Substitute Teacher Experiments with Combinatorics

Here is a puzzle from Just a Substitute Teacher:

Lesson plan entry: “Hand out worksheet packets and have students staple before starting. They know what to do.”

Sounds simple enough! Four numbered sheets, eight total pages, printed front and back. What could go wrong?

Do you know how many possible combinations four pieces of paper can be arranged for stapling?

This might be a fun hands-on activity to try with next year’s crew of beginning MathCounts students. (This year’s team members are all moving on.) Of course, we would need to start with a few simpler problems, like dinner menus and pizza topping combinations — or perhaps work through this lesson by Keone Hon. Then we can play around with printed pages:

With 1 printed sheet, how many ways can we orient it?

How many different 2-sheet stacks could we make?

A 3rd and 4th sheet will increase the number of possible arrangements beyond what we have class time to make — so how can we count up the combinations?

Finally, we will wrap the lesson up with a review of my Counting and Probability Basics handout.