University of Arizona
Institute for Mathematics and Education

The Tucson Teachers' Circle Session

Session     October 8, 2008

Dr. Nate Carlson and Ginny Bohme encouraged exploration with Pascal's Triangle .


Pascal's Triangle has been named for Blaise Pascal who published his understanding of the triangle in 1665. Records dating back to the eleventh century verify that Arab and Chinese mathematicians had also known of and explored the properties of the triangle.

Although most teachers are familiar with many of the triangles properties, new discoveries are still being made by current day mathematicians.

  • Notice the patterns in each diagonal.
  • Notice the sum of the numbers in each row.
  • If you alternate the signs of the numbers in any row and then add them together, there is a pattern in the sums of the rows.
  • The “hockey-stick rule”: Begin from any 1 on the right edge of the triangle and follow the numbers left and down for any number of steps. As you go, add the numbers you encounter. When you stop, notice the number found by taking a 90-degree turn on your path to the right and stepping down one.
  • Notice patterns in prime-numbered rows.


    • The connection between the triangle and the binomial theorem was discussed along with the connection to mathematical combinations. The group explored the number of possible poker hands in a standard 5-card stud deal.



    Click here to access a partially-filled 10-row triangle.



    Some of the AIMS Standards addressed include:
    2.2.1 Determine theoretical probabilities within a given compound scenario.
    2.2.3 Use all possible outcomes (sample space) to determine the probability of dependent and independent events.
    2.3.1 Represent, analyze, and solve counting problems with or without ordering and repetitions.
    2.3.2 Solve counting problems and represent counting principles a;gebraically including factorial notation.
    3.1.1 Recognize, describe and analyze numerical sequences using words or symbols; make conjectures about these sequences. 5.2.1 Analyze a problem situation to determine the question to be answered.
    5.2.2 Identify relevent, missing, and extraneous information related to the solution of a problem.
    5.2.3 Analyze and compare mathematical strategies for efficient problem solving; select and use one or more strategies to solve a problem.
    5.2.6 Communicate the answer to a question in a problem using appropriate representations, including symbols an informal and formal mathematical language.