Pascal's Pumpkins

Pascal's Pumpkins   encourages students to look for patterns in Pascal's Triangle.   The handout develops awareness of this important mathematical pattern through a timely seasonal activity.

Students who take higher math courses will meet Pascal again in many different applications, including probability.   This handout is an outgrowth of the Rutgers Discrete Math Institute.

Download Pascal's Pumpkins handout and solution.

Pascal Paths

How Many Ways Can You Make Snow? encourages students to apply the patterns in Pascal's Triangle.   A teacher instructional plan with mathematical background, answer and challenge is included to explain how to present this problem.   A recording sheet was also included as part of the teacher packet so that students are able to record all different solutions in a "systematic" way, which is a goal of discrete mathematics. 

• Download How many different ways can you make the word SNOW? problem, recording sheet and answer key. 

Extension:
Extend this lesson using the word winter to form Pascal paths.  How many possible different paths are there for this word?  How does it relate to Pascal's Triangle?  

How Many Winter Paths  involves systematic counting of the different Pascal paths in this arrangement of the word WINTER.   Students will be challenged to relate this activity to Pascal's triangle as they analyze their solutions to see if they have found all of the ways to reach each R in the bottom row.   A teacher instructional plan with mathematical background, answer and this challenge is included to explain how to present the problem.   The recording sheet encourages students to trace one path in each frame, making it easy to see if students use a systematic counting approach to solving the problem.

• Download How many winter paths do you see? problem, recording sheet and Pascal's link.  This PDF also contains instructional suggestions for presenting this problem to students.

 Note:  Systematic counting is an important concept in discrete mathematics.  Be sure to model using a systematic approach to finding all of the different paths, as presented in the instructional suggestions for this problem.  Students will benefit from adding this logical and systematic counting approach to their problem-solving repertoire.
 

How many ways can you spell HEART?

Heart Paths

Students will enjoy this discrete math activity that challenges them to find all of the possible ways to spell heart.  Each path starts on the H and moves down to an E, then an A, then an R, and finally a T.  You may only move to one of the two letters directly below the letter you are on. 

This activity is directly related to Pascal's Triangle, so it would be beneficial if students have previous experience with this famous pattern.  Try Pascal's Hearts which challenges students to try to figure out how to fill in the missing rows.  The downloadable file contains a notated solution which teachers may use to lead a class discussion of the solution and any patterns students can find in Pascal's Triangle. 

The downloadable Heart Paths file contains a student recording sheet so that students can record their paths in an orderly fashion.  This enables students to check that they have not duplicated a previous path.  Classroom teachers might make a transparency of this handout to be used on the overhead.  This way different students can draw paths as the class discusses the possible paths.

Be sure to check out the Mathwire Discrete Math collection for additional discrete math Pascal's path activities.  The SNOW activity download includes a teacher step-by-step orderly solution that relates the rows to the rows of Pascal's Triangle. This same process is easily extended to the 5-letter HEART paths.