Monthly Archives: September 2011

Diving into Something New

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A numeration system is a logically organized collection of numerals. In my everyday life I use numbers to represent numerals, but numerals can also be written as symbols.  Egyptian numeration, Roman numeration, Babylonian numeration, and Mayan numeration systems all use written symbols for numerals. These systems have a unique organized collection of written symbols.

The Eqyptian and Roman numeration systems use symbols in a base-ten system. This means that each symbol stands for a power of ten (10, 10×10, 10x10x10, …). Although the Roman’s used a base-ten system, they modified it so that there are symbols for 5, 50, and 500. The Egyptian numeration system has seven symbols: the astonished man, tadpole, pointing finger, lotus flower, coiled rope, heel bone, and the stick. Each of these symbols represents a different number. For example the astonished man = 1,000,000, the tadpole = 100,000, the pointing finger = 10,000, the lotus flower = 1,000, the coiled rope = 100, the heel bone = 10, and the stick = 1. The Roman numeration system also has seven common symbols: I, V, X, L, C, D, and M; each symbol equals a different number. For example the I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1,000. Visualizing the symbols and the number they represent helps to understand the organized collection of numerals in the Egyptian and Roman numeration systems.

    The Babylonian numeration system is unlike the Egyptian and Roman systems because it uses symbols to represent base-sixty. A base-sixty system means that each symbol represents a power of sixty (60, 60×60, 60x60x60, …). There are basic symbols for 1-59; a martini glass looking symbol stands for 1 and a boomerang looking symbol stands for 10. The basic symbols for 1-59 and place value help to write numbers greater than 59. The standard way to read Babylonian symbols is from left to right. For example if there was two martini glass looking symbols on the left and one boomerang looking symbol plus five smaller martini glass looking symbols this would equal 135. In this case the two martini glass looking symbols on the left stand for 120 because each symbol alone stands for 60. This is where the concept of place value is important because now there are two meanings for the same symbol. The two larger martini glasses stand for 60, the boomerang symbol stands for 10, and the smaller martini glass symbols stand for 1.  The picture provided helps to explain the previous sentences.

The Mayan system is also very different because it uses a modified base-twenty system (20, 20×20, 20x20x20, …).  There are 20 basic symbols for the numbers 0-19; there is an oval that stands for 0, a dot that stands for 1, and a horizontal line that stands for 5. You stack the symbols vertically to add them together. For example 19 would be represented by three horizontal lines (5×3=15) plus four dots (1×4=4). To write numbers greater than 19, they used their basic symbols and place value. They wrote the symbols vertically with one symbol above another and had the powers of the base increasing from bottom to top. This means that the symbols in the bottom position stands for the number of units, the symbols on the second position stands for the number of 20’s and the third position stands for 18×20 instead of 20×20 because the Mayan calender had 18 months of 20 days each. This can be confusing because of the complex rules, but a picture of an example will help to explain.

 These numeration systems take time, effort, and patience to fully understand!

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

Fun Functions

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Functions can be a confusing topic in math; they include different sets that can be easily mixed up. A class discussion helped me to visualize and understand the true meaning of functions. The two sets that a function includes are the domain and the range. The domain is simply the first set of data and the range is the second set of data. A function must also include a rule or relation between the sets that assigns each element of the first set to exactly one element of the second set. This may sound confusing, but if you draw an arrow diagram it helps to break down the relation. The topic of our class discussion was grocery shopping; we chose items for our first set (domain) and prices for our second set (range). The items included ramen noodles, milk, eggs, and lucky charms. The prices included $0.10, $3.59, $3.59, and $2.99. An arrow diagram is used to show the relation between the items and the prices; therefore, the ramen noodles has an arrow to $0.10, the milk and eggs have an arrow to $3.59, and the lucky charms has an arrow to $2.99. The arrow diagram is a great way to picture the relation between items (domain) and prices (range).

Function notation can be another confusing topic, but it is actually a simple concept if you break down what it means. A function notation is relating an input to an output. For example there are two functions: the scanner at Meijer grocery store and the scanner at Walmart. Although they are both working as a “function” the outputs can be different. This means if I scan a gallon of milk at Meijer the output (price) can be different than the output (price) at Walmart. The function notation for this example is f: scanner at Meijer and g: scanner at Walmart; f(milk)=$3.59 and g(milk)=$3.49. The milk is the input and the $price is the output.

An activity we did in class that helped me to better understand function notation was “guess the function”. There were three volunteers that were given a function and the class had to guess what the function was.  For example a function was f(x)=3x-1, the class continued to guess the inputs and the volunteers wrote the outputs; the (x) is the input and the 3x-1 is the output. This looked something like:   f(2)=5, f(6)=17, f(4)=11, f(10)=29, and f(3)=8. The activity helped to show that function notations can be simple!

Avoiding Confusion

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 There are two different properties of equality. They are the addition or subtraction property of equality and the multiplication or division property of equality. The addition or subtraction property of equality is to add or subtract the same element on both sides of the equal sign. The multiplication or division property of equality is to multiply or divide by the same element on both sides of the equal sign. One important reminder when using the multiplication or division property of equality is that you never divide by zero.

Although the addition or subtraction property of inequality is the same concept as the property of equality, the rule for multiplication or division property of inequality is different. The rule for this is that we can multiply or divide by a positive number on both sides…no problem, but if we multiply or divide by a negative number the inequality must be reversed. These rules are used for problem solving and are fairly easy to remember.

It feels like just yesterday when I was sitting in my 3rd grade classroom learning about inequalities. My teacher taught my class the “alligator trick.” The “alligator trick” is the idea that the alligator eats the larger number, but I have learned that the “alligator trick” is not a very good technique to teach to elementary students. This is because the idea that the alligator eats the larger number can become very confusing. An example of this is -5<-8 (-5 is less than -8) or -5>-8 (.5 is greater than -8). -5 is actually greater than -8 on a number line, but because most students look at the number before the sign it can get confusing.  The picture below is also confusing because the 3 is written smaller than the 2 in size, but the 3 is actually greater than the 2 on a number line. Avoiding this trick is a good idea to avoid any confusion.

“Alligator Trick”

Learn Something New Everyday

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Venn diagrams and sets of elements can become frustrating and confusing if there is no understanding of key words, like intersection and union. The symbol for intersect is ∩ and the symbol for union is U. Intersect means there is a overlap in a set of elements or a overlap in a Venn diagram. Union means the joining in sets between a set of elements or in a Venn diagram. 

An example of a Venn diagram is separating elements into sets by shape (hexagonal) and color (red). This means in the hexagonal area of the circle the hexagonal elements that are not red will be placed and in the red area of the circle the red elements that are not hexagonal will be placed. This means that the red hexagonal elements will be placed in the center of the Venn diagram.  It is pretty easy to separate elements into these groups, but understanding what each area means can be more challenging.

In order to understand the areas of the Venn diagram it is necessary to put the elements into sets. The elements are categorized by size, color, and shape; for example, the LYH stands for Large (shape), Yellow (color), Hexagonal (shape). Hexagonal: {LYH, LBH, SYH, SBH, LRH, SRH} Red: {SRS, LRC, LRT, SRC, LRS, SRT, LRH, SRH} Hexagonal ∩ Red: {LRH, SRH} Hexagonal U Red: {LYH, LBH, SYH, SBH, SRS, LRT, SRC, LRS, SRT, LRH, SRH, LRC} These sets can clearly be explained by the Venn diagram activity I did in class. Understanding Venn diagrams and patterns can be difficult at times, but listing elements and drawing a picture are great problem solving techniques that can help to understand the question.

 

Class Activity

Visualizing Problems

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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.

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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