Figuring out Figurative Numbers--a mini unit

Last week, my Honors Problem Solving Seminar students wrote a collaborative guest blog about their unit on Figurate (or figurative) numbers here.

Here are the handouts. They are mostly from a book I wrote 16 years ago called Investigating Discrete Math. Unfortunately, I did not publish the book. Fortunately, I use it a lot as a resource. 

Most of the students had already seen The Locker Problem, which I blogged about here. This is the first example of a figurate number investigation, though they may not know it at the time.

I gave them the handout below and had them work in pairs, investigating the patterns in oblong and triangular numbers. Interestingly, they can get the oblong number explicit formula and the triangular number recursive formula quickly, but they have a hard time with the oblong recursive and the triangular number explicit. 

What I like about the worksheet is it leads right into the sums of  counting and even numbers, and that the nth triangular number represents the sum of the first n counting numbers, and that the nth oblong number represents the sum of the first n even numbers.

Many of my students needed a hint on finding the explicit formula for triangular numbers. I drew a diagonal through the middle of an oblong and soon they discovered triangular numbers were half the oblong numbers. So oblong number formula is n(n+1) and triangular is that divided by 2.

Since they remembered the locker problem, we reviewed square numbers and showed that the nth square number represents the sum of the first n odd numbers.


I also talked about the story of Gauss whose teacher was trying to keep him out of trouble as a very young boy.  She told him to add the numbers from 1 to 100, thinking it would take him quite a while to do so.  As the story goes, Gauss had the answer within moments.  How did he do it?  Problems 1 and 2 demonstrate two quick ways to do it:

1.         Shown below is the list of numbers he was told to add.

            Notice it you add together the first and last number, you get 101, as shown above.

            a.  Add together the second and the second to the last number.

            b.  If you continue adding (i.e., 3 and 98), you continue getting the same number of 101.  
                 How many pairs of numbers sum to 101?

            c.  Find the answer to the problem using the information above.

            d.  Generalize this for adding together the numbers from 1 to n.

2.  Another interesting way to find the sum of the numbers from 1 to 100 is to list the numbers from 1 to 100 and then underneath it, list the numbers from 100 to 1, as shown below.  Let both sums equal X.

          X =   1 + 2 + 3 + 4 +…+ 97+98+99+100

          X = 100+99+98+97+…+  4  +3 + 2 + 1   

Now add each of the pairs of numbers that are underneath each other; for example, add together the 1 and 100, the 2 and 99, etc.  How many pairs of the same number do you get?  Note that this answer equals 2X.  Why?  Use this new method to solve for X, the sum of the numbers from 1 to 100.

Once we had all of that "figured" out, I gave them the Handshake Problem. Many knew it, but could not remember the formula. I try to encourage students not to remember the formula, but to get it from scratch.

It was pretty cool when they eventually realized that the sequence for the number of handshakes with n people: 0,1,3,6,10,15,... (i.e., for 1 person there are 0 handshakes, for 2 people there is 1 handshake, etc) is one off of the triangular numbers: 1,3,6,10,15, .... Thus, they developed the formula n/2(n-1) by substituting n - 1 into n for the triangular numbers formula. Or, they came up with the formula by realizing that for n people, each person shakes hands with n-1 people, but you need to divide that by 2 since when I shake your hand and you shake my hand, that counts only once not twice.

Next, I gave them the Pentagonal Numbers Investigation.



This was much harder, and again, I encourage you to look at the blog the students wrote here to see their thinking. But one student figured out that pentagonal numbers (by looking at the 3rd figure) is just the sum of the two nth triangular numbers + one (n-1)st triangular number - 1. Pretty cool!!

Finally, I gave them the Pizza Problem.


Students discovered the sequence to be 2,4,7,11,... which most recognized as the triangular numbers + 1, which was pretty cool. I did not do the Pyramid Numbers Investigation, but next year I will, as it is a nice intro to the Method of Finite Differences, which demonstrates how to solve explicit polynomial formulas without looking at patterns.

Here is the Pyramid Number Investigation:



And the Method of Finite Differences plus homework handout. Keep in mind that this was made 16 years ago and so the typeset is a little "old!"

Students really enjoyed that this was out of the normal "math" that they see in their other math classes. We also went to https://www.desmos.com/ and did some regression there as well. There are many ways to solve for the formulas!!

The highlight of this was twofold. One was the collaborative blog they wrote. But the second was when they were the teachers for the Algebra 1 classes, and taught them the classic handshake problem with a twist: The Fist Bump Problem, a la Sam Shah. It was amazing. I will blog about that soon.

~Lisa