I have this written on the board:
“We want to see if figurate numbers are making sense to you. Grab the mild and medium figure numbers sheet and grab a random step number from the step number bin. Write about how many dots there would be for that step.”
Students get started and analyze the square and triangular step number that they chose. I circulate and see what their approach is. If a student is struggling I offer them a lower step number. The idea is that every student needs to be ready for this challenge eventually, but if they need more time or support, then I will give that to them. In my class, I believe that every student should get what they need to succeed. For students who are done with the activity early (I give them about 10 minutes), they can start the spicy problem (the pentagonal numbers).
I collect their work and review it for the following lesson. It is a type of formative assessment. It tells me who is ready to work on the final figurate number problem of this series (the hexagonal numbers).
“Today everyone is going to have a change to work on a spicy number pattern. You can work on it on paper, model it with shapes or build a sculpture. The goal is to analyze this pattern in any way you can. I encourage you to set up a table, graph and equation to help.” I always offer class challenges as part of our “spice rack” system. Mild is the easiest, Medium is in the middle and Spicy is extra difficult. Its important to have moments where students choose their challenge level and to have moments where everyone is encouraged to try the hardest type of challenge. The pentagonal numbers are particularly challenging. Not only is the function more complex, but the actual look of the shape requires students understand why they are considered pentagonal. This part of class is meant to give students that chance. I circulate and ask simple questions about the numbers. I might talk to students about the formulas they are creating around these numbers, but mostly I will check for basic reasoning. I want to make sure they have an efficient way of counting the dots in each figurate number (as opposed to counting each dot one at a time). I want to make sure they can explain why these are called pentagonal (this is tough on every step accept the second, which has 5 dots.) If students are ready to move past the basic questions (in most classes only a few are ready) then I remind them to sort their data into a table and graph. They need to figure out if this pattern is linear and they need to look at first and second differences (second difference/2 = coefficient of x^2 term).
This summary is really a teacher led discussion around the concept of deriving the pentagonal formula from the square and triangular numbers. I start by gathering the clues they gathered in their work and let students lead the majority of this discussion if they have a viable solution. If no students were able to write and explain a formula, then I lead the conversation by giving them a new perspective on figurate numbers. I start by rearranging the pentagonal numbers and clustering the dots into shape like a house. I take a different color marker and outline the square and triangular numbers hidden inside the house shape. If it is the third pentagonal number, I help them realize that it contains the 3rd square number and the 2nd triangular number. In algebraic language, students realize that the nth pentagonal number is the sum of the nth square and n-1 triangular number. I give them a few minutes to try and write this into a formula and then we simplify it as a class. This is a big moment for them. They are expressing their observations and intuition through the language of algebra. I take this part of class as slow as I can and constantly have them rephrase and summarize the steps involved in understanding this process. It is a big moment and can’t be missed.