Factoring Quadratics

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.

Reflections on Blogging

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.

5.D.2: Applets

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. 

For 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. 

For 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!

The Magic of Proportions

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:    

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″.

Every 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!

Evaluating our Definitions: Equations and Functions

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.

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

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 2^nd 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.

My Reflections on Math Myths

Through out the years we have bumped head on in to various math myths that distracts our students from believing in their own abilities.  Below are two examples that I have and continue to encounter:

Myth #5:  There is always one best way to do a math problem.

I do not remember encountering this myth as a student, maybe because a lot of my math education involved understanding the meaning of concepts before learning the process.  Mr. Mims, my high school math teacher, encouraged us to do math our way, and I have extended that model in my own classroom.  However, I have encountered this myth as a teacher – from my peer teachers.  Often, I have been chastised for demonstrating and accepting different processes for solving problems, especially since that means I have deviated from the student “textbook.”    I think the best way to dispel this myth is to demonstrate more than one way to my students, and then allow my students to choose which way works best for them.

Myth #11:  There is a “math mind” – some people have it and some don’t (very similar to Myth #2 – Math requires only a very logical mind).

This is probably one of my biggest pet peeves; I have heard this myth in the form of “I can’t do math, or my mind is not wired for math.”  Being able to do math is just as critical in everyday life as being able to read.  Again, my Dad was a huge influence in keeping me from ever believing this myth.  Just as it took effort on my part to learn to read, he ensured that I understood that it took effort on my part to do math. 

 

I think the best way to dispel this myth is find good role models and exemplify them in our classroom to the students – for example, Danica McKeller. 

 Both of her books, Math Doesn’t Suck and Kiss My Math, are classroom staples. 

 

 

The movie, Stand and Deliver, is a must for the classroom too.  Showing students individuals who the “world” least suspects as being able to do math is a great way to dispel this myth. 

Playing with Pascal II

Pascal’s Triangle (formal desciption):  The arrangement of binomial (algebraic expression containing two terms that are not like terms) coefficients in a pattern of a triangle.  Named after French mathematician Blaise Pascal, each number in the triangle is the addition of the two numbers above it.  Each row in the triangle starts and ends with 1.  Pascal’s triangle can be extended infinitely.

Playing with Pascal

Looking at Pascal’s triangle, note that the outer edge always starts with 1.  That is because the numbers that form the triangle are equal to the sum of the two numbers above it – and of course the apex of the triangle is 1.  For the outer edge, there is only one number above it:  1.  And as we know from the identity property of addition, any number plus zero, is equal to that number (1+0=1).  Now look at the third row which 1, 2, 1.  There are two ones above the 2 … 1+1=2.  The fourth row follows the same pattern; above the 3 is a 1 and a 2 … 1+2=3.  And so the triangle expands infinitely. 

Once you have built the triangle to your self-determined length, you can identify numerous other number patterns, for example, the edge right underneath the outer edge increases sequentially by 1.   Notice too, that if you exclude the middle row of numbers (perpendicular to the bottom edge of the triangle), that the sides on each side of the middle row are mirror images of each other.  Also notice, the sum of each diagonal row forms the set of Fibonacci numbers.  If one highlights all of the number divisible by 5, inverted equilateral triangles are formed with each side 4 numbers long.  Having students play with this triangle would be a great exploration activity.

Web Quest: Non-linear Patterns

Lets go on a Web Quest!

 http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html  

(by Dr. Ron Knott – fibandphi@ronknott.com)

The above link takes you to a site titled Fibonacci Numbers and Nature.  The site is broken down into sections focusing on family tree patterns in regard to Fibonacci Numbers and then the Golden Ratio in nature.  Each section has activities that can be use in your classroom as well and all sections cite the resources.  

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

I am an avid reader of all kinds of genre.  I remember thinking that the first time I had ever heard of fractal patterns and the Mandelbrot set was when I read Fractal Mode by Piers Anthony, in 1992.  The term fractal was coined by Mandelbrot in 1975.  Now I understand why this term was unfamiliar to me – I finished my math degree in 1980, just five years after Mandelbrot’s publication and way too soon for the term to have been in any of my college texts.

2.  What images did you find particularly striking?

 All, retrieved 10/30/09, are from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

“Fibonacci numbers can also be seen in the arrangement of seeds on flower heads.  The reason seems to be that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage, all the seeds being the same size, no crowding in the centre and not too sparse at the edges.

The spirals are patterns that the eye sees, “curvier” spirals appearing near the centre, flatter spirals (and more of them) appearing the farther out we go.  

So the number of spirals we see, in either direction, is different for larger flower heads than for small. On a large flower head, we see more spirals further out than we do near the centre. The numbers of spirals in each direction are (almost always) neighboring Fibonacci numbers!”

“The leaves here are numbered in turn, each exactly 0.618 of a clockwise turn (222.5°) from the previous one.  The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.

If we count in the other direction, we get a different number of turns for the same number of leaves.

The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers!”

“Romanesque Broccoli/Cauliflower (or Romanesco) looks and tastes like a cross between broccoli and cauliflower. Each floret is peaked and is an identical but smaller version of the whole thing and this makes the spirals easy to see.”

Actually, I really liked this example because not only does it represent Fibonacci numbers in nature, but this is also an example of a fractal pattern (enlarge this picture and look closely)!

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

In addition to all of the examples in my garden, you can see the nonlinear patterns on the scales of the fish in our fish tank, on the brick patio out back, the fractal print we have in our loft, and so on.  It is also fun to explore the idea of Phi with our own bodies, although caution should be taken because there are exceptions to the Golden Ratio throughout nature and ourselves.

4.  How can you adapt this webquest activity for your classroom?

For this type of webquest, I would recommend that the students individually research a few websites that have been pre-identified, to provide the students the opportunity to determine their own areas of interest.  Then, have each student identify what they would like to explore further (prioritize top three).  Based on their areas of interest, thoughtfully form groups of four and have them go on a webquest to gain more in-depth information on their particular topic of interest with the outcome being that of preparing a presentation for the entire class.