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Stepping Stone Game Pascal’s Triangle: The Stepping Stone Game How many different routes are there from the Start stone to the Finish stone? Rules: You can only walk East or South from any stone. We will start by looking at 5 possible routes (be careful how you walk)

How many routes are there to: Pascal’s Triangle: The Stepping Stone Game

How many routes are there to: Pascal’s Triangle: The Stepping Stone Game

How many routes are there to: Pascal’s Triangle: The Stepping Stone Game

How many routes are there to: Can you see all 6 of the routes? How could you have calculated the 6 routes without the need to draw or visualise them? 1 Pascal’s Triangle: The Stepping Stone Game

How many routes are there to: Can you see all 6 of the routes? How could you have calculated the 6 routes without the need to draw or visualise them? 3 routes to this stone What do you have to do to get the number of routes to any stone? Why must there be 6 routes to here? 1 Pascal’s Triangle: The Stepping Stone Game

How many routes are there to: Can you see all 6 of the routes? How could you have calculated the 6 routes without the need to draw or visualising them? What do you have to do to get the number of routes to any stone? 4 5 Calculate the total number of routes to the finish stone Pascal’s Triangle: The Stepping Stone Game

Do you notice anything about the numbers produced by the routes through to the finish stone? The numbers are symmetrical about the diagonal line. Counting numbers Triangular numbers Tetrahedral numbers Square base Pyramid numbers Pascal’s Triangle: The Stepping Stone Game

Pascal’s Triangle Pascal’s Triangle =2 0 =2 1 =2 2 =2 3 =2 4 =2 5 =2 6 =2 7 =2 8 =2 9 =2 10 =2 11 =2 12 = Find the sum of each row. 1. Complete the rest of the triangle. 3. Write the sum as a power of 2. R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 Blaisé Pascal ( ) Counting/Natural Numbers Triangular Numbers Tetrahedral Numbers Pyramid Numbers (square base)

Fibonacci Sequence Add the numbers shown along each of the shallow diagonals to find another well known sequence of numbers The sequence first appears as a recreational maths problem about the growth in population of rabbits in book 3 of his famous work, Liber – abaci (the book of the calculator ). Fibonacci travelled extensively throughout the Middle East and elsewhere. He strongly recommended that Europeans adopt the Indo-Arabic system of numerals including the use of a symbol for zero “zephirum” The Fibonacci Sequence Leonardo of Pisa

Activity: There are 5 books on a shelf. Complete the table to find the number of ways that you can choose 0, 1, 2, 3, 4 and 5 books. ChoosePossibilitiesN o of Ways 0 books book A,B,C,D,E 5 2 books AB, AC,AD, AE BC, BD, BE CD, CE 3 books ABC, ABD, ABE ACD, ACE ADE BCD, BCE BDE, CDE 4 books ABCD, ABCE ABDE ACDE BCDE 5 books ABCDE You will need to think systematically! A B C D E Books 5

ChoosePossibilitiesN o of Ways 0 books book A,B,C,D,E 5 2 books AB, AC,AD, AE BC, BD, BE CD, CE DE 10 3 books ABC, ABD, ABE ACD, ACE ADE BCD, BCE BDE, CDE 10 4 books ABCD, ABCE ABDE ACDE BCDE 5 5 books ABCDE 1 Activity: There are 5 books on a shelf. Complete the table to find the number of ways that you can choose 0, 1, 2, 3, 4 and 5 books. You will need to think systematically! A B C D E Relate these numbers to entries in Pascal’s triangle.

A B C D E R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R Books The entries in row 5 give the number of combinations of choosing 0,1,2,3,4 and 5 books respectively. Choose 0Choose 1Choose 2 Choose 3 Choose 4 Choose 5 So if you wanted to know the probability of choosing one particular combination of 2 books at random, then the probability is 1/10

Activity: There are 6 balls in a box as shown below. Complete the table to find the number of ways that you can choose 0,1,2, 3,4,5 and 6 balls from the box. ChoosePossibilities N o of Ways 0 balls ball 1, 2, 3, 4, 5, 66 2 balls 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56 3 balls 123, 124, 125, , 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, balls 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 5 balls 12345, 12346, 12356, 12456, 13456, balls You will need to think even more systematically!

Activity: There are 6 balls in a box as shown below. Complete the table to find the number of ways that you can choose 0,1,2, 3,4,5 and 6 balls from the box. ChoosePossibilities N o of Ways 0 balls ball 1, 2, 3, 4, 5, 66 2 balls 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, balls 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, balls 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, balls 12345, 12346, 12356, 12456, 13456, balls Relate these numbers to entries in Pascal’s triangle. You will need to think even more systematically!

R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R Balls The entries in row 6 give the number of combinations of choosing 0,1,2,3,4,5 and 6 balls respectively. Choose 0Choose 1Choose 2 Choose 3 Choose 4 Choose 5 Choose 6 So if you wanted to know the probability of choosing one particular combination of 3 balls at random, then the probability is 1/20

Balls 12 R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R Balls Use Pascal’s triangle to determine how many ways there are of choosing: (a)2 balls (b) 5 balls (c) 9 balls Choose The probability of choosing one particular combination of 5 balls is 1/792 Find the combination key on a scientific calculator and evaluate. nCrnCr 12 C 2 12 C 5 12 C 9 12 choose 2 12 choose 5 12 choose 9 n choose r

Books 8 R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 Use Pascal’s triangle to determine how many ways there are of choosing: (a)3 books (b) 4 books (c) 6 books Choose Books nCrnCr 8C48C4 8C68C6 8C38C3 Now do it on a calculator. The probability of choosing one particular combination of 4 books is 1/70

5-a-side R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R nCrnCr 10 C 5 In how many ways can a 5-a-side team be chosen from a squad of 10 players? Choose 0 The probability of choosing one particular combination of 5 players is 1/252

Remember: The top row is Row 0 Use Pascal’s triangle to determine the number of combinations for each of the following selections.

Mix A B C D Choose 3 books Choose 5 players Choose 7 cards Choose 4 balls

Lottery National Lottery Jackpot? 49 balls choose 6 ?

National Lottery Jackpot? 49 balls choose 6 Choose 6Row There are ways of choosing 6 balls from a set of 49. So buying a single ticket means that the probability of a win is 1/ C 6 Row 0

Pascal’s Triangle on a Spreadsheet ABCDEFG… Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds. Spreadsheet

Pascal’s Triangle on a Spreadsheet ABCDEFG… Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds. 2. In cell B2 enter the formula =B1+A2 then fill right.

Pascal’s Triangle on a Spreadsheet ABCDEFG… Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds. 2. In cell B2 enter the formula =B1+A2 then fill right. 3. Fill down as far as row 50. Remember that row 1 on the spreadsheet corresponds to row 0 in Pascal’s triangle.

Pascal’s Triangle on a Spreadsheet ABCDEFG… Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds 2. In cell B2 enter the formula =B1+A2 then fill right. 3. Fill down as far as row 50. Remember that row 1 on the spreadsheet corresponds to row 0 in Pascal’s triangle. 4. Use a step-up procedure to see each row more clearly. Row 3 Row 6

Pascal’s Triangle on a Spreadsheet Row 49 6 th entry

Historical Note Blaisé Pascal ( ) Pascal was a French mathematician whose contemporaries and fellow countrymen included Fermat, Descartes and Mersenne. Among his many achievements was the construction of a mechanical calculating machine to help his father with his business. It was able to add and subtract only, but it was a milestone on the road to the age of computers. He corresponded with Fermat on problems that led to the new branch of mathematics called Probability Theory. The two problems that they examined concerned outcomes when throwing dice and how to divide the stake fairly amongst a group of players if a game was interrupted. These investigations led Pascal to construct tables of probabilities that eventually led to the triangle of probabilities that bears his name. Pierre de Fermat (1601 – 1675)

The Binomial Expansion Pascal used his triangle to find the coefficients in the expansion of (a + b) n The coefficients of each term correspond to the entries in the n th row. (a + b) 2 = a 2 +2ab +b 2 (a + b) 3 = a 3 +3a 2 b +3ab 2 + b 3 (a + b) 4 = a 4 + 4a 3 b +6a 2 b 2 + 4ab 3 + b 4 (a + b) 5 = a 5 + 5a 4 b +10a 3 b a 2 b 3 + 5ab 4 +b 5 (a + b) 8 = a 8 + a 7 b + a 6 b 2 + a 5 b 3 + a 4 b 4 + a 3 b 5 + a 2 b 6 + ab 7 + b 8 (a + b) 8 = a 8 + 8a 7 b + 28a 6 b a 5 b a 4 b a 3 b a 2 b 6 + 8ab 7 + b 8 Binomial Expansion

Worksheet 1 Pascal’s Triangle: The Stepping Stone Game

Worksheet 2

Worksheet

Worksheet 4 ChoosePossibilitiesN o of Ways 0 books book A,B,C,D,E 5 2 books AB, AC,AD, AE BC, BD, BE CD, CE 3 books ABC, ABD, ABE ACD, ACE ADE BCD, BCE BDE, CDE 4 books ABCD, ABCE ABDE ACDE BCDE 5 books ABCDE

Worksheet 5 ChoosePossibilities N o of Ways 0 balls ball 1, 2, 3, 4, 5, 66 2 balls 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56 3 balls 123, 124, 125, , 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, balls 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 5 balls 12345, 12346, 12356, 12456, 13456, balls