Discrete Math - Day 4

Today we looked at the problem of adding the squares of two consecutive triangular numbers. Students were having trouble see patterns. I wrote out the first five sums that students found so that all of the students were clear on the values being squared and summed. I then asked students to look for patterns and connections in the first nine sums. This provides students with three values that connect to each other. Most students still struggled with making connections but one student noticed that the sum of T₁² and T₂² was the same as T₄ and that the sum of T₂² and T₃² was the same as T₉. Someone asked if the sum of T₃² and T₄² equaled a square number and if it was T₁₆. Students verified this and the next one by using the Gaussian summation formula to verify. The use of the Gaussian summation formula came from the students and it was encouraging to see them make use of this in reverse to validate the conjecture.

I briefly mentioned the idea of polygonal numbers and figurate numbers as being numbers that could be represented by dots arranged in the form of figures.

We then moved to pentagonal numbers. I asked students what a pentagonal number (see Wolfram Math - Pentagonal Number for more information) would look like and gave them a couple of minutes to think about how they might arrangement dots. Students were reluctant to share so I said that I'd get things going. I drew a single dot, labeled it as P₁ and set it equal to 1. I said that now that I got things going someone could draw the next figure. Someone came up and drew five dots in the shape of a pentagon, labeled it as P₂ and set it equal to five. A student had used their smartphone to google the next result of P₃ = 12. We were a bit more challenged to draw the figure but finally had it drawn correctly. We also managed to get P₄ drawn and get its value of 22.

The challenge was to find a general formula for P_n, the n^th pentagonal number. I asked students to look for patterns and try to make connections. A student found a relationship between the value of n and the difference between P_n=1 and P_n. Specifically, he found P_n+1 = P_n + 3n + 1. This provided an opportunity to discuss recursive formulas. I related this back to the triangular numbers and the idea that most had found that T_n+1 = T_n + n + 1. I also mentioned that theoretically every recursive formula could be converted to a closed formula, such as T_n = n(n+1)/2. The next challenge was to find the closed formula for P_n.

Students have little experience converting non-linear relationships into formulas. In addition, even with linear expressions, many students are unsure and hesitant about creating a formula. I use this as a stepping off point to explore finite differences. It leads to many aha moments and provides a foundation for students to develop formulas from their table values.

Today was the first day that a student asked about when figurate numbers are used. I told them that the purpose of use working with these is for them to become better at reasoning and problem solving. These are unfamiliar items and they need to be able to work through the uncertainty, making connections to things they already know and to find patterns that they can explore and try to make sense of in the context of the problem. They seemed comfortable with the response. What they will come to realize is that this material is some of the easier material they will face and that building their confidence and abilities to explore, reason, and persist in problem solving now will serve them well as we dive into more complex problems.

Visit the class summary for a student's perspective of the day's lesson.