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Pascal's Triangle

EDTE 203. Pascal's Triangle. 9 th grade math. By: Ryan Montgomery. Table of Contents. Introduction 4 Essential Question 5 Background History 6 How to Build 7 Patterns 8 Practical Uses 9 Careers 10 How to Find the Variables 11

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Pascal's Triangle

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  1. EDTE 203 Pascal's Triangle

  2. 9th grade math By: Ryan Montgomery

  3. Table of Contents • Introduction 4 • Essential Question 5 • Background • History 6 • How to Build 7 • Patterns 8 • Practical Uses 9 • Careers 10 • How to Find the Variables 11 • How to Find the Coefficients 12 • Lesson 3 13 • Assessment 14 • Contract 15-18

  4. Introduction The information you will be learning from this lesson will help you solve binomials to the nth power. The information you will be learning will allow you to find the probability of two possible outcomes.

  5. Why is Pascal’s Triangle important? Essential Question

  6. History Pascal’s Triangle has a long line of history behind it. It might be understood that because it’s named after the mathematician Blaise Pascal, that he was the one that invented it. This however is not entirely true. There were many a people that had worked with and even created a Pascal’s Triangle long before Pascal was even thought of. Don’t believe me? Click on the picture and learn more about the history of Pascal’s Triangle……. Or click here to see some more history….. Blaise Pascal

  7. How to build a Pascal’s Triangle A Pascal’s Triangle rows are determined by the power to which a binomial is taken to. For example, a binomial to the 5th power will have the 6th row’s coefficients. This is because the triangle starts as a binomial to the “0” power and increases by one each row down. With this knowledge we know that Pascal’s Triangle’s first row is just one because anything to the “0” power is just one. The next row is to the first power, and we know anything to the first power is just itself. So the next coefficients are one and one. EX: (x+y)1 = 1x + 1y Learn how to find the variables Learn how to find the coefficients

  8. Patterns in Pascal’s Triangle There are many patterns that are hidden in Pascal’s Triangle. This is one thing that makes it quite interesting. Here are a few: If you fold Pascal’s Triangle in half from top to bottom, you will see that it is symmetric. Another is the summation of the rows…. 1 =1 1+1 =2 1+2+1 =4 1+3+3+1 =8 1+4+6+4+1 =16 Notice anything unique? Each row is 2 to the power of the row. 20=1 21=2 22=4 23=8 24=16 More patterns can be seen here and here

  9. Practical Uses There are a lot of things that Pascal’s Triangle can be used for. One example is the probability of a sequence with two variables and the number of times tested. For example: If 10 coins are flipped, then the 11th row of Pascal’s triangle will give you the total of possible outcomes. Another example is solving for binomials of course. Just what we’ve been talking about. One last example is finding triangular numbers. There is a diagonal in Pascal’s Triangle that gives all the triangular numbers. It’s one of the many patterns hidden within the Pascal Triangle.

  10. Careers with Pascal’s Triangle Other than the obvious of being a high school math teacher or a college professor, there are a few jobs that also concern the knowledge of Pascal’s Triangle. It is found that any job with chemistry has a real use for Pascal’s Triangle with some of the patterns found within. Some computer programming jobs need some help with Pascal’s Triangle. A very highly recommended job is becoming an actuary. An actuary is one of the world’s best jobs. All they do is compute math problems and you can bet that they use some of Pascal’s Triangle in there!

  11. How to Find the Variables The variables are found by going left to right with the power increasing or decreasing. EX: (x+y)5 The line of Pascal’s Triangle with a binomial to this power would look like this: cx5y0 + cx4y1 + cx3y2 + cx2y3 + cx1y4 + cx0y5 Where c is just a constant that we will learn later. We know anything to the zero power is one and anything to the first power is just itself, so let’s rewrite this: cx5 + cx4y + cx3y2 + cx2y3 + cxy4 + cy5 Notice that the binomial’s first variable’s exponent starts with the binomial’s exponent “5” and decreases as it moves left. The binomial’s second variable’s exponent starts with exponent “0” and ends with the binomial’s exponent.

  12. How to Find the Coefficients The coefficients in a Pascal’s Triangle are found like what was beginning to be explained on How to Build Pascal’s Triangle back on slide 7. The coefficients are found with this simple formula: 0 power 1 1st power 1 1 2nd power 1 2 1 3rd power 1 3 3 1 The arrows are showing what is being added together. With this information, you can now find the coefficients for a binomial that is to the 17,406,034,061st power!!!

  13. Putting It All Together Now that we know how to find the coefficients and the variables of binomials to a certain degree, we can now expand the problem. So with given information from the previous slides we know that: (x+y)3= x3 + 3x2y + 3xy2 + y3 What if we were to throw in some numbers? How would those work? Let’s take this example and solve for it: (2t+8)3= ? The first number will have the 3rd power distributed to both: 23t3= 8t3 Now the next will have the coefficient times the first number to one less power, which is 2, times the second number to first power: (3)(22t2)(8)=24(4t2)=96t2 The next number will have the next coefficient times the first to, again, one less power than before and the second number’s power will increase by one: (3)(2t)(82)=6t(64)=384t Now the last number will have the first number to the zero power which is just one, and the second number to the 3rd power: 83=512 Now put them together: 8t3 + 96t2 + 384t + 512

  14. Learning Contract assessment

  15. Learning Contract • You must earn a minimum of 50 points for the unit. • The activities in the gray squares are compulsory, or required. They must be completed by everyone, and are worth a total of 25 points. • You may choose the remainder of your activities. • Know & Understand activities are worth 5 points each. • Apply & Analyze activities are worth 10 points each. • Create & Evaluate activities are worth 15 points each.

  16. Teacher’s Materials • http://pages.csam.montclair.edu/~kazimir/history.html • http://milan.milanovic.org/math/english/fibo/fibo0.html • http://science.jrank.org/pages/5059/Pascal-s-Triangle-History.html • http://people.bath.ac.uk/mds21/history.html • http://people.bath.ac.uk/mds21/applications.html • http://mathforum.org/dr.math/faq/faq.pascal.triangle.html

  17. Objectives • Students will know how to create a Pascal’s Triangle to the 16th power. • As a class, students will demonstrate how accurate Pascal’s Triangle is by taking an n number of coins and counting the number of heads. • With a poster board and two different colored markers or crayons, students will color the odd and even numbers of their triangles. • Students will write a one-page report on the history of Pascal’s Triangle.

  18. Content Standards

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