Concept Byte: Pascal's Triangle

For Use With Lesson 5-7

You are standing at the corner of the grid shown below (point A1). You are only permitted to travel down or to the right. The only way to get to point A2 is by traveling down one unit. To get to point B1, travel to the right one unit. There is one way to get to point B1. There is also one way to get to point A2, however there are 2 ways to get to point B2.

Exercises

1. What are the two different ways you can get from point A1 to point B2?
2. Copy the grid. Travel only down or to the right. In how many ways can you get from point A1 to point A3?
3. Use your copy of the grid from Exercise 2. In how many ways can you get from point A1 to point C2?
4. In how many ways can you get to point B5 from your starting point, A1?
5. Mark the number of ways you can get to each point on the grid from point A1.
6. Reasoning Describe any patterns you see in the numbers on the grid.
7. Describe how the numbers increase in each row. How can you tell the next number in each row without counting the number of paths from A1?
8. 1. Make a copy of your completed grid. Color a shaded square around every number that is a multiple of 2. (You may need to extend the grid to see a pattern.)
   2. The pattern you see in part (a) resembles the Sierpinski triangle. Look at page 10, exercise 42 to see a Sierpinski triangle. Find another way to describe how to obtain this pattern.
9. Writing The completed grid is called Pascal's Triangle. Rotate your copy of the grid 45 ° 45 degrees  clockwise so that point A1 is at the top. Explain why the grid is called a triangle.
10. Find the sum of the rows of the rotated grid. (These would be diagonals in the original grid.) What pattern do you find?

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