Post on 10-Feb-2016
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Pascal’s Arithmetic Triangle
Kelly ShattuckMAT 2009
Pascal’s Triangle
Rows Diagonals
ElementsTriangle Terminology
Patterns in the RowsSum 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’sIf 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
Patterns in the DiagonalsTriangular 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
Hockey Stick PatternThe 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
Now…
Let’s Color!!
Coloring MultiplesEven Numbers
Coloring MultiplesMultiples of 3
Coloring MultiplesMultiples of 4
Coloring MultiplesMultiples of 7
What is the probability of tossing 2 Heads if you toss
4 fair coins?
ApplicationsIt 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 4HTTTT HTTT HHTT THHH HHHH
THTT HTHT HTHHTTHT HTTH HHTHTTTH THHT HHHTTHTHTTHH
1 4 6 4 1
Pascal’s Triangle saves the trouble of using this tedious formula
Example: 1 4 6 41
Pascal’s Triangle Video
Applications
ApplicationsThe 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
Lessons and ActivitiesPattern Exploration
Middle School level exploration of the triangle
Coloring Multiples Exploration
Coin Tossing ActivityExploring theoretical and experimental probability
Pizza ProblemDiscovering the number of combinations of pizza topping
Binomial CoefficientsRelates the triangle to the Binomial Theorem