Please choose one of the following problems from your textbook or one of the 6 discussion questions that accompany the videos and respond fully. Please use full sentences and use a minimum of 50 words in your responses to each question.
Section 1.1 63, 66, 68, 70
Section 1.2 61, 65, 66, 68, 76
Section 1.3 56, 58, 60
Video 5: Pascalâ€™s Triangle â€“ Part 1
This video uses a bean machine, also known as a quincunx, to demonstrate both the central limit theorem as well as Pascalâ€™s Triangle. Students drop marbles from the top of the quincunx and see which slot they land in at the bottom. After numerous marbles have been dropped, it is observed that a bell curve forms and the presenter explains that this happens any time random processes occur. This reasoning helps explain how Pascalâ€™s Triangle is formed, where each number represents how many paths the marble can take around that peg. All the numbers within the triangle also have different meanings, such as the rows representing the powers of 11, adding the rows giving the powers of 2, and the diagonals representing how shapes are organized in space.
Class discussion questions
1. Using what you know about Pascalâ€™s Triangle, what is (a+b)4 ?
2. The quincunx is said to represent how random processes occur and that it will result in the bell curve. What does the presenter mean by a bell curve and when does this become more accurate?
3. At the end of the video, the presenter mentions that the diagonals represent how shapes are organized in space. What do you think he means by this?
Video 6: Pascalâ€™s Triangle â€“ Part 2
Going along with the previous video on Pascalâ€™s Triangle, the presenter goes into more detail on the diagonals of the triangle. He first explains that in order to create the triangle, you have to start with a triangle of ones and then add up two numbers on one row to create a new number on the row after. How shapes are organized in space is the basis for the diagonals of this triangle. It is explained that the third diagonal (1, 3, 6, 10, etc.) are the triangle numbers. The fourth diagonal numbers (1, 4, 10, 20, etc.) are the tetrahedral numbers of the third dimension. The fifth diagonal, however, can be described using the intersection points of different shapes. Fractals are then described by shading in the even numbers within Pascalâ€™s Triangle to form the Sierpinski Triangle, a series of triangles within each other.
Class discussion questions
1. What is Pascalâ€™s triangle? How do you find the numbers in any row of the triangle? Describe some patterns you notice within the triangle.
2. It is discussed that the entries in each diagonal represents different properties of shapes. What do you notice when you add up the numbers in each diagonal?
3. The Sierpinski Triangle is created by a series of triangles within each other. These shapes are called fractals. What other fractals can you think of?