Category Archives: Patterns

Visualizing Problems


At first glance inductive reasoning appears as a hard concept, but it is not. I have learned that inductive reasoning is a simple vocabulary word that means forming a conclusion based on patterns, observations, examples, or experiments. Recognizing patterns is a very useful problem solving strategy. Observations, examples, and experiments are all great ways to help visualize patterns and form conclusions. An example of inductive reasoning is 2+2+4=8    4+5+7=16    8+12+4=24 all of the sums of the  problems are divisible by 4. This is an example of inductive reasoning because I used patterns and observations to solve the problem.

Counterexamples are another concept involved in problem solving. Counterexample is a very basic term used to describe an example that shows a statement to be false. An example of counterexample is that the opposite of a number is always positive. This is obviously false because the opposite of 2 is -2.

In many cases visualizing problems help to understand the problem.  Visualizing a problem can simply be writing out the answer in a chronological order. For example if a question asks how many 3 digit numbers can be written using 2, 5, and 8 only once. I start by writing all the ways that the 3 digit number can be written with 2 being the first number, then with 5 being the first number, and lastly 8 being the first number. Breaking down the problem helps to clearly see that there are 6 ways to write a 3 digit number using 2, 5, and 8 only once.



Understanding Problems


While working on my first set of text questions I realized that confusion truly is a sign of understanding. There were multiple questions where I had to stop, reread, and unclutter the information before I even began the question. This part of my problem solving is what I would call confusion. Although problem solving can take me a while to figure out, I feel relief and happiness when I find the answer. I think throughout my first couple homework problems I was able to understand the questions more because they started off confusing. The article entitled “Confusion is a Sign of Understanding” allows me to not stress when I am confused, but to continue through frustration to really understand the homework questions and feel confident with the end result.

The terms arithmetic sequence, geometric sequence, and finite difference were all intimidating at first glance, but as I began to study them they were quite simple. Arithmetic sequence is a term used to explain a common difference in a set of numbers. The common difference is simply adding or subtracting the same number from each number in the sequence. Geometric sequence is a term used to explain a common ratio in a set of numbers. Common ratio is multiplying or dividing the same number from each number in the sequence. Finite difference is a term used to explain a “hidden” pattern. The “hidden” patterns vary from problem to problem. For example the sequence: 1, 2, 4, 7 do not seem to have a pattern, but it does. Step one: take the difference of the numbers which is 1, 2, 3. Step two: take the difference of the difference which is 1, 1, 1. This was a simple finite problem, but it gives a good idea to the definition of “hidden” pattern.

I used problem solving to help me understand the terms arithmetic sequence, geometric sequence, and finite difference. I wrote down multiple examples of each term and worked them out until I truly understood them. This helped me to look at the differences between the terms. The hands on activity we did in class also helped me to understand the terms on a greater level. The hands on activity consisted of categorizing different sequences into the three terms. Although a couple of the sequences were complex, we were able to place them in the correct category. This activity was a great way to visualize the terms and understand the differences.

Class Activity