Sheri's Math Blog

8-B-2: Reflections on Blogging

sabellas — Wed, 22 Jul 2009 16:11:44 +0000

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

This course allowed me my first blogging experience. I was very nervous at first when we had to do a blog. It took me a few entries to get used to the idea. After a while, I became comfortable with it. I am not sure if I will continue using my blog or not. I see the greatest positive of using a blog is the ability to get feedback from other people. Without an audience to respond to my blog as I have had in this class, I don’t see the benefit.

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

I mostly just confirmed my knowledge in math and algebra in particular. Algebra has always been my most favorite branch of mathematics. It has been great to get a fresh approach to teaching it.

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.

My two favorite activities we did with our blogs were looking at applets and the webquest. I can’t wait to do more exploring with the applets and use some of them with my classes in the future. They are a great tool to practice and reinforce various concepts in a way that is fun and interesting to students. I also found the webquest a useful project. Giving students the freedom to research various concepts online (with certain guidelines) is a powerful way to have them take ownership in their own learning. I hope to incorporate this project idea in my room this year. For myself, doing the webquest gave me new insight into certain concepts. I did not fully understand the relationship between Fibonnaci’s sequence, the Golden Ratio, and the logarithmic spiral before.

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

I would love to use journals with my students. Through this class I have really come to appreciate the value of having students write, whether it be a reflection of what they have learned or to do some research and explain the meaning behind certain problems. Blogs would be a great tool to have students share their journal entries. I would love to have my students do a blog. It is not always possible to have students on computers in my classroom. Because of this they would have to write in their blog and respond to others’ blogs from their homes. I could then share some of the better entries with the whole class on my SMARTboard.

8-B-1: Factoring Quadratics in My Own Words

sabellas — Tue, 21 Jul 2009 20:45:38 +0000

There are several steps to follow in order to factor a quadratic equation. We need to remember that when we multiplied two binomials to get a quadratic, the product of the first two terms in the binomials become the first term in the quadratic. The sum of the last two terms in the binomials becomes the coefficient of the second term in the quadratic, and the product of the last two terms becomes the last term in the quadratic. We see this happening when we multiply two binomials as such.

(x + 2)(x + 3)

x² + 3x + 2x + 6

x² + 5x + 6

Knowing how we created a quadratic will allow us to understand how to factor them.

Example: Factor 2x² + 6x – 36

1) Factor out the Greatest Common Factor, if it exists, between all terms.	2 is the GCF 2(x² + 3x – 18)
2) Factor the first term in the quadratic. This is the square root of the first term. Each factor will become the first term in each binomial.	x² = x ∙ x So far, our answer is 2(x )(x )
3) Determine the operation symbols in the binomials. If the last term is positive, the symbols are either both positive or both negative (product of terms). If the last term is negative, then one will be positive and one will be negative.	Since the last term is negative, one will be positive, the other negative. 2(x + )(x - )
4) To find the last two terms of the binomials, we need to know which two numbers multiply out to get the last term, but add to become the coefficient of the second term.	Factors of -18 and their sums -1 x 18 sum +17 -2 x 9 sum +7 -3 x 6 sum +3 * -6 x 3 sum -3 -9 x 2 sum -7 -18 x 1 sum -17 Our final answer is 2(x + 6)(x – 3)

Did paraphrasing the words help you internalize the concepts more?

Putting the directions in my own words helped me to really think about the steps. I had to really think about the reasons for each of the steps and describe my steps in such a way that the reasons were explained as well.

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

As we have previously learned, having students write helps them to understand. By having them think about the steps to solve a problem and put it in their own words, they will better understand it. Also, they better remember how to solve the problem if they have had to write out the steps in their own words. Therefore, I think I will have my students explain the steps to solve a problem when practicing the problems themselves.

5-D-2: Applets

sabellas — Thu, 02 Jul 2009 19:51:49 +0000

I reviewed the game “Line of Best Fit” on the Illuminations site http://illuminations.nctm.org/ActivityDetail.aspx?ID=146 I really liked this game. Students can put their own points on a scatterplot, then draw their own line of best fit. Finally, the computer plots what it believes to be an appropriate line of best fit. It also gives an equation for the line in slope/intercept form. This will be great with my classes when we are studying lines. It would work really well on my SMARTboard. It will give practice in several areas.

Drawing lines of best fit. It will be great for students to see how their lines compare to the computers lines. They can then discuss with peers why their lines may or not be appropriate.
Writing the equation of a line from 2 points. When we study lines, I have my students create scatterplots, draw lines of best fit, and write the equation for that line. This will be great for students to compare equations to the equation the computer writes.
Making predictions from data. Although the applet doesn’t do this, students could use the line and/or the equation to make predictions about the data.

5-A-4: Evaluating Our Definitions: Equations and Functions

sabellas — Thu, 02 Jul 2009 16:04:34 +0000

After reviewing the definitions of equation and function posted by my other classmates, I would change my definitions slightly. One change I would make is to include the word “sentence” when describing an equation. Matt used this wording in his definition. I often use this word when describing equations to my class, but I left it out in my own definition. I like to compare equations to sentences as a complete thought. The equal sign is the verb of the sentence. What I would keep the same about my definition is the fact that equations have equal signs. Many other of my classmates included this fact in their definition as well. As for my definition of functions, I didn’t really define it, more describe what it does. Again, I refer to Matt’s definition as a relationship between two numbers. I think this definition describes a function well. I also liked Bridget’s description. She included the fact that this relationship creates a pattern. I think my examples were fine, as were the definitions provided by my classmates.

There is a fine line between equation and function, as equations often are functions. What is key to remember, is that a function will come from an open sentence (equation with a variable) and has a defined relationship between the variables. Functions also have only one output for each input. I think the best way for students to understand the difference is to have them look at examples of equations that are functions and those that are not. By having them identify the differences, they should be able to better grasp the concept.

5-A-3: Equations and Functions in My Own Words

sabellas — Wed, 01 Jul 2009 12:42:33 +0000

Equations – to me, the word equation relates directly to its root word: equate or equal. Anything that has an equal sign is an equation. Sometimes, as in x = 4 + 5 ∙ 3, you only need to follow the order of operations to solve the equation. In other cases, as in 4x + 3 = 11, you need to use opposite operations to solve the equation. Of course, some equations don’t need to be solved, they are simply fact, as in 4 + 3 = 7. Equations can be thought of as a balance scale. To be in balance, the pans have to be equal. If you do one thing to one side, you have to do the same thing to the other for them to remain equal.

illuminations.nctm.org

Functions – to me, the word function means something is working. If an open equation, such as y = x + 3, creates a set of numbers that all “work” for that equation, then those numbers are the “function” of the equation. We could put all the numbers that “work” for that equation into a table of values to easily see the numbers that make up the function.

For y = x + 3

X	Y
0	3
1	4
2	5
3	6

In all of these pairs of numbers, the y value is 3 more than the x value. Therefore they “function” for the equation. We could think of a function machine, where we put the x value into the machine, the machine does some “work” to the number (in this case, adds 3) and it gives us an answer. If it were to give us some other answer that does not “work” for the equation (like putting in 2 and getting 17), then the machine is not FUNCTIONing.

instruction.aaps.k12.mi.us

5-B-1: The Magic of Proportions

sabellas — Mon, 29 Jun 2009 22:47:14 +0000

Proportions are one of the most useful tools in problem solving. Any time you have two numbers that you can compare, and you need to convert them, you can use a proportion. I tell my students to start by writing their original information as a ratio written in fraction form. I also require that they include any appropriate units with the numbers. This will allow them to line up corresponding bits of information in the other ratio. The next step is to place the other known information in the 2^nd ratio, so that it corresponds to its matching unit, along with a variable for the missing information. Finally, students use cross multiplication to solve the proportion. We will have already done experiments with cross multiplying known proportions, so we know that the cross products of proportions are equal to each other. Then we solve our equation to find the value of our variable (in other words, solve the proportion).

Example problem 1:

A 354 gram box of cereal has 20 grams of fat. How many grams of fat will be in a 55 gram serving size of the cereal?

354 g (cereal) = 55 g (cereal) 354 ∙ x = 20 ∙ 55 20 g (fat) x 354x = 1100

X ≈ 3.1 grams of fat

Example problem 2:

When surveyed, 34 out of 50 high school seniors said they work on weekends. How many students in the senior class of 324 would you expect to work on weekends?

34 work on weekends = ___ x 50 ∙ x = 34 ∙ 324 50 students 324 students 50x = 11,016

X ≈ 220 work on weekends

4-C-1: Nonlinear Pattern Web Quest

sabellas — Sat, 27 Jun 2009 12:29:47 +0000

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

Of course I had heard of and even used the Fibonacci sequence of numbers in my classroom. What I did not know was the extent to which you can find this sequence in nature. According to http://www.daviddarling.info/encyclopedia/F/Fibonacci_sequence.html, the Fibonacci sequence can be found in the petals of daisies, seeds of a sunflower, whorls on a pinecone, rings on the trunk of palm trees, and snail shells. In fact, the pattern in floral patterns of this sequence is known as phyllotaxis. There is also something called “the logarithmic spiral, which emerges when seeds on a plant grow and space themselves according the Fibonacci sequence. The logarithmic spiral is approximated by the rule: start at the origin of the Cartesian coordinate system, move F(1) units to the right, move F(2) units up, move F(3) units to the left, move F(4) units down, move F(5) units to the right, and so on. By growing in this way, on structures as such sunflowers, pine cones, and pineapples, seeds are able to pack themselves together most efficiently.” The Fibonacci sequence is also related to the Golden Ratio.

I have heard of the Golden Ratio, the decimal equivalent of approximately 1.6, but I had no idea of how it occurs so often. This ratio is naturally occurring, and generally considered pleasing to the human eye. Many paintings were made using this ratio (example: The Mona Lisa). In fact, picture frames are typically sized to this ratio (example: 5 x 8 inch frame). The ratio exists in the human body as well. Your finger joints are the ratio, as well as the ratio of the size of your face to the distance your eyes are spaced. DaVinci studied this ratio in his famous painting of the human body.

What images did you find particularly striking?

I am including some images that I mentioned in the previous section of my webquest: Fibonacci’s sequence in a shell, the logarithmic spiral, finger joints measured to the Golden Ratio, and DaVinci’s painting.

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

I see a nonlinear pattern in the back of my dining room chairs. They consist of a pattern where a circle is surrounded by other irregular shapes radiating away from the center. Looking outside my window, I am observing a tree. From the 1 trunk of the tree, several large branches shoot out. From each branch, grows a greater number of smaller branches. This nonlinear pattern continues through the smallest twigs.

How can you adapt this webquest activity for your classroom?

I could definitely see myself using this webquest in my classroom. For 7^th graders, I would have to simplify it by giving them only one or two ideas to look up. I love how we were given specific questions to answer to guide our search. I would use the same procedure, which would allow me to easily assess their learning in the activity. I could use the idea of a webquest for almost any concept I wanted to introduce. By having students sharing and commenting on each other’s webquests, students could learn from each other as well as what they discovered on their own.

4-C-3: Formal Vocabulary for Pascal’s Triangle

sabellas — Thu, 25 Jun 2009 14:53:04 +0000

Pascal’s Triangle is created by a pattern of numbers, the first of which is a 1. Each succeeding row is created by finding the sum of the two numbers adjacent to it in the preceding row. When no other number is there, we use the identity property, and add a zero. This created an affect of 1’s being extended along the sides of this equilateral triangle. Each row then has a symmetrical pattern, creating an overall vertical symmetry within the triangle.

4-D-1: My Reflection on Math Myths

sabellas — Tue, 23 Jun 2009 20:54:07 +0000

I have come across several of the math myths mentioned in the document. One that stands out to me is the myth that boys are better at math than girls. I have heard this myth, but I have not seen a lot of proof of it. A majority of students in my advanced level math class are usually girls. I have also found that most of my top students overall are girls. In addition, of the eight math teachers in our building, only one is a male, all others are female. I could chalk that up to the fact that most teachers in our building our female, but seven out of the eight Social Studies teachers are male, so that does not hold true. I believe the myth goes back to the days when women were not given the same opportunities as men, and therefore, most of the math related jobs were filled by men. Times have changed, and that is no longer the case. An organization that I am involved with, called Math Options, works to dispel this myth. Every year, they hold a math and science career fair for 7^th grade girls at Penn State Brandywine. Many women scientists and mathematicians from the area hold workshops to help these students see the options they have available to them in the areas of math and science.

Another myth is the one that states that there is a “math mind”, and some students have it and some don’t. I must admit that reading about this myth was an eye-opener for me. I am guilty of believing this. I have always felt that some students have more natural ability toward math, just like some are more musically talented, artistically talented, or athletic, etc. I know that other factors, like prior knowledge, experiences, work ethic, and parental support go into the success a student has in math class, but I also always thought that a certain natural ability was involved as well. Reading about this myth has made me realize that what I believed is untrue and that all my students have the ability to be successful. I just need to be able to help them reach their potential.

4-A-2: Working with the Definition of Linear Patterns

sabellas — Mon, 22 Jun 2009 14:45:28 +0000

NON-TRADITIONAL PATTERNS:

Non-traditional patterns are patterns that do not follow a repetitive format.

LINEAR PATTERNS:

“Kid” definition: Linear patterns follow a consistent, repetitive pattern. When written as an equation, there is a defined relationship between the variables. For example, in the equation y = x + 3, y is always 3 more than whatever the value of x is. When graphed, a straight line if formed, hence the word “linear”.

Formal definition: If the plotted points make a pattern, then the coordinates of each point may have the same relationship between the x and y values. In such a case, the x and y values are connected by a certain rule.

A linear pattern is said to exist when the points examined form a straight line.

SIMILARITIES/DIFFERENCES:

Both my definition and the formal one talk about patterns, relationships, and straight lines. The formal definition talks about the coordinates of each point forming the relationship, whereas I talked about the relationship between the variables in the equation. In truth, the relationship exists in both places, I just think it is easier for students to see the relationship if they are written as an equation, not just a series of points. The formal definition also talks about the x and y values being connected by a certain rule. That rule is the equation I discussed in my definition. Both definitions acknowledge that the points will form a straight line when graphed.

When teaching this to my students, I always start by asking them what they think the root word of linear is. Some students say ear , but most figure out it is line. One way to have them remember what the definition is, is to have them make a connection like with the root word. We also look at the graphs of some linear equations, and what exactly causes the lines to be straight. Students see that, for example, e very time we increase the x value by 1, we also increase the y value by 1. This creates a step pattern, which is a straight line when graphed. By putting these values in a table, we can see the relationships between the x and y values. Now they understand the part of the definition that discusses the relationship between the variables. Finally, we write equations that connect these values. Students can then see that the values are connected by a rule, which is the equation.

Reference: www.mathsteacher.com