Analyzing the Circle of Life

Image retrieved from here.

     The circle is one of my favorite shapes. I enjoyed remembering how I learned to find the circumference and area of circles as a teen while I watched the lesson online for my math class his week. I enjoy algebra, and plugging numbers into formulas to find answers gives me a feeling of satisfaction. Knowing that, it is only reasonable that I would enjoy the concrete process of finding area and circumference of circles.

Image found here.

      I'll begin with the circumference of the circle. The circumference is the measurement of the curve that makes the circle. To find the circumference of a circle, we use the equation C=2πr, where r is the radius or distance from the center of the circle to any point on the perimeter. This means if we have a circle with a radius of 5, our circumference will be 10π.

     When we look for the area of the circle, we are measuring the space contained within the perimeter of the circle. We find this information with the equation A=πr². This means if we are going to find the area of the circle we worked with above, with a radius of 5, we will end up with an area of 25π.

Just in case you need help remembering the formulas for circumference and area of a circle, you will never forget this catchy tune!

Dilation for my old eyes...

     When I saw we were learning dilation, my mind first went to my annual eye exam and having my pupils dilated. I was hoping the math version would be more fun than the eye exam version, and I am still on the fence with my decision. They both leave me with a bit of a headache. A dilation in geometry is a translation of an image that creates an image similar to the original, but bigger. Occasionally, the resulting image will be smaller than the original, and in that case we refer to it as a reduction. In order to perform a dilation, you must have a scale and a center of dilation. For the purposes of illustrating a dilation, I found this image here.

     As you can see, the two shapes are similar in that their corresponding angles are all congruent, but they are different in that their corresponding sides are different lengths. The image gives us the center of dilation, but it does not give us the scale. To find the scale, we must look at two corresponding sides. In this example, we have a 6 as the original image and a 12 as the dilated image. Because our original image is smaller than the dilation, we know the scale factor will be greater than 1. (If we reversed the roles and our larger image was our original, our resulting image would be smaller, a reduction, and have a scale of less than 1.) To find our scale we can use the equation 6*X=12 and solve for X. This dilation must have a scale factor of 2. This means that the dilated image sides should all be twice the length of the original image. If we were looking for the center of dilation, we would need to know the scale. Since our scale is 2, that means that the distance from our center of dilation to our dilated image needs to be twice as far as the length from the center of dilation to our original image. 

     I think typing out the rules for dilation helped me to understand them a little better, maybe reading them will have the same affect for you! If not, there is an awesome video that takes dilation and explains it in its simplest form and why we need to understand it here.

In the mirror

         All those old 9th grade frustrations are resurfacing again. I did the math today, and it's been 23 years since I've had to think about any of these geometry concepts. I was able to do them okay then and now, but I have never felt confidence in explaining them, or even confidence that I fully understand them. Anyways, back to the mirror. Reflections are a funny thing. When I look in the mirror, I think I see myself, and don't even consciously consider that what I am actually seeing is a reflection. I had to alter my thoughts for this lesson.
          Within the realm of math and geometry, a reflection is thought of as a flipping or folding of a shape over the line of reflection. That means that when I am asked to perform a reflection, the most important things for me to consider are the line of reflection (or mirror line), and where the shape's vertices fall on the grid in respect to the line of reflection. This means that if I have a shape with vertices at (-5,2), (-3,-3) and (-1,2) and I want to do a reflection of this triangle across the y-axis, I'm going to need to look closely at those points. A basic rule for reflecting over the y-axis will be that each point's mirror will be as follows: (x,y) mirrored to (-x,y). That means for the triangle above, my reflection vertices would be (5,2), (3,-3), and (1,2). If I was reflecting across the x-axis, the rule would be (x,y) mirrored to (x,-y).  I found this awesome chart to help with a few of the other common mirror lines and the rules for finding new coordinates here.

Outside of these common lines of reflection, a person would have to see where the image sits in relation to the mirror line in order to find the mirror image across a different line of reflection. If all else fails, students could try folding their paper along the line of reflection to see where placement for the mirror image should be. Good luck! If you need me I'll be staring at my reflection in the mirror, admiring all my new grey hairs thanks to geometry!

I Really Love Polygons!

My brain has been in overdrive the last few days trying to learn totally new (to me) geography concepts, so I thought it would be nice to think about the most enjoyable concept in class so far. I enjoy shapes, and I was reminded in the lesson they are also called polygons. I feel like most things in this world can be broken down into multiple shapes to help me understand the more complex shapes. I happen to be a very visual oriented person. This lesson was enjoyable to me because shapes are something my mind can easily manipulate. I found I had no problem classifying triangles, but quadrilaterals had me all confused. I found this awesome parody on Youtube.com, and I thought the words were catchy and do a pretty good job of helping me to remember how to classify my quadrilaterals. 

https://www.youtube.com/watch?v=kzJ6l-Hr7lc

A quadrilateral has four sides and the interior angles will always add to 360 degrees. 

Squares and rectangles must both have ninety degree angles, but a rectangle must have two sets of parallel lines that can have differing lengths while a square has two sets of parallel lines with congruent lengths. A kite must have two pairs of congruent sides, but they will always be adjacent. A parallelogram must have two pairs of congruent sides, but they will always be opposite sides. A trapezoid may not have any congruent sides, but two sides will always be parallel. A rhombus has four sides equal in length, and the opposite sides will be parallel as well. I found an awesome chart at This Site that organizes the quadrilateral polygons into a nice flow chart for those visual learners like myself. Have fun finding and sorting the polygons in your world!

Variance and Standard Deviation

     This is one of those topics that I genuinely don't remember from my 9th grade Geometry class, or any of my other high school math courses either. I was hoping that looking up the definition in Webster's would help. I learned that variance is the fact or quality of being different, divergent or inconsistent. As a special needs mom, I can totally understand the need for different or divergent parenting, but how does this relate to math?
      The overall theme of the lesson was interpreting data. I do remember from high school that mean is the average number you get from a data set when all points are added together and then divided by the number of points. I also remembered that the median is the middle value of an organized data set, and that the mode was the number seen most often in the data set. So how does variance fit into classifying data with these concepts I do understand? The variance shows me how far each number in the set is from the mean.
The steps to finding variance are as follows
     1. Find the mean
     2. subtract the mean from each data point
     3. Square each difference from step 2
     4. Find the sum of the squares from step 3
     5. divide by the number of data points

This means if i take the teat scores of my friends, 65, 74, 79, 83, 87, and 95, I find the variance by

     1. 65+74+79+83+87+95= 80.5 (mean)
     2. (65-80.5)+(74-80.5)+(79-80.5)+(83-80.5)+(87-80.5)+(95-80.5)
          (-15.5)   +   (-6.5)   +   (-.5)     +       2.5   +    6.5     +      14.5
     3. 240.25 + 42.25 + .25 + 6.25 + 42.25 + 210.25
     4. 541.5
     5. 541.5/6= 90.25

To find how close the data points are to the mean I should do one more step and find the standard deviation. To do this I find the square root of the variance. In the case of my friend's test scores, the standard deviation is 9.5. I interpret this to mean that the vast majority of my friends scores were within 9.5 percentage points of the mean (80.5).

https://mathbitsnotebook.com/Algebra1/StatisticsData/normalgrapha.jpg

I was sure I was going to lose my mind trying to figure this concept out, but my hard work eventually paid off. On a positive note, I now feel fairly confident with standard deviations and the new grey hairs I gained figuring this concept out were easily covered with some cheap box color.

Perils of Probability

As of January 2019, I am officially a math student again. It has been a glorious 15 year break, but we are officially back together to experience the lows of confusion and the highs of understanding, and the resulting madness that can occur somewhere in between.

As it has been so long since I participated in a math class, it would be an understatement to say I felt like I was drowning in confusion before the end of week one, PROBABILITY. After I struggled through the homework, where tears may or may not have been shed, I dove into an assignment called Toads and Vines. This happened to be an online game much like Chutes and Ladders, only you need to answer probability questions relating to the game board correctly to earn a roll of the dice or a spin of the spinner. Because I had persevered with the homework, I had what I thought was a decent understanding of probability. Level one began with a spinner equal sections numbered one through three. With questions like, "What is the probability that you will end on the vine?", and knowing there was only one spot I could possibly hit this turn to land on a vine, I rocked the level. When I won (reached the end of the game board), I was almost arrogant in my ability to answer probability questions.

Lucky for me, my wonderful teacher assigned level one AND level two. I was super excited to show the game how smart I am! Level two began not with a spinner, but with two dice. I really thought it wouldn't be a big deal because I totally understand probability at this point. I could not have been more wrong. The first question was "How many outcomes are possible if you roll two dice?" Me, in my arrogance didn't even think twice before thinking "each dice has six possibilities, so there must be twelve possible outcomes." What was I thinking? That began level two with an incorrect answer and a very humbled student. After continuing to get the next few questions wrong, I finally decided it was time to get back to learning. I immediately hit Google in an attempt to find a video that might shed some new light on the concept for me. I found this gem!

Video found on YouTube.com, but produced by the Kahn Academy

After watching the video, I realized that addition was the wrong kind of problem for determining the total possible outcomes for two dice, and that the Toads and Vines game level two had a chart of all possible sample set at the bottom. Through trial and error I learned to use the chart, count the number of possible outcomes per each individual question, and use my prior understanding that the probability is the number of favorable outcomes over the total possible outcomes. I was eventually able to finish level two. We won't say how long it took, or how many questions I missed, only that I finished and won. I gained a few new grey hairs, but haven't completely lost my mind yet.