Shaun Errichiello Middle School 255 SALK SCHOOL OF SCIENCE, NEW YORK, NY
8th Grade Math : Unit #6 - Math Exploratorium : Lesson #10

Skeletons and Hexagons

Objective: Students will be able to two and three dimensional interpretations of a figurate number.
Standards: MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8
Subject(s): Math
60 minutes
1 Start Up - 15 minutes

I have this written on the board:

“Yesterday we combined triangular numbers and square numbers to form pentagonal numbers. Take the figurate combinations number sheet, label the triangular and square numbers and draw the first pentagonal numbers as a combination of triangular and square numbers.”

This is a tough start, but students have already had some support in this process during the last lesson. I circulate and check for understanding as I go. I usually carry a few color pencils to encourage color coding and prompt students to go beyond simply drawing the pentagonal numbers. That process is meant as a simple command that gives all students access to the problem, but I want students to generalize with the algebra and see how they can combine like terms and simplify the equation for pentagonal numbers. When all students have had a chance to process these numbers, I share my observations and have students demonstrate the structure and formula that defines pentagonal numbers.

When they are done, I ask students to take out their skeleton towers worksheet from a previous lesson. I ask them to read out the first group of skeleton numbers and I put them into a list: 

Step number

1

2

3

4

Number of dots

1

6

15

28

 “These numbers are usually represented as a two dimension figurate number. It is one we haven’t talked about yet. What figurate number is next on our list?”

 As a class, we recall that the figurate numbers started with triangular, then square, then pentagonal and therefore the next should be hexagonal.

“That’s right, and there is no limit to the figurate numbers we study. We can always add another side to the polygon we are looking at.”

2 Constructing the Hexagonal Numbers - 30 minutes

I tell students that it is their goal to draw the first four hexagonal numbers. I leave the diagrams out for pentagonal numbers and remind them that the skeleton tower is their guide. They know that the hexagonal numbers will be structured like the pentagonal numbers and total to the same number of dots (or whatever) as the skeleton towers. I encourage groups to work together and travel with a diagram of the pentagonal numbers and skeleton tower numbers to help them. Instead of guiding them directly on their hexagonal drawings, I constantly help them describe the structure of the pentagonal number. Students need to understand that the pentagonal and hexagonal numbers are almost like concentric circles, but they often anchored around a single point.

 

For students who are finished early, they grab the spicy challenge, which is something like, “what’s next?” and “can you show how other figurate numbers combine to make the hexagonal numbers?”

3 Summary - 15 minutes

This is one of my favorite conversations in this unit. The hexagonal numbers are the most difficult of all. Students often need about 25 minutes just to accurately draw four of them. They struggle with the process of simply defining these numbers. So this discussion is the most rewarding when it works, because it is a really great learning moment for students. I can see them “getting” this topic. It is such a great switch. They go from this confused and frustrated state and then jump to a state of real excitement. This is the reaction I always hope for. Students should be excited about making progress in math. It is difficult, but that is part of the lure. Math is both challenging and rewarding. Great problems bring out this process and the hexagonal numbers certainly seem to fit in this area of great problems.

 I start by celebrating the process of drawing. Students need to know that the time it took to draw the shapes was a real success. They need to know that the act of drawing is also a form of problem solving. Drawing and doodling is not just a side of mathematics, but a critical step in understanding.

I try and build on their drawings by again reshuffling the dots in the pentagonal shape into a sort of house made from square and triangular numbers. Students remind each other that the nth Pentagonal number is made from the nth square number and the n-1th triangular number. I ask them if pentagonal numbers and other figurate numbers can be combined to form the hexagonal numbers. To help them see how this could work, I draw the second hexagonal number below the second pentagonal number. This helps students see that there is only a one dot difference. Students then hypothesize that one of the figurate numbers first steps is that missing piece. I repeat the process with the third hexagonal number. This time the difference is obviously a triangular number.

From this process, students create two working conjectures:

  1. nth hexagonal numbers = nth Pentagonal numbers + n-1th triangular numbers.
  2. nth hexagonal numbers = nth square number + 2(n-1)th triangular numbers. 

Since these two conjectures always seem to appear, I must ask, “are these two the same? How do we know?”

The intuitive argument tends to revolve around the transitive property: since a pentagonal number is the nth square number and the n-1th triangular number, then then nth Pentagonal numbers + n-1th triangular numbers  = the nth square number + n-1th triangular number + n-1th triangular number = nth square number + 2(n-1)th triangular numbers.

Of course I would never write it like this, but break it down neatly and with concrete examples from the pattern. We follow this argument with the algebra. I give the students two minutes to begin both algebraic approaches and then show as a class that since both descriptions are algebraically equivalent that they are equal. In other words, we can break down the hexagonal numbers in multiple ways. This gets students excited about higher figurate numbers. I usually get some of the following questions, and each of them seem to reflect the habits of mind that I always hope to cultivate in my classroom:

  • Do Octagonal numbers have even more ways of being solved?
  • What about a decagon?
  • What if I worked on even higher polygons?
  • How many ways could I use previous figurate numbers to make any n-gon?
  • Since all of these formulas build into each other, could we write one formula for all of them?