Reflections on Blogging

Describe your blogging experience in this course. Do you think you will continue using your blog? Why or why not?
Prior to taking this course I had little to no experience with blogging although I had considered setting one up as a way to keep in touch with my family in the Midwest.  I had spent some time investigating using blogs in my pre-algebra class, but aside from using them for journals or personal reflections I had not really found any applications I liked.  This class showed me that I can use blogs as a way to provide additional resources to my students.  I think I will continue to use them in the future for journaling as well as for things like having my students put concepts in their own words, discuss topics from class through the comments features, and as a glossary.  Now that I know how easy it is to set up a blog, I think I will also finally get around to setting up that blog for my family.

What did you learn about yourself and your abilities or interests in Math or Algebra?
This class reaffirmed quite a few of the methods I use for teaching Algebra.  However, it also caused me to realize that I tend to focus on the processes of the course and not enough on the meaning behind the math.  Using the blogs to paraphrase these processes helped me to realize how integral understanding the ideas behind them are to actually learning them.  The best thing I derived from this class was the time it gave me to sit down and brainstorm activities that would help me to accomplish this goal.

Did you learn or discover anything you found particularly interesting through your course actives or your own internet research? Describe one interesting discovery and why you found it fascinating.
I always find the real-life applications of Algebra to be the most interesting thing about the subject.  This class showed me quite a few of those.  In particular, I found the articles on the history and applications of quadratic equations to be fascinating.  Although I was familiar with many of the items that were listed, I had never taken the time to think about how big a role quadratic equations play.  I found the discussion on the ancient Babylonians to be particularly interesting.

Do you think you will use journals with your students? Do you think you will use blogs? Why or why not?
I am currently using journaling for self-reflection and goal-setting purposes.  In addition, I am able to use a blogging format through the BlendedSchools.net blackboard platform.  I have had a large amount of success with this practice, and I plan on continuing to do this in the future.  However, I do think I will have each student set up his or her own journal for daily reflection.  This will allow them to truly reflect on what they learn each day.

Factoring Quadratics…In My Own Words

To factor a trinomial into its two factors, look at the coefficients of the three terms and take the following steps:

1) Determine the different factors of the third coefficient.  Pay attention to whether the third coefficient is positive or negative.  Remember that your factors could also be positive or negative.
2) Choose the two factors that add up to the middle coefficient.
3) Split the first term into its individual factors.  For instance, if the first term is x2, put one x in the first binomial and the other x into the second.
4) Put one of the factors that was determined in step two into the first binomial and the second factor into the second binomial.

Paraphrasing these steps for factoring really helped me to streamline the process in my own mind.  Since I have never taught factoring, I had not thought ever about how this process could be taught to middle school students.  I hope that I now have a better handle on how to make factoring clearer.

Having my students complete an exercise like this could be useful for any process that I am teaching my students.  I firmly believe that you never truly understand something until you can put it in your own words.  I would probably have them do this exercise as closure at the end of class or as part of a review.  Doing so would allow me to really know if the students understand the material for that day.

Evaluating Our Definitions: Equations and Functions

After reviewing several of my classmates’ definitions and seeing how they defined “equations” and “functions,” I feel that I could change some of the wording in mine.  One of the things teaching middle school students has taught me is the importance of using “student-friendly” vocabulary.  If students can’t understand the words that are used to define a concept, how can they be expected to understand and implement the concept itself?  My definitions now read as follows:

Equation – An equation is a statement in math that has an equal sign in between two mathematical expressions. These two expressions may or may not have variables. Algebraic equations contain variables while numeric equations have only numbers.

Function – A function is a mathematical rule that describes what mathematical operations should be used to change one number into another number. When one number is entered into the function, a different value is outputted.  There is no limit to the number of numbers that can be entered, but the number of outputs depends on the function.

I feel that the examples I provided are good examples of functions and equations.  However, I could probably provide some examples of different types of quadratic or logarithmic functions.

To determine if my students know the difference between an equation and a function, I could provide them with a list of both and have them identify whether each was an equation or function.  Students should recognize that functions always have variables, while equations may or may not.

6-B-1: The Magic of Proportions

Proportion Example #1

As a teacher, I hate taking sick days. Unfortunately, working with middle school students does not present a very sanitary environment so I’m always looking for ways to stay healthy. Drinking water is something simple I can do to help me accomplish this goal. We’ve always been told that we should drink eight 8-ounce glasses of water every day (8 x 8 = 64 ounces). I have a Nalgene bottle that I take everyday to work that holds 32 ounces. I can use the following proportion to find out how many full Nalgene bottles I will have to drink to get my daily water intake.

8 oz. glass = x Nalgene bottles daily
32 oz. bottle    8 glasses daily

32x = 8 · 8
32x = 64
32      32

x = 2 Nalgene bottles daily

Proportion Example #2

(Numbers in this problem are not actual numbers)

I just received an offer in the mail to lock in a fixed rate for my natural gas from Dominion People’s. They told me the new rate through October of 2010 would be $15 per Mcf (whatever an Mcf is). I looked at my most recent bill and noticed that I was charged $12 per Mcf, and my total bill was $70 (although this rate could change). I can use the following proportion to determine what my approximate bill would be under the locked in rate (assuming that taxes do not change…fat chance).

$12 per Mcf = $15 per Mcf
$70                   b

12b = 70 ·15
12b = 1050
12       12

b = $87.50

6-D-2: Applets

Algebra Balance Scales from the National Library of Virtual Manipulatives

http://nlvm.usu.edu/en/nav/frames_asid_324_g_3_t_2.html

This uses a seesaw to represent an equation. The student is given an equation and begins solving it by placing the appropriate terms on right and left sides of the seesaw. For instance, for the equation -3x + 2 = -8 the student would place three “-x” balloons and two positive “1″ blocks on the left side of the fulcrum (i.e. equal sign). They would then place eight “-1″ balloons on the right side of the fulcrum. Once they do this and the seesaw is balanced, they are prompted to select addition, subtraction, multiplication, or division to solve the equation. They then type in how much they would like to add, subtract, multiply, or divide from both sides. The game then removes the appropriate balloons and tiles and shows the beam to be either balanced or unbalanced. If it is unbalanced, the student knows that they did not do the same things to both sides of the equation and therefore the two sides are no longer equal. The students then continue to add, subtract, multiply, or divide until there is an “x” on one side and a certain number of “-1″ balloons or “1″ blocks on the other side.

This manipulative does not just reinforce the fact that when solving, the same thing must be done to both sides. It also reinforces the concept of a variable and like terms. I would probably not be able to use this program until the end of the year, since quite a few of the equations have a variable on both sides of the equation (something I do not teach until the end of the year). To implement this game, I would have each of my students use a laptop to practice solving equations using the program first. After several rounds, I would then have them transition to solving these equations with a pencil and paper. I would have each student write a problem for the person next to them. Students would then trade problems and solve them. This game would provide a great introduction and reinforcement for the concept of solving equations.

6-A-3: My Definition of Equations and Functions

In this post, I will provide my own definitions for the terms “equation” and “function.” These terms are not research based and are derived from my own experiences in mathematics. I’ll post a revision later.

Equation – An equation is a statement of equality between two mathematical expressions. These statements may or may not have variables. Algebraic equations contain variables while numeric equations have only numbers.

• Example 1: Numeric Equation

4(6+2) = 24 + 8

• Example 2: Algebraic Equation

7x + 3 = 24

Numeric Equation

Function – A function is a mathematical rule that describes the relationship between two numbers. When different numbers are entered into the function, different values are outputted.

• Example 3: Function

For y = 3x – 12 if x = 4, y = 0; if x = 8, y = 12; if x = 0, y = -12, etc.

References

http://openclipart.org/people/jetxee/jetxee_people_near_a_blackboard.png

http://www.clipartof.com/images/clipart/xsmall2/14118_simple_math_equation_written_in_chalk_on_a_blackboard.jpg

Non-Linear Pattern Webquest

For my webquest, I chose to explore the Fibonacci sequence and fractals. Both of these concepts provide great examples of observable mathematical patterns in nature. The following post will provide a brief description of both concepts, some interesting points and pictures from each, and ideas for how this webquest could be modified for middle school students.

Fibonacci Sequence

Fibonacci first described this now famous sequence of numbers during the early part of the 13th century. He posited this pattern by considering the following problem. Two rabbits are placed in a pen and made to reproduce. How many rabbits will there be after n months. Each consecutive number of rabbits in the sequence can be derived by finding the sum of the previous two entries. So the first two rabbits make a sum of two (1 + 1 = 2). They then reproduce and make three rabbits (1 + 2 = 3). These three rabbits can then make five if there are two pairs with one left over (2 + 3 = 5). The sequence reads 1, 1, 2, 3, 5, 8, 13, and so on.

When limited to promiscuous bunnies, this list of numbers is not very interesting. However, what is interesting are the items in nature that follow this pattern. Things like the number of ridges on a pine cone, the distance between leaves on plants, the centers of sunflower seeds, and the sides of pineapples all follow the Fibonacci sequence and something called a logarithmic spiral. The logarithmic spiral is a method for graphing points that represent the Fibonacci sequence on the Cartesian plane. When this is done, the shape followed by most plant life is followed. By orienting themselves this way, plants are able to receive the optimal amount of sunlight.

Fractals

Fractals are a relatively new concept in mathematics and science. A fractal is defined as anything that “self-symmetry.” In other words, if you zoom in on part of the object, you will find repetitions of the larger pattern. Prior to their discovery it was thought that all objects in nature have some amount of chaos. However, through close study it has been found that even the most chaotic objects seem to contain these fractals. One exercise that supports this theory is to take randomly generated numbers and plot them. After several repetitions of this activity the image of a fern “appears.” Fractals have also been observed in such “random” objects as a desert landscape, broccoli, and bacteria.

Reflective Questions

1. Were there ideas or concepts you were not familiar with? What were they?

While I was familiar with the concept of the Fibonacci sequence and its applications in nature, I had never taken the time to study fractals. I was only aware that there was such a thing, even though I wasn’t entirely sure of the definition. I was also unaware that Fibonacci may not have been the man’s actual name (it may have actually been Leonardo de Pisa), or of how he first came up with the idea of the Fibonacci sequence. I had never seen either graphing activity for the fractals (fern iterations) or the Fibonacci primes (graphing the primes derives the shape for optimal acquisition of lights by plants).
2. What images did you find particularly striking?

I found the graphing activities to be the most fascinating, because I could see the direct relationship between these mathematical patterns and the real world (kind of like John Nash in A Beautiful Mind). I also found the fractal pictures to be absolutely stunning. To see the patterns in something so small as bacteria and something so strange as a stalk of broccoli and to realize that there was math to be found there was amazing.
3. Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?

As I look around I can see quite a few repeating patterns in my own home. For instance, the globes on my lights seem to follow a certain pattern. While I cannot specifically prove that the arrangement follows the Fibonacci sequence, it would make sense to use this pattern for such purposes. If light is most efficiently obtained in this arrangement, it would make sense that it would most efficiently be projected in this arrangement. It also seems that there may be some fractals in the patterns of my carpet and in the leaves that are quickly piling up in my yard outside (someday I may actually have time to rake them).
4. How can you adapt this webquest activity for your classroom?

I would most likely have my students search first for images of the Fibonacci sequence without looking at the definition. I would then have them trace the objects on a coordinate plane and see if they could find any patterns from the ordered pairs or the points’ distances from the axes. From this activity, students should be able to work together to derive the numbers in the sequence.

For fractals, I would have students download pictures of the Sepinski triangle (seen above) and make their own in a drawing program. I would then have them download pictures of snowflakes to see if they can find fractals within the object. I feel that both of these activities would provide students with hands on experience that would help them further understand these topics.

References

http://www.daviddarling.info/encyclopedia/F/Fibonacci_sequence.html

http://www.geom.uiuc.edu/~zietlow/defp1.html

http://www.lifeinitaly.com/heroes-villains/fibonacci.asp

http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

My Reflection on Math Myths

Here are two common myths that I have encountered as a student and a teacher and my reflections on each.

There is a “math mind” – some people have it and some don’t.

One argument that people make when they encounter difficulties in math is that they just don’t have a mind for math. People reference Dr. Gardner’s theory of multiple intelligences as evidence that certain people just cannot learn math. The other side of this assumption is that if a student is not skilled in math, they are better at reading and writing. As a student, I used this as an excuse to under-perform in my English classes. I did not enjoy writing, and since I was gifted in the math and sciences I assumed this was because I had a math mind and not a writing mind. It wasn’t until later that I began to develop an appreciation for other academic areas.

I had been told about the theory of multiple intelligences in high school, but I didn’t actually study them until I took an educational psychology class in college. There I learned that while we each have a dominant intelligence that we naturally gravitate toward, we are all capable of growing the other intelligences. In fact, studies have shown that as people grow older the differences between their dominant intelligences and the others become less pronounced. This is usually described as becoming more “well-rounded.”

As a teacher, I can dispel this myth by reminding my students that hard work can overcome a lot of obstacles. I can also remind them that it is important to improve on all areas of their lives, not just ones that they enjoy. Above all, pointing out the real-life applications of other content areas will provide my students with the motivation they need to improve.

Boys are naturally better at math than girls.

I grew up in a small farming community in the Midwest. We had very small class sizes (96 in my graduating class), and not everyone went to college. Quite a few of the girls in my school knew that the expectation for them was to stay in the area, get married, and start a family. Even the girls who did take the same math classes as me in my freshman and sophomore year seemed to lack ambition. The boys in these classes dominated the discussions and propelled the class forward. It wasn’t until I took calculus as a senior that I found myself in the same math class with girls who were driven to not only learn but to excel in math. This group of seven or eight girls had decided that they were going to make it out of Starke County and go on to places like Purdue, Butler, and Evansville Universities.

In college, I found myself working shoulder to shoulder with females that were much smarter than me in the fields of mathematics and science. These women made me wonder how anyone could think that boys have a natural talent for math that girls do not have. My wife, in particular, is a brilliant mathematician who has a natural intuition for the field. I now work on a staff that has an equal amount of men and women in the math department. Clearly, the difference between the women I observed during my first years of high school math and the women I worked with during my senior year and beyond can be linked to expectations and environment. The girls who felt that they could not succeed in math were acting under the presupposition that they would not need math to fulfill the expectations that had been placed on them by the environment of our small community. Those women who succeeded realized that they wanted something more out of life. They pushed through the boundaries that were set for them and succeeded in realizing their real potential. In the end, this made it abundantly clear that there are no true differences in the potential that both men and women have to excel in mathematics.

As teachers, it is our job to help all of our students realize their potential. This includes both boys and girls. I think the best way to end the perpetuation of this myth is to show our female students real examples of other women who have excelled in this academic arena. Having female teachers is not a prerequisite for this task, but it certainly does help. When my female students try to tell me that they are not as gifted in mathematics as their male counterparts, all I need do is point to the other two female instructors who teach the same thing as I do. I also try my best to create an environment where all students feel safe to speak their mind and contribute to class discussions. Many days I make an explicit point to make sure that every single student answers at least one question. This is one way that I can hold myself accountable to ensure that I am pushing my female students to succeed as much as I push the male students.

Pascal’s Triangle

Pascal’s Triangle is a great example of a non-linear pattern that has applications in the world of algebra and equations. The numbers from each row can be used when factoring polynomials. The triangle is formed in the following way.

The outside edges of the triangle are made up of ones that form an isosceles triangle. The apex of the triangle is based on the concept of raising a variable to the zero power. Since anything raised to the zero power is one, the number one is placed at the top of the triangle and we call this the “row zero.” Within the outside edges, the individual numbers are found by taking the sum of two numbers directly above it. These numbers form symmetrical rows that read the same forward and backward.

When using this triangle, the numbers in each row represent the coefficients when a polynomial is raised to the power that corresponds to the number of that row. For instance, row four reads “1, 4, 6, 4, 1.” This means that if you are multiplying the polynomial (x + y)4, you would get the following polynomial:

x4 + 4×3y + 6×2y2 + 4xy3 + y4

Working with the Definition of Linear Patterns

Non-Traditional Patterns (Official Definition): any pattern that does not follow a repetitive or symmetrical pattern.

Linear Patterns (in 7th grade language): a group of numbers or shapes that look like they repeat in the same order.

Linear Pattern (Official Definition): any pattern that appears to contain repeating sequences of shapes or numbers.
(from: http://www.wgquirk.com/NCEEES.html )

These two linear pattern definitions differ only in their verbiage. This is an important point to attend to when introducing any new topic to students. The first definition contains “kid-friendly” terms that are age appropriate. In the second definition, students may not realize that the word “sequences” refers to the order in which the numbers or shapes appear. After researching the term a little bit more, I think I would change my “7th grade” definition to read like this:

Linear Pattern – a group of numbers or shapes that repeat in a certain order. This order follows a “rule.”

This word “rule” is used in quite a few PSSA problems, so I feel it is important to introduce that term from the very beginning. By repeating the phrase over and over, students will become familiar with it.

As far as teaching the definition goes, students need not memorize it verbatim. It may be easier for them to remember what a linear pattern is by remembering patterns that repeat (e.g. 1, 3, 5, 7, 1, 3, 5, 7… or @, $, #, @, $, #…). Examples from real life can also be used. If students can remember that a tile floor or that a brick wall are examples of linear patterns, it should be very easy for them to remember that linear patterns contain repeating sequences.