Category Archives: Division

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!

Describing Division

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Division can be a tricky, but understanding some basic terms can help. There are three basic terms to describe a division problem; they are dividend, divisor, and quotient. An example of this is 24/6=4. In this example the 24 is the dividend, 6 is the divisor, and 4 is the quotient.

There are two different concepts to find the solution to a division problem. They are the sharing concept and the measurement concept. The sharing concept uses the divisor to divide the problem into number of groups, and uses the quotient to determine how many parts are in each group. A great way to visualize this is by using fruit. For example, we have a total of 24 apples, and we want to divide them equally between 6 groups. How many apples will each group receive? According to the sharing concept the divisor, 6, is used to find the number of groups, and the quotient, 4, is used to find the number of apples in each group. This means we will have 6 groups with 4 apples in each group. On the other hand, the measurement concept uses the quotient to divide the problem into number of groups, and the divisor to determine how many parts are in each group. If we use the apple problem again, there would be 4 groups with 6 apples in each group. The picture provided may help to clear up any confusion about the concepts.

Drawing pictures is a great way to practice division, but I have also found a fun game that is great for practice. The game is called Alien Munchtime; in the game it allows you to choose the fact families that you need help with. The choices include 2 through 12. After you have chosen your fact families, an alien pops up and says she needs help serving lunch. Once you click the “lets go” button the game begins. Hungry alien students come through the lunch line with a division question like 88/8 and you have to serve them the answer like 11. This is a great way to get practice with division and have fun at the same time!

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