Grade level: 4th or higher

Objectives:
1. The students will generate a pattern to solve a problem.
2. The students will demonstrate how this pattern creates Pascal’s triangle.
3. The students will find other patterns within Pascal’s triangle.

Materials:
1. Paper (grid paper if possible)
2. Pencils

Procedure:
1. Draw the following street map on the chalkboard: At the top in the middle is the main corner, or starting point. The two boundary streets run at 45-degree angles down from the starting point, one to the right, and one to the left. Inside these boundaries draw a square grid, fairly widely spaced, of lines parallel to, and ending at, these boundary streets. If this description confuses you, take a piece of graph paper, rotate it 45 degrees so that one corner is at the top, and copy a 6-by-6 square of it. I space the "streets" about 6 inches apart.
2. Pose the problem. Starting at the top of the grid, you want to know how many different paths there are from the starting point to any corner, subject to the following conditions. All paths must follow streets, and you are only permitted to move downward, either to the right, or to the left.
3. Begin with the first row of intersections below the starting point. Show that there is only one path to each of these intersections. Write the number 1 on each intersection.
4. Move on to the next row of three intersections. Ask the students how many paths there are to the leftmost intersection in that row. Ask them to describe the path by telling you which direction to take at each intersection. Starting at the top, you move down to the left, which takes you to the leftmost intersection in row two. Then you move to the left again, taking you to the intersection you are trying to reach. This path is therefore "left-left."
5. Have the students describe the paths leading to the middle intersection in row three. The two legal paths are "left-right" and "right-left." Next, have them find the only path to the rightmost intersection, "right-right."
6. Proceed to the fourth row, with four intersections. This is where the puzzle starts to get interesting. By this time, most students will realize that there is only one legal path to any intersection on the right or left boundary. Don't tell them how many paths to look for in the middle. As they describe paths, write them down using the letters L and R, for example LLR for "left-left-right." Have the students continue to look for paths until both you and they are satisfied that they have found them all. (You, of course know that the three legal paths for the second intersection in the row are LLR, LRL, and RLL.) The students may use symmetry to conclude that the number of paths to the third intersection in the row is equal to the number of paths to the second intersection, or they may need to list them (LRR, RLR, and RRL.)
7. Continue with the next row. The numbers of paths to the fifth row intersections are 1 4 6 4 1.
8. Summarize by writing in the form of a triangle the numbers discovered so far:
------------1-------------
----------1---1-----------
--------1---2---1---------
------1---3---3---1-------
----1---4---6---4---1-----
Ask the students if they see a pattern that will generate the next row of numbers. Most students quickly figure out that the first two numbers in the row are 1 5, and the last two are 5 1. Some students can guess the rule: add the two numbers above to get the next number. Thus 1+1=2, 2+1=3, 1+3=4, 3+3=6. Applying this rule, the middle numbers of the sixth row are 10. If students do not figure the rule out in a few minutes, you can tell them.  Have the students search for other patterns
9. Tell the students this pattern is called Pascal's Triangle. Tell them it comes up in a number of mathematical contexts, the two most common areas being the probability of coin tosses, and the expansion of binomials in high school algebra.

Assessment:
The students will be evaluated on their participation throughout the activity and also their ability to find and continue patterns within Pascal’s triangle.

Follow-Up Activity: Probability of Coin Tosses

Materials:
1.  Four pennies per student

Lesson Plan:
1. Ask the students if they know what "odds" are. If they don't, explain that odds are numbers that give the relative likelihood of events. For example, when tossing one penny, the chances of getting heads or tails are equally likely. Thus the odds are 1:1.
2. Test the prediction by having each student toss one coin. Count the numbers of heads and tails. Explain to the students that experimental results do not always match the theoretical probability.
3. Now ask the students to list the possible outcomes of tossing two coins. (They should come up with two heads, two tails, and one head/one tail.) Explain that order matters here, so that HT and TH add together to double the odds. Thus the odds for HH HT/TH TT are 1:2:1.
4. Test the prediction by having each student toss two coins.
5. Have the students figure out all the possible outcomes of tossing three coins. (They are HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.) Write down the theoretical odds 1:3:3:1. Test the prediction by having the students toss three coins.
6. Figure out the possible outcomes for tossing four coins. (HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT.) The theoretical odds are 1:4:6:4:1. Test the prediction by tossing four coins.
7. Write the odds in triangular format:
----1:1----
---1:2:1---
--1:3:3:1--
-1:4:6:4:1-
Ask the students where they have seen this before. They will recognize Pascal's triangle from the street map puzzle.