Today in math class I presented an activity. I had the class divide into groups of two, and each group was given a shape; there were squares, triangles, and rectangles. The measurements of the sides were labeled to help find the area and perimeter of each shape. One person in each group was in charge of finding the perimeter, and the other person had to find the area. I did have a formula sheet under the projector to help if the students got stuck on a problem. The class did a great job with area and perimeter and understood how to find the solutions. Before the activity, we discussed circumference and diameter.
We went on a circle scavenger hunt! This was fun because we grouped into partners and went around the college measuring circles. Each group had a ruler, pencil, piece of string, and a worksheet. To measure circumference we wrapped the string around the circle and measured the length of the string with the ruler. After we found the circumference, we found the diameter. This was done by placing the string on one side of the circle’s face and measuring the length across. Some objects that my partner and I found were a master lock, door knob, and elevator button. The scavenger hunt was fun and productive because it helped me understand how to find the circumference and diameter of a circle.
Although the scavenger hunt was productive and helped me understand the meaning of circumference and diameter of a circle, there is a formula for circumference. Circumference = ∏ x diameter or ∏ x 2radius. (∏=pie) Radius can be found by taking half of the diameter. If you have a circle with a diameter of 2, you can multiply it by ∏ and get 6.28; therefore, 6.28 is the circumference of your circle. If you have a circle with a radius of 3, you can multiply it by 2 then multiply it by ∏ and get 18.85; therefore, 18.85 is the circumference of your circle. I have included a picture of these two problems below to clear up any confusion you may have.
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!
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!
What is the difference between rational and irrational numbers? A rational number can be written in a simple fraction, and an irrational number cannot. I think rational numbers are easier to understand because they can be written in a ratio. I have provided you with an example of a rational number below. On the other hand, irrational numbers can be tricky.
Irrational numbers cannot be written in a ratio; therefore, it can be hard to determine what a number is equal to. In the picture above I have given an example of an irrational number. Nonrepeating decimals are an example of irrational numbers. 8.08008000800008… is an example because there are no repeats in the numbers.
Rational and irrational numbers are both real numbers. There are also non-real numbers; these numbers are usually not learned until higher grade levels, but they do exist. An example of a non-real number is the square root of -4. Having a negative number under a square root is a sign that the number you are dealing with is non-real. This is a good tip to know if you have to categorize numbers into different groups. I experienced this while I was working on a worksheet in my math class. Understanding the difference between real and non-real numbers helped me on my worksheet and can help you too!
There are multiple ways to look at percents. One way to look at percents is by using the “easy 10%” method. The “easy 10%” method simply means to move the decimal to the left one spot. If you are teaching this to a group of young students it may be easier to understand this method by saying you are making the number smaller. A couple of examples are 10% of 53 = 5.3 and 10% of 249 = 24.9. In both of these cases we move the decimal to the left one spot which makes the number smaller.
Understanding the “easy 10%” method can help you figure out other percentages. If something in 30% off, you need to find 10% off of the item, then multiply it by 3. An example of this could be if an item costs $25.00 and it is 30% off. We can start by finding 10% of the item which is $2.50, then multiply that by 3; this will equal $7.50. I think percentages are very important to learn because they are common in our lives. Everytime we shop we can use percentages to figure out our discounts, coupons, and the total of our purchase!
Another way to look at percentages is by using percent charts. If we are taking 50% of 200, we can set up a 10×10 grid with each unit equalling 2. By setting the grid up like this we are essentially multiplying 2×100 which makes 200. In order to get 50% of this grid we would fill in 50 units. In this case 50 units would equal 100 because each unit is equal to 2. This example is drawn in the picture provided.
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!