By taking this course I learned that online courses are no more difficult than face to face classes. I rather enjoyed doing my course work when I wanted to in the comfort of my own home. Because this is a career change for me, this course helped me become more comfortable with the subject of Math in general. I’m becoming confident that I could teach Math if I choose to next year.

I am amazed at how interrelated math is with nature. I especially enjoyed the nonlinear patterns web quest and learning about Fibonacci, the Golden Rule and Fractals. I think these theories help us with design principles that work when decorating our homes, designing a quilt pattern or putting together a coordinated outfit.

Yes I will use journals with students. I see the value of students taking the time to write down their thoughts, feelings and questions about Math. I also think it would be a great way to assess how students are grasping the content material in classrooms that are larger than ideal size. My writing responses to the students would be a way to give the students a little more individual attention, not always possible during the class period. I especially like the idea of a blog. Students would be more open to writing if they can do it on the computer. I also think blogging would be more convenient because I could read the blogs while at home without dragging a stack of journals home.

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Steps: #1. Determine the factors for “c”.

Ex. Factors for 8 are 1×8 and 2×4

#2. Which set of factors will add up to “b”.

Ex. 2+4=6

#3. Determine the factors for “a”

Ex. (x) (x) = x²

#4. Write the following binomial

Ex. (x + __) (x +__)

and fill in the correct factor from steps 1 and 2.

Ex. (x + 2) (x + 4)

I have always been the type of person who has to write things down in order to reinforce my learning. Writing involves thinking about the concepts just learned, putting those thoughts into my own words and writing those thoughts down. By taking the time to organize the information and make connections to what I already know I can better understand the new material. Many of my students would benefit from the same process. I will ask students to keep a journal or blog when the computer cart is available to our classroom. The writing prompt I could give to students is to explain how to factor a quadratic equation to a younger sibling or friend.

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The purpose of this game is to solve simple linear equations by placing virtual manipulatives on a balance beam. When the beam is balanced, the player then solves for x. Students soon realize that any operation done to one side must be done to the other to keep the beam balanced.

I think this applet would be excellent for students who need more practice with solving linear equations. The site also has another version of the Algebra Balance Scales game that has problems with negatives. I find many children and adults who need extra help when working with negative numbers. My district doesn’t have math manipulatives, so this Applet would be an inexpensive way to give those students, who learn best by working with concrete examples, the opportunity to do so.

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I would add the following example for functions:

Find f(4) if f(x) = 3x +7 f(4) = 3(4) + 7 f(4) = 12 + 7 f(4) = 19

When assessing the student’s knowledge I like to use pinch cards. Each student is given a 5×7” card with the words equation and function written in big letters. I would ask a series of question such as “Is f(x) = -5x – 4 an equation or function?” The students pinch near their answer on the cards and hold them up over their heads. This technique works best if the students are conditioned to allow for a short wait time and then everyone shows their response at the same time, on a signal given by the teacher.

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To do a price comparison I will figure out the unit cost for 50 clips at the $5.49 price for 10.

My equation: 5.49 = x 5.49 x 50 = 10x 274.5 = 10x 274.5 = 10x $27.45

10 50 10 10

Answer: I will purchase the 50 count package because the savings is $27.45 – $24.99 = $2.46 plus the 50 count package comes with a free storage container.

I have just started a new knitting project. The sock pattern instructed me to knit for 8 ½ inches. I knit 5 rows and it measures ½ inch. How many rows will I have to knit to reach 8 ½ inches?

My equation: 5 = x 5 x 8.5 = .5x 42.5 = .5x 85 = x

.5 8.5 .5 .5

Answer: I will need to knit 85 rows to reach 8 ½ inches.

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3 + x = 10

V = ∏ r²h

**My definition** of a function: Input a number, the number goes through a math process and produces an outcome.

f(x) = 3x² + 5 (Read by saying f of x equals 3 times x squared plus 5.)

Supplementary resources that reinforce equations and functions:

Kid friendly definitions:

http://www.mathsisfun.com/definitions/equation.html

http://www.mathsisfun.com/sets/function.html

Students are given simple word problem and they practice writing the equation.

http://www.math.com/school/subject2/lessons/S2U1L3GL.html#sm1

This link has another version of the function machine. Students put in an input of their choosing and the machine spits out the output. Students then come up with the rule.

http://www.shodor.org/interactivate/activities/FunctionMachine/

This site has a simple game to play with the function machine.

http://mathwire.com/games/guessmyrulegame.pdf

Journal entries: To determine background knowledge I would have students record their definitions of equations and functions before the lesson. Following the lesson they would go back and revise their definitions and provide examples.

Reference: Pizza image came from http://mathspig.wordpress.com/category/lists/10-best-maths-funnies-of-2010/

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While I was in elementary and middle school the method of teaching was rote learning. We memorized multiplication tables and formulas. Our homework consisted of completing drills. Because of that, I can often get the correct answer but may not be able to explain the logic behind or process of how I arrived at my answer.

This myth goes back to our discussion earlier in this course about meaning vs. method. Students will be more successful if they understand how to perform a math skill rather than memorize a rule. This author feels that students need both intuition and logic to be successful in math.

*Myth #10: To solve a difficult problem, work intensely, and don’t stop until the problem is solved.*

When my sisters and I came home from school we had to sit right down and complete our homework. This had a negative affect on my assignments. I would rush thru the problems, and be careless so I could go outside to play. On the nights my mother checked our work I would sit there struggling over a difficult problem and become very frustrated with the amount of time it was taking.

Now I often find myself leaving a problem and coming back to it later. I did this yesterday with a level 5 Sudoku puzzle. After a break I can see the problem with fresh eyes and was able to complete it without peeking at the answer.

Giving students the permission to leave a difficult question on a test and come back to it, is a good test taking strategy. Sharing the strategies we use when dealing with difficult problems, may help the students handle their problems more affectively.

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Fibonacci is a simple series of numbers named after Fibonacci. The sequence is made by adding the last two numbers to get the next number. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21… I’ll best remember this pattern because of the spiral it can make shown in the diagram below.

Phyllotaxis refers to how leaves grow on a stem. The two main ways leaves grow on a stem are opposite and spiral. Opposite is 2 leaves growing from the same level on the stem. Basil is the example I found in my home that grows with an opposite growth pattern. Spiral leaves alternate at different points on the stem. My African violets grow in a spiral. http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm shows great photo’s of Fibonacci numbers in nature. One example is my Shasta daisy’s having 21 petals.

**“The Golden Ratio” and “Pentagrams”**

The golden ratio refers to the idea of proportions that are pleasing to the eye. The ratio of our forearms to our hand is an example of the golden ratio. Pentagrams are five pointed stars that fit inside a pentagon.

The Parthenon in Greece may have been built using the Fibonacci number sequence or golden ratio.http://www.hypermaths.org/cropcircles/chapter5/ shows pictures of artists who made crop circles using pentagram designs.http://www.mathsisfun.com/geometry/pentagram.html is a site I would like to remember if I ever need to draw a perfect 5 pointed star for quilting. This shows two methods. The site also uses kid friendly language to relate the pentagram to the golden ratio.

**“Fractals” and “Nature” and “Patterns”**

My definition of a fractal is it’s a geometric shape that is repeated at different scales to produce an irregular shape. They are figures with lots of detail. There are many fantastic examples of fractals in nature as seen in the following website. http://www.youthedesigner.com/2011/07/29/interesting-patterns-and-fractals-from-nature/

**Questions:**

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

The only term on this list I might have heard of was fractals but I would not have been able to tell you what they were. I am amazed at how interrelated the Fibonacci and golden ratio are with math, nature, art, architecture, music and design.

2. What images did you find particularly striking?

My favorite images are of fractals. They are so intricate and interesting to look at. I also enjoyed looking at the flowers and plants that show how leaves, branches and petals grow in spirals.

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

The following are examples of nonlinear patterns in my home.

Fibonacci: Petals on amaryllis, pinecones, and skin on pineapple.

Golden Ratio: plaid pattern in quilt, picture frames, and photo arrangement on wall.

Phyllotaxis: Cactus, violet, and aloe plant

Fractals: Snowflakes, ice crystals on car window, sugar crystals when making candy, fern, cut open cabbage, broccoli, and shells.

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

I think students would enjoy doing a web quest like this one. I would assign groups different sections to research rather than the entire web quest because of the time issue. The students can share what they learned with the class. I have found students can really take a long time when using the computers. The idea of using the quotes is an excellent technique to narrow the search results. I looked up pentagrams without quotes and found sites on witchcraft.

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Right and left sides of the triangle are formed by one’s. To find each inside number, add the two numbers above it. Rows are numbered beginning with the top row which is zero.

Other interesting patterns I found while researching Pascal’s Triangle:

- The sums of the diagonals form the Fibonacci sequence. 1, 1, 2, 3, 5, 8. 13 …
- The diagonal rows next to the row of one’s are counting numbers.
- The product of six number’s surrounding any number (a ring) is always a perfect square. The six numbers’ I worked with were 1,1,5,10,6,3 that surround the number 4 from the 4
^{th}row. The product is 900 with a square root of 30. - The multiples of a number form inverted triangles. When two numbers next to each other have the same factor, the number below them will also have the same factor. For example: 4 and 6 each has a factor of 2. The 10 below also has a factor of 2. The shape these three numbers make, if shaded in, would be an inverted triangle.
- Pascal’s Triangle can be used to answer questions involving combinations. For Example: Find the number of combinations of 4 things taken 2 at a time. Refer to row 4 of Pascal’s Triangle. (Remember the 0 element of every row is 1) There are 6 combinations of 4 things taken 2 at a time.

Number Taken at a Time |
0 | 1 | 2 | 3 | 4 |

Row 4 | 1 | 4 | 6 | 4 | 1 |

- Prime numbers: If the first element in a row is a prime number, all of the numbers in that row are divisible by it. Example: Row 5 is 1, 5, 10, 10, 5, 1.
- Hockey Sticks: If a diagonal of numbers of any length is selected starting at any of the 1’s bordering the sides of the triangle and ending on any number inside the triangle, the sum of the number selected is equal to the number below the diagonal. 1+3+6=10. This number combination is circled on Pascal’s Triangle below and looks like a hockey stick.

- Sums of the Row: The sum of the numbers in any row is equal to 2 to the nth power with n = the number of row. I looked at the 3
^{rd}row. 2³ = 1+3+3+1=6 - The most fascinating pattern to me is that when all the odd numbers on Pascal’s Triangle are colored in it forms a Sierpinski Triangle.

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**Formal Definition**: Not in a straight line.

http://www.thefreedictionary.com/nonlinear

Linear Pattern

**Kid Friendly Definition**: A linear pattern repeats itself over and over again and forms a straight line when plotted on a graph.

**Formal Definition**: When a pattern in a number sequence is added or subtracted by the same number every time. http://wiki.answers.com/Q/What_is_the_definition_of_a_linear_pattern

Comparison: The formal definition may be clearer on how a pattern repeats but fails to mention the number sequence will form a straight line when plotted on a graph.

How to help student learn the definition: To help students remember the definition I would ask them identify the root word for linear which is “line”. Having students complete a few sample questions and then come up with patterns on their own will help the student develop there own sense of the definition.

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