Reflections on Blogging

I really liked putting together the idea of a math blog.  I think it would be exceptionally helpful for homework help.  However, I’d much prefer they use their notes from class with multiple examples.  A better use for the students would probably be the links to sites for math practice on different topics.  I may continue to build my math blog, if time allows; it just takes so much time to type everything up.  I hesitate to make it available on the internet for student use, though.  Unless, of course, I could make it so that no comments could be left; I’d hate to open up a can of worms in today’s world.

 

Not having a secondary degree in math (I’m middle school certified), I was expecting to be a bit behind my classmates.  Being an elementary teacher first, then middle school math teacher second, I was a bit worried.  However, I do believe that I have a strong foundation of math knowledge.  So, I learned that I should be confident enough in my abilities and teaching abilities to discuss the topic with anyone.  It was also nice to look at math from so many different perspectives and get different “takes” on teaching strategies.  I really learned a lot from my classmates.

 

I learned about a lot of internet resources that are very kid friendly and educationally valuable.  One interesting discovery I made was from reading my classmates “mathographies.”  After having early success in math, at some point most people struggled to find their way, and it wasn’t until they truly understood the ideas of math that it began to make sense.  I also found it interesting that there is, in everyone’s past, one teacher that made a difference in their lives, not just mathematically, but personally.  From reading my classmates’ work and responses, I can see how much everyone absolutely loves math and wants to create that love of math in their students.

 

I love the idea of using journals with my students, but I just don’t know where I’d find the time to read them.  So, I really doubt that I will.  I might if there comes a time when I only teach math and not language arts (reading/English/spelling) as well. 

 

I would really like to incorporate student blogs into my curriculum.  I think it would be a great way for students to show a true understanding of concepts.  However, our firewall blocks all blogs at our school, and our technology is so poor that only about 50% of the computers work.  Therefore, having students access my blog from school wouldn’t work either.   

Blogging was fun…maybe I should create a Pirates blog this summer when I have more time?!  What do you think, Pirates fans out there?  Well, then again, maybe we’ve suffered enough…

 

Factoring Quadratics

Generic Format of a Quadratic Equation:     y = ax² + bx + c

 

Example: x² + 5x + 4

 

1.  Find the third term:   c = 4

 

2.  List factor pairs for c:           4 = (4 * 1) or (2*2) 

 

3.  Which set, (4 * 1) or (2*2), has the sum of b, 5?                  (4*1)

 

4.  Find the first term.  Factor it:            x² = (x * x)

It only has one factor, x.  Therefore, both binomials will start with x.

 

5.  In one binomial, add 4 (from the factor pair in Step 3).

 

6.  In the other binomial, add 1 (the other factor in Step 3).

 

7.  Therefore, we will have (x + 4)(x + 1)

 

8.  Check your work using FOIL (First Outer Inner Last/aka the Distributive Property):

           

(x + 4)(x + 1)

            (x * x) + (x * 1) + (4 * x) + (4 * 1)

            x² + x + 4x + 4

            x² + 5x + 4

 

 

Paraphrasing did help me internalize the concepts.   I was able to better break down the process of creating the correct binomials by listing each factor pair before creating the binomials.  I think creating this step-by-step process will allow for each student to have framework in place that will yield more success.

 

I think paraphrasing processes would be a great activity to reinforce computation, especially for computation of fractions.  This is a weak area for most students; therefore, this would be an opportunity for them to internalize the rules themselves rather than just “following the rules of the teacher.”

Welcome!

Hello everyone!  My name is Shawn Sowers and I’m a sixth grade teacher.  I have two goals for this class.  I’m currently teaching two math courses, one eighth grade level and one seventh grade level.  By taking this class, I’m hoping to better prepare myself to move to our new middle school (it is supposed to open in 2010) and hopefully move up in age/grade level.  I’m hoping to some day teach Algebra I as well as the normal middle school math curricula.  I’m also hoping to get a better insight into how middle school teachers teach for meaning when time is such an issue.  Right now, we have math for 70 minutes a day, and I’m able to accomplish quite a bit in that time.  However, in the regular middle school day, classes are 45 minutes.  I wonder how I’ll “be as good” with 25 minutes less per day?!

6-D-2: Applets

Applet: Pan Balance – Numbers

 

I really enjoyed the Pan Balance – Numbers applet.  It was an activity in which students create balanced equations using numbers.  When numbers are entered on one side, they must be balanced out on the other.  When a balanced equation is achieved, the equation is listed for the student.  Students should have ample time to play with this using all operations.  This applet was actually located in the Grades 3-5 activities and is a great visual to reinforce the concept of equivalence and also operations. 

 

However, the concepts that I would like to explore with this applet are the Properties of Equality.  Once students have created balanced equations, they would be directed by the teacher to add (subtract, multiply, divide) one side by a number and then do the same to the other side.  Students could then record the results of those operations.  The result, obviously, would be another balanced equation.  They could visually see that when something is done to only one side, an unbalanced equation occurs.  (This is the common mistake when students are learning procedural rules for solving equations.)  

 

After trying this many times, students will be able to discover the properties of equality rather than being told the vocabulary term and then its application.  They would truly see the properties in action and how they work.  This reminds me of the Countryman article that we read regarding problem solving and vocabulary needing to be experienced before noting specific mathematical terminology.  

 

To access this applet: http://illuminations.nctm.org/ActivityDetail.aspx?ID=26

Math Myths

Math Myths:    

“There is a ‘math mind’ – some people have it, some people don’t.”

“Math requires only a very logical mind.”

 

The myth that reminds me of my days as a math student was “There is a ‘math mind’ – some people have it, some people don’t.”  This one really hit home because at one time I thought I had a math mind.  A few years later, I was pretty much convinced that I didn’t have a math mind at all.  If I had a math mind to begin with, I wouldn’t have just lost it (along with the litany of other items at the bottom of my locker).  Now, I’m back to having a math mind.  Obviously, there is no such thing as a math mind, just the math experiences that we’ve had along the way that make us believe that we are either good at math or bad at math.  I’m lucky; I got a second chance at math.  As they say on TV, “Myth Busted.”

 

I never believed “Math requires only a very logical mind.”  I have a very logical mind, and I was a mathematical failure.  Some classmates of mine didn’t have an ounce of common sense among them, but were tremendous math students.  As a teacher, I’ve found this to be true as well.  Some people are just mathematically intuitive, not logical in any manner, but intuitive and creative.  These are the most important qualities for those who discover math, not just do math.

 

To help dispel these myths for my students, I constantly try to engage them in different thinking activities through problem solving.  Students need to try to solve problems rather than just do exercises.  There is a world of difference between the two.  It’s surprising to see the difference between thinkers and doers.  The students who run through homework assignments usually struggle with the problem solving activities and vice-versa.  By varying the strategies and activities, every student has a chance to excel in some area of math.  I am a true believer that success breeds success, so we need to optimize our efforts at assuring some kind of success at some level for every student.

Nonlinear Web Quest

I was somewhat familiar with the mathematicians, Leonardo Pisano (a.k.a. Fibonacci) and Pierre de Fermat.  Fibonacci was familiar for obvious reasons, his sequence, but I was amazed by the amount of study done on just this one sequence.  I was really, really surprised to see that a journal is devoted to Fibonacci, The Fibonacci Quarterly.  As my students would ask, “Don’t these people have anything better to do?”  I learned a few things about the sequence that I didn’t know before: 1.) the ratio of consecutive numbers approaches the golden ratio (http://www.lifeinitaly.com/heroes-villains/fibonacci.asp); and 2.) you can use this application of the golden ratio to convert from miles to kilometers because in 1 mile there are about 1.6 kilometers. For example, look at the 5.  The next number is 8.  In 8 miles, there are about 8 kilometers.   (http://www.daviddarling.info/encyclopedia/F/Fibonacci_sequence.html) 

 

I was also familiar with Fermat, though it was in name only.  To introduce probability I always give the activity, “The Problem of Points.”  (http://mathforum.org/isaac/problems/prob1.html)

Years ago I had heard of Fermat’s Theorem, so this was a nice reminder of that. (http://en.wikipedia.org/wiki/Fermats_Last_Theorem) I was also familiar with the golden ratio, even though I had not seen some of the applications with pentagons and pentagrams.

 

I found the images in nature particularly interesting and also those that were created by computers using fractals.  What I found most interesting was the theory of chaos.  I remember that theory from Jurassic Park, but had never really put much though into it because it is Hollywood.  Just the fact that chaos is not really chaos, but does follow some type of small “generalizable” pattern is interesting to try to wrap your mind around.  The fact that so many things in nature follow these patterns is astonishing.  Nature picks the easiest path; that must mean that math is the easiest path one can take to making sense of the world around us.  (http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html)

 

I cannot honestly say that I was able to identify any applications of nonlinear patterns in my home or school.  I’ve looked, but maybe my mind just isn’t trained quite well enough to notice nonlinear patterns.

 

I certainly would like to adapt this type of web quest for the classroom.  I think I would just webquest mathematicians to begin with and see where this takes us.  It opens up so many doors into math for the children to dive into.  I did this many years ago within a data unit that incorporated technology and was amazed at the results and at how much the students learned.  They were excited, really excited that these “old Greek guys” I told them about were actually people and did a lot more than just what they are remembered for.  Of course, the web quest will only work if our computers do (about a 50-50 proposition these days).  Another application of this that would tie in well with the gender issues in math is web questing for female mathematicians and their accomplishments in the field.  We all could stand to learn more in this area of math.

Pascal’s Triangle: Formal Language

Pascal’s Triangle is a very useful tool to have in your mathematical toolbox.  Image available at: http://www.math.umass.edu/~mconnors/fractal/generate/pascal2.gif

 

The very top triangle in Pascal’s Triangle is an equilateral triangle.  (For this assignment, I use equilateral triangle to mean that there are equal amounts of numbers on each side, i.e. four numbers on each side).  There are two more triangles below this that, coupled with the one above, create a larger equilateral triangle with eight numbers on each side.

 

Below this 8-8-8 triangle to the left and right are two more triangles congruent to this one.  Within each of these, there are smaller triangles congruent to the smaller triangles at the top.

 

If we look at Pascal’s Triangle as a whole, we see triangular numbers.  For example, look at the rows of numbers.  In Row 0 there is one number.  In Row 1 there are 2 numbers; add those to the one number above, and you have 3 total.  In Row 2, there are 3 numbers; now a total of 6.  In Row 3, there are 4 numbers; now a total of 10.  These are the first numbers in the triangular number sequence, 1, 3, 6, 10, 15, 21…  This pattern is neither geometric nor arithmetic because it follows the sequence, +2, +3, +4, …  These totals (1, 3, 6…) are also specifically listed for us diagonally top to bottom (third diagonal row from the top in each direction).

 

Also in looking at the horizontal rows, I see a probability application.  These rows are the solution to a problem that I use in class, “A pizza shop offers 5 different toppings.  If you order one pizza per week, how weeks could you order a different pizza before you’d have to order one that you’ve already had?  (Sauce and cheese is a standard plain pizza.)”  The solution to this is 32, the sum of the digits in Row 5 (1 + 5 + 10 + 10 + 5 + 1).  The results are summarized in the table below (with combination notation below):

 

# of Toppings	0	1	2	3	4	5
# of Pizzas	1	5	10	10	5	1
	5C0	5C1	5C2	5C3	5C4	5C5

 

The natural extension to this is to increase the amount of toppings, thus creating a use for Pascal’s Triangle; it works for any amount of toppings.  It’s a nice tool for the students to have to solve problems quickly, though they must discover this application of Pascal’s Triangle for it to make sense and not just be an unfounded shortcut.

Working with the definition of linear patterns

Non-traditional Patterns (Formal Definition): patterns that do not follow a repetitive format

 

Linear Pattern (Kid Language): is a sequence of terms with a recurring pattern (common difference); if the results are graphed, they form a line

Linear Pattern (Formal Definition): In a sequence of numbers, each number is called a term.  Linear patterns have the same difference between the terms.  http://www.studyit.org.nz/subjects/maths/math1/1/subjectcontent/linearpatt.html

Their actually is not much difference between my definition and the formal one.  I attribute this to the fact that I was an elementary teacher before becoming a middle school math teacher (I still teach language arts as well as math).  Understanding and using vocabulary is an absolute must at this age (11-12).  Without understanding vocabulary, one doesn’t understand what they are reading and studying.  To facilitate vocabulary acquisition, my main strategy is to get them thinking in terms of everyday synonyms and learn to paraphrase dictionary definitions.  This makes the vocabulary terms useful.

 

As with math, students need to understand concepts, not memorize procedure only.  The same goes with vocabulary.  Students should understand that a pattern is just a sequence (which they are familiar with through reading).  They should also recognize that the change from one number to another is what creates a pattern.  This is called a common difference.  I like the term common difference because it links them back to related topic (common multiples, common factors).  By noting commonalities, students can internalize the true meaning of pattern and not just find them, but create them (inductive/deductive reasoning).  Simple graphings of these numbers will create the line for them, thus the root word of linear, line.

PUMAS Site Review

I really liked the PUMAS site for its real life math connections.  One activity caught my eye because I (and I’m sure every other teacher) invariably have the same mistake made in class.  The activity is called “Dollar$ and Cent$.”  Dollar$ and Cent$ centered around real-world math mistakes such as grocery store coupons and sales ads.  The mistakes are rooted in which unit they are trying to measure.  For example, .99 ¢ is meant to be 99 cents, but the value is actually less than one penny.  Students make this same mistake when they mix their units.  [I’m sure it couldn’t be math people making this mistake when they design their ads…let’s blame it on the graphic designers or those dreaded English-type folks =)]  This is a great learning activity for students.  They think it’s great that they should be able to go into a store and pay less than a penny for a two-liter bottle of pop.  However, when I put them into the store owner’s shoes and I’m the customer, they have a bit of a different attitude.  They really change their mind when I demand change from the penny that I give them for the pop.  Overall, I think it’s a pretty neat site, though most of the activities are above my student’s heads, but they certainly wouldn’t hurt to try.  I’m often surprised with how far some kids can get when asked the right questions.  Learning through discovery is the most powerful form of learning!

Inverse Properties

Additive Inverse: 1.)  any two numbers whose sum is zero; 2.)  the sum of a number and its opposite always equals zero.

Arithmetic Example:  4 + (-4) = 0; -7.5 + 7.5 = 0

Algebraic Example:  n + (-n) = 0; -x + x = 0

 

Multiplicative Inverse: 1.)  any two numbers whose product is one; 2.) the product of a number and its reciprocal always equals one.

Arithmetic Example: 4 x 1/4 = 1; 2/5 x 5/2 = 1

Algebraic Example: n  x  1/n = 1; 4/m  x  m/4 = 1

Order of Operations

Most people use the mnemonic device, Please Excuse My Dear Aunt Sally and theorganizer, PEMDAS.  However, I do not because some students believe that multiplication comes before division because M comes before D in the organizer; the same with addition and subtraction.  Here is what I use to alleviate this problem: Gary Eats Mc Donalds After School.  (Thank you Lucas!)  Here is the visual organizer:

G – Grouping symbols: brackets, parentheses, absolute value bars, fraction bar

E – Exponents

MD – Multiplication or  Divsion from left to right; I put an arrow above these going from left to right.

AS – Addition or  Subtraction from left to right; also an arrow above these letters.

Order of Operations Practice: http://amby.com/educate/ord-op/