Pascal Triangle
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This document contains information from a mathematics class. It discusses expanding binomial expressions using Pascal's triangle, worked examples of finding specific terms in expansions, and an activity for students to practice expansions. It also presents the "monkey donkey paradox" word problem and asks students to find the number of cupcakes eaten on subsequent days if the pattern continues.
Pascal Triangle
- 1. Dr. Fe L. Faz Dr. Rey S. Guevarra Dr. Meredith P. Romero Dr. Emelita D. Bautista Mathematics Teachers Pedro E. Diaz High School
- 3. Direction: Answer the following How many terms are there in the expansion of (x+y)0 What is the second term in the expansion of (a+i)3 Find the indicated term in the expansion of each given expression: 4th term; (x + y)5 2th term; (p + q)6 1st term; (x + y)2
- 4. Problem of the day!!! Monkey Donkey Paradox On the first day, monkey donkey ate 1 piece of cupcake. On the 2nd day, monkey donkey ate 1 cupcake at the morning and 1 more during nighttime for a total of 2 cup cakes. On the third day, monkey donkey ate 1 cup cake at the morning, 2 at lunch time and 1 more during night time for a total of 4 cup cakes. On the fourth day, monkey donkey ate 1 cup cake, then 3 cup cakes and 3 more, then 1 more at the end of the day, for the total of 8 cup cakes. If this pattern continues, how many cup cakes will monkey donkey eat on the 5th day? On the 6th day?
- 7. BLAISE PASCAL (1623 - 1662) A French mathematician, who discovered a pattern row known as Pascal’s Triangle of Coefficients.
- 8. PASCAL TRIANGLE used to find the coefficients of the expansion of any integral power.
- 9. The coefficients may be written in either (x + y)0 (x + y)1 (x + y)2 (x + y)3 (x + y)4 (x + y)5 (x + y)6 (x + y)7 way 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1
- 10. (x + y)0 (x + y)1 (x + y)2 (x + y)3 1 1 1 1 2 1 1 3 3 1
- 11. 1 9 36 84 126 126 84 36 9 1 0 1 2 3 4 5 6 7 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 10 45 120 210 252 210 120 45 10 10 1
- 12. Illustrative Examples: 1.What is the fourth term when (x + y)7 is expanded? Solution: 8th row: 1, 7, 21, 35, 35, 21, 7, 1 x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7 4th term: 35x4y3
- 13. 2. What is the third term in the expansion of (a + i)5 ? Solution: 6th row: 1, 5, 10, 10, 5, 1 a5 + 5a4i + 10a3i2 + 10a4i3 + 5a3i4 + i5 3rd term: 10a3i2
- 14. 3. What is the sum of the numerical coefficients when (x + y)6 is expanded? Solution: 7th row: 1, 6, 15, 20, 15, 6, 1 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 The sum of the coefficients: 64
- 15. Group Activity
- 16. Activity: Direction: Write the expanded form of each binomial expression and identify the term asked: 1. (x + y)4; second term 2. (x + y)8; fourth term 3. (x + y)10; sixth term 4. (x + y)6; third term
- 17. Summary What are the characteristics of the product of the binomial expression (x + y)n , where n represents the integral exponent?
- 18. Let’s try some challenge… Choose the letter of the best answer.
- 19. The essence of Mathematics is not to make things complicated but to make complicated things simple.
- 20. Agreement Think of this: Add the terms in each of the first five rows of the Pascal’s Triangle. Compare the sum and find a pattern for this sequence. Make a general formula to express this relation.