Reflections on Blogging

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

I found the blogging experience to be difficult at first, but pretty simple as the course progressed. I struggled to get everything arranged, but once I had that finished, I thought the blog was very easy to use. I don’t see myself using the blog too much in the future. I already have a couple technology-related activities I need to stay current with, and I don’t see adding the blog to that list. I don’t think it would be hard, if I decided to add it, but I don’t feel that I need it for my classroom at this point.

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

Through this course, I learned that I didn’t grasp the “whys” in too many areas. I know how to do the problems, but this course challenged me to understand why problems were solved in certain ways. The course also gave me a student’s perspective on math, and I believe this perspective will really help me with my teaching.

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 found the Function Machine and graphing websites to be very valuable. Both could complement my current instruction and would be very easy to organize. They are not huge projects, but they would provide students with many examples in a relatively short period of time. As far as course material, I was interested in the “why” behind the FOIL method of factoring quadratic equations. Although I don’t teach the material, I had never actually learned why you FOIL. So, that was a positive experience for me in this class.

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

I really want to use journals with my students. I have thought about using them before, but I have had to incorporate different philosophies into my class. I wanted to make sure I could implement ideas properly, and now that I have everything else established, I feel more comfortable including journals into the curriculum. I don’t, however, envision using blogs for my students. In a perfect world, I would use them, but it is very difficult to get computer access for my students, and I have other activities, such as webquests, that would be more beneficial for them. I don’t feel that blogs are necessary, particularly if I use journals.

Factoring Quadratics

Factoring a quadratic equation is essentially breaking the function down into smaller parts. At this level, the equations will be broken into binomials, which are simply two terms that are, in this case, being added or subtracted. In order to check the factoring, students will use the Distributive Property. You may also recognize this action as FOIL, which stands for first, outside, inside, and last.

To help us understand the idea of factoring a quadratic equation, we will look at a specific example: x² + 5x + 4. In order to factor, we need to think of two binomials that, when multiplied, will equal x² + 5x + 4. Since there is no coefficient for x², the variable x needs to begin both binomials. Next, we need to decide which factors of 4 we are going to use. We can choose 1 x 4 or 2 x 2. In order to make this determination, we need to see which set of factors, when combined, will equal 5. In this case, 1 and 4 will be the factors we will use. Now, we need to add or subtract, and since 1 + 4 equals 5, we will use addition to reach the middle term in the quadratic equation. Our answer will then be (x + 1)(x + 4). If we use the Distributive Property to check, we will get x² + 5x + 4, which means that we correctly factored this quadratic equation.

Questions:

Did paraphrasing the words help you internalize the concepts more?

Paraphrasing did help me to internalize the idea of factoring, because I was able to use language I understood. This activity also forced me to develop an example, which was helpful to see how everything tied together. I also learned what FOIL actually represents. I just learned the steps when I was younger, and did not worry about the why. Now, however, I can see that FOIL is simply a convenient way to remember to use the Distributive Property.

How can you apply this type of exercise in a lesson for your own students?

I wouldn’t use this exact concept with my class, but I would like to have them journal about solving equations. My students usually understand one-step equations, but they get confused with two-steps. If they were required to write how to solve an equation in their own words, it would help them internalize the information.

5-D-2 Applets

I enjoyed the Pythagoras Mystery Tablet Applet (http://mathforum.org/escotpow/puzzles/irrationals/applet.html). This applet combined many different topics that we cover throughout the year and would be a perfect fit towards the end of the year. The goal is to multiply two numbers, one a decimal and one a fraction, in order to find the area of a square. Different numbers are listed on the tablet, so you try to get the product to equal the number on the tablet. We discuss squares and square roots, and students always struggle with this concept. I think this applet provides a good visual, and the square actually changes size to match the number inside. The applet also provides students with an opportunity to practice converting fractions and decimals. Although the computer takes one number and converts it, students would still be able to see equivalent fractions and decimals. Students should have mastered these conversions earlier in the year, but that doesn’t always happen. In summary, the Pythagoras Mystery Tablet Applet provides great visual examples of a new concept while also reviewing concepts from earlier in the year. I would certainly use this applet during the introduction of squares and square roots.

Evaluating our Definitions: Equations and Functions

After reviewing your classmate’s post, would you alter your definition? Why or why not? Would you provide different examples?

I wouldn’t alter my definition of an equation, but I would change the definition for a function. I would add the idea that functions generally have two variables (an input and an output). I think this addition would help students separate the two concepts. In seventh grade, we don’t solve many equations with two variables, so it would make a clear distinction between a function and an equation. In future classes, there would be equations with two variables, but I could tell my students they will see that in the future.

How can you evaluate whether or not your students grasped the difference between the two?

I would have students complete two activities in order to differentiate between the two terms. First, I would give them examples and ask them to identify each example as an equation or a function. Second, I would ask them to write an explanation or complete a Venn Diagram comparing and contrasting equations and functions. I believe this approach would show me which students can simply identify the two and which students can differentiate the subtle differences between an equation and a function.

5-A-3 My Definition of Equations and Functions

Equation: A mathematical problem in which two expressions are equal. For example, 5 + 5 = 10 is an equation, as is x + 5 = 10.

Function: A mathematical equation where there is only one output for each input. For example, y = x + 1 is a function, because each input (whatever number you put in for x) results in only one output (the number y will equal). You can check to see if an equation is a function by graphing it. When you graph a function, no vertical line will hit more than one point from the function. If a vertical line goes through more than one point, then the graph does not show a function.

5-B-1: The Magic of Proportions

Problem 1: Tiffany goes to the store to buy a shirt. The shirt’s original coast is $20.00. Today there is a sale of 30 % on all items in the store. How much will Tiffany need to pay to buy the shirt today?

Solution #1 (Proportion): You can solve this problem by using a proportion. Thirty percent can be written as 30/100. Since 100 % of the original price is $20, that needs to go in the denominator of the other fraction. Now, 30/100 = x/20. To solve this proportion, take 30 times 20, which equals 600. Then, take 600 divided by 100, which equals 6. Tiffany does not pay $6 for the shirt, but that is the discount on the shirt. Now, take $20 – $6 and Tiffany will need to pay $14 for the shirt.

Solution #2: If the sale is for 30%, that means you need to find 30 % times $20 to find the discount. You need to rewrite the percent as a decimal, and it becomes .30. Take .30 times $20, and the product is $6.00, which represents the discount. Now, take $20 – $6 and Tiffany will need to pay $14 for the shirt.

Problem 2: Greg earned a 32 out of a possible 35 on his most recent quiz. What was his percentage and letter grade?

Solution: Since we need to find Greg’s percentage on the quiz, we are going to use the number 100, because that always goes along with percents. We know he earned a 32/35, so we are going to make a proportion. For this problem 32/35 = x/100. In order to solve proportions, you first multiply the two numbers which are diagonal from each other. In this case, you take 32 times 100, which equals 3200. Then, you divide that product by the number you haven’t used, which is 35. 3200 divided by 35 is 91.43 (rounded to the nearest hundredth in decimal form). Now, we need to decide which letter grade, Greg earned on the quiz. Since 91.43%, does not round up to 92%, Greg earned a B on this quiz.

Non-linear Pattern Webquest

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

Fibonacci: I was not familiar with the idea that plants following Fibonacci’s Sequence. I think I may have heard something about it before, but I hadn’t thought about it. It’s interesting that following the sequence is the best way for a plant’s leaves to maximize their exposure to light. I never thought that the sequence had any application to nature.

Fractals: I had heard the term fractals, but never learned what they were. From my understanding, fractals are patterns that occur in smaller measurements relative to another feature. I didn’t realize so many things in nature are fractals. From riverbeds to ferns, fractals can be seen in many different places.

What images did you find particularly striking?

Source: http://kiwitobes.com/wiki/Fibonacci_sequence.html

The coastline of Chile

Source: http://fractalfoundation.org/category/natural-fractals/

I really enjoyed using this website. It has great pictures!

Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?

I have a poster that has no repeating pattern. It shows baseball memorabilia aligned in no specific order. I also have clothing that has no particular pattern. It has designs, but they do not repeat in the same order.

How can you adapt this webquest activity for your classroom?

I think my students would really enjoy looking at fractals. They are very visual learners, and this activity would stimulate their curiosity. I would also look for websites so that students could create their own fractals. I have done that with tangrams, and the students really enjoyed the experience. My only concern would be relating it to my curriculum.

For Fibonacci’s Sequence, I could have students investigate where this sequence occurs in nature, and perhaps pair it with a science activity to develop some cross-curricular ideas

My Reflection on Math Myths

I chose to focus on the following myths:

1. People who are good at math do problems quickly, in their heads.

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

Did you encounter any of these myths in your own experience with Math education as a student? If so, which ones?

I definitely encountered the first myth while I was going through school. Once I reached middle school, I rarely solved problems in my head. I liked to go through every problem step-by-step and check the work to make sure I had no mistakes. Many of my friends were faster at math than I was, and it was a little embarrassing to take more time than most people, but I usually answered the questions correctly, which to me was more important than finishing quickly. The second myth didn’t affect me too much as a student. People would tell me that I had a good math mind, but they didn’t always realize that I had that mind because I completed the homework and got extra practice on the topics we covered in class.

What has happened since to dispel or perpetuate your understanding of the myth?

In my class, I am supposed to do a mental math activity every day. While many of the activities are useful, they can also turn students off to math. They feel like if they can’t solve the problem mentally, they aren’t smart enough, while the students who can solve them are the smarter ones in the class. While that is sometimes the case, it isn’t always. The “math mind” myth drives me crazy! Every year, I have about 20 students tell me that they just have never been good at math and that’s why they struggle. On the other hand, I have students who have been told they have great “math minds” and they try to coast through class by doing all of their work mentally, which usually catches up with them.

How can you help dispel any of these myths for your students?

I only use the mental math questions for a portion of the year. They are helpful for basic facts, and we can discuss different ways to solve the same problem. As the year progresses, however, I replace those questions with review from earlier in the year, so that all of our material stays fresh in their minds. I also require students to show work on all of their tests, which encourages them to write down steps to the problems. Depending on the type of problem, there may be times when they do not need to solve their arithmetic work. So, for those students who enjoy mental math, they have that choice. For those students who don’t enjoy mental math, it is just one more step that they will be writing. As for the “math mind,” I view it as a convenient excuse for students. The easiest way to convince students that anyone can have a math mind is to have success early in the year. I sometimes wish that my class were easier, so that students would build confidence. At the same time, I don’t want them to get an inflated ego by not doing top-notch work. It is still a balance for which I am striving. I do encourage all of my students to ask questions and see me if they need help, but a limited number actually take advantage of that option.

Translating Pattern Narrative into Formal Math Language

The first thing I notice with Pascal’s Triangle are the 1′s that line the sides of the triangle.

The next thing I notice are the second and next-to-last digits in each row. You can fill in these digits by adding 1 each time. If you think of it as a diagonal line, the digits will increase by 1 every time.

I also see that the sum of two numbers which are next to each other horizontally result in the number below those two.

Each row has one more digit than the row before it.

The odd numbers, which are highlighted, completely fill in the rows where the number of digits is equal to an exponential form of 2. So, in the triangle we have, the rows with 1 (2⁰), 2 (2¹), 4 (2²), 8 (2³), and 16 (2⁴) digits are filled with odd numbers. The next row you would expect to see all odd numbers should have 32 (2⁵) digits.

Working with the definition of linear patterns

Formal definition of a non-traditional pattern: A pattern that does not follow a repetitive format.

My definition of a linear pattern: A listing of numbers or objects where the same thing happens between each number or object.

Formal definition of linear pattern: A linear pattern is said to exist when the points examined form a straight line. (Source: http://www.mathsteacher.com.au/year8/ch15_graphs/02_linear/patterns.htm)

Comparison of the formal versus informal definitions: My definition does not include anything about a straight line, while the formal definition does. I also mentioned objects, whereas the formal definition focuses solely on numbers. I think these differences come form the content I teach. My students are not experts with coordinate planes, so for me to tell them a line is formed will not necessarily make sense to them. As the year goes on, we spend more time working with the coordinate plane, but by following our curriculum, I teach patterns before coordinate planes. We also talk about patterns with objects, so I felt it was important to mention that term in my definition.

First of all, I don’t think it would be difficult to memorize the formal definition, but I could also go through some examples where students see a line being formed on a coordinate plane. As students see the line, they can make the connection that a linear pattern will make a line on a coordinate plane.