Drawing Shapes

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Drawing shapes may sound simple, but it can be challenging if you are trying to draw a regular shape. The difference between a shape and a regular shape is that a regular shape has equal sides and equal angles. For example, a pentagon may have 5 different side lengths and 5 different angle degrees, and a regular pentagon would have 5 equal side lengths and 5 equal angles.

In my math class we had to draw a regular hexagon and a regular decagon. In order to do this we had to use a protractor, which allowed us to make equal angles and straight sides. First, we determined the central angle of the shape. Central angle = 360/number of sides; therefore, to find the central angle of a hexagon we divided 360 by 6. 60 degrees is the central angle. This means that every line we draw in the hexagon must be drawn at a 60 degree angle. The picture below is the worksheet that includes the drawings of a regular hexagon and a regular decagon.

As well as finding the central angle, it is helpful if you find the vertex angle. Vertex angle = 180 – central angle. I have highlighted the vertex angle in the worksheet above. This helps to determine the degrees of every angle in the shape. If you were to draw a regular decagon you must find the central angle first (360/10 = 36 degrees). Once you have the central angle you can find the vertex angle (180-36 = 144 degrees). Understanding how to find the central angle and vertex angle can help you determine any angle of a shape; I think these two formulas are great to know!

Tricky Significant Digits

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Significant digits can be a frustrating concept if you do not understand the rules. Significant digits are the digits needed to have a precise answer. If you were to take a chemistry class, then you will be familiar with them. Recently, I began working with significant digits in my math class. I think the more you work on a concept the more you understand it. There are different rules when dealing with significant digits. Adding and subtracting numbers have different rules compared to multiplying and dividing numbers.

Adding and subtracting numbers focuses on the accuracy of the answer. The number with the least amount of decimal places is the “winning” number. For example if we are adding 5.36+2.2, the “winning” number is 2.2 because it has the least amount of decimal places. The answer to this problem is 7.56, but since we need to have 1 decimal place the answer is 7.6. In the picture below I have provided an example of a subtraction problem.

Multiplying and dividing numbers focuses on the least number of significant digits. Instead of looking at the amount of decimal places, we look at the whole number and count the amount of significant numbers; therefore, the “winning” number is the number with the least amount of significant digits. The solution to the problem should be rounded to the same amount of numbers as the “winning” number. For example if we are multiplying 82.1×3.2, the “winning” number is 3.2 because it only has 2 significant digits. In the picture above I have included a division problem and some tricky problems dealing with zeroes. I think with effort and determination anyone can understand the rules and differences between adding and subtracting numbers compared to multiplying and dividing!

Refreshing Real Numbers and Decimals

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Understanding the different types of numbers is great way to learn more about math in general. Real numbers are the main type of numbers that elementary students learn, but there are imaginary numbers that are learned during higher grade levels. Although real numbers are the main type of numbers, there are five different types of real numbers. Making a diagram of these different types is a great way to understand the differences.

There are also irrational numbers which are not in the diagram above. They are kind of in their own circle. These numbers are nonrepeating decimals that do not terminate and have no pattern. An example of an irrational number is the square root of 2. This number does not come to an end (not terminating) and has no pattern. If you type the square root of 2 into a calculator you will get something like 1.414213562 and it continues on and on. Clearly, you can see that this is an irrational number.

Adding and subtracting decimals are very similar concepts. In order to add and subtract numbers you need to line up the decimals in the problem. An example of adding two decimal numbers is 1.2+2.4. If we line up the decimal numbers it will be 3.6. Subtracting two decimal numbers like 3.6-2.1 uses the same method. You need to line up the decimals and subtract the numbers; therefore, the answer is 1.5. Lining up the decimal numbers can be hard if the numbers are super long. A great way to eliminate this problem is by lining up your decimal numbers on graph paper. I think this is a great tip that may create less confusion!

Digging Deeper into Fractions

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Did you know fractions are easier to picture in your head if you estimate? I have learned from experience that fractions truly are easier if you use estimation. An example of this is 7/8 + 1/3, this may look tricky if we actually were to work out the problem and find the same denominator. If we were to attempt this problem using estimation, we could say that 7/8 is almost equivalent to 8/8 (1), and 1/3 is a little larger than 1/4. If we add 1 +1/4, the answer is 1 1/4. Although this may not be the exact answer, it gives us a good idea of what the fraction will be. This is an easy way to compare fractions and gain a better understanding of them!

Another great way to help gain an understanding of fractions is by using fraction bars. Fraction bars are rectangular shaped pieces of paper that are divided up into different amounts. The different amounts are the fractions that the shape equals. For example, I have 5 different fraction bars: twelfths, sixths, fourths, thirds, and halves. The rectangles that represent twelfths are divided into 12 sections; the rectangles that represent sixths are divided into 6 sections, and so on. These are great tools to work out fraction problems. Finding a fraction between 2/6 and 3/6 is easier with fraction bars. This may sound easy, but it is hard to figure out if you do not have a good understanding of fractions. Let’s use fraction bars to help us out; for starters we will need a 2/6 fraction bar and a 3/6 fraction bar. If we set these fractions bars side by side, we can easily see that 5/12 is between both 2/6 and 3/6. I found this by looking for a fraction bar that was in between 2/6 and 3/6.

When dealing with whole numbers and fractions, it is easier to set them up into improper fractions. To set up an improper fraction you need to multiply the denominator by the whole number and add the numerator. We do this because we are essentially looking for a common denominator. An example of this is 4 1/2, we start with 2(denominator) x 4(whole number) = 8, then we add 1 (numerator) = 9/2. 9/2 is an improper fraction because the numerator is larger than the denominator. We can also find the improper fraction by finding the common denominator. 4 1/2 can also be written as 4/1 + 1/2; in order to find the common denominator we need to multiply the 4 by 2. Once this is done we can add the two fractions together which gives us the sum of 9/2. This idea is helpful when adding and subtracting fractions: 1 2/3 = 5/3 – 1/3= 4/3. Take your time to learn fractions because I think it is a great tool that we can use in our daily lives.

Investigating Integers

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The word integer may seem a bit simple, but as I have learned it can actually be tricky to understand. The first thing that one must know about integers is that the number can be either positive or negative. For example, a set of integers looks like this: {…-3, -2, -1, 0, 1, 2, 3…}. This shows that the negative numbers will continue infinitely as well as the positive numbers. Once the idea of integers is understood, adding and subtracting begins.

A simple way to understand the meaning of an integer is to name the number n. So if n=the number, -n=the opposite of the number, and -(-n)=the opposite of the opposite of n. The opposite of the opposite of n would simply mean the number is positive. An example of n+n is 2+2=4. In this example there are two positive numbers which result in a positive answer. An example of -n+n is -2+2=0. This shows that 2 negatives plus 2 positives cancel each other out resulting in zero. An example of -(-n)+n is -(-2)+2=4. In this example the -(-2) is equivalent to 2, so when we add 2+2 it equals 4.

A great hands on activity that helped me to understand integers was using color counters. Color counters are little square shapes; one side of the square is red and the other side is black. The red side stands for the negative integer and the black side stands for the positive integer. For example to show 2+2=4, you would have 4 black color counters. The example -2+2=0 you would have 2 blacks and 2 reds, which cancel each other out. To show -(-2)+2=4 you would simply have 4 black color counters like in the example 2+2=4. The picture provided is a great representation of the examples I have explained.

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”

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