# Gaining Knowledge

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Adding and subtracting are math concepts most people learn at a young age. Although we start by learning simple math problems, we still develop an understanding of the concepts. For example, if there are 5 sheep on a farm and 2 get lost there are now 3 sheep on the farm; if there are 2 cows on a farm and the farmer buys 2 more cows, then there are now 4 cows on the farm. These simple math problems help students to understand the idea of adding and subtracting. As we get older we begin to learn more complex problems like 159+253=412. This may be easy to understand for most people, but may be very confusing for a student who has no idea what they are doing. How does 9+3=2? This is a question a student may ask because clearly 9+3 does not equal 2; it equals 12. In this case we use the idea of “carrying” numbers, and if it were a subtraction problem we may have to use the idea of “borrowing” numbers. What is a good way to teach a student to “carry” and “borrow” numbers? How can we help a student understand how to add and subtract numbers? I have learned great ways to answer these questions and gained a better understanding for adding and subtracting numbers.

A great way to teach a student to “carry” and “borrow” numbers is to use manipulatives. Using a manipulative is a hands-on-way to get students involved with math. For the standard adding and subtracting problems we use base-ten units; therefore, we will use base-ten pieces for a manipulative. The base-ten pieces include: flats (100 units), longs (10 units), and units (1 unit). For the problem 159+253=412, we will use 1 flat, 5 longs, and 9 units to show the number 159; we will use 2 flats, 5 longs, and 3 units to show the number 253. After we decide what base-ten pieces we need for the problem, we then begin “carrying” units. In this manipulative we show “carrying” by exchanging units for longs and longs for flats. The first exchange in this problem is to exchange 10 units for 1 long; the second and final exchange is to exchange 10 longs for 1 flat. Once the exchanges are done, we are left with 4 flats, 1 long, and 2 units. 4 flats is equivalent to 400, 1 long is equivalent to 10, and 2 units is equivalent to 2; therefore, if we add 400+10+2 we get 412. This is time consuming, but it is a great way to explain addition and subtraction. Although this is an example of addition, the same idea is used when using manipulatives to show subtraction. The pictures provided may help to clear up any confusion about adding with base-ten pieces.

Hands-on-activities are great ways to get students involved in the math problems, but there are also other options. Technology is growing very quickly; therefore, there are dozens of online websites that have interactive games and manipulatives to get students involved in math problems. This is a great way to stay updated with new opportunities available for teaching math concepts. National Library of Virtual Manipulatives is a great website that has a bunch of activities. This website allows you to download different manipulatives for free! This is a wonderful way to enhance any classroom.

# Logically Thinking

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The topic deductive reasoning is complex and very intimidating, but with patience and effort it is nothing more than using logic to solve a problem. Deductive reasoning is the process of making conclusions from one or more given statements. The statements that are used to form the conclusions are conditional statements, contrapositive statements, converse statements, and inverse statements. Conditional statement is if p, then q; contrapositive statement is if not q, then p; converse statement is if q then p; inverse statement is if not p, then not q. The letters p and q simply stand for a sentence (statement). An example will help to understand these meanings.

An example is p: The power is out at school and q: They cancel school. In this problem the conditional statement is if the power is out at school, then they will cancel school. The contrapositive statement is if they do not cancel school, then the power is not out at school. The converse statement is if they cancel school, then the power is out at school. The inverse statement is if the power is not out at school, then school is not cancelled. The conditional and contrapositive statements are both true; the converse and inverse statements are both false. The converse and inverse statements are false because for example there could be a snow day that causes school to be canceled. Deductive reasoning is just one example of logical thinking; there are many more.

Logically thinking is a great way to get your brain working! There are multiple ways to improve your skills; according to the article “Top 10 Ways to Improve Your Brain Fitness”, the number one way to improve your brain is by playing games. The games that the article included were Sudoku, crosswords, and online games. All of the games listed rely on logic and math skills; we played some logic games in class. Sudoku was one of the games we played in class. The game has 3 rules; every row must contain all the numbers (example if you are dealing with a 4×4 square then you must use the numbers 1, 2, 3, and 4) and no number can occur more than once, every column must contain all the numbers and no number can occur more than once, and every inner square must contain all the numbers and no number can occur more than once. This may sound confusing, but it gets your brain thinking logically!

Sudoku

# Understanding Problems

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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