Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 6 Math Circles October 18 & 19, 2016 Pascal’s Triangle Warm Up Before we begin, let’s do a few exercises! 1. Rewrite the following expressions using exponents: a) 2 × 2 × 2 × 2 × 2 b) 5 × 5 × 5 c) 12 × 12 × 12 × 12 × 12 × 12 × 12 2. Solve the following and state whether it is a square, cube, or neither: a) 3 2 b) 5 × 5 × 5 c) 2 0 Who is Pascal? Blaise Pascal (1623-1662) was a French mathematician with many published works in different topics of mathematics. He is best known for his work in 1653 on Trait` e du triangle arithm` etique which is more famously known as Pascal’s Tri- angle. 1
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Faculty of Mathematics Centre for Education in
Waterloo, Ontario N2L 3G1 Mathematics and Computing
Grade 6 Math CirclesOctober 18 & 19, 2016
Pascal’s Triangle
Warm Up
Before we begin, let’s do a few exercises!
1. Rewrite the following expressions using exponents:
a) 2× 2× 2× 2× 2
b) 5× 5× 5
c) 12× 12× 12× 12× 12× 12× 12
2. Solve the following and state whether it is a square, cube, or neither:
a) 32
b) 5× 5× 5
c) 20
Who is Pascal?
Blaise Pascal (1623-1662) was a French mathematician with
many published works in different topics of mathematics. He
is best known for his work in 1653 on Traite du triangle
arithmetique which is more famously known as Pascal’s Tri-
angle.
1
Pascal’s Triangle
row 0 =⇒ 1
row 1 =⇒ 1 1
row 2 =⇒ 1 2 1
row 3 =⇒ 1 3 3 1
row 4 =⇒ 1 4 6 4 1
row 5 =⇒ 1 5 10 10 5 1
row 6 =⇒row 7 =⇒
To draw Pascal’s triangle, start with 1. In the next row, we have 1, 1. Then in the next row,
1, 2 (⇒ 1 + 1), 1 and so on. Each number within the triangle is the sum of the two numbers
in the row above. Pascal’s Triangle can go on for however long you like!
We call the first row, “row 0” followed by “row 1”, “row 2”, ...
The first number, or term, in each row is called “term 0”, followed by “term 1”, “term 2”,
and so on. For example, let’s take a closer look at row 3.
row 3 =⇒ 1 3 3 1
⇑ ⇑ ⇑ ⇑term 0 term 1 term 2 term 3
Example 1 Find the number given the row number and term number:
a) row 2, term 1 b) row 4, term 2
Example 2 Find the row number and term number of the following:
a) 5 b) 35
2
Patterns in Pascal’s Triangle
Diagonals
Check it out. The first diagonal, the term 0 of each row, are all 1s! Not very surprising. The
next diagonal are the counting numbers (1, 2, 3, 4, ...). And the third diagonal contains the
triangular number sequence.
1 ⇐ 1s
1 1 ⇐ Counting Numbers
1 2 1 ⇐ Triangular Numbers
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
The triangular number sequence is made from a pattern of dots that form a triangle. We
can find the next number of the sequence, or the next number of dots for each new triangle,
by adding a row of dots to the previous triangle and counting all the dots.