Every couple of weeks or so, Danny will give us a
problem-solving exercise to complete (or not) in class. Almost every week I try
the problems, work in circles for a while, and then eventually wait it out
until we can look at a hint, or Danny presents a possible solution to this
class. This week, however, I feel like I was finally able to make good progress
in my solution without being guided through it. 'BUT HOW DID YOU DO IT?' I hear
you ask. Worry not my friend, I'll walk you through my process of solving the
problem.
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| The original problem |
Understanding the
Problem
I took
several approaches to trying to understand the problem presented to me: drawing
diagrams, discussing the problem with my partners, and rewording the problem in
my own words. This gave me a clear goal that I needed to reach, and allowed me
a clear vision of what my solution was supposed to look like (for example, this
question required some sort of equation or set of equations that represented a
pattern of the number of squares the diagonal line passed through).
Devising a Plan
This
step can be very easy to overlook, but it's definitely important to plan out
your work before doing anything. Randomly guessing and checking can be
extremely time consuming, and was likely the reason I spent the last couple of
problem-solving classes making circles around nothing. This problem felt like it was closer to home, as the analysis of the properties of shapes was familiar territory to me. Based on my past experience with analyzing properties of shapes, I started devising a general plan.
What I decided to do was
to begin by drawing lines on the provided grid and recording the results in a
chart. My method of recording data would be to change single variables at a
time, and observing what happened when one dimension changed. Following this
step I would either expand the grid and collect more data, or attempt to
identify a pattern given the data (I was anticipating having to do the former).
What actually happened when executing the plan was a different story...
Executing the Plan
Things don't always exactly go as planned...and that's not
necessarily always a bad thing. Upon creating the chart and attempting to
analyze the limited data, I started noticing... pretty much no discernible pattern that applied across all values of m and n. At this point, I thought it was time to split up the problem into cases, as I started to consider the possibility that the number of squares depending on the property of the dimensions (i.e. odd numbers, even numbers, etc.). After further analysis and some amounts of experimentation, I came up with the following equations:
When n = m + 1 or m = n + 1 (or there is one even and one odd value), # of squares = n + m - 1
When n = m, # of squares = n or m
When n and m are even and not equal, # of squares = n + m - 2
When n and m are odd and not equal, # of squares = n + m - 1
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| This is my work and thought process for the solutions I came up with |
Looking Back
Looking back at my thought process and solutions, I'm actually quite proud and happy with my work. I do think that discussing my thoughts with other students would have allowed me to come to my solution quicker, and perhaps would have even made my solution and more efficient. I'd love to hear what other people did, and any alternate solutions people came up with. If anyone has any advice or tips they'd like to offer based on my solution, I'd love to hear them!
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