Content Pedagogy Math
Content Pedagogy Math EDU 654
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This 3 page Class Notes was uploaded by Ila Krajcik IV on Wednesday October 28, 2015. The Class Notes belongs to EDU 654 at Wake Forest University taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/230759/edu-654-wake-forest-university in Education and Teacher Studies at Wake Forest University.
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Date Created: 10/28/15
Pascal s Triangle and Binomial Expansion Student Teacher Lindsey Bakewell School Course Algebra 11 Date Period Materials Strategies 1 Objective Students will be able to use Pascal s Triangle to writing the coefficients of a binomial expansion 2 Warm up Begin by expanding the following binomials raised to a power a x y 0 1 b xv1 My c 2 xZlnyJHyz d x y a x3 3 ny 3xy2 313 e x yquot x4 431 612312 4mg J x 3915 x5 5x4 loxayz l xzya 5x374 395 Prompt students in looking for patterns Students may say For x y l the first term is x and the last term is 339quot Each of these terms has a coefficient of l 2 the exponents of x decrease by 1 from term to term The exponents of y increase 3 the sum of the exponents of each term is n 4 the coefficients of the terms increase and then decrease in a symmetrical pattern 3 Lesson Development After noticing these patterns students may wonder if there is a shortcut for expanding binomials Ask students to predict the terms of 311539 x6 x5y 7x43 72633 7162 x315 ys Now we need only to find the coefficients Tell the students that there is a pattern for writing binomial coefficients The first historical reference to this pattern dates back to the 10111 Century as it was found in an ancient Indian book written by Pingala At around the same time other mathematicians in Iran studied the numbers of the coefficients However it was not until 1655 that a French mathematician Blaise Pascal 16231662 compiled the pattern in its now known form of a triangle In addition to being a mathematician Pascal was physicist and a religious philosopher He also is a known for his contributions to the mechanical calculator He died very young at the age of 39 However in his young life he made a lasting impact on mathematics and science The triangle that he is known for is characterized by 1 s down the left and right side coinciding with the l coefficient for x and y After the l on the left side of the triangle the following number comes from adding the numbers above and diagonally to the left and above and diagonally to the right This process of adding from the preceding row continues to the form what is known as Pascal s Triangle Row0 gt 1 Row1 gt 11 Row2 gt 121 Row3 gt 1331 R0W4gt 14641 R0W5gt 15101051 R0W6 I 1615201561 R0W7 172135353271 Row n of Pascal s Triangle corresponds to the coef cients for l y 4 Guided Practice As a class nd the numbers for Row 8 of Pascal s Triangle Then write the binomial expansion of x 13 a Practice another example 2139 3 NOTE For a binomial difference there is an alternating pattern of signs Solution 23 3813 8x3 12x2 y 6x312 ya 5 Independent Practice From the textbook Heath Algebra 2 An Integrated Approach assign for homework p 655 29 7 36 Use Pascal s Triangle to expand the binomials 6 Assessment Use the following activity as a ticket out the door Use Pascal s Triangle to expand the binomial 55 Solution x5 25x4 250x3 1252 3125x 312539 7 Re ection Blaise Pascal httpenwikipediaorgwikiBlaise Pascal Accessed Oct 16 2008
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