Non-Linear Pattern Web Quest

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Fibonacci” and “Phyllotaxis” and “Prime Numbers”

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.

Translating Pattern Narrative into Formal Math Language

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PASCAL’S TRIANGLE IN FORMAL MATH LANGUAGE

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 4th 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 3rd 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.

 

Working with the Definitions of Linear Patterns

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Non-traditional Linear Pattern

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.

My review of PUMAS

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PUMAS is a mnemonic for Practical Use of Math And Science.  Many of these activities were written by scientist and engineers.  I looked at dozens of these activities.  One suggestion I have for the website is to group the activities by age group.  I became frustrated when I thought titles were interesting and found the activity to be geared for 9-12th grade.

            My first impression of these activities is that they are too advanced for my students.   Maybe it’s because my science background is rusty.  I would not feel comfortable in doing most of these activities written with the 6-8th grader in mind.  I looked at some 3-5th grade activities and they appeared beyond my abilities too.

            I did find one activity I was comfortable with.  It’s called The Fall of the Ruler.  The materials needed are inexpensive.  The students would like the hands-on lesson.  The science part of this activity deals with human reaction time.  The math part is recording the results on a scatter plot or calculating the mean.

            Before the experiment, the ruler is converted to a “time keeper” using a chart supplied by PUMAS.  I would do this before the experiment, rather than have the students do the conversion, because the decimals involved are confusing.

            For the experiment, students work in pairs.  One holds the ruler up vertically and drops the ruler without warning.  The other student holds their fingers below the ruler, at the 0, and tries to pinch the ruler quickly as it is coming down.  Students can record their results on the scatter plot or calculate the average.

Inverse Property

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There is an inverse property of addition and multiplication.  Lets look at the inverse property of addition first.

Definition of Additive Inverse:  When a number and its additive inverse (same number with opposite sign) are added together, the result is always 0. 

Examples:  35 + (-35) = 0                                  -44 + 44 = 0

Definition of Multiplicative Inverse:  When a number and its multiplicative inverse (reciprocal) are multiplied together, the result is always 1.

Examples:  25 * 1/25 = 25/1 * 1/25 = 25/25 or 1         

-66 *  1/-66 =  -66/1 *  1/-66  =  -66/-66  or  1

When will we use this information?  When solving equations.

Use the additive inverse property and solve for x.     x + 6 = 10

To eliminate the +6, the additive inverse property of (-6) can be used.  Add the additive inverse to each side of the equation.

                                                           x + 6 = 10

                                                                -6        -6

                                                           X + 0  =  4

                                                                    X = 4

Use the Multiplicative Inverse property and solve for x.  3x = 9

To eliminate the +3, the multiplicative inverse property of 1/3 can be used.  Multiply each side of the equation by the multiplicative inverse.

                                                     3 x = 9

                   1/3 * 3x = 1/3 * 9     =     1/3 * (3/1)x = 1/3 * 9/1 =    (3/3)x = 9/3   =    1x = 3  or   x = 3

 

 

My Mathography

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To be honest I am having trouble remembering my school years.  It was ages ago.  I pulled out my old report cards to see if they would jog my memory.  My kindergarten records states I could count and understand numbers to 10 but, I was not checked off for recognizing numbers to 10.  It’s hard to believe children now a day have to be able to count to 100 before entering school.  My three year old granddaughter can recognize numbers at least until 10 and almost counts to 20. She consistently skips 13. 

            My early memories of math are lots and lots of worksheets.  Many of the worksheets were busy work to keep us quiet and well behaved while the teacher was running reading groups.  One early thing I learned in math and enjoyed were patterns.  The problems would show a shape that is moving or changed and you would select the example that represents the next in the series.  I enjoy puzzles and these problems were similar to puzzles. I still enjoy puzzles, especially Sudoku.

            Mr. Winslow was my favorite teacher.  He taught 9th grade algebra.  He had a good sense of humor and could relate well to students.  My self esteem was low in middle school but I felt comfortable in his class.  Mr. Winslow’s patience and encouragement allowed me to have the attitude that I could be successful in his classroom.  I really amazed myself by getting a 98 on the New York State Regents exam that year.  I am not a good test taker and up to this point, a high B was my personal best.

            Math was not my favorite subject but I’m not sure I had a favorite subject.  School was not my favorite place to be.  I much preferred being home.

Math Stories for the Everyday

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MATH STORIES FOR THE EVERYDAY

 

My son and daughter-in-law moved to a new home last weekend.  For a housewarming gift my husband and I decided to surprise them with a portable dishwasher.  Having a dishwasher was high on their wish list but this home does not have one.

I researched dishwashers on the internet.  The kitchen does not have room for a built in.  The two options available are a small counter top model or a full sized portable. This is a family of 4 so the small countertop model would not be practical. It only fits 6 place settings at a time. The other negative issue is it would sit on the countertop.  My daughter-in-law likes to decorate cakes as a side business and needs all the countertop space possible.  The full sized portable model is the best option.  The full sized model will hold 14 place settings.  There is an ideal area in the attached dining room to store the dishwasher when not in use.  It will actually hide the hook ups to a stackable washer/dryer that used to be in the space.

 

We visited two stores and had one model to choose from at each store.  Following is a chart I kept while comparison shopping.

 

 

hhgregg

Miles Maytag

Color

White

White/Black

List Price

609.99

599.99

Repeat customer discount

NA

20.00

Delivery

69.00

NA

Tax

36.60

34.80

Total Price

715.59

614.79

Price after Rebate

646.59

NA

We decided to purchase the model from Miles Maytag because it was cheaper and we like to buy local.

 

Portable dishwashers are more expensive than built in models.  I asked the salesman why.  I thought he would say it’s because the sides and top are finished.  He explained it only costs them about $8 on the extra finishes.  The main reason is supply and demand.  There is little demand for these dishwashers and it is expensive to keep the assembly line open and running.

 

My husband and I bought a dishwasher for our home early in Jan.  It was a Maytag and cost $477.77.  I decided to calculate the percent of change.  Original (built in) cost $477.77.  New (portable) cost 614.79. 

First I determined the difference:  614.79 – 477.77 = 137.02

Next I applied the percent proportion.  P = r

                                                              B    100 

137.02 = r__             Cross multiply:   13702 = 477.77r

477.77    100        Divide each side by 477.77:     28.679071 = r

The portable dishwasher was a 29% increase over the built in.

 

The last thing I am curious about is volume of the countertop model vs. the full sized portable.  Here’s a chart of the measurements.  (Can you tell I like charts?)  I will use the formula: V = lwh

 

 

 

Counter top

Full size

Height in inches

17.25

37

Length in inches

21.63

24.12

Width in inches

19.69

26.50

Area in cubic inches

7346.68

23649.66