Reflections on Blogging

Describe your blogging experience in this course. Do you think you will continue using your blog? Why or why not?

I preferred using the forums since everything was in one place and it was easier to figure out if there were updates. The blog seemed redundant and not as user friendly. For these reasons, I will not use my blog in the future. I think it is useful to write about your thoughts and have others provide support and suggestions, but this blog does not seem like the best place to do that, mostly because few people would ever see it.

What did you learn about yourself and your abilities or interests in Math or Algebra?

This class was very helpful, especially as I see others write about how they struggle with the same things that I do. At least I’m not alone. And there have been some great suggestions for those problem areas we all have in common.

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 really liked the activities using blocks which require students to create tables and define functions. I have used this with some of my students that I have been tutoring over the summer and I believe it is a great way to visualize how things change and how that relates to an equation. I will definitely continue to use this activity with my regular classes and expand on it to include graphing.

Do you think you will use journals with your students? Do you think you will use blogs? Why or why not?

I will use journals…or at least journal entries as a part of their homework. I’ve been thinking a lot about how to revamp my homework policies and one thing I’d like to do is eliminate daily homework. I have found that it really only benefits a few students. I will be assigning only one big homework each week which will require the students to write. I liked the idea of students keeping a journal, but I’m worried that I’ll run out of steam and by November forget about it all together. With the homework, I can start slow and do more with it in the following years.

Factoring Quadratics in your own words

Use these steps to factor a quadratic, as long as the x² term has a coefficient of 1 (i.e. none at all). First, look at the last number that is all by itself. Think of all of the factors that make that number. Which of these factors, when added together, give you the coefficient for the x term? Once you find them, these are the numbers you will use in your binomials. Write (x + first number)(x + second number) and you’re done.

 I feel like I’ve taught factoring quadratics so often that my explanation here sounds so familiar. I don’t feel that this exercise helped me gain any new insights about factoring. However, I think it is beneficial to see what the simplest way is that you can present something. I don’t like teaching by only providing the algorithm, but some students still need this in the end as that is the only way they can be successful.

Evaluating our Definitions: Equations and Functions

• After reviewing your classmates post, would you alter your definition? Why or why not? Would you provide different examples?
• How can you evaluate whether or not your students grasped the difference between the two?

I believe that almost everyone had similar definitions for equation and function. Many used some of the same kind of examples and graphics to further illustrate these concepts. However, no one (including myself) gave an example of what a function is not. I think it is easier to understand what a function is if you also focus on examples that are not functions. It may help to show one of these anti-examples where one input value has two different output values. Of course, I think more people would have included these examples in their definitions if we were allowed to look up other references. It certainly did not occur to me when I was making my first draft.

 I think it would be difficult to teach my students the difference between equations and functions in one lesson. In my curriculum, functions are introduced before we talk about equations that are not functions. The best I can do is help my students understand the difference between functions and non-functions with tables and graphs. They would have difficulty looking at some equations (y² = x) and deciding whether or not it is a function.

 I also think my function may be a bit too limiting where it sounds like x and y are the only variables we use. Some textbooks also make this mistake.

Exploring the World of Applets

I believe that the scatter plot applet in the National Library of Virtual Manipulatives would be very useful for my class. When I’ve worked on scatter plots in the past, I had to spend some time teaching my students how to use some of the more difficult applications on Microsoft Excel. While I believe that learning Excel is very important for the student, it always felt like I had to spend much more time teaching Excel than teaching scatter plots. Unlike Excel, this applet is very easy to use. All you have to do is input your data. The applet will plot the data, create a line of best fit, give the equation for that line, and provide the correlation coefficient. It even automatically sets the scale based on the data you have inputted. This allows the student to quickly create and study multiple scatter plots. I believe this is a much better way to study scatter plots as compared to creating them by hand.

5B1 The Magic of Proportions (and Cheese snacks)

Fatty gets hungry for Cheez-It’s

 I am currently on a diet and I am watching my calories. I’m starting to get hungry but it’s only 4:00. Two whole hours until dinner! I figure out that I can have a snack with about 100 calories. I could have bought the 100 calorie snack packs at the grocery store, but I figured out that the unit price wasn’t as good as just buying a large box. So I’m stuck with the Cheez-It’s that are not individually packaged into nice snack sized pouches. If I want only 100 calories, I’m going to have to figure out how many Cheez-It’s are needed by reading the nutrition facts.

 It says that a serving (29 crackers) is equal to 130 calories. I need to know how many crackers are 100 calories. I set up my proportion making sure that the number of crackers is on the top and the number of calories is on the bottom for each ratio (see below).

 

When I cross-multiply, I come up with the new equation:

2900 = 130x

 Finally, I divide each side by 130 to get 22.3 crackers. I look in my box for 22 crackers and take comfort in knowing that I will stay under 100 calories by not eating that 0.3 cracker, which I’m sure exists somewhere at the bottom of the box.

 

Late night snack

 I’m hungry again. This time, cheese flavored crackers won’t cut it. I need some actual cheese. I bought a block of cooper sharp the other day and I have yet to dice it up. The block weighs 0.75 pounds and the nutrition facts say that a serving size of 1 ounce is 100 calories.

Before I can start cutting, I have to figure out the total number of calories in the whole block. I know that 1 pound is equal to 16 ounces. I need to know how many ounces are in 0.75 pounds. For this, I set up a proportion. I keep pounds on top and ounces on the bottom:

 

After a quick cross-multiplication, I figure out that I have 12 ounces of cheese. (I cheated a bit to figure this out without using decimals. I knew that 0.75 was the same as ¾. All I had to do was multiply 16 by ¾. First, 16/4 is 4 and 4 times 3 is 12.) Since each ounce is 100 calories, the total number of calories in the chunk is 12 times 100: 1200 calories.

 Now I’m not sure how much cheese I want…definitely less than 100 calories. I figure that I’ll just chop it up until the chunks are a reasonable size. I start by cutting the block in half to get two pieces. Way too big! I then cut each of these blocks in half to get four pieces. Still too big. I chop these up into eight pieces. Closer, but still too big. It would look like a decent size if I cut the remaining strips into thirds. So I do so and end up with 24 pieces of cheese. Due to the Law of Conservation of Cheese Calories, there are 1200 calories in 24 chunks of cheese. To figure out how many calories are in each chunk, I set up another proportion.

 

By cross-multiplying, I derive the following equation:

 24z = 1200

 Finally, by dividing each side by 24, I determine that each chunk has 50 calories. I eat one, decide that that is not enough, and have another for a total of 100 calories.

5-A-3: My definition of Equations and Functions

An equation is a mathematical expression that states that one quantity is equivalent to another quantity. This can take a numerical form (only numbers) such as 5+3=8 and it can take an algebraic form (with variables) such as 3x+4y=15.

 A function is a type of equation where every input has one and only one output. They can take the following forms where x is the input and y is the output: y = 2x+4, y = x²-3, y = |x|.

Pascal’s Triangle: Formal Description

Pascal’s Triangle is an ever-expanding list of series of numbers which begins with a row consisting of only one number (the number one). Each successive row has one more number than the one preceding row. To determine the value of each number in the new row, add the two numbers from the preceding row that appear diagonally above the new number. If there is only one number diagonally above the new number, then add this number to get zero. This will only happen on the edges of the triangle.

My Reflection on Math Myths

I reviewed several math myths as identified by the article, “Add-Ventures for Girls: Building Math Confidence” by Margaret Franklin et al. While I could probably have something to say about each of the 12 myths, I chose two for which I held strong beliefs.

 

Myth #1. Boys are naturally better in math than girls.

Between 7^th and 12^th grades, I was always in the accelerated math classes and the boys in the class always seemed to be the best at math. For a long time, I believed that boys were better at math and girls were better at talking good and, ya know, stuff like that. Case in point, I became a math teacher, and my wife became an English teacher. There are two other married couples in my school district who match this profile. So I even carried this commonly held belief into my teaching career.

However, I quickly found out that this was not the case at all. Every quarter, I put a list of the top ten students from my three eighth grade sections and my one seventh grade section. It is not uncommon for the list to be primarily made up of girls or to have a girl at the top of the list. Last year, 45% of my eighth grade lists were made up of girls and 70% of my seventh grade lists were made up of girls. For both grades, a girl was at the top of the list 7 out of 8 times. The cumulative average for my 8^th grade girls was 3 points higher than the boys. The cumulative average for my 7^th grade girls was 6 points higher than the boys. (And no, I’m not going to do any hypothesis testing on this stuff.)

So I don’t believe that boys are better in math than girls. I think my initial perception may have had more to do with the fact that boys tend to be more vocal in a math class. Girls, as I’ve read, may not be as vocal in math class because this is not a subject girls are traditionally supposed to be at. It’s just not cool to be a girl that’s good at math.

I believe that posting these results can be a motivator for the girls in my classroom. I think many of them just don’t realize that they’re just as good as the boys, if not better. I think it is also important to call on and  encourage girls in class.

 

Myth #2/3. Math is not creative; it is simply hard work.

When I was an electrical engineer, one of my primary duties was to use computer simulations to predict the electrical properties of power transistors. For the semiconductor illiterate, transistors are these “switches” that either allow or not allow current to pass through a circuit. They are found in computers, cars, cell phones, and any other piece of electronic equipment that starts with a c. These transistors are created by etching trenches into a silicon surface. These trenches are similar to the trenches you would think of in trench warfare. They are deep (a couple of millions of a meter) and rounded on the bottom. The problem was, simulation software did not have a good program to define the shape of these trenches. The shape was very important for determining a bunch of electrical properties (breakdown voltage, Miller capacitance, on-state resistance). So I had to create a program that would define the shape of the trench and would be user-friendly to that dimensions could be easily varied.

I spent at least a few days working on this problem. It required knowledge of the ellipse formula, some trigonometry, and a strong foundation in algebra. (Algebra was necessary to make the program user-friendly, so that dimensions could be easily changed.) The program was full of large equations which had to be typed exactly right and there was a lot of trial and error and debugging. The whiteboard in my cubicle was covered with drawings and equations.

What I loved about working on this program was that the solution did not present itself right away. It did take a long time to solve and I really enjoyed working on it. And when I finished the first revision, there were still things for me to improve. I had to develop creative solutions to make the program run faster and produce better results.

I think this was the first time in my life that I really enjoyed math. I didn’t see math as a series of problems to plow through. It was necessary to solve a very real and very difficult problem. I believe that myths 2 and 3 have been completely eradicated in my view. Math can involve a lot of creativity, and it is the problems that take a long time to complete that are the most interesting.

There are two ways that I can help dispel this myth for my students. The first would be to relate this story to them. The second would be to provide problems that give them similar experiences in math. This can be difficult though as not all students are so easily enthralled with figuring things out. These kinds of problems may be best suited as enrichment activities.

Non-Linear Pattern Web Quest

 “The Golden Ratio” and “Pentagrams”

I was not familiar with the golden ratio, although I’ve seen it mentioned in my textbook. It is not part of my curriculum so I never paid much attention to it. Nor do I remember ever learning this when I was in grade school.

My textbook talks about the golden spiral and I think it is interesting that the golden ratio can be applied to other shapes. It used to seem kind of trivial, but now that I see it appears in different places, it seems pretty cool.

I thought it was really interesting how the sides of the pentagram could be simplified to either 1, Φ or 1/Φ. The image and text that I am referring to is shown below and can be found at http://www.contracosta.edu/math/pentagrm.htm

In this next figure (to the left) is a regular pentagon with an inscribed pentagram. All the line segments found there are equal in length to one of the five line segments described below: The length of the black line segment is 1 unit. The length of the red line seqment, a, is Ø. The length of the yellow line seqment, b, is 1/Ø. The length of the green line seqment, c, is 1, like the black segment. The length of the blue line seqment, d, is (1/Ø)2, or equivalently, 1 – (1/Ø), as can be seen from examining the figure. (Note that b + d = 1).

I also found this image of Venus’s “orbit” around Earth interesting. It is explained that this path takes the form of a pentagram and nearly repeats itself (you can see the shift near the dot representing Venus).

http://en.wikipedia.org/wiki/File:Venus_pentagram.png

I also liked reading about all of the historical and religious references that are made to the pentagram. I wondered about why there was this fascination and if it had anything to do with the golden ratio. Or was the golden ratio more of an afterthought? After reading some things on Wikipedia, it sounds like it was more of an afterthought.

 

“Nonlinear Patterns” and “Fermat’s Theorem”

        I recently read about Fermat’s Theorem somewhere and I was interested in knowing more about it. Suite101.com explains Fermat’s Theorem below:

The Theorem

Consider the Pythagorian theorem:

a² + b² = c² (or, a squared plus b squared equals c squared). This is one of the most revolutionary discoveries in the history of mathematics.

In its simplest instances, this theorem can be solved using only non-zero integers. For example a = 3, b = 4 and c = 5. This, and other examples, hold true: 9 + 16 = 25.

But what happens, Fermat asked, if this theorem is written as a^n + b^n = c^n? In other words, if instead of squaring each of these numbers, they are raised to a power higher than two? Fermat’s hypothesis was that in any case where n > 2 (n is greater than two), it is mathematically impossible to find three non-zero integers, a, b and c that satisfy this equation.

Read more: http://math.suite101.com/article.cfm/fermats_last_theorem#ixzz0JNKY3qJg&C

 

I think I find this theorem interesting for the same reason that so many mathematicians did over the years. It looks so simple and its proof is so difficult. It’s nice to know that not everything has been figured out. That there’s still a bit (or perhaps a lot) of mystery left. The other funny thing about all of this is that Fermat acted like he had the proof but didn’t bother telling anyone what it was. Part of me wonders if he was full of himself. It reminds me of a story I remember hearing about Newton, about how he knew that the orbital paths of planets were ellipses and he proved it, but he lost the proof. (He actually did reprove it later I believe…unlike Fermat, Newton was the man.)

Non-linear Patterns at Home

I’ve been something of a musician for a good chunk of my life and my primary instrument is the tuba. I was trying to figure out some kind of pattern while looking at my tuba, such as the lengths of the tubing, but nothing really came to mind. I then looked at my acoustic guitar (which I am terrible at) and immediately noticed the distances between the frets getting smaller as you work your way down the fret board. I know that the frequency of the note played is dependent on the length of the string and that the next fret up will provide the next note in the chromatic scale. I’m sure that if I were to measure the distances between these frets, I would see some kind of pattern.

Adapting the Webquest to my Classroom

What I really liked about this webquest is that it leads to other learning (and not necessarily all related to math). I think it is really helpful to understand all of the contexts of how something is used to appreciate it. While I may not do a webquest as described here in my class, I like Countryman’s idea of a formal paper and I could see using some of these topics as part of that assignment. Naturally, the students would be doing some kind of webquest when they research the topic for their paper.

Working with the definition of linear patterns.

Non-traditional pattern (formal definition from module 4, topic A): “patterns that do not follow a repetitive format”

 Linear pattern (informal/kid-friendly/my definition): A linear pattern is a pattern that repeats and the numbers increase or decrease by the same amount each time. One example is 3,7,11,15,.… Each number is four more than the last number.

 Linear pattern (formal definition): “Linear patterns repeat indefinitely in either direction along a line. Beads on a necklace, the weave of a fabric or basket, wallpaper borders, stripes on clothing, zippers, walking footprints, musical rhythms, the meter of poetry, the passage of a day, and the changing seasons are all examples of linear patterns that are created or extended by the regular repetition of units, sounds, or events.”   from http://www.brooklynkids.org/patternwizardry/pattern_linear.html

 When I first wrote a definition of a pattern, I wrote “Patterns are sequences of numbers, words, or objects that are predictable as they change from one to the next”. When I wrote the informal definition of a linear pattern, I did not consider non-numeric patterns. The big difference between my definition and the formal one is that I did not consider these other patterns. However, for the purposes of teaching middle school students mathematics, my definition seems more appropriate.

 To help students understand the definition of a linear pattern, I believe that they need plenty of examples. The website above gives plenty of non-numeric examples and a textbook would provide plenty of numeric ones. The only trick for students would be to understand the differences between non-linear and linear patterns. Again, examples are required to observe the differences.