Hello world!

Hello, I am an eighth grade mathematics teacher who is entering the world of blogging for a graduate class assignment.  I have enjoyed teaching middle school for the past six years and when I am not teaching I try to be outdoors.  The three top things I’d rather be doing are motorcycle riding, hiking, or Frisbee golfing.

By taking on online mathematics course I hope to be able to steal some new ideas from other math teachers that will make my classes more relevant to my students.  I am also hoping to find an easier way to use technology in the mathematics classroom.   I would like start using PowerPoint and would love to have a smart board and be able to upload class notes to Edline for my students to access.

Non-Linear Pattern Web Quest

Non-Linear Pattern Web Quest

“Fibonacci” and “Phyllotaxis” and “Prime Numbers”

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

I knew that the Fibonacci sequence is often described as a pattern for the population of rabbits:

Start with a single pair of rabbits. Any pair of rabbits of one generation will produce a pair for the next generation, and then another pair of rabbits for the generation after that. But then they will die. How many rabbits will be produced in the n-th generation?

The answer is the famous Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368…

http://www.lifeinitaly.com/heroes-villains/fibonacci.asp

I was unaware that Fibonacci’s actual name was Leonardo Pisano and his sequence can be found in nature’s phyllotaxis, an arrangement of leaves.  The sides of pineapples, ridges of pines cones, and centers of sunflowers are examples.

2.  What images did you find particularly striking?

I really liked going beyond the example of rabbits and looking at the cut away of a nautilus shell, sunflower and pinecone.  I really feel that there is an artistic touch that nature employs on these items.  I am mostly impressed that they can be modeled by mathematics.

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

At work I follow a non-linear pattern of time throughout the day.  I have blocks of time designated for instruction, but that also have three minute intervals in between.  During the middle of the day the times are slightly different to accommodate lunches.

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

This web quest could be used to help students see that mathematics is truly a quest to explain what we see in nature.  I think that the students would enjoy discovering the different phyllotaxis that appears in nature.  If students start to see the sequence displayed in real-life then they should be more motivated to learn it.

“Fractals” and “Nature” and “Patterns”

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

I knew that fractals are made of a recursive pattern and I was most familiar with the Sierpinski triangle:

I soon found that the pattern can be found in plants, animals, and lightening.  All of these natural phenomenons are part of nature’s beauty.

2.  What images did you find particularly striking?

Fractals are found anywhere from photographs of earth, lava, lightening, a tree, the blood vessels of a kidney, an image of the Mandelbrot set, to manmade art of a tiled floor.

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

I can picture a nonlinear pattern of my paint chipping away from my front right fender of my car and creating what seems to be a fractal pattern of rust.  I can look out my window at home and see the leaves falling to the ground on a maple tree, thus showing the non-linear pattern of limbs and how the spread out and away from the trunk.

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

This idea of linking what we are learning in mathematics class to what we see in nature would be a motivating activity.  Students would greatly benefit from being curious about how mathematics can be used to explain something in nature.

My Reflection on Math Myths

I have encountered the myth that “it’s always important to get the answer exactly right.”  During my math education it seemed that you always had to work out problems for several decimal places.  In chemistry class, it makes sense that you use the correct significant figures when working on a lab.  I have found that in real-life, an adult mostly needs to use estimation in their daily lives.  I think that students need to realize that rounding and estimation are two very important skills that they can use in everyday life.  Students need practice when and where estimations are appropriate to use.  I may estimate the number of paper plates I’ll need for a family picnic or I may estimate the tip on a bill when eating dinner out.  Students need to recognize that these two cases do not need an exact answer.

“People who are good at math do problems quickly, in their heads,” is a myth that I believed in throughout elementary school, because I associated the smartest person in class with the student who won the flash card competitions.  I soon understood that even though I could not answer my multiplication tasks as quickly as others, I could still do well in mathematics.  I knew that I still needed to have my facts memorized in order to complete tests on time.  Now, as a teacher, I do not place time restrictions on my students.  I offer them the option of finishing a quiz or test during a studyhall at a later time.  I have also seen students competing with the class to be the first person to turn in their quiz and make minor mistakes that I believe the student would never had made if they would have taken their time.  I would ask my students what bridge over a dangerous river they would prefer to cross: a bridge built as quickly as possible with no plans or a bridge that someone created with carefully drawn out blue prints, computer models, and then built to the exact specifications of the blue prints.  Students need to realize that speed is not as important as quality when it comes to mathematics.

 

Add-Ventures for Girls: Building Math Confidence

Pascal’s Triangle

Pascal’s triangle is an isosceles triangle that is generated from the top down.  The top of the triangle is row zero and has one element, which is 1.  Row one is directly below row zero and has two elements, which are 1 and 1.  Row two contains the numbers 1 3 3 1, where each element is generated from the sum of the two numbers directly above it.  The outer left and right side of the triangle consists of all 1’s.  This pattern of adding the two numbers above to get the number below is continued throughout the triangle.  When you dissect the triangle row by row you start to see that the row is symmetrical can creates a palindrome.  For instance, row 5 is: 1, 5, 10, 10, 5, 1 and can be read forwards and backwards.  A line of horizontal symmetry can be see on the triangle.  Each row of the triangle, as you move down, has exactly one more number in it.  The zeroth and last element of every row is always a 1.

Working with the Definition of Linear Patterns

Non-traditional Pattern:

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

Linear Patterns:

Personal definition:

Linear pattern-a pattern that seems to grow or decay in a predictable manner.

Formal definition:

Linear pattern-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.

http://www.mathsteacher.com.au/year8/ch15_graphs/02_linear/patterns.htm

The difference between my personal definition and the formal definition is that the formal definition defines the pattern in terms of a coordinate grid.  I would have the students try to memorize the definition by looking at the word “line” in linear and remember that linear patterns create straight lines.

Practical Uses of Math and Science

Practical Uses of Math and Science

http://pumas.jpl.nasa.gov/

I gravitated towards a lesson “Metric Wins” and I enjoyed what I read.  As a math teacher who also liked science I always tell my students that we should get rid of our standard measurement system and work with the rest of the word using the metric system.   I also took at look at “Teaching Radicals in Less Than 5 Minutes”, which showed how to simplify the cubic root of an algebraic expression by having students play with a deck of cards and practice making pairs or three of a kind.  I thought that this same idea could be used at the middle school level when students are looks for greatest common factors or least common multiples when looking at the bottom line of the factor trees.  The concept of pairing up numbers like when they play go fish would make a great connection.

Inverse Properties

Additive Inverse:

a + (-a) = 0 or (-a) + a = 0

The sum of a number and its additive inverse is zero.

Any number plus its opposite is zero.

Example:

9 + (-9) = 0

4x + (-4x) =0

Your mother gave you $10 to buy lunch on the field trip and you spent $10.  Now you have no money.

10 + (-10) = 0

Multiplicative Inverse:

a * (1/a) = 1  for all real numbers, except a=0

The multiplicative inverse of a number a (written as 1/a or a^-1) is the number that, when multiplied by a, gives the result of 1.

Any number times its reciprocal is 1.

Example:

2 * ½ = 1

You bought 4 hotdogs at the baseball game and you split them between you and three friends.  You only get ¼ of the order, so you get one hotdog.

4*(1/4) = 1

My Mathography

My earliest memory of elementary mathematics was in first grade adding numbers.  My mother helped me to remember that we had an old chalkboard in the basement that we used to draw on as kids.  After my first couple days of first grade my mom told me that I dragged my younger sister down to the basement to show her what Mrs. Kennedy taught us that day.  Apparently I still have the same effect on some of my students, because she was not interested.   I enjoyed mathematics better than English class because if I missed a problem on a test I could get the answer and work backwards to figure out what I was doing wrong.  The fact that there was an actual “right” answer to the problem is what attracted me to mathematics.   When I wrote a paper for English I would wonder why I got a “B” or “C.”  The lack of feedback turned me off from English until I reached college and I finally got the feedback I needed to correct my writing.

I remember the fresh smell of the ditto machine and all the practice.  Since I knew my facts, it was gratifying to complete a sheet with ease.  It felt good to just get it and understand mathematics without studying.  I remember working a lot with my mother to memorize my vocabulary words for Friday’s spelling test.  The practice we did in class was enough for me to do well and understand it.

The two most vivid math memories I had were the two math tests I failed in elementary school.   The first was in second grade and I had to write down what time it was if it was quarter till four.  Since the entire class did so poorly on the test, Mrs. Gammel retaught telling time and I did well on the retest.  The second test that I did poorly on was the first test I took for Mr. Turner in sixth grade.  I made my four with the triangle (ex. 4), but he marked all of the questions with a four in it wrong because he only accepted square fours with the open top.  All of the problems were solved correctly, but to this day I make the square open topped fours.

Mr. Sharick, my seventh grade Pre-Algebra teacher, was my favorite math teacher.   I remember the classroom being very calm and comfortable.  He had very clear expectations for his students and I only remember one occurrence of student misbehavior.  I do not remember learning a specific topic or lesson, just that everything was clean, clear, and orderly.  I also greatly appreciated when a large portion of students would miss a test question; he would remove it from the test score, reteach the topic, and retest the topic on the next test.

Mathematics was always one of my favorite subjects growing up.  The reason I enjoyed it was because I could easily identify the mistakes I made and learn to correct them.  The analytical processes involved worked well for my learning style.  I did not mind learning a rule because once you memorized it; the rule could be applied to similar problems.

Math Stories from the Everyday-Hiking

After graduating college in 2003, a college friend and I decided to try backpacking for the first time.   We figured it would hard to get vacation time off once we got our first “real” jobs.  Instead of starting off small with a couple of weekend backpacking trips to prepare, we just winged it and set off for one week in August on the Appalachian Trail.  We started off in Harpers Ferry, West Virginia and ended in the Shenandoah National Park.   We made little progress hiking up our first mountain out of Harpers Ferry in a down pour.  We had to set up our tent in the rain and it was a cold dinner on the first night.  We had a hiking guide of the trial that showed us points of interest and mileage in relation to these points.   Along the trail there are shelters that you can sleep in and stay dry.  The second day we did not plan appropriately and had to set up the tent again.  We needed to know how many miles we could hike in a day so that we could get the most amount of miles in before dark and make it to a shelter before night.

I used distance = (rate)(time) to find our hiking rate so that we could finally sleep off the ground.

The distance was calculated using the hiking guide’s points of interest.

d = 5.2 mile r = ? t = 4 hours

 

 

 

 

Knowing that I was hiking up and down the Appalachian Mountains at approximately 1.3 miles per hour, I was able to calculate that I could hike about 13 miles per day if I hiked for 10 hours.  Knowing this I was able to sleep in shelters for the remaindered of the trip.

Integers

Integers-

Personal definition:  the set of whole numbers and their opposites

Formal definition: The set of integers contains zero, the natural numbers, and the negatives of all the natural numbers:  …, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, … an integer is a real number that does not include a fractional part.  The natural numbers are also called the positive integers, and the integers smaller than zero are called the negative integers.

Downing, Douglas. (1995). Dictionary of Mathematics Terms. Second Edition. Barrons.

The following is a link to a website with examples and practice problems with integers.

http://www.mathgoodies.com/lessons/vol5/intro_integers.html