Stretching to Fit or not to Fit: That is the Percentage

<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> Stretching to Fit or not to Fit:  That is the Percentage</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> Educators' Science and Mathematics Institute Series (ESMIS)</div> <div> Fall, 2002</div> <div> <I>Artistic Expression Lesson Plan</I></div> <div> <I>(Manuscript)</I></div> <div> Submitted by:  Denise Wilder</div> <bR> <bR> <div> <I><B><U>Stretching to Fit or not to Fit:  That is the Percentage</U></B></I></div> <bR> <div> After attending the ESMIS workshop during the summer of 2002, I was challenged with the obstacle of how to incorporate art into my 7th grade mathematics classroom in the fall of 2002.  Since the current curriculum focused on the relationship between fractions, decimals, and percents I chose to find a project that would deal with at least two of these numerical formats.  Not being artistically inclined I turned to the computer as a medium.  On the computer I was able to manipulate digital photos of my children to send to family and friends.  (Also, after doing a poetry project with my English students last year I learned that many students enjoyed scanning photos and altering their image with the aid of computer software.)  So, because of my familiarity with arranging photographs on the computer using Microsoft® Power Point® and my belief that students would also enjoy this type of activity I attempted to overcome the obstacle of using art in my math classes.  </div> <div>      When looking into the capabilities of Power Point® I discovered that right clicking on a digital image brings up a menu that includes “Format Picture.”  When left clicking on “Format Picture” I gain access to the “Size” of my image's height and width in both inches (to hundredths place) and percentage.  Percentage will always be 100% x 100% unless you check the box that says “relative to original picture size.”  So, due to the nature of this unit, you do not want students to use this option - do not check this option box!  Another way to avoid students having the computer do their work for them is to have them resize their original image after inserting it into their slide.  The new `resized' preimage will be considered 100% x 100% while the computer will base all percentages on the image they first inserted prior to resizing.  With these abilities of Power Point® in mind I chose to focus my lesson on decimals and percents.  Students will use a digital image (they can scan one in or insert one from the internet or clip art), resize the image using the diagonal arrow to resize it proportionally, and then distort their `new preimage' using the vertical and horizontal arrows.  The “Size” menu will automatically give them their new measurements for height and width to the nearest hundredth of an inch.  Finally, students will calculate the percentage by which they changed their preimage in order to create their distorted images.</div> <div>      On day 1 I introduced and talked about changing decimals to percents and percents to decimals.  I did this using the textbook lesson (Lesson 2-6 in Transition Mathematics ©1998) along with my own discussion of the process.  Points I stressed included:  1) Percents are based out of 100, so when converting we will be using this fact; 2) To go from percent to decimal the `shortcut' is to move the decimal point two places to the left; and 3) So to do just the opposite and to go from decimal to percent we do the opposite - the `shortcut' would be to move the decimal point two places to the right.  The students then completed as homework a worksheet I had chosen.  Assessment occurred the following class day using:  number correct/number possible.</div> <div>      On day 2 I discussed photo distortions and prompted them to discuss where they see distorted images.  I also demonstrated on the computer/projector system how to distort images on Microsoft® Power Point®.  I then provided them with a page of images.  One photo was labeled the original (or preimage) and others were distortions (images) of the original.  (See Attachments 1-3 for a sample of this type of worksheet.)  As a class we took each image and calculated the percent of change from the preimage.  (NOTE:  This assignment was their pre-test.  They were not notified of this!  I simply made note of who needed assistance and to what extent with completing this assignment.  By my record 1 of 11 students was able to complete with little assistance.  I would equate this one student with a score of 4 on the rubric used on the post-test or project.  8 of 11 students needed assistance and were able to complete.  I would equate these 8 students with a score of 2 on the rubric used on the post-test or project.   2 of 11 were not able to complete without major assistance.  I would equate these 2 to a score of 0 on the rubric used on the post-test or project.)  The basic process was:  1) subtract decimals ® ôlengthPre - lengthIô and ôwidthPre - widthIô; and 2) calculate percent of change ® (change in decimal amount/original decimal value) * 100.  While working through the images together I tried to challenge them to first visually inspect the image and estimate the percent of change.  The decimal amounts should have given them clues and any student who could verbally explain their estimates were encouraged to do so.  Finally, I hoped that someone would discover that non-distorted images have either increased or decreased in both length and width at the same percentage of change and I had 1 student make this discovery independently.  After this student made the discovery and shared it with the rest of the class I had other students join in a conversation on this quality. The students then received another worksheet similar to the one just discussed.  Assessment occurred the following class day using:  number correct/number possible.</div> <div>      Day 3 was spent in the computer lab.  Students created their own photo distortion worksheet and answer sheet.  Each worksheet was required to have:  1) Preimage labeled;  2) 5 images - 4 distorted and 1 non-distorted but changed in size;  3) all images labeled with decimal value of length and width;  and 4) answer sheet.  Assessment occurred on day 4.</div> <div>      Day 4 relied on students completing their day 3 assignment, therefore I had additional worksheets prepared in case students were absent or failed to complete their work.  (This was not needed with this class - all attended and completed this assignment.)  Students then traded their own photo distortion worksheet with another student and completed each other's worksheet.  They then graded each other's work - discussions were brought to me for settlement.  (I had to settle 4 disagreements.)  I assessed them on completion of their day 3 assignment, accuracy in filling out another student's worksheet, and accuracy in grading their worksheet that was completed by another student.</div> <div>      For day 5 I offered them three project options.  These projects served as the post-test for this lesson plan.  Projects included:</div> <div> <IMG BORDER="0" SRC="Symb075c00b700f000000000.png" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Take a preimage and measure to nearest eighth of an inch.  (Can also use preimage from clipart or other image taken from computer.)  Determine the percent of increase/decrease needed for height and width in order to have the image fill an 11" x 8½" sheet of paper while leaving a ½" border around the paper.  Scan their preimage and enlarge/decrease to their predetermined size and print out on 11" x 8 ½" sheet.  Write a paragraph explaining why or why not your image reflects the ideal increase/decrease for the 11" x 8 ½" sheet with ½" border.</div> <div> <IMG BORDER="0" SRC="Symb075c00b700f000000000.png" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Have them find a natural object such as an insect, leaf, or flower.  They will then measure it and determine percentage of change for a poster of pre-determined size. I used an 18” x 18” poster.*</div> <div> <IMG BORDER="0" SRC="Symb075c00b700f000000000.png" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Have them select a poster of a natural object.  They will need to determine average size of pictured object (using reference book, internet or other reliable source) and calculate the percentage of change based on their   measurements of the object on the poster. *</div> <div> *(Please note:  I had the same written paragraph assignment with each of these projects as I assigned with Project 1.)</div> <bR> <div> The project was due one week from the assignment date and most class time after day 5 was focused on the next lessons in our text.  Any free time was spent working on their project.  (I worked in 2 other days where a student could go to the library for approx. 20 minutes if necessary.  Most students used the library time to finish their projects.)  Assessment of the project was done using a rubric similar to those used on the MEAP test.  The rubric I used for project 1 follows:</div> <div> 4 points     Measurements and percentage of increase accurate for `10" x 7 1/2"' sized paper.  Written defense for image relies on accurate information and shows complete understanding of how photos are distorted.  Well written.</div> <div> 3 points     Minor errors in either measurements, calculations, or written defense.  However written response shows significant understanding of how photos are distorted.</div> <div> 2 points      Some understanding of photo distortion is apparent however errors in calculations or written statement show carelessness or lack of understanding.</div> <div> 1 point     Major errors in project completion or missing part of project.  Part that is completed should show some understanding of photo distortion.</div> <div> 0 points     No attempt at project completion or only one part of project done with major errors or directions not followed.</div> <bR> <div>      I completed this project with my advanced 7th grade math class in October.  All of the students (11) chose project option 1.  I believe the fact that they were able to print out a finished project on the computer and that pictures on the computer are readily available was the</div> <div> attraction of project 1.</div> <div>       My post-test results had 4 students earning a 4, 4 students earning a 3, 1 student earning a 2, and 2 students earning a 1.  Overall I was </div> <div> very impressed with their final projects.  Even the students scoring 1 or 2 had a basic understanding of calculating percent of change but</div> <div> either failed to communicate their understanding in their writing or failed to follow directions of the project.  The 4 students earning a</div> <div> score of 3 made the mistake of only subtracting ½” from 11” and 8 ½” instead of 1” (to allow for ½” border all the way around). </div> <div>           At the completion of this unit students should be able to:  1) subtract with decimals; 2) calculate percentage of change; 3) convert decimals to percents (and percents to decimals); 4) estimate percentage of change from preimage to image given decimal measurements of height and width; 5) accurately measure with a ruler to the nearest eighth of an inch (with project 1 I found most students used the computer's measurements but still measured with a ruler their final image measuring 10” x 7 ½”); 6) use Microsoft® Power Point® to insert and resize a digital image; 7) explain in writing how an image is resized and distorted (height and width are increased or decreased equally or unequally); and 8) talk about how mathematics is involved in graphic displays such as pictures on posters.                                    </div> <div>      The Michigan Content Standards for Mathematics that are addressed and the manner in which I see them being addressed are as follows:</div> <bR> <div> Strand I, Content Standard I.1 - Patterns:</div> <div> <I>As students examine their images and the mathematical values that describe them they should see that the distorted images have an unequal percentage of change </I>for their height and width.  When they estimate (or calculate) the percentage of change they can predict <I>whether the image will be distorted or not.</I></div> <bR> <div> Strand II, Content Standard II.3 - Measurement:</div> <div> When students complete their day 5 project they will have to measure (to nearest eighth of an inch) and then <I>determine what percentage of change their preimage must experience in order to fit the new area available for their preimage (or vice versa for project 3).  In writing they will have to defend their final decision.</I></div> <bR> <div> Strand IV, Content Standard IV.2 - Representation and Uses of Numbers:</div> <div> Students will be able to give the equivalent percent or decimal for a given decimal or percent respectively.  They will see how percents are the appropriate way to express a change in an image.</div> <bR> <div> Strand IV, Content Standard IV.3 - Number Relationships:</div> <div> Students will determine which decimal value is greater in order to carry out the subtraction of two decimal numbers (this will avoid getting into an early discussion of absolute value which doesn't occur until Lesson 5-3 in the Transition Mathematics text).  Students will look at the equality in percentage of change in images that are similar to the preimage (and at the inequality in percentage of change when images are distorted from the preimage).</div> <div>       </div> <bR> <div> <B>References and Other Resources</B></div> <div> Michigan Department of Education. (1996).</div> <div> Michigan Curriculum Framework Standards and Benchmarks.</div> <div> Lansing, MI.  Author.</div> <div> (<FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><A HREF="http://www.michigan.gov" TARGET="_top" TITLE="http://www.michigan.gov"><U><U>www.michigan.gov</U></U></A></FONT>)</div> <bR> <div> Microsoft® Power Point®  2000 SR-1 software.</div> <div> © 1987-1999.  Microsoft Corporation.</div> <bR> <div> The University of Chicago School Mathematics Project (1998).</div> <div> Transition Mathematics.</div> <div> Glenview, Illinois:  Scott Foresman Addison Wesley.</div> <bR> <div> Michigan Department of Education</div> <div> Michigan Educational Assessment Program</div> <div> (<FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><A HREF="http://www.michigan.gov/mde" TARGET="_top" TITLE="http://www.michigan.gov/mde"><U><U>www.michigan.gov/mde</U></U></A></FONT>)</div> <bR> <bR> <div> <TABLE BORDER="2" CELLPADDING="1" CELLSPACING="1" WIDTH="605" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="195"><div> Student</div> </tD> <tD VALIGN=TOP WIDTH="195"><div> Pre-test </div> </tD> <tD VALIGN=TOP WIDTH="195"><div> Post-test</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> A</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>     </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 4</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> B</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 3</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> C</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 4</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> D</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 0</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> E</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 3</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> F</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 1</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> G</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 4</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><div>  </div> <div> H</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 3</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><bR> <div> I</div> </tD> <tD VALIGN=TOP WIDTH="195"><bR> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="195"><bR> <div> 3</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><bR> <div> J</div> </tD> <tD VALIGN=TOP WIDTH="195"><bR> <div> 0</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 1</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 39 WIDTH="195"><bR> <div> K</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 4</div> </tD> <tD VALIGN=TOP WIDTH="195"><div>  </div> <div> 4</div> </tD> </tR> </TABLE></div> <bR> </td> </tr></table></body>