Monthly Archives: February 2012

Applet Review

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I’ve decided to review Algebra Balance Scales found at http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2.html

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.

Evaluating My Definitions of Equations and Functions

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After reviewing my classmate’s definitions I need to revise mine.  To my definition of an equation I would add that equations have equal signs.  To my function definition I would add the rule that there is one output for every input.  I think these additions make my definitions more accurate.

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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There is a new quilting notion I would like to purchase.  It comes in two sizes.  I will use a proportion to determine how much of a savings the larger size will be.  The wonder clips come in packaged of 10 for $5.49 or 50 for $24.99.

 

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.

My Definition of Equations and Functions

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My definition: An equation is a math sentence used to solve a problem.

            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/

My Reflection on Math Myths

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Myth #3:  In order to find a correct answer, you must always know how you got that answer.

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.

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.