Three-Dimensional Objects in the Classroom

April 28th, 2016

I'm currently doing Service Learning in my old fifth grader teacher, Mrs. Nowicki's, classroom. Right now in math, the class is discussing three-dimensional shapes and their names. On Friday the 22nd, Mrs. Nowicki began math and asked the class to point out and name 3D objects in the classroom. One student pointed out a bin holding filing folders and said it was a rectangular prism. Another student pointed out a stack of sno-cones in the back of the classroom used for science experiments and said it was a cone. One final student pointed out the clock on the wall and said it was a cylinder, which another student immediately disagreed with. A discussion began among all the students. Mrs. Nowicki asked the second student why she didn't think the clock was a cylinder. The student replied, "It's not tall enough to be a cylinder." Mrs. Nowicki then said, "If you took the face of the clock off and all of the gears out of it, it would be empty on the inside. Couldn't you then fill it with something else like water?" The student replied that someone could do that and said "I guess the clock is three-dimensional because you could fill it with something, so it must be a cylinder." The students then discussed as a class the different types of three-dimensional objects identified in the room and and why they could be classified as a certain object.

I really liked the way Mrs. Nowicki handled this situation. I thought it was interesting how she helped this student understand three-dimensional objects by asking if she could fill the clock with something if it were empty. It was a nice early introduction to volume, one that I think will help her students understand the concept of volume very well. I also really liked how Mrs. Nowicki asked her students to determine 3D shapes in their classroom. The shapes that they found will be examples that they can remember when they are struggling with a certain object. For example, the student who struggled to understand that the clock was a cylinder will remember what a cylinder looks like because the time was taken to explain why the clock is in fact a cylinder.

This video is useful for coming up with every day examples of three-dimensional shapes and helping students remember the names of several of the most common 3D shapes. The original source of the video can be found here.

Surface Area and the Bigger Picture

April 26th, 2016 

Surface area has always been a concept that I've really struggled with. When I first learned about it, I was in ninth grade Honors Geometry, and it was the most confusing thing I'd ever heard about. The formulas were so difficult for me to remember (in fact, I still don't like using formulas when finding the surface area of a given object), and my teacher insisted we only use the formulas in her class, so I didn't do too well on the test over the subject. I couldn't think of any shortcuts to find surface area, so I just accepted the fact that it wasn't my strong suit and hoped I would never see it again. 

Fast forward to the second semester of my sophomore year of college (also known as this past semester). As I've mentioned, I'm currently taking a math class called Investigating Geometry, Probability and Statistics, which is an Education credit. One day I walked into a class period where we were learning about surface area. A feeling of dread washed over me as I prepared to revisit one of my worst memories associated with math (seriously, I did not like surface area). However, instead of launching straight into formulas with no indication of where they came from, our teacher gave us a worksheet to work on. On the worksheet was a single problem, in which we were asked to find the surface area of a house that was a rectangular prism and had a triangular prism for the roof, with several windows, two garage doors, and one regular door. Because we weren't given any formulas for surface area, I decided to find the area of the walls and the roof, and subtract the areas of the windows and doors as needed. 

Here is the problem. 

I ended up finding the correct surface area of the building and correctly determined how much paint would be needed to paint it. I couldn't believe it took me five years to see that when you're finding the surface area of a three-dimensional object, you can find the area of each of its faces and add them together. After completing this problem, it seemed like such an obvious method of finding surface area, and I was actually mad at myself for not realizing it sooner. 

Surface area stressed me out. Because of this, I had a very narrow mindset on how to solve a surface area problem. I didn't let myself think outside the box and consider different ways to solve a problem. 

As a future educator, I've learned about the importance of being open-minded when it comes to math. Not all of my future students will solve a problem in the exact same way, and that's okay. We all think differently; we all see the world differently. Looking at the big picture and expanding your view when solving a math problem can help lead you to discover new ways to solve things you never thought you've ever come up with, whether those ways are obvious or hidden. 

If you're still struggling with surface area, this website may help you. 

Parallel Lines Cut By a Transversal

April 24th, 2014 

Parallel lines are my personal favorite kind of lines. They blow perpendicular lines right out of the water for me. I first learned about parallel lines in depth in ninth grade Honors Geometry, and they quickly became one of my favorite topics. Parallel lines, for anyone who is unsure about what they are, are lines on the same geometrical plane that do not intersect or cross at ANY point on the plane. They are always the same distance apart from each other on the plane that they are located on. 

Image found here. 

(Perpendicular lines are lines on the same place that intersect at one point to form four 90 degree angles. Not nearly as exciting.) 

Sometimes, a pair of parallel lines is intersected by a third line, which is called a transversal. 

Image found here.

When this happens, several kinds of angles are formed. These angles are: corresponding angles, alternate interior angles, alternative exterior angles, same-side interior angles, same-side exterior angles, and vertical angles. 

Before I define these angles, we need to understand that when two angles are congruent, they have the same measurement, and when two angles are supplementary, their measurements add up to 180 degrees. 

Corresponding angles are angles that are located at the same relative position at an intersection, and are congruent. In the image above, angles 1 and 5, angles 2 and 6, angles 3 and 7, and angles 4 and 8 are corresponding angle pairs. 

Alternate interior angles are angles that are located on opposite sides of the transversal and inside of the parallel lines. They are congruent. In the image above, angles 3 and 6 and angles 4 and 5 are alternate interior angle pairs. 

Alternate exterior angles are angles that are located on opposite sides of the transversal line and outside of the parallel lines. They are congruent. In the image above, angles 1 and 8 and angles 2 and 7 are alternate exterior angle pairs. 

Same-side interior angles are angles that are located on the same side of the transversal line and inside of the parallel lines. They are supplementary. In the image above, angles 3 and 5 and angles 4 and 6 are same-side interior angle pairs. 

Same-side exterior angles are angles that are located on the same side of the transversal line and outside of the parallel lines. They are supplementary. In the image above, angles 1 and 7 and angles 2 and 8 are same-side exterior angle pairs. 

Vertical angles are angles that are on opposite sides of two intersecting lines (in this case, a parallel line and the transversal line). They are congruent. In the image above, angles 1 and 4, angles 2 and 3, angles 5 and 8, and angles 6 and 7 are vertical angle pairs. 

Oftentimes when there is a pair of parallel lines cut by a transversal, we can figure out the measure of each of the angles if we know the measurement of one angle. Let's do an example using the image below (the same image above). 

Let's say we are given that the measure of angle 1 is 135 degrees. Angle 5 is congruent to angle 1 because they are corresponding angles, so we know the measure of angle 5 is 135 degrees. Angle 4 is congruent to angle 5 because they are alternate interior angles, so we know the measure of angle 4 is 135 degrees. Angle 8 is congruent to angle 4 because they are corresponding angles, so we know the measure of angle 8 is 135 degrees. 

Angle 7 is supplementary to angle 1 because they are same-side exterior angles. 180 - 135 = 45, so we know the measure of angle 7 is 45 degrees. Angle 2 is congruent to angle 7 because they are alternate exterior angles, so we know the measure of angle 2 is 45 degrees. Angle 6 is congruent to angle 2 because they are corresponding angles, so we know the measure of angle 6 is 45 degrees. Angle 3 is congruent to angle 6 because they are alternate interior angles, so we know the measure of angle 3 is 45 degrees. 

We were able to determine the measurements of each of the angles using the measurement of just angle 1. 

Remember, if two angles that are supposed to be congruent are NOT congruent, then at some point the two "parallel" lines cross, and they aren't really parallel. The same is true when two angles that are supposed to be supplementary, are NOT supplementary. 

If you still struggle with the concept of parallel lines cut by a transversal, check out this website.

The Triangle Trios

April 22nd, 2016 

I love triangles. Actually, I love all polygons, but triangles are my favorite. Three has always been one of my favorite numbers, and as weird as it sounds, there's something satisfying about looking at a three-sided figure. I also like the fact that the three angle measures in a triangle always add up to 180 degrees. Finding the measure of a missing angle isn't too difficult if you can remember that, especially if you already have the measures of the other two angles. For me, it's one of the easiest concepts in geometry.

There are three different kinds of triangles concerning side length, and three different kinds of triangles concerning angle size.


SIDE LENGTH: 

1. Equilateral - An equilateral triangle is a triangle whose side lengths are all the same size. Equilateral triangles also have same angle measures. Each angle of an equilateral triangle is 60 degrees no matter what, because 180 degrees (the sum of the measures of a triangle) divided by 3 (the number of angles a triangle has) is 60.

Image found here.

2. Isosceles - An isosceles triangle is a triangle who has two sides that are the same length and a third side that is a different length. Two of the angles of an isosceles triangle are also the same size. 

Image found here.

3. Scalene - A scalene triangle is a triangle whose three sides are all different lengths. None of the angles in a scalene triangle are the same size. 

Image found here.

ANGLE SIZE: 

1. Right - A right triangle is a triangle that has one angle that measures at 90 degrees. 

Image found here.

2. Acute - An acute triangle is a triangle whose angles all measure less than 90 degrees.

Image found here.

3. Obtuse - An obtuse triangle is a triangle that has at least one angle that measures greater than 90 degrees. 

Image found here.

If you struggle with the different classifications of triangles, click here.

Volume: It's Not Just a Setting On Your Television

April 20th, 2016

As I've mentioned before, I'm currently doing Service Learning in a fifth grade class. Today during math, the students were put into groups of three, and each group was given a section out of their math workbook that they were to study and later teach to the rest of the class. Each of the sections dealt with different concepts regarding volume. As I was going around the classroom and observing the groups study the concept they were assigned, I noticed one group struggling with the idea of finding volume using cubes (similar to what is shown in the figure below that can be found here). 

  

 I went over to them to assist them, because I remembered struggling with this same idea when I was in elementary school. One girl, Laura, said, "Miss Haley, I don't understand where the cubes come in. I know the cubes are stacked on top of each other and I know the sides of each cube are each one centimeter long but I don't understand how I can find volume using them." 

I replied, "Imagine the cubes aren't there. Imagine that instead, there's only a container that's in the shape of how the cubes are stacked. Now, count the cubes one at a time and pretend that each time you count a cube, you're putting it into the container." Laura proceeded to do this, and once she finished I asked her how many cubes she "placed" into the container. She responded, "Forty cubes." (There were five layers of cubes, and each layer consisted of two rows of four cubes.) 

I then asked Laura, "What is volume?" She replied, "Volume is the amount of something a container can hold." "Something like cubes?" I asked her. Her face lit up and she exclaimed, "Yeah! Like cubes or maybe water or something like that. I get where the cubes come into it now!" "Good job!" I told her. "What is the length of each side of a cube?" "One centimeter," she replied. "So what would the unit of volume be for this figure?" I asked. She paused for a minute and replied, "One cubic centimeter, because each side is a centimeter long and the figure is a cube." 

Volume was always a tricky subject for me in school. The formulas always confused me, and it was difficult for me to imagine filling a container with something as insubstantial and changeable as water. However, when my teachers began using tools such as cubes, it became easier for me to see where numbers were coming from. Cubes also helped me to understand why volume is measured in cubic units, just as area is measures in units squared. They're an excellent tool for teaching volume, especially in the beginning of teaching it. 

You can find more practice with volume using cubes here. 

Protractors

April 18th, 2016 

I've always hated protractors. They're some of the most confusing pieces of plastic I've ever had to use. I mean, just look at all those numbers and lines. 

When I was in the fourth grade, my teacher thought it would be a good idea to familiarize us students with protractors and teach us how to use them early. I remember struggling more than the other students to line up the protractor correctly to the angle I wanted to measure; somehow, I always ended up being at least twenty degrees off. I was convinced I would never successfully learn how to measure angles using a protractor, and I was content with telling myself I would never really use them in high school or college. 

When I was in ninth grade, my geometry teacher, Mrs. Coleman, dedicated an entire week of class to teaching us how to correctly use a protractor. During this time, she used a giant protractor to demonstrate the correct technique to use. Each of the students were required to go up to the board and use her giant protractor until he or she correctly measured an angle. I went up to the board and couldn't seem to measure the angle she had drawn correctly. Finally, I got frustrated and asked, "Mrs. Coleman, can I please sit down?" Mrs. Coleman replied, "Haley, you're overthinking it. Just relax. Remember to line up the bottom of the protractor with the side of the angle you're measuring. You can do it." I took a deep breath and remembered how I had been taught to use a protractor, and I measured the angle correctly. It was an oddly exhilarating moment, finding the right value of that angle. 

Since that day in the ninth grade, I've enjoyed using my protractor. Although I sometimes have to take a moment to remember how to line up the protractor to the angle, I no longer panic at the thought of that little piece of plastic. I'm currently taking a math class for education majors called Investigating Geometry, Probability, and Statistics, and I actually look forward to the times we have to measure angles! 

In the classroom I'm doing my Service Learning for, the students are learning how to use protractors. I was walking around today and I could see the same panic I once felt on the faces of many of the students. I sat down with as many struggling kids as I could to help them understand that protractors aren't intimidating or scary; they're actually pretty simple and kind of fun. I can't wait to teach my own students how to measure angles. 

If you're struggling with using a protractor, this  website may be helpful to you.  

About Me!

April 16th, 2016

Hello! My name is Haley Smith. I'm currently a 20 year old college student at Mesa Community College studying Elementary Education. I want someday to be a fifth grade teacher; I love the curriculum taught at that level, as well as that age group. Fifth grade was one of my all time favorite grades in school and I can't wait to teach it.

I've lived in Mesa, Arizona my entire life and love it here. I find the desert beautiful and never grow tired of it, although I do want to live somewhere else once I graduate college to experience something new. I'm the fourth of sixth children, and I have eight (very soon to be nine) nieces and nephews. I've always been around kids, and I love working with them.

I'm an avid reader, and I enjoy going to all kinds of concerts in my spare time.

I work with a ten year old boy who has Prader-Willi Syndrome, which is a genetic disorder. If you want to know more about PWS, click here.

I love my job and have considered going into Special Education on more than one occasions because I love working with special needs kids! However, I feel like my true passion is teaching the fifth grade. Who knows? Maybe one day I'll decide to go into Special Education, but for now I'm content with just the fifth grade.

I've created this blog to discuss my thoughts on geometry, as well as document any instances I come across geometry being used in a classroom or in my job. It's interesting to see how much geometry is referenced in the "real world," and just how it's being taught in classrooms. Geometry was never my strong point when I was in school, but I've grown to appreciate it. It's an important part of daily life and can be found even when it's least expected.

I'm excited to share my thoughts with you, and I hope you enjoy!