Students will be given the first few rows of Pascal's triangle, and asked what they notice. They will likely notice it's symmetric. They might also notice the counting numbers and triangular numbers. Next, ask students what the next row is, or in other words, what is the pattern to go from each row to the next? If students can't figure out how each row is generated, show them. If students haven't already done so, have them generate the next row in the triangle, to verify their understanding of the rule. After that, if students haven't already done so, ask them to locate the counting numbers and triangular numbers in Pascal's triangle. Ask your students why these diagonals must continue as the counting numbers and triangular numbers respectively. After that, have students sum each row, and ask them what they notice about the sums. They should notice each row is double the previous. Ask students to explain why. If they fail to come up with the reason, provide the explanation as seen in the first part of this video. Conclude by giving your students this challenge: In the figure below, you must start at the S, and end at the star. Each step, you can move only right or down. How many ways are there of going from the S to the star?
Give each student several copies of each figure. If students are having trouble getting started, ask them for the number of ways of getting to the square directly to the right of the S. Then allow them time to think. Then ask them for the number of ways of getting to the square directly below the S. Allow them time to think again. Then ask them for the number of ways of getting to the square one step right and one step down from the S. Continue in this way until students find the pattern. Here's the solution.
Next, give your students this challenge:
Bugs in the Garden by NRICH
Conclude by leading this investigation: