My Weblog

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

I hope to continue using my blog. At first I wanted to continue using it to keep in touch with out of town friends and relatives, but now I think it may be a useful tool for the classroom. So many kids are into technology. That is how they communicate. Instead of journals, blogging would be a great way for students to write in math class.

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

I don’t know if I learned anything about my abilities in math or algebra, but I did find other useful ways to present lessons and many useful real-world activites and lessons to help relate math to something that my students may eventually come across. Relating math to real life is vital for understanding.

• 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 many terrific websites that have manipulatives that I can use in my classroom. One website in particular was the national library of virtual manipulatives. (http://nlvm.usu.edu/en/nav/vLibrary.html) This site is great because of my new smartboard. There are many different things I can pull up on the board and have the students manipulate.

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

I have tossed around the idea of using journals with my students, but now that I have been introduced to blogging, I think that may be the way to go. It saves on paperwork and I can check their blog entries from home without having to carry home 70 journals.

Sample Quadratic: x² + 8x + 16

Look at the last term (the constant-16). List its pairs of factors:

1 x 16

2 x 8

4 x 4

We need the pair that will add up to the coefficient of the second term (8).

This pair is 4 x 4 because 4 + 4 =8.

Since the first term is x² it will factor into x * x.

Therefore, we can factor the trinomial into the following binomials: (x + 4)(x + 4).

We could check this by using FOIL (First, Outer, Inner, Last)

F: x * x = x²

O: x * 4 = 4x

I: 4 *x = 4x

L: 4 * 4 = 16

Then we add them to get: x² + 4x + 4x + 16 = x² + 8x + 16 (the original quadratic).

• Did paraphrasing the words help you internalize the concepts more?

Yes, I really had to think about what I was doing instead of just going through the motions of solving the problem.

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

Unfortunately, it is not in my curriculum to teach factoring quadratics, but I could easily apply this to another concept. I could have students paraphrase the steps to solving a multi-step equation, or for adding or subtracting fractions with unlike denominators, etc…

1. You are planning a trip from Erie, PA to Pittsburgh, PA to go see a Steelers game! On your map, 2 inches represents 50 miles. You measure the distance to be about 4.5 inches. About how many miles is it from Erie to Pittsburgh?

Solution:

We can set up a proportion:

Number of inches in scale = Number of measured inches

 Number of miles in scale       Number of actual miles

Plugging in numbers, we get:

  2   =  4.5

 50       x

Solving the proportion gives us:

2x = 50(4.5)

2x = 225

x = 112.5

It is approximately 113 miles from Erie to Pittsburgh.

2. Your basketball coach bought 20 bottles of water for your team for $25.80. How much did each bottle cost?

Again, we can set up a proportion:

Price of total bottles = Number of total bottles

  Price of one bottle              one bottle

Plugging in numbers gives us:

25.80 = 20

    x        1

Solving the proportion gives us:

20x = 25.80(1)

20x = 25.80

x = 1.29

Each bottle of water costs $1.29.

• After reviewing your classmates post, would you alter your definition? Why or why not? Would you provide different examples?

After reading my classmates posts, I relly liked how they defined functions and equations, but I do not think I would alter my definition much. For me, this is the best way I can describe the concepts to the students, but if they are still unsure after my explanation, I now have many more ways of explaining the concepts to them.

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

There are many ways I can evaluate if students understand the difference between functions and equations. I could have them explain it verbally to me, and also have them write the definitions in their own words. Of course I would use daily class work and quizzes to check understanding as well.

While searching the various sites containing applets, I found several that I would like to use with my smartboard, but one in particular I found that I really enjoyed was the factor tree applet on the NLVM (National Library of Virtual Manipulatives) site. This applet can be used for several different concepts. It can be used to show prime factorization, to find the GCF or the LCM. It is great because the computer will generate the numbers for you, or you can type in the numbers for students to solve. Not only does it help do the factoring, but you type in the answers to the GCF and LCM and it tells you whether or not you have the correct answer.

http://nlvm.usu.edu/en/nav/frames_asid_202_g_3_t_1.html?from=category_g_3_t_1.html

Equation: a mathematical sentence that includes an equal sign.

Examples: 2 + 7 = 9 (true equation)

                   13 – 6 = 9 (false equation)

                   2 + x = 15 (open sentence)

Function: a statement in which each input (x ) is paired with exactly one output (y). If we think of the x values as people and the y values as places, we can say that there can be more than one person at the same place (function), but you cannot be in more than one place at one time (not a function).

Examples:

Function:

Not a Function:

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

There are a few of the myths that I have encountered, not only as a student, but I’m seeing these in colleagues when they teach.

1. There is always one best way to do a math problem.

2. It’s wrong to count on your fingers.

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

I was always taught how to do math problems one way, but as I continue in my career, I’m finding that students who struggle often find it helpful to have things explained to them another way. A colleague of mine wants me to teach students a certain way to work with fractions, so that when they get to her they are used to doing it that way. But I have found that as long as they understand the process and are finding the answer, there usually is more than one way to solve the problem.

I have heard many math teachers complain that students in middle school and high school are still counting on their fingers. In fact, I used to be one of those teachers who would complain, but this goes along with the myth that there is only one way to do the problem. Some students need the visual of their fingers to help them add or subtract. Right now my daughter is learning addition and subtraction facts in 2nd grade and I have told her that if she needs to use her fingers to help count up or down, then she should go ahead and do that.

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

I can encourage students to figure out why we are solving problems, not just emphasize to them that they need the right answer. I need to tell students that there are different ways of solving problems and to encourage them to think of more than one way to solve a problem. This will help with their critical thinking skills as they get older.

Pascal’s Triangle is an isoceles triangle that is created by starting with the number 1. This 1 at the top makes up Row 0. The first row (1 & 1) contains two 1′s, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0 (all numbers outside the Triangle are 0′s). Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this way, the rows of the triangle go on forever.

There are many uses of Pascal’s Triangle. For example:

• The sum of the numbers in any row is equal to 2 to the n^th power or 2ⁿ, when n is the number of the row. For example:

• If the 1^st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1′s) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.

• The triangular numbers can be found in the diagonal starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the fourth is 10, and so on.

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

• I am pretty familiar with Fibonacci and the Golden Ratio, but I was not familiar with Fermat’s Theorem or Phyllotaxis. I have heard of fractals before, but did not realize how often in nature they appear.

What images did you find particularly striking?

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

• My bouquet of flowers on my kitchen table at home has nonlinear patterns in the leaves and petals.
• The patterns in the carpet at work and the patterns in the curtains and pillows at home all have nonlinear patterns.

How can you adapt this webquest activity for your classroom?

• I think students would find these pictures fascinating! They don’t realize how math is conveyed in nature. I found an interesting activity where students are able to draw their own fractal pattern, creating a snowflake.

http://www.schools.manatee.k12.fl.us/boehm/snowflakecurve/index.html

Non-traditional Patterns: patterns that do not follow a strict repetition of numbers, shapes, or figures.

Linear Pattern (kid language): a number sequence that follows a strict pattern. When graphed, linear patterns form a straight line.

Linear Patterns: have the same difference between terms. This gap between terms is called the common difference.

There are several differences between my definiton and the actual definition. I would stress to students that linear patterns form a straight line, but we don’t always talk about graphing these patterns, so they would need to know how to write a rule for the pattern by finding the common difference.