Formal Definition of Non-Traditional patterns

Non-traditional or Nonlinear patterns have varying rates of change and do not form straight

lines when graphed.

Example -

A nonlinear pattern: 1, 4, 9, 16,…

Common nonlinear patterns include patterns such as square numbers, cube numbers,

and growing patterns.

 

Kid Friendly Definition for Linear Patterns

 

Linear Patterns- Patterns repeat over and over. Patterns can be formed by objects, actions, or characteristics. Patterns can be arranged or occur naturally.

Examples-

A combination of repeated lines, colors, letters, numbers, shapes, forms, figures such as A B C A B C A B C

 You can find many patterns in nature such as in leaves and snowflakes.

Formal Definition for Linear Patterns

 

Linear patterns have constant rates of change and form straight lines when

graphed.

Example  – A linear pattern: The sequence 3, 5, 7, 9, 11, 13,… is a linear

pattern because each term increases by 2.

My definition for linear patterns emphasized the repetition that is found in a pattern as well as what elements can be contained in a linear pattern. The formal definition stresses the rate of change between the objects in a pattern and points out that graphing a pattern results in a straight line. As students begin to work with patterns, they look for the repetition within the pattern. By asking guiding questions, students can be focused on the rate of change and then introduced to the formal math vocabulary during discussions. Students can create their own simple patterns that can be graphed to show them how linear patterns form straight line graphs.

For example, the following data set can be studied and graphed.

The rate of change for this data set is  y= 2x.

The graph that results from this data shows students that connecting the dots creates a straight line.

Combining this  hands-on practice with the formal terms of the definition will help students internalize the definition without relying on memorization since they will have  concrete experience with the rate of change and with a graph.

Credit:

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

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

 

My experience with blogging in this course is not the first experience I have had with blogging. I would concur that there are many educational benefits to blogging:

As noted in Education World,

As David Warlick points out on his Web site, however, the blog has evolved rapidly into something more:

Number 1: A blog is a Web-publishing concept that enables anyone — first graders, political pundits, homeless people, high school principals, presidential candidates — to publish information on the Internet.

Number 2: Blogs (a shortening of weB LOG), or blogging has become a journalistic tool, a way to publish news, ideas, rants, announcements, and ponderings very quickly, and without technical, editorial, and time constraints. It essentially makes anyone a columnist. In fact, many established columnists now publish their own blogs.

Number 3: Blogs, because of their ease of use, and because of the context of news and editorial column writing, have become a highly effective way to help students to become better writers. Research has long shown that students write more, write in greater detail, and take greater care with spelling, grammar, and punctuation, when they are writing to an authentic audience over the Internet.

In terms of my own use maintaining a consistent blog always seems to become an issue. So while I would say that I intend to continue using my blog, it might not happen in all honesty. Continuing the blog would allow my students to see me model the process of problem solving and explaining what I thought and did. I could also use the blog to post resources such as videos, podcasts, online readings, or links to helpful tools so that my students could access the links in one site from home or school.

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

I learned that I need to review concepts before teaching them to my students. This would allow me to craft lessons that engage kids and also to provide them with real world examples to which they can relate. The more prior knowledge I can activate the better grasp my students will have by connecting new skills to previously learned one.

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.

What I found especially interesting were the number of internet resources available. I can use the National Library of Virtual Manipulatives (http://nlvm.usu.edu/en/nav/vLibrary.html) for a number of units in my class.

I also think the Math Trail activities (http://www.nationalmathtrail.com) will give me the inspiration to create a math trail through the school’s neighborhood. Right across the street there is a cemetery from the mid 1800′s and an old church that have possibilities already coming to mind.

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

 

I already use journals with my students since there has been a real push in our building to encourage reading and writing in all disciplines. If I were to require my students to create a blog to replace their interactive notebook that would provide them with a sense of reaching a real audience beyond the classroom. In reality, to insure safety using the Internet I would use a host site that restricts access to group members to keep our privacy. Many students are intimidated by the physical process of writing. Using blogs removes that obstacle and allows them to express their thoughts without fear. All students are then given an equal voice and might interact with students they would choose to ignore in the face to face environment.

 

Here is a good collection of examples for using blogs in education:

http://www.slideshare.net/suziea/blogging-examples-in-education-oct-08-presentation

Quadratic Factoring

To factor a quadratic expression written in the general form of ax² + bx + c,  means to factor a trinomial into two binomials . The binomials are expressions that when multiplied using the distributive property will result in the trinomial. A trinomial has three terms, an x² term, an x term, and a constant. After factoring, the expression will show the multiplication of two binomials, each with two terms that are added or subtracted.

To break down a trinomial the FOIL method is used in reverse. The steps in the FOIL method are:

First – Multiply the first term in each set of parentheses

Outer – Multiply the outer term in each set of parentheses

Inner – Multiply the inner term in each set of parentheses

Last  – Multiply the last term in each set of parentheses

 Example 1: x² + 5x + 6

 Set up parentheses to show the 2 binomials like this:

 

( __ + ___ )( ___ + ____)

 

Start by listing the factors of the first and last term. 5x will be the trickiest one to find so it will be last.

 

Factors of x² : x

Factors of 6: -6,-3,-2,-1,1,2,3,6

The only way to get x² is to multiply x by itself.

 

( x__ + ___ )( __x_ + ____)

The next step is to find two factors of 6 that add up to 5 for the term 5x. If the terms in the binomials were negative the middle term would also be negative so we need a positive pair of factors that add up to 5. The only possible factors that fit are 2 and 3.

 

( x__ + __2_ )( __x_ + _3___)

Now check using the FOIL method.

 

( x__ + __2_ )( __x_ + _3___)

 

First: (x)(x)= x²

Outside: (x)(3)=3x

Inside: (2)(x)=2x

Last: (2)(3)=6

x² +3x + 2x + 6= x² +5x+6 This is the original trinomial. All the terms are positive in this example but this is not always the case.

 

Example 2:

x² -6x +8

Factors of x²: x

Factors of 8: -8,-4,-2,-1,1,2,4,8

The second term is negative so the 2 factors must be negative. Two negative integers added together will equal a negative and multiplied will equal a positive.

( x__ -__2_ )( __x_ -_4___)

Use FOIL to check:

First: (x)(x)= x²

Outside: (x)(-4)=-4x

Inside: (-2)(x)=-2x

Last: (-2)(-4)=8

 

x² -4x – 2x + 8= x² -6x+8

 

Example 3:

x² -x -12

Factors of x²: x

Factors of 12: -12,-6,-4,-3,-2,-1,1,2,3,4,6,12 The constant in this example is negative so it must be the product of a negative integers and a positive integer. The 2 factors must add up to -1, the b value.  -4 and 3 are the only factors that add up to -1 while also resulting in -12 when multiplied.

( x__ -__3_ )( __x_ +_4___) 

 

 

 

 

Did paraphrasing the words help you internalize the concepts more?

Paraphrasing was very helpful because I needed to clearly picture the steps before I could translate them into language my students would understand. I also had to create my own examples and then apply the steps of both factoring and then checking using FOIL to make sure they were accurate examples. That let me think through the process so that I was better prepared to teach this as a lesson.

How can you apply this type of exercise in a lesson for your own students?

Students could create a demonstration for teaching factoring to students in another class using technology such as PowerPoint or iMovie where they can visually show the steps. They could then describe the development process as well as the presentation in their interactive math notebooks or in a journal.

After reviewing your classmates post, would you alter your definition? Why or why not? Would you provide different examples?

After reviewing a number of definitions I see that most of us have the same information. Some people chose to provide more specific formal language in their definitions as well as a number of visuals or examples. When I was writing my definitions I was keeping in mind the students I teach. I was using simple math language and easy to follow examples because that is what I would present to my class. Through a variety of activities and guided questions I would then develop the concepts of one step,  two step, and multiple step equations as well as equations containing only variables. I think I would include additional examples of equations as well as graphs of both functions and non functions. This would allow students to actually see how the straight line test works. After working with and creating their own examples, I would then introduce the formal definitions and have my students compare them with their own understanding by making a journal entry.

  How can you evaluate whether or not your students grasped the difference between the two?

I would use a pair/share to determine if students grasped the difference between equations and functions. Each partner would create 5 problems where you need to identify if whether each is an equation or a function and explain why. To identify functions they could use a graph, a data table, or a chart, whatever they are comfortable using. This could be done in the students’ journals so that I could collect and read them after the students correct each other’s work.

In exploring the site, http://mathforum.org/escotpow/puzzles/   I discovered the applet called Fractis. It opens up a game where students try to create bars by putting together fractional parts to fill up a box and earn points. One fractional part of the bar drops in and then students add fractional parts trying to equal one whole bar. If the fractions add up to one whole, the bar disappears and the player earns 10 points. They try to form as many bars as possible in 250 seconds.

My students often have difficulty trying to identify equivalent common fractions. After working with fraction tiles as a whole group and working with small groups, I would allow students to use the  applet for independent practice. When they have had time to master the game, I would then challenge them to create their own game for identifying equivalent fractions. Students could exchange games and play them. As a final step in the process students would make a journal entry in their interactive notebooks describing their steps to master the game, how they developed their own game, and then analyze their classmates’ games. I would follow up the lesson with a discussion about how the games worked to help them learn equivalent fractions.

There are many examples where proportions could be used to solve everyday problems. Here are two from my own experience.

 

I am baking a  vanilla birthday cake for my father’s party on Saturday. There will be  33 people at the party who want a piece of cake. The recipe will make a cake with 8 servings. The recipe calls for 3 cups of flour. If there are 4 cups of flour in a pound, how many pounds of flour will I need to buy in order to give all people a piece of cake?     

  1.Set up a proportion:

                                Cups of flour in recipe      =  cups of flour needed

                              servings of cake in recipe        servings needed

 

                                                3 cups                   =         X       

                                                   8 servings                      33

   2. Cross multiply

                                3 X 33 = 8X                             

                                99 = 8X

    3. Divide both sides by 8

                                99÷ 8 = 8X ÷ 8

                                12 .375 = X

 4. I need 12 3/8 or 12.375 cups of flour. Since a pound contains 4 cups, I divide 12 3/8 by 4.           

                                12.375 ÷ 4 = 3.09 pounds

 5. Since I need a bit more than 3 pounds, I will have to by at least 4 pounds of flour.

 

 

 

My family is painting two bedrooms in our home this week and we need to buy paint.  The paint we chose says that 1 gallon will cover 350 square feet. The first bedroom measures 21 feet long by 18 feet wide and the second is 24 feet long and 17 feet wide.

1. Set up a proportion:

 gallons of paint/square feet covered = gallons I need/square feet I need to cover

 To finish my proportion I need to know how many square feet are in the  bedrooms. The formula for area is A = lw or length X width.

                                Bedroom 1    A = 21 X 18               

                                                          A = 378 sq ft

                                Bedroom 2     A = 24 X 17

                                                           A = 408 sq ft

                                Total square feet = 378 + 408 = 786 sq ft

   

2. Cross multiply

                1X 786 = 350X

                786 = 350X

3. Divide each side by 350

                786÷350 = 350X÷ 350

                2.2457142857142857142857142857143 = X                                            

 Round to 2.25 gallons are needed so I must buy 3 gallons of paint.

An equation is a number sentence that represents two equal quantities . An equation can use numbers and letters called variables.

                                                                                               

1 + 6 = 7   (true equation)                            

Y – 4 = 10 ( open equation)

12 + 4 = 21 (false equation)

 

A function is a statement where one specific input, X , is matched with one specific output, Y. The results can form a pattern. Think of this like a machine where you take one quantity, perform a math operation, and get another quantity.

For example:

X + 4 = Y

 

X	Y
2	6
3	7
4	8

                                                                                                                     

 

Can you figure out the equation?

X	Y
2	4
4	8
6	12
8	16
10	20

 

References:

image of scale from tulanelink.com

image of grinder from mincer.en.alibaba.com

I have seen many of these math myths in action in my own math education. One that strikes an especially significant chord for me is “It’s always important to get the answer exactly right.”  In all the years of math education from k-12, every teacher I had was ultimately concerned with getting the answer exactly right. In high school courses, such as Algebra, Geometry, and Calculus, not only was it necessary to get the answer exactly right, I also was required to get all the steps written correctly with the answer. After teaching math in grades 6-8, especially in pullout resource room sections, I saw that I could learn more from my students mistakes. Requiring them to write the steps out as they were developing concepts was an important tool for them and for me to assess their thinking process. Did they reach an incorrect answer because they did not understand the problem or because of a simple computational error. By examining the problem solving process , they could learn the how and the why of the concept.

Additionally, there are also instances where an exact answer is not necessary. For example, if one of my students is shopping for school clothes with a $100 budget, being able to estimate or approximate a total will help them stay under budget before reaching the checkout. This gives kids a real life application for learning mental math skills of rounding and addition within a practical context.

“Math requires a good memory, and memorizing formulas and rules is the best way to learn it.” I am an ultimate example of why this statement is a myth. Because rote learning came easily to me, I could plug numbers into rules and formulas and get the answers. This resulted in being placed in advanced classes but really having no concept of why the formulas worked. I did not really learn or retain much from Calculus and Trigonometry because there was no context or application I could associate the concepts with. Working with special ed students helped me dispel this myth since they often have poor rote skills and simply memorizing could not work for them. I had to find other strategies that allowed them to construct meaning. Trial and error and as many real world applications as I could include allowed my students to develop math skills. I also allowed them to create their own reference sheets with formulas and rules and use them when problem solving. Students wrote in their math journals to describe their trial and error process, their results, and propose changes they would make after examining the errors. This encouraged them to keep making efforts to learn new concepts.

Each row in Pascal’s Triangle has one more number than the previous row. Each row begins and ends with a 1. The first row, containing simply a 1, is referred to as row 0. The next row, row 1, has two 1s in it. Row 2 consists of 1  2   1. The 2 is obtained by adding together the two 1s in the previous row.

                                1                                              Row 0

                1                              1                              Row 1

1                              2                              1              Row 2

The pattern continues row by row, with each row being a palindrome reading the same forward and backward.

 

There are simple patterns within Pascal’s Triangle. The outer diagonals consist solely of 1s. The second diagonals are made of the natural or counting numbers ( 1, 2, 3…). The sum of the numbers in each row equals 2 to the nth power. For example, the sum of the numbers in row 3 is 8 and 2³ equals 8.

Sum
								1						1
							1		1					2
						1		2		1				4
					1		3		3		1			8
				1		4		6		4		1		16

 

The powers of 11 are also evident in Pascal’s Triangle.

			1									1	Row 0
		1		1								11	Row 1
	1		2		1							121	Row 2
1		3		3		1						1331	Row 3

 

By reading across the row, we see 11 raised to the nth power where n is the number of the row.

 

Fibonacci numbers are also found in Pascal’s Triangle by adding the diagonals.

 

 

 

 

Graphics from http://themathforum.org.

“Fibonacci” AND “Phyllotaxis” AND “Prime Numbers”

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

Although I have certainly worked with Fibonacci numbers, I never realized how many places they appear in nature or that there was a specific term, phyllotaxis, referring to the arrangement of leaves in plants. Additionally, I did not know that Fibonacci was born Leonardo Piso and that he had a lifelong interest in numbers. He wrote books but unfortunately, because the printing press had not yet been invented, very few copies survived.

 

Fibonacci was a wealthy merchant born in Pisa, Italy in 1170 and named Leonardo Piso. In his book entitled Liber Abaci, Fibonacci first posed the question:  If you start with a single pair of rabbits in a generation and each pair produced a pair in each succeeding generation,  how many rabbits will be produced in the n^th generation?

 

This resulted in the now 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…

The rule for this sequence can be stated as follows: After two starting values, each number is the sum of the two preceding numbers.

 

In fact this sequence appears frequently in nature on the sides of pineapples, centers of sunflowers, ridges of pinecones, and in the phyllotaxis or arrangement of leaves in many plants.

 

 Images

What images did you find particularly striking?

 Looking at the following pictures from nature shows me just some of the places where Fibonacci numbers appear. It’s pretty amazing to think that of the seemingly endless variety of plants in this world, that it is rare to find any with leaf arrangements of 4 and that almost all are arranged in a pattern matching one of the numbers in the Fibonacci sequence.

The leaves are arranged in spirals of five.     

The lily has petals

in groups of three.        

                                                                        This poppy seed head has 13 ridges on top.     

   The pine cone has seven spirals.         

           

Many of the Fibonacci numbers are also primes . The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …

 The largest one that has been discovered  to date is 604711, discovered in 2005 by Henri Lifchitz.

 

 The following link opens up a slide show of Fibonacci numbers in nature.

http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

These You Tube videos provide a great explanation of the Fibonacci sequence.

http://www.youtube.com/watch?v=N6RMQy7pBtM

http://www.youtube.com/watch?v=wS7CZIJVxFY&feature=related

 

Fractals, Nature, and Patterns

 

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

 

I never knew the precise definition of a fractal. One definition  is that a fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are described as  generally self-similar. The structure on all scales might not be exactly the same but must be the same type of structure.  The field of fractal geometry developed as an answer to  the question “How long is the coast of Britain?”  The answer depends on how fine a ruler  you use to  measure it.  The smaller the ruler, the longer the length will be.

 

 

Fractals are simply patterns that repeat at different scales of space and/or time. These patterns  appear throughout nature’s structures and processes. Nature uses what are called Chaotic physical processes, meaning that tiny inputs of energy into a living system can lead to huge transformations in how it behaves. These processes are occurring in structures that  have Fractal architectures, hence are fractal processes. 

 I have associated fractals with the work of MC Escher. Some of his artwork consists of patterns of shapes repeated to form an image and as you look closer the shapes reveal more detail. But what I learned from research is that our planet is full of fractals. Mountain ranges, such as the Rocky Mountains, the Andes, the Alps and the Himalayas contain fractals.                            

  

 

River networks formed by repeated erosion from numerous

 rainstorms over many years create fractal canyons.                                                          

 Spirals can occur over a wide range of scales, from tiny seashells to entire galaxies. 

  This spiral occurs in the melting ice shelf in Greenland.

                                                      

What images did you find particularly striking?

 

 These images contain additional examples of how fractals help to create what is perceived as beauty in nature. Some of the most vivid natural images contain fractals.

    

 

     

 

     

       The following link contains great resources for teaching about fractals.:

http://argento.bu.edu/ogaf/

A fractal game is located at the following site:

http://www.geom.uiuc.edu/java/LeapFractal/

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

This year I have planted a small vegetable garden as well as some trees and perennials. Each plant variety has its own arrangement of petals and leaves correlating to different Fibonacci numbers. I also use time in a nonlinear each day because I teach online and so can access my work at various times throughout the day. I also exercise each day by completing a walking routine. When I walk outside, although I may be in the same few parks I take different routes that create unique nonlinear patterns.

How can you adapt this webquest activity for your classroom?

Whatever concepts are being explored in a unit, students could conduct their own web search for facts and pictures. These could then be shared in a blog, a wiki, or even through slide shows. The advantages of using a blog or wiki are that students can leave comments creating a discussion about the concepts, students can easily share pictures, videos, or podcasts, and finally an ePortfolio entry could document the students’ growth and concept development along with their self-reflection.