I learned through the use of my blog, that I am somewhat computer literate.  I was a bit apprehensive at first, but realized that I was comfortable with my technical skills to succeed at blogging.  I mentioned this to my students as motivation to persevere through your doubts and they understood that (after they stopped laughing at me for being scared to blog!).  In mathematics, I always knew that this was my strong subject and that I enjoyed it very much and this course just reaffirmed those feelings.  Algebra continues to be my favorite part because it is so much like a puzzle. 

I enjoyed looking at the many applets and sites that are out there for Algebra, in particular.  I plan to use many of them with my students, both in exploration and in review.  It was nice to be “forced” to search through, since I normally don’t have the time to look them all.  The research on the Fibonacci sequence was particularly interesting to me.  I also heard about it and glanced over it several times, but to really read about it and see how often it comes up was remarkable.

I plan to use journals in my class at the start of the second semester.  I think it is a great way for students to reflect on their learning and to perhaps rephrase topics in their own words to internalize each concept.  I believe it might also reduce the math anxiety if students are able to freely write their feelings about math.  As I stated before, I will attempt to continue to use blogs to communicate more with students and parents at home.  Any way to incorporate parental involvement is a positive addition to my classroom.  I truly believe that these two techniques allow me to interact with students on an entirely new level that I have never done before. 

Example 1:  x² + 11x + 30 (the constant, c, is positive and the b is also positive)

1.  Find the key factors of the constant, c, that will have a sum equal to b.

            (15 x 2, 10 x 3, 30 x 1, 5 x 6)  5 x 6 = 30 5 + 6 = 11

2.  Separate the key factors into a binomial.  Each binomial will also have x, since the leading coefficient, a, is 1.

            (x    5)(x   6)

3.  Determine the correct signs for the factors.  Since both c and b are positive, the signs of the key factors should be positive.  This step follows the rules of multiplication, addition, and subtraction of integers.

            (x + 5)(x + 6)

4.  Mentally multiply the binomials to ensure the factors and signs are correct.

            (x + 5)(x + 6) = x² + 5x + 6x + 30 = x² + 11x + 30

Example 2:  x² – 12x + 32 (the constant, c, is positive and the b is negative)

1.  Find the key factors of the constant, c, that will have a sum equal to b.

            (32 x 1, 8 x 4, 16 x 2)  8 x 4 = 32, 8 + 4 = 12

2.  Separate the key factors into a binomial.  Each binomial will also have x, since the leading coefficient, a, is 1.

            (x    8)(x   4)

3.  Determine the correct signs for the factors.  Since c is positive and b is negative, the signs of the key factors should be negative.  This step follows the rules of multiplication, addition, and subtraction of integers.  When multiplying two negatives, the answer is positive, and when adding two negatives the sum is negative.

            (x – 8)(x – 4)

4.  Mentally multiply the binomials to ensure the factors and signs are correct.

            (x – 8)(x – 4) = x² -8x – 4x + 32 = x² – 12x + 32

Example 3:  x² + 8x – 20 (the constant, c, is negative and the b is positive)

1.  Find the key factors of the constant, c, that will have a difference equal to b.  You must subtract since the product of a positive and a negative is negative, and when adding different signs, you must subtract.

            (4 x 5, 10 x 2, 20 x 1) 10 x 2 = 20 10 – 2 = 8

2.  Separate the key factors into a binomial.  Each binomial will also have x, since the leading coefficient, a, is 1.

            (x   10)(x   2)

3.  Determine the correct signs for the factors.  Since c is negative and b is positive, the signs of the key factors should be different.  This step follows the rules of multiplication, addition, and subtraction of integers.  When adding different signs, subtract and keep the sign of the number with the largest absolute value.  Since b is positive, 10 should also be positive..

            (x + 10 )(x – 2)

4.  Mentally multiply the binomials to ensure the factors and signs are correct.

            (x + 10)(x – 2) = x² + 10x – 2x – 20 = x² + 8x – 20

5.  Similar steps are followed if c is negative and b is negative.

Paraphrasing the directions was very helpful to me.  I was able to internalize the instructions and put it into my own words that made sense to me.  It really made me think about the step by step process that I occasionally take for granted after teaching it for so many years.  It made me realize the little things that I might not have stressed in the past and I will begin to stress this year.  I think it will help my students tremendously.

I will use this activity of paraphrasing the directions with my students for the same reasons stated above.  They will be able to internalize the material and put it into words that they will remember.  I would certainly start with many examples of FOIL and then begin factoring by explaining that it “undoing” FOIL.  The object is to get back where you started.  I try to present it as a puzzle because the students react positively to that.  I would also discuss the GCF of quadratic equations as well in the beginning.  That would be a review and a common, safe place to start.

The following is the link to the game for solving equations using a balance.

http://www.nlvm.usu.edu/

Click on grades 6-8 Algebra topics.  Click on algebra balance-negatives.

I went to the store today to take advantage of a store special on soup.  I wanted to compare the sale price at the grocery store to the price at Wal-Mart.  The grocery store was selling the soup 8 cans for $10.  I set up a proportion to find the unit price of the soup.

$10      =     x

8 cans       1 can

8x = $10

8          8

x = $1.25 per can of soup at the grocery store

The soup at Wal-Mart costs $1.50 a can, so this was definitely a deal.

________________________________________________________________________

Narrative 2:

I bake cookies for Christmas and sell them to coworkers and friends.  I use 18 ounces of M and M’s for a double batch of cookies.  The double batch makes about 7 dozen cookies.  I sold a total of 25 dozen so far.  I needed to know how many bags of M and M’s I should buy for this weekend. 

18 ounces =    x

7 dozen        1 dozen 

7x = 450

7          7

x  is approximately 64.2 ounces

Therefore, I would need to buy 5 14 oz. bags of M and M’s.  Gee, I guess I’ll have to eat the rest of the M and M’s.

With the definition of function, I would show more examples of functions and distinguish between equations and functions with more detail.  I am sometimes fearful of too much detail in a definition for middle school students.  I think that it can be overwhelming.  Therefore, I would split up the definition of function into a couple of segments, or least introduce the vocabulary, such as domain, range, independent, dependent, etc. at different times.  I would show them many examples of the vertical line test and tables to stress the idea of one and only one output for each input.  The more they see this, the better they will understand a function. 

To determine if my students grasp the distinction between equations and functions, I would start with a ticket out the door.  They would give an example of an equation and a different example of a function.  I may also add an example of a non-function.  The next day, I would have a bell ringer for the students to complete.  They would have to take examples of equations and functions and classify them accordingly.  I would use this as a starting point for the follow up discussion. 

Examples:  2 + 3 = 5                x + 2 = 5          2x – 1 = -10

Linear Equation:  y = x + 4  x is the independent variable and  is the dependent variable

                                    When graphed on a coordinate system, the solutions will be a

                                    straight line.

Function:  A function is a rule that for every input ( domain, independent variable) there is one and only one output (range, dependent variable)

This is very similar to an equation, which can be a function.  However, some equations can have more than one output for one input.  Example:  y =  is an equation, but not a function because for x = 4, y = 2, and -2. 

Examples of functions in equation form and table form.

F(x) = x + 4    

 To determine if a graph is a function, you can use the vertical line test. 

  If a vertical line can pass through a graph and it intersects the graph

  only once, then it is a function.

The way I was taught math in elementary and middle school is that there is only way do solve certain problems.  I was very lucky in that I liked to follow the rules to please the teachers.  The farther I went in mathematics, however, I discovered that some problems can be solved a multitude of ways.  I originally had difficulty with this concept when a specific teacher asked me to show at least two different ways to solve one problem.  The more I did this, the more I realized it was more fun to be different than everyone else.

In my classroom, I often give problems that can be solved many different ways.  I then have the students explain their process to the rest of the class.  I hear “I didn’t think of that.  That was so much easier than the way I did it.”  I also try to show more than one way to do a problem on the board and have the students choose which way they are most comfortable.  I find that modeling the multiple ways encourages the students to do the same.

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

In high school, I often felt that I was not good at math because I needed pencil and paper or a calculator to work out a problem.  The other kids in my class, primarily boys by the way, could do a lot of calculations in their heads.  It wasn’t until I started teaching that I was able to do computation in my head.  I saw many students like me in college and I felt much more comfortable.  Although, my students and parents feel the way I did in high school.  They tell me all the time that they are not good in math because they can’t do it as fast as everybody else.  Due to these conversations, I employ a long wait time when asking for answers in class.  I remind my students that they have many tools to choose from:  their minds, their paper, their calculator, or anything else they think they can use.  I often model doing some simple computation on the board so the students feel more comfortable using paper, if they need to, to find the answer.  If the teacher sometimes has to do it that way, I must not be so dumb after all. 

“Fractals” and “Nature” and “Patterns”

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

I knew of the Fibonaccio sequence, but I was unfamiliar with all the ways it is found in nature and mathematics.  Examples of it in mathematics are things I deal with everyday:  the product of any four numbers in the sequence is the area of a Pythagorean triangle, the conversion of miles to kilometers can be applied to the sequence because the actual conversion is close to the golden ratio, the connection between the diagonals of Pascal’s triangle and the Fibonacci numbers.  In nature, the number of petals in certain flowers and plants are Fibonacci numbers.

I never really understood fractals before this Webquest activity.  I found it so interesting that I probably spent too much time looking at all the sites.  It was so “Chaotic” that I looked up the formal definition of fractal to determine that the “pastern” that occurs is that a piece of the whole is usually a smaller version of the whole. That helped me distinguish a better picture of a fractal.

What images did you find particularly striking? There were so many to choose from, but these are just a few sites I found particularly striking during my search.

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

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

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

I looked around my house and my work and I found several examples of the Fibonacci Sequence and fractals.  The most gross was probably the mildew spots down in the basement(my hot water tank was leaking for several weeks and there is no ventilation) that resembled a fractal.  I also found that many of the plants I have followed the idea of the Fibonacci sequence.  I take a daily walk with my dog and I now find myself looking at nature in a whole new light, mathematically speaking.

How can you adapt this webquest activity for your classroom?

I am definitely going to use this webquest in my classroom.  Initially, I will have my advanced learners complete this webquest as an area of interest for them.  I will then use it when I teach a mini unit of geometry, with very specific questions for them to answer.  The students will find it very interesting to see math outside of the classroom.  They seldom think of geometry outside the traditional polygons.  This would definitely spark interest.

Formal definition:  Non-traditional patterns are simply patterns that do not follow a repetitive format.

Linear Pattern

“Kid friendly” definition:  A linear pattern is a repeated operation or operations done to the independent variable that creates a predictable dependent variable.  The operations in a linear pattern are restricted to addition, subtraction, multiplication, and no power greater than 1.

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. (www.mathsteacher.com.au/year8/ch15_graphs/02_linear?patterns.htm)

The differences between the “kid friendly” definition and the formal definition of a linear pattern are slight.  The formal definition refers to the fact that a linear pattern will create a straight line when graphed.  It also is more straight forward (pardon the pun).

In my classroom, I would start with making tables to establish the relationship between two variables.  I would continue this and examine the rules that establish the pattern.  The students enjoy finding the “rule” for the table of values.  I think this would help them connect the relationship between the two variables.  I would then introduce the idea of independent and dependent variables.  I would give concrete examples that they could relate to in order for them to retain the concept.