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Happy Mother’s Day!

April 27, 2018

via Barbara & Bunny—Ladies of the Greatest Generation

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Factoring Quadratics

November 22, 2009

When factoring a quadratic equation (ax2 + bx + c), one should remember that you are simply breaking a trinomial into its binomial factors.  Following these steps will help you factor a quadratic equation:

Ex:  x2 + 9x + 20

1.  Look at the x2 term.  If it does not have a coefficient, then each of the binomial factors will just have an x term, like such:  (x +    )(x +   ).

2.  Look at the constant (or c) in the equation, and determine each set of factors, like such:  Factors of 20 = (1&20, 2&10, 4&5).  Determine which set of factors’ sum is equal to b, the coefficient of the x term.  In the above example, 4+5 = 9

3.  So, one of the binomial factors will have a positive 4 and the other will have a positive 5, like such:  (x+4)(x+5)

Paraphrasing the instructions gave me the opportunity to think through the process and make sure that I was not missing a step somewhere along the way.  I think that by having your students journal their own understanding about a process helps them to realize if they have a strong understanding of the process and helps them to identify areas where they may still have some questions.

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Reflections on Blogging

November 22, 2009

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

I doubt that I will continue to use my blog established as part of this course.  I found it easier to use the Forum in this course for dialoguing with my classmates.  I do see where there is a place for blogging on my own website, especially for my students and their parents.  But I would envision the blogging to be only one aspect of my website.

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

I have always enjoyed math … learning it, doing it, and teaching it.  What I learned though, is that there are so many more resources and tools available for math teachers today than 15 years ago.

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 enjoyed exploring the various websites that have math applets that can be used in the classroom with the students.  I even found an applet for downloading onto an ipod touch or iphone on “number lines.”  I downloaded this onto my own ipod touch, and sure enough, as soon as my younger daughter found it, she started playing with it.

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

I know for certain that I will incorporate journaling in my classroom.  I have found that journaling is a wonderful way to see into our students’ minds and to use that as a means for guiding my future lessons.  I will have to think about the use of blogs.  First, know that I am putting aside for now the issue of internet safety and school regulations.  Blogging pro – students love using technology and they may be more willing to blog than to journal.  Blogging cons – feedback to student is more public and may not be wanted.  Blogging is not as conducive for drawing diagrams/pictures; while this can be done it takes significant time on the part of the blogger.  Accessing, reviewing, and providing feedback on blogs is more time consuming for me as a teacher.

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5.D.2: Applets

November 8, 2009

The available resources for teaching math concepts to students at all levels are amazing as compared to 10, 20, even 30 years ago.  Two of my favorite online sites with free applets for teachers and students to explore are Illuminations (http://illuminations.nctm.org) and National Library of Virtual Manipulatives (NLVM) (http://matt.usu.edu/nlvm/nav).

Both of these sites allow you to choose grade level as well as topic area to help narrow down the hundreds of tools available. 

balance beamFor those students who are still struggling with the concept of keeping equations balanced as they seek to solve them, I suggest the Algebra Balance Scales applet on the NLVM site.  This applet has the students set up a balance beam based on an equation, and then solve for x.  As the students solve the equation, (by adding, subtracting, multiplying or dividing), they must keep the beam balanced.  Each step taken is represented in a number sentence as well as with manipulatives. 

Proof no wordsFor your visual learners, check out the Proof without Words: Pythagorean Theorem applet on the Illuminations site.  This applet proves Pythagoras’s theorem visually – without any words!

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The Magic of Proportions

November 7, 2009

My daughter loves to watch the CSI and NCIS programs (all of them) and is fascinated with the idea of being a crime scene investigator.  As we talked last night, she pointed out that the CSI investigators, as part of their investigation for the recent shooting at Fort Hood, would have to draw the crime scene as documentation of physical evidence locations, as well as measurements showing pertinent size and distance relationships in the crime scene area.  So, if the area of the shooting was 50 ft x 60 ft, how large of a piece of a paper would they need to use if the scale was 1in:5ft?

For this you would first need to set up two proportions and solve: CSI body on ground   

1 in/5 ft = x in/50 ft 

50 in/5 = x in 

10 in = x                                 

and                 

1 in/5 ft = x in/60 ft

60 in/5 = x in

12 in = x

The steps used to solve both of the proportions are:

1.  Multiply both sides of the proportion by the amount of the denominator where x is the numerator (multiplicative identity property)

2.  Reduce the resulting fraction to simplest form (divide the numerator by the denominator).

The CSI investigator will need a sheet of paper that is at least 10” by 12” to draw the crime scene for the investigation.  I would add a one inch border, so the paper size would need to be 11″ by 13″.

SpaghettEvery year the youth group at my church conducts a spaghetti dinner fundraiser.  Last year, the youth made 20 batches of homemade spaghetti sauce and served 140 people.  This year, the group has already sold 200 dinner tickets.  They also feel that they will have approximately 40 people who will buy tickets at the door.  They are now trying to determine how many batches of spaghetti sauce they will need to make to serve 240 people.

First you would need to set up one proportion: 

20 batches/140 people = x batches/240 people

4800 batches/140

34.29 batches = x

To solve the proportion:

1.  Multiply both sides by the denominator where x is the numerator (240).

2.  Simplify the resulting fraction by dividing the numerator by the denominator.

In order to be sure that all 240 people can be served, the youth group will need to make 35 batches of spaghetti sauce.  WOW, that is a lot of sauce!

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Evaluating our Definitions: Equations and Functions

November 7, 2009

After looking at my classmates’ postings and reflecting on my own definition, I realized that I had approached my definitions as if my students were just learning about equations, functions, and the similarities/definitions between the two – I did not go into points – (x,y) or graphing – for functions.  I think I should further expand my definition of functions to include points and graphing.  Then, I would be able to add the “vertical” line test, which I think would be key to understanding that there are times when there are equations that are not functions.

I love using graphic organizers, and I think a great way to evaluate whether my students have grasped the difference between functions and equations is through the development of a Venn Diagram.  I also like to challenge my students to write “equations” that are not functions.

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

November 5, 2009

Equation:  A number sentence that contains numbers and/or a variable, an operation and an equal sign, such that the value(s) for the variable (if there is one) can be determined.  Equations may be linear (straight line) or nonlinear (polynomial equations of the 2nd degree and up). Equations are “solved.” 

Examples include:

 Ex. 1:                                                               Ex. 2:

 2 + x = 9                                                         x2 = 4

-2 + 2 + x = 9 + -2                                         √x2 = √4

 x = 7                                                               x = 2  AND   x = -2

Function:  An equation, where for each value of x (input) there is one and only one y (output).  Functions are usually notated as f(x) [read verbally as “f of x”].  Functions can be linear and nonlinear.  Functions have domains (the input values for x) and ranges (the resulting output value or y). 

Examples include:

                                 f(x) = x + 2

x x + 2
1 3
2 4
3 5

                                  f(x) = x2

x x2
1 1
2 4
3 9

There are several activities that I would use to reinforce the concepts of functions, equations, variables, and linear patterns/equations.

1.  Compare & contrast, using a Venn diagram, functions and equations (this could also be done as a journal enty).

2.  Journal entry:  Solve for x a given equation, e.g.  x  + 2 = 7.  Solve for x + 2 = 9.  Equals 10.  Equals 12.  Describe what happens when the sum changes.  How would you define x?  Why?

3.  The function machine applet set up on a SmartBoard as a math activity.

4.  Journal entry:  Explain why “x2 = 4” is an equation and “f(x) = x2” is a function?

5.  For given table(s), write a formula that defines the relationship between each of the values.

6.  “Guess My Rule” calculator game in groups of four.

7.  Webquest, done in groups of four, on Fibonacci Numbers, Golden Ratio, and Fractals with outcome a formal presentation to the class on results of the webquest.