the crabs make it out of the water and once again all eight crabs are safe because of their friendship. Every section in the book follows the same format, which is why there is no sequential plot.

Throughout the book, many mathematical concepts we discussed are used; there are also concepts that are more basic. Tang utilizes combinations to add up numbers together, the book attempts to teach young children how to count. This can also be related to functions because functions can be added and subtracted, it also shows that specific inputs will get you an output it also highlights algebraic skills. For example with the crab story, Tang would use different combinations of how we can add to get the number eight or how we can subtract from the number 8. He would subtract one and get 7 crabs and a single crab or would divide them by two, to get two sets of four crabs. Cahoon used illustrations to supplement what Tang’s writing and Math Fables. The illustrations have parabolas and use shifts to show how an animal can move from one point of the page to another. For the section titled Gone With The Wind, a butterfly shifts down a branch in the same way it would shift along the X-axis if it were a function of a graph. Other mathematical concepts used by illustrations consist of symmetry and other forms of concavity and parabolas. The third section has illustrations of turtles and there are two turtles adjacent to each other that appear to be symmetric to each other across a vertical line. Additionally, in the sixth section, otters are seen jumping into the water and the water that splashes up has concavity to them as many look to be concave down and increasing.  

I think literature is vital to teaching mathematical concepts and even our textbooks can be considered literature because it uses text to help explain the concepts and examples set forth. Without the text, it would be much harder for us to learn the material. This is used as an example for how other books or forms of literature can be used as an aide to learn mathematical concepts. Clearly, textbooks have dry text, which can be hard to follow. In contrast, other forms of literature can be engaging and can draw readers in, in a more effective manner. Literature can creatively give students another point of view when learning a new concept. Some students may prefer a different method to solving problems or students may approach problems in a different way. With a different point of view, students can grasp the same material or concept in a way that is more conducive to them. 

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.