Last December 2016, I was chosen by the Math teachers of our school along with my two schoolmates to attend a 3-day DIVISION MATHEMATICAL INVESTIGATION (MI) TRAINING-WORKSHOP (SECONDARY MATHEMATICS) at Pavia National High School. On the first day at the workshop, I listened very well to the lessons discussed especially the how's of conducting a mathematical investigation. Before the first day of the workshop ends, we were told to prepare a mathematical investigation from the copy of situations or handouts they gave us. This mathematical investigation will be done in the school during the second day.

After looking at the situations thoroughly, we decided to investigate pentagonal numbers. Today, I'm going to show you how we did it so that you may also have the knowledge to learn how to conduct a mathematical investigation.

Introduction

So for introduction, you need to cite what got you interested to investigate this situation/problem and a little background about the situation/problem. And here is our introduction,

"A pentagonal number is a type of figurate number represented by an equal number of dots on each side that form the pentagons and the pentagons are overlaid such that they share one vertex."

As you can see it very simple (without why we got interested) because honestly we just copied that with a little paraphrasing from the handouts they gave us. I can remember that the three of us that time didn't have any mobile phones so we just used the resources available for us.

Statement of the problem

After creating your introduction, you need to form your statement of the problem. These problem/s should be clearly defined and generated from the situation and as much as possible, a problem that is challenging.

Here is our statement of the problem:

"What is the total number of points needed to form an n-pentagonal array?"

For our statement of the problem, we wanted to find a pattern which will enable us to find the number of points needed to form a 116th pentagonal array (for example) without drawing or counting them.

Conjecture

For conjecture, you only need to present or state a concise mathematical statement clearly generated from the data you gathered.

Like this:

"The total number of points needed to form an n-pentagonal array is given by Pn = n(3n-1)/2"

Remember that our conjecture is only an opinion or conclusion derived with incomplete data or information and without concrete proof or evidence meaning it is not yet accepted to be true.

Data gathering

For data gathering, you only need to gather everything you need for the investigation and they way you present it should be well-organized. In our case, we just counted all of the points needed to form an n-pentagonal array and use a table. Like this:

Deriving Formula

For deriving formula, you only need to show how you were able to come up with that. State any relationship between numbers or anything that helped you derive your formula. Here's an example of how we derived our formula.

The image you can see above is an example of a Finite Difference Chart used generate a quadratic polynomial. We are going to use this to make a conjecture regarding the relationship between the data we collected and this Finite Difference Chart.

As you can notice, we replaced the values under Tn=an^2+bn+c with the data we gathered which 1,5,12,22, and 35. Then we took their differences just like in the Finite Difference chart.

After doing this, we only need to equate the values we obtained to the Finite Difference chart starting on the right side.

Given 2a = 3 then a = 3/2

Given 3a + b = 4 and a= 3/2 then b = -1/2

Given a + b + c = 1 and a= 3/2, b= -1/2 then c = 0
Substituting the values of a,b, and c in Pn = an^2 + bn + c;

Pn = 3/2 n^2-1/2n

Pn = 3n^2-n/2 or Pn = n(3n-1)/2

Justification of Conjecture

In writing your justification of conjecture, as much as possible, provide a deductive justification or a formal proof, the mathematical ideas involved are accurate, and statements are logically presented.

In our case, to justify our conjecture, we used the same number of pentagonal arrays (n) from the data we collected to test our conjecture.

The investigators use the formula Pn = n(3n-1)/2 to test whether the conjecture can be accepted or not.

After proving that the results of using the formula are the same to the data gathered, we need to test it in other examples to check if it is really correct.

Summary

In writing the summary, you need to provide a critical review of the investigation you conducted and it should be long enough to highlight the major ideas and phases of the investigation, yet short enough to be manageable in a limited time. This is our very short summary that we submitted,

The total number of points needed to form an n-pentagonal array is denoted by Pn= n(3n-1)/2.

Final thoughts

At the end of the second day, we were able to finish our investigation with joy in our hearts for conquering such challenge. We presented our outputs the next day and then get some useful feedbacks that we may be able to use in our next journey.

Well, I hope you also learned a lot from me though honestly I am not good at teaching things but still I hope I was able to share my knowledge with you. Always smile and have fun every time there is new learning.

God bless you all!

I would like to thank my fellow mathematical investigators for giving me the permission to post our output.