Phi - Natures Special Number

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=56 BGCOLOR="#003333" TEXT="#F7F7FF" bgproperties="fixed" background="011a0a03.jpg"> <div> Phi - Natures Special Number</div> <div> </div> </td> </tr><tr> <td valign="top" width=135 BGCOLOR="#003333" TEXT="#F7F7FF" bgproperties="fixed" background="21fa0c01dduajq.jpg"> <div> <A HREF="index.htm" TARGET="_top" TITLE="Educators Science and Math Institute Ser"><U><B>Educators' Science and Math Institute Series</B></U></A></div> <bR> <div> <A HREF="id2.htm" TARGET="_top" TITLE="About the Series"><U><B>About the Series</B></U></A></div> <div> <A HREF="id117.htm" TARGET="_top" TITLE="Goals"><U><B>Goals</B></U></A></div> <div> <A HREF="id3.htm" TARGET="_top" TITLE="Institutes"><U><B>Institutes</B></U></A></div> <div> <A HREF="lesson_plans_prepared_by_esmis_participants_.htm" TARGET="_top" TITLE="Lesson Plans"><U><B>Lesson Plans</B></U></A></div> <div> <A HREF="id53.htm" TARGET="_top" TITLE="Links"><U><B>Links</B></U></A></div> <div> <A HREF="id138.htm" TARGET="_top" TITLE="Photo Albums"><U><B>Photo Albums</B></U></A></div> <bR> <div> <A HREF="http://www.ed.mtu.edu/search.htm" TARGET="_top" TITLE="http://www.ed.mtu.edu/search.htm"><U><B>Search</B></U></A><A HREF="http://www.ed.mtu.edu/search.htm" TARGET="_top" TITLE="http://www.ed.mtu.edu/search.htm"><U><B> ESMIS</B></U></A></div> <bR> </td> <td valign="top" height=424 BGCOLOR="#EEEEEE" TEXT="#080000"> <div> <B>ISLAND HOPPING ACROSS THE CURRICULUM</B></div> <bR> <div> <B>Phi (</B><FONT SIZE="3" COLOR="#000000" FACE="Symbol" >f</FONT><B>) - Natures Special Number</B></div> <bR> <div> <B>Jon Stasiuk</B></div> <bR> <div> <B>Introduction:</B></div> <div> Mathematics is a wonderful subject for students interested in solving problems but seems to only be loved by a few.  Students seldom get to appreciate the beauty and creativity that mathematics has to offer.</div> <bR> <div> This unit makes an attempt to show the student that mathematics isn't a subject to be studied by a few and actually exists in the real world.  </div> <bR> <div> Students will explore the Fibonacci Sequence and the Golden Ratio using plants, pictures, algebra, geometry, and music.</div> <bR> <div> <B>Rationale for the Project:</B></div> <div> “Geometry has two geat treasures:  one the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio.  The first we may compare to a measure of gold; the second we may name a precious jewel”  - Johann Kepler ( 1571 - 1630)</div> <bR> <div> For years I have been using parts of this project at different times in my classes.  I have never taught the golden ratio as a mini - stand alone - unit</div> <bR> <bR> <div> Overview of the Project:</div> <div> I have included many activities and non-traditional assignments that actively engage the students. I have found that this works well for the students that I am targeting for this lesson.</div> <bR> <div> Introductory Lesson - Discovering the Fibonacci sequence</div> <div> Observations in Art, Nature, Music, Architecture</div> <div> Activity - Most pleasing rectangle</div> <div> Lesson 1 - Historical background of Fibonacci sequence and the Golden Ratio</div> <div>      Activity - Calculating the Fibonacci sequence and the golden Ratio using EXCEL</div> <div>      Assignment - measuring body parts of family members/friends</div> <div> Lesson 2 - Pentagram/Golden Ratio Constructions ( optional Computer )</div> <div>      Activity -  Constructions</div> <div>      Assignment - NCTM Handout</div> <div> Lesson 3 - Algebra - derive golden ratio</div> <div>      Activity - Mozart math</div> <div>      Assignment - explorations with phi</div> <bR> <div> Wrap up - Cover Miscellaneous material</div> <div>      Fibonacci math, Complex fractions, Golden spiral, Golden Triangle</div> <div>      Activity - Video of Donald Duck in Mathemagic Land.</div> <bR> <div> Resources</div> <div> <I>Discovering Geometry</I>, Michael Serra,  Copyright 1993 by Key Curriculum Press</div> <bR> <div> <I>Algebra in the Real World</I>,  LeRoy C. Dalton,  Copyright 1983 by Dale Seymour Publications</div> <bR> <div> <I>Mathematical Puzzles and Diversions</I>, Martin Gardner, Copyright 1961 by Simon and Schuster</div> <bR> <div> <I>Donald in Mathmagic Land,</I> Walt Disney,  Copyright 1959 by Buena Vista Home Videos</div> <bR> <div> <I>The Golden Section and the Piano Sonatas of Mozart</I>, John F. Putz, <U>Mathematics Magazine</U>, October 1995</div> <bR> <div> <I>Pentagrams and Spirals</I>, Lew Douglas, Mathematics Teacher,  November 1996</div> <bR> <div> <I>The Golden Ratio: A Golden Opportunity to Investigate Multiple Representations of a Problem</I>, Edwin M. Dickey<U>, Mathematics Teacher</U>, October 1993.</div> <bR> <div> <A HREF="http://www.contracosta.cc.ca.us/math/pentagrm.htm" TARGET="_top" TITLE="http://www.contracosta.cc.ca.us/math/pen"><U><U>http://www.contracosta.cc.ca.us/math/pentagrm.htm</U></U></A></div> <bR> <div> <A HREF="http://www.markwahl.com/golden-ratio.htm" TARGET="_top" TITLE="http://www.markwahl.com/golden-ratio.htm"><U><U>http://www.markwahl.com/golden-ratio.htm</U></U></A></div> <bR> <div> <A HREF="http://homepage.mac.com/efithian/Geometry/Activity-02.html" TARGET="_top" TITLE="http://homepage.mac.com/efithian/Geometr"><U><U>http://homepage.mac.com/efithian/Geometry/Activity-02.html</U></U></A></div> <bR> <div> <A HREF="http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibat.html" TARGET="_top" TITLE="http://www.ee.surrey.ac.uk/Personal/R.Kn"><U><U>http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibat.html</U></U></A></div> <bR> <div> <A HREF="http://www.pass.maths.org.uk/issue3/fibonacci/index.html" TARGET="_top" TITLE="http://www.pass.maths.org.uk/issue3/fibo"><U><U>http://www.pass.maths.org.uk/issue3/fibonacci/index.html</U></U></A></div> <bR> <bR> <div> Standards Addressed</div> <div> <I>From the Michigan Curriculum Framework (Math)</I></div> <div> <I>I.1.1, I.1.5, II.3.1,II.3.5, II.3.6, III.1.1, III.1.2, V.2.1, V.2.3, V.2.5</I></div> <div>      I. Patterns, Relationships and Functions</div> <div> Students recognize similarities and generalize patterns, use patterns to create models and make predictions, describe the nature of patterns and relationships, and construct representations of mathematical relationships.</div> <div> Analyze and generalize mathematical patterns, including  sequences, series, and recursive patterns.</div> <div> Use patterns and reasoning to solve problems and explore new content.</div> <div> II  Geometry and measurement</div> <div> 3   Students compare attributes of two objects, or of one object with a                                                     standard (unit), and analyze situations to determine what measurement(s) should be made and to what level of precision.</div> <div> Select and use appropriate tools; make accurate measurements using both metric and common units, and measure angles in degrees and radians.</div> <div> Use proportional reasoning and indirect measurements, including applications of trigonometric ratios, to measure inaccessible distances and to determine derived measures such as density.</div> <div> Apply measurement to describe the real world and to solve problems.</div> <div> III  Data Analysis and Statistics</div> <div> Students collect and explore data, organize data into a useful form, and develop skill in representing and reading data displayed in different formats</div> <div> Collect and explore data through observation, measurement, surveys, sampling techniques and simulations.</div> <div> Organize data    using tables, charts, graphs, spreadsheets, and data bases.</div> <div> V  Numerical and Algebraic Operations and Analytical Thinking</div> <div>      2   Students analyze problems to determine an appropriate process for                                          solution, and use algebraic notations to model or represent problems.</div> <div> Identify important variables in context, symbolize them and express their relationships algebraically.</div> <div> Solve linear equations and inequalities algebraically and non-linear equations using graphing, symbol-manipulating or spreadsheet technology; and solve linear and non-linear systems using appropriate methods.</div> <div> Explore problems that reflect the contemporary uses of mathematics in significant contexts and use the power of technology and algebraic and analytic reasoning to experience the ways mathematics is used in society.</div> <bR> <div> Introductory Lesson</div> <bR> <div> <I><B>Introduce the pattern</B></I></div> <div> 1  1  2  3  5  8  13  21  . . .</div> <bR> <div> Many students at one time or another have come across the Fibonacci sequence but do not recognize it by name.  </div> <bR> <div> <I><B>Observations in Art, Nature, Architecture, Music</B></I></div> <div> One of the fascinating relationships of this pattern is that it exists everywhere around us.  <B>Nature</B> contains several examples including:</div> <div> Pine cones, sunflowers, pineapples - in the way that their surfaces grow</div> <div> Flowers - many flowers have petals that number in one of the Fibonacci numbers</div> <div> Humans - have 2 hands each with 5 fingers separated into 3 parts by 2 knuckles. </div> <div>      Phyllotaxis - the botanical term for a topic which includes the arrangement of leaves on the stems of plants.  Many plants show the Fibonacci numbers in the arrangement of the leaves around their stems.    If you look down the stem of a plant,  leaves are often arranged so that leaves above do not hide leaves below.  This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots.</div> <bR> <div> Several of these objects will be available and the patterns explored.  The following is a write up for one of these explorations.</div> <bR> <div> <B>Pineapple</B></div> <div> There are three directions of rows.  Count the rows in each direction.  This takes a while to see because every pineapple is not perfect.  If you are careful, you can do it.  The number or rows in the three directions on the pineapple are 8, 13, 21</div> <bR> <div> <B>Music </B></div> <div> Many students are not aware of the many links between mathematics and music.  The Fibonacci sequence has many.</div> <div> Piano - each octave has 13 keys, 8 which are white and 5 which are black. The black keys are broken into groups of 3 and 2.  Every black key is separated by 1 (or more) white keys.</div> <bR> <div> Scott Joplin and Mozart exhibit the Fibonacci sequence in their work.  The patterns in limericks also contains this pattern.</div> <div> Last line of Maple Leaf Rag</div> <div> Di dum di di dum</div> <div> Di dum di di dum</div> <div> Di dum di</div> <bR> <div> <B>Limerick</B></div> <div> Di dum di di dum di di dum          8</div> <div> Di dum di di dum di di dum          8</div> <div> Di dum di di dum               5</div> <div> Di dum di di dum               5</div> <div> Di dum di di dum di di dum          8</div> <bR> <div> <B>Art and Architecture</B></div> <div> The Parthenon in Athens, Mona Lisa, and other famous structures, paintings and statues were constructed using quantities derived from Fibonacci numbers.</div> <bR> <div> Assignment :  Pick your favorite rectangle.</div> <div> Students will examine a group of rectangles that is most appealing.  Then they will ask the same of 5 other people outside of class.  The results will be tabulated the next day.  Usually the rectangle that is most commonly selected has sides that form the Golden Ratio.  These results will be used to introduce the topic of the golden ratio.  This is a very common activity and is available on many of the websites listed in the bibliography.</div> <bR> <div> <I><B>Lesson 1</B></I></div> <div> <B>A Brief Historical background of Fibonacci sequence and the Golden Ratio</B></div> <div> This will include the pythagoreans, Leonardo de Pisa ( Fibonacci) and many other interesting historical topics related to the Golden Ratio.</div> <bR> <div> Available on the markwahl website located in the resources is a picture of the head of a famous greek statue.  The dimensions of the various body parts are in the golden ratio.  I had the students calculate these and also describe the results from last nights assignment.</div> <div> This is a lead in to their take home assignment.</div> <bR> <div> <B>Activity:  Calculating Fibonacci Numbers using a spreadsheet</B></div> <div> Objective:  Students will use Excel to calculate Fibonacci numbers and Golden Ratio</div> <div> <U>Step 1</U>     Start Microsoft Excel and create a new worksheet.  In cell A1 enter 1, in A2 enter 1, in A3 enter 2, in A4 enter 3, and in A5 enter 5.  These are the first 5 Fibonacci numbers.</div> <div> <U>Step 2</U>  In cell A6 write this formula: = A4 + A5.  This formula will add the contents of cells A4 and A5 and put their sum in A6</div> <div> <U>Step 3</U>  Place the pointer on the tiny square in the lower right corner of cell A6; it changes to a + sign.  Hold the left button down and drag the pointer down through cell A39.  This will copy the formula to all of these cells.  In column A there is now a list of the first 39 Fibonacci numbers.</div> <div> <U>Step 4.</U>  In cell B2  Write this formula: = A2/A1.  This formula will divide the contents of A2 by A1 and put the result in B2.  Place the pointer on the tiny wquare and drag it down to copy the formula down through cell B39.  In column B we now have the ratio of each Fibonacci number to the one before it.  This ratio approaches the Golden ratio as Fibonacci numbers increase.</div> <bR> <div> Assignment:  Measuring body parts</div> <div> This take home assignment will be for students to measure lengths of various given body parts and calculate their ratios.  They need to do this for 5 people ( not students ) including family members, neighbors, etc.  The average of these ratios should approach the golden ratio.</div> <bR> <div> <I><B>Lesson 2</B></I></div> <div> <B>Golden Ratio/Pentagram constructions</B></div> <bR> <div> <B>Activity:</B><B><U>  Golden Ratio Constructions ( Computer )</U></B></div> <div> This activity can be done using the drawing tools in Microsoft Word</div> <div> Be sure that the Drawing toolbar is displayed.  ( From the View menu, choose Toolbars, and then click Drawing).</div> <bR> <div> 1)  This activity can be done using the drawing tools in Microsoft Word</div> <div> Be sure that the Drawing toolbar is displayed.  ( From the View menu, choose Toolbars, and then click Drawing).</div> <bR> <div> 2)  Click on the rectangle tool.  As you move the pointer, be sure to hold down the Shift key;  this will allow you to draw a perfect square.</div> <bR> <div> 3)    After you draw the square, select it and then choose Copy and then Paste to create an exact duplicate.  Click on this copy to select it; then right-click, choose Format Auto Shape, and then under Scaling make the width 50%.</div> <bR> <div> 4)     Move this half-square so that it overlaps the original, and then draw the diagonal of the second half-square as indicated.</div> <bR> <div> 5)     Copy this diagonal and rotate it so that it extends the base of the original square.</div> <bR> <div> 6)     The resulting rectangle will be a “Golden Rectangle,” whose width and length are in the ratio 1 : 1.62</div> <bR> <div> <B>Activity:</B><B><U>  Golden Ratio Constructions ( compass and ruler )</U></B></div> <div>                                              </div> <div> 1)    Construct a square.  Label it GOEN.  Extend <FONT SIZE="3" COLOR="#000000" FACE="Symbol" >¾</FONT>GO and <FONT SIZE="3" COLOR="#000000" FACE="Symbol" >¾</FONT>NE.</div> <div>           </div> <div> 2)     Bisect <FONT SIZE="3" COLOR="#000000" FACE="Symbol" >¾</FONT>GO.  Label the midpoint M.  With ME as your radius and M as center, construct an arc intersecting <FONT SIZE="3" COLOR="#000000" FACE="Wingdings 3" >n</FONT>GO at point L.  Construct the rectangle OLDA.  GLDN is a Golden Rectangle.</div> <bR> <div> Assignment:  Triangles in a Pentagram and The Golden Ratio and the Golden Rectangle.</div> <div> This handout available from the Mathematics Teacher November 1996 will be used to reinforce the constructions and topics discussed earlier and also introduce the algebraic definition of the golden ratio which will be developed the next day.</div> <bR> <div> It uses a Pentagram - 5 pointed star and discovers that the Fibonacci sequence and Golden Ratio are embedded in this figure.  The Golden rectangle is reinforced and the golden spiral is discovered.</div> <bR> <div> <I><B>Lesson 3 </B></I></div> <div> <B>Algebraic definition and derivation of the Golden Ratio</B></div> <div> Using pictures of the golden section and manipulating symbols will be used to derive and prove the golden ratio.  This will be performed by the teacher with the help of the students.</div> <bR> <div> <B>Activity</B><B><U>: Mozart Math</U></B></div> <bR> <div> Music and mathematics have always been closely related.  In this project students discover a curious relationship between the structure of Mozart's piano sonatas and the golden ratio.</div> <bR> <div> Sonatas can be naturally divided into two parts:  the exposition, which introduces the musical theme, and the development and recapitulation, which develop and repeat the theme.  The data below shows all of Mozart's piano sonata movements that have these two parts.  </div> <div> <TABLE BORDER="2" CELLPADDING="1" CELLSPACING="1" WIDTH="611" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="96"><div> <B>Sonata</B></div> </tD> <tD VALIGN=TOP WIDTH="96"><div> <B>X</B></div> </tD> <tD VALIGN=TOP WIDTH="96"><div> <B>Y</B></div> </tD> <tD VALIGN=TOP WIDTH="96"><div> <B>Sonata</B></div> </tD> <tD VALIGN=TOP WIDTH="96"><div> <B>X</B></div> </tD> <tD VALIGN=TOP WIDTH="96"><div> <B>Y</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 279, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 62</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 38</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 310, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 84</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 49</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 279, II</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 46</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 28</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 311, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 73</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 39</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 279, III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 102</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 56</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 330, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 92</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 58</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 280, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 88</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 56</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 330, III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 103</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 68</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 280,II</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 36</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 24</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 332, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 136</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 93</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 280,III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 113</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 77</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 332, III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 155</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 90</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 281, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 69</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 40</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 333, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 102</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 63</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 281, II</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 60</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 46</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 333, III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 50</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 31</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 282, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 18</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 15</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 457, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 93</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 74</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 282, III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 63</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 39</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 533, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 137</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 102</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 283, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 67</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 53</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 533, II</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 76</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 46</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 283, II</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 23</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 14</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 545, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 45</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 28</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 283, III</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 171</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 102</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 547a, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 118</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 78</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 284, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 76</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 51</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 570, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 130</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 79</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="96"><div> 309, I</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 97</div> </tD> <tD VALIGN=TOP WIDTH="96"><div> 58</div> </tD> <tD VALIGN=TOP WIDTH="96"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="96" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="96"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="96" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="96"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="96" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <bR> <div> 1    Enter the points (x,y) in a graphing calculator and sketch a scatter plot of the points.  Describe the relationship between x and y.</div> <bR> <div>  2    Use the statistical capabilities of a graphing calculator to find the least squares regression line for the data.  How well does the line fit the data?  Explain.  Interpret the meaning of the slope and the y-intercept.</div> <bR> <div> 3      The golden section was defined by Euclid as the point B on a line segment AC such that:</div> <div> AB/BC = BC/AC.  Assume that AC = 1 and BC = r</div> <bR> <div> 4     Then this proportion can be written as  (1-r)/r = r / 1.  Use a graphing calculator to graph y<FONT SIZE="1" COLOR="#000000" FACE="Arial" >1</FONT> = (1 - x)/x and y<FONT SIZE="1" COLOR="#000000" FACE="Arial" >2</FONT> = x/1.  Find their point of intersection.  How does it compare to the line in part 2?</div> <bR> <div> <B>Assignment: Algebraic explorations with Phi</B></div> <div> On pages 169 - 196 of <U>Algebra in the Real World</U>, several smaller assignments were combined into one larger assignment.  These use continued fractions, quadratic equations, and end with how the golden ratio is discovered and subsequently proved.</div> <bR> <div> Wrap up</div> <div> Wrap up the unit by drawing everything together and showing that not only can math be creative and fun, but it exists across the curriculum.  A few last topics will be discussed showing how the golden ratio is involved</div> <bR> <div> <B>Activity:  </B><B>Donald Duck in Mathemagic Land ( Video )</B></div> <div> This classic video uses Donald to explore through a ficticious land discovering many unusual mathematical facts along the way, one of which is the golden ratio.  This video should be viewed by all.</div> <bR> <div> Sample of Pre and Post Assessment Questions</div> <div> These questions will be graded using a predetermined rubric.</div> <bR> <div> Describe the Fibonacci sequence.</div> <div> What is the Golden Ratio?</div> <div> What is a Golden Rectangle?</div> <div> Are all Golden Rectangles similar?  Why or Why not?</div> <div> Describe how the Fibonacci sequence exists in nature?</div> <div> How does the golden ratio exist in nature?</div> <div> What is a pentagram?</div> <div> How does the pentagram relate to the Fibonacci sequence and the golden ratio?</div> <bR> <bR> <div> Overview of the Project:</div> <div> My main concern as it became time to do the project was for students to see that math exists in many forms and places.  I think it was moderately successful.  In the process, and due to curriculum time constraints, some of my project was not implemented.  The omitted items are in italics.</div> <bR> <div> Introductory Lesson - Discovering the Fibonacci sequence</div> <div> Observations in Art, Nature, Music, Architecture</div> <div> Activity - Most pleasing rectangle</div> <div> Lesson 1 - Historical background of Fibonacci sequence and the Golden Ratio</div> <div>      <I>Activity - Calculating the Fibonacci sequence and the golden Ratio using EXCEL</I></div> <div>      Assignment - measuring body parts of family members/friends</div> <div> Lesson 2 - Pentagram/Golden Ratio Constructions <I>( optional Computer</I> )</div> <div>      Activity -  Constructions</div> <div>      <I>Assignment - NCTM Handout</I></div> <div> Lesson 3 - Algebra - derive golden ratio</div> <div>      Activity - Mozart math</div> <div>      Assignment - explorations with phi</div> <div> Wrap up - Cover Miscellaneous material</div> <div>      Fibonacci math, Complex fractions, Golden spiral, Golden Triangle</div> <div>      Activity - Video of Donald Duck in Mathemagic Land.</div> <bR> <div> Pre and Post Assessment Questions</div> <div> These questions were graded as part of prompts for a writing assignment about this unit.  Students submitted papers answering most of the questions below.  Pre test questions were assigned individually in a traditional format of question and answer.  I did this informally as I could correctly predict their success.  No student was able to answer any of the questions in their pretest.</div> <bR> <div> Example of post test responses in bold</div> <bR> <div> Describe the Fibonacci sequence</div> <div> <B>Everyone was able to give the pattern and explain how it is calculated.</B></div> <bR> <div> What is the Golden Ratio?</div> <div> <B>In some form, everyone could describe the Golden Ratio.  Very few could explain it in mathematical terms but all could relate to it.</B></div> <bR> <div> Describe how the Fibonacci sequence exists in nature.</div> <div> <B>Instead of describing how, they described where.  Some also described the golden ratio instead of the Fibonacci sequence.</B></div> <bR> <div> How does the golden ratio exist in nature?</div> <div> <B>All were able to describe how the golden ratio exists in nature.</B></div> <bR> <div> What is a pentagram?</div> <div> <B>All could describe it as a five pointed star but only some described it in more detail.</B></div> <bR> <div> How does the pentagram relate to the Fibonacci sequence and the golden ratio?</div> <div> <B>This proved to be rather difficult for them to explain using words.  They all experienced this in class but had difficulty explaining it afterwards.</B></div> <bR> <div> Are all Golden rectangles similar?</div> <div> <B>They did not have enough background in the term similar as it relates to geometric figures to answer this question.</B></div> <bR> <div> Because of my knowledge of my student's background, I tried to keep the pretest and posttest nonthreatening - designing it more like a survey of knowledge.  I knew that their background was very limited in this topic, and I was concerned with their self-image and self esteem which for my students is already very low.</div> <bR> <div> I have 18 Math  2 students,  50% of my classes are labeled as at risk and receive additional help in their daily schedule.</div> <bR> <div> They seemed to understand why they had to fill out the survey two times.  I instructed them to be brutally honest.  I did administer the surveys in the correct procedure with the only adjustment being that when they saw science on the survey to think of math and science.  They  thought that taking a survey of science in math class was confusing.</div> <bR> <div> <B>What worked well</B></div> <div>      DNA Decorations - although it took them two days to build them.</div> <div>      Completing the activities.  </div> <div>      Donald Duck in Mathemagicland  Video - This video really summed up a lot of what we had discussed in class.  They were getting it the second time.</div> <bR> <div> <B>What didn't work well</B></div> <div> Understanding the activities.  They did them well but I am not sure what they took from this experience.  </div> <div> Lack of appreciation for the subject.  Many were eager to get `back to the book'.  Some appreciated it.</div> <div> The computer related activities.  I substituted more hands on items.  For the Excel spreadsheet activity, I had them perform it on paper and then enter the information into the lists on the TI-83 plus calculators.    I also had them perform the constructions by hand rather than on the computer.</div> <div>      </div> <div> I think that I picked a tough topic and tried to relate it to a population of students who did their best.  It proved to be very challenging.  I plan on using this with my Algebra students in the winter and my Calculus students in the spring.  I think the calculus students will enjoy the beauty of mathematics.</div> </td> </tr></table></body>