Are there more EVEN or ODD numbers in Pascal’s Triangle?
1. What pattern do you see in the placement of even numbers? Odd numbers?
2. Which rows have no evens? How can you represent these rows in general?
3. What are “convenient stopping places” in your counting process? Why?
4. Can you find a pattern for the total number of odds at the “convenient stopping
places?” The total number of evens?
5. What is the ratio of even numbers to odd numbers after each of the “convenient
stopping places?”
6. Investigate patterns with other multiples. Try multiples of 3, 5, and 7.
Notes
To begin the investigation, have students count the number of evens and odds in
each row and keep cumulative totals. They will soon observe that rows 3, 7, 15, 31,
…,2n-1 consist of only odd numbers and are therefore convenient stopping places in
the counting process. At each of those rows all the triangles of even numbers above
are complete and counting the totals is easier.
All the even numbers form inverted triangles and that, after each row of all odd
numbers, a bigger triangle of even numbers is formed, with the previous pattern of
even numbers duplicated to the right and left of the bigger triangle of evens in the
middle.
It is beneficial to complete rows 16 – 31 to more fully investigate the pattern.
Encourage students not to spend time determining each specific number in these
rows, only generate whether the number is even or odd (even + even = even, etc.).
Have students find the sum of even numbers in each triangle. These are the
triangular numbers, which can also be represented as the sum of 1+2+3+…n, which
equals n(n+1)/2.
The cumulative sum of odd numbers at the convenient stopping places 2n-1 is 3,
and the cumulative sum of even numbers can found either by subtracting the
number of odds from the sum of all numbers in the triangle (the easier way!) or by
generalizing the pattern to the following sum:
i=1 3n-1-i[(2i – 1)(2i)/2].
Extension
Instead of looking at multiples of 2, look at multiples of 3, 5, 7, etc. Determine
patterns for each, and then generalize across primes. For example, the “convenient
stopping places” for multiples of 2 are 2n-1 and for multiples of 3 they are 3n-1.