Discrete Math - Day 2

Today we continued to build norms for the classroom by working with figurate numbers. Figurate numbers come from ancient Greece and were an attempt to connect geometric figures to numbers that would somehow unlock secrets of nature and the universe.

The first problems looked at square numbers. The class was asked to look at these without using calculators and to try to make direct connections between the figures and the values. I used the physical connections made between Pascal's triangle and the number of names said in the Circle-Name game as an example.

The square problems are hard for students to think of from a fresh perspective since they are so familiar with square numbers. I push them to look for patterns and make connections to the figures. Typical patterns include that the number of dots in a side match the figure number and the difference in square values are odd numbers that increase by two. The discussion around the first square number shows that it is a logical extension of the patterns but in fact creates degenerate square.

Next came triangular numbers. This forces students to look at patterns and make connections. Most realize that you need to sum up values. You may see other interesting patterns and recursive formulas that crop up. Student presentations are always interesting because you typically can have students present in such a way that the story about triangular numbers builds upon itself. The discussion of what the first triangular number should be is similar to the first square number.

Most students will not be familiar with the Gaussian summation formula and this is a good time to tell an anecdote, introduce the Greek capital letter sigma as a symbol for summation, and to connect the formula to the work that was done.

The presentations of these simpler problems help develop classroom community and establish the norms and expectations as to how the classroom operates. Students are starting to realize that solutions and insights come from them.

Visit the class summary for a student's perspective of the class and to view a copy of the day's lesson slides.