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Pascal’s Arithmetic Triangle

Pascal’s Arithmetic Triangle. Kelly Shattuck MAT 2009. Pascal’s Triangle. Triangle Terminology. Elements. Rows. Diagonals. Patterns in the Rows. Sum of the Rows The sum of the numbers in each row is equal to a power of 2 where n is the row number. Powers of 11’s

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Pascal’s Arithmetic Triangle

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  1. Pascal’s Arithmetic Triangle Kelly Shattuck MAT 2009

  2. Pascal’s Triangle

  3. Triangle Terminology Elements Rows Diagonals

  4. Patterns in the Rows • Sum of the Rows • The sum of the numbers in each row is equal to a power of 2 where n is the row number. • Powers of 11’s • If a row is made into a single number by using each element as a digit, the number is equal to a power of 11 where the power is the row number. 20 = 121 = 1+1 = 222 = 1+2+1 = 423 = 1+3+3+1 = 824 = 1+4+6+4+1 = 16

  5. Patterns in the Diagonals • Triangular Numbers • Triangular numbers can be found on the diagonal starting with row 3. where stands for the term and . 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, etc

  6. Hockey Stick Pattern • The diagonal of numbers of any length starting with any of the 1s bordering the side of the triangle and ending on any element inside the triangle is equal to the number below the last element of the diagonal not on the diagonal

  7. Now… Let’s Color!!

  8. Coloring Multiples • Even Numbers

  9. Coloring Multiples • Multiples of 3

  10. Coloring Multiples • Multiples of 4

  11. Coloring Multiples • Multiples of 7

  12. What is the probability of tossing 2 Heads if you toss 4 fair coins?

  13. Applications • It shows you the results of heads and tails when a fair, 2-sided coin is tossed Example: Toss a fair coin 4 times. 0H 1H 2H 3H 4H TTTT HTTT HHTT THHH HHHH THTT HTHT HTHH TTHT HTTH HHTH TTTH THHT HHHT THTH TTHH 1 4 6 4 1

  14. Applications • Pascal’s Triangle saves the trouble of using this tedious formula Example: 1 4 6 4 1 • Pascal’s Triangle Video

  15. Applications • The numbers in each row of the triangle are precisely the same numbers that are the coefficients of binomial expansions. Example: Expand 1 4 6 4 1

  16. Lessons and Activities • Pattern Exploration • Middle School level exploration of the triangle • Coloring Multiples Exploration • Coin Tossing Activity • Exploring theoretical and experimental probability • Pizza Problem • Discovering the number of combinations of pizza topping • Binomial Coefficients • Relates the triangle to the Binomial Theorem

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