<html> <head> <title> Module 8.2 - Areas of Polygons - SpaceMath@NASA</title> <meta http-equiv="Content-Type" content="text/html; charset=us-ascii"> </head> <body bgcolor="#FFFFFF" link="#0099FF"> <table width="1195" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr bgcolor="#000000"> <td height="171" valign="top" colspan="21"><img src="Banner0.jpg" height="152" width="1195"> </td> </tr> <tr> <td height="45" valign="top" colspan="2" bgcolor="#000000"> <div align="center"> <h4><font size="5" face="Arial, Helvetica, sans-serif"><b><font color="#0099FF" size="4"><a href="http://spacemath.gsfc.nasa.gov/SpaceMath.html"><font color="#00CCFF">SpaceMath Home</font></a></font></b></font></h4> </div> </td> <td width="186" valign="top" bgcolor="#000000"> <div align="center"><font size="5"><b><font face="Arial, Helvetica, sans-serif" color="#0099FF" size="4"><a href="http://spacemath.gsfc.nasa.gov/5EIntroduction.html"><font color="#00CCFF">5E-Introduction</font></a></font></b></font></div> </td> <td width="154" valign="top" bgcolor="#000000"> <div align="center"><font size="5" face="Arial, Helvetica, sans-serif" color="#0099FF"><b><font size="4"><a href="http://spacemath.gsfc.nasa.gov/grade35.html"><font color="#00CCFF">Grade 3-5</font></a></font></b></font></div> </td> <td valign="top" colspan="2" bgcolor="#000000"> <div align="center"><font face="Arial, Helvetica, sans-serif" size="5" color="#0099FF"><b><font size="4"><a href="http://spacemath.gsfc.nasa.gov/grade67.html"><font color="#00CCFF">Grade 6</font></a></font></b></font></div> </td> <td valign="top" colspan="4" bgcolor="#000000"> <div align="center"><font size="5" face="Arial, Helvetica, sans-serif"><b><font color="#0099FF" size="4"><a href="http://spacemath.gsfc.nasa.gov/media.html"><font color="#00CCFF">Grade 7 Modules</font></a></font></b></font></div> </td> <td valign="top" colspan="5" bgcolor="#000000"> <div align="center"><font face="Arial, Helvetica, sans-serif" size="5" color="#0099FF"><b><font size="4"><a href="http://spacemath.gsfc.nasa.gov/algebra1.html"><font color="#00CCFF">Algebra I</font></a></font></b></font></div> </td> <td width="113" valign="top" bgcolor="#000000"> <div align="center"><font size="5" face="Arial, Helvetica, sans-serif"><b><font color="#0099FF" size="4"><a href="http://spacemath.gsfc.nasa.gov/geometry.html"><font color="#00CCFF">Geometry</font></a></font></b></font></div> </td> <td valign="top" colspan="2" bgcolor="#000000"> <div align="center"><font face="Arial, Helvetica, sans-serif" size="5" color="#0099FF"><b><font size="4"><a href="http://spacemath.gsfc.nasa.gov/algebra2.html"><font color="#00CCFF">Algebra II Textbook</font></a></font></b></font></div> </td> <td valign="top" colspan="3" bgcolor="#000000"> <div align="center"><font size="5" face="Arial, Helvetica, sans-serif" color="#0099FF"><b><font size="4"><a href="http://spacemath.gsfc.nasa.gov/calculus.html"><font color="#00CCFF">Calculus</font></a></font></b></font></div> </td> </tr> <tr> <td valign="top" height="172" colspan="21"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="350" height="82" valign="top"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="330" height="51"> <div align="center"><font face="Arial, Helvetica, sans-serif" size="6" color="#333333"><b>Module 11</b></font></div> </td> </tr> <tr> <td height="31" valign="top"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="350" height="31"> <div align="center"><font face="Arial, Helvetica, sans-serif" size="5"><b>Areas of Polygons</b></font></div> </td> </tr> </tbody> </table> </td> </tr> </tbody> </table> </td> <td width="85"></td> <td width="759" valign="top"> <div align="center"><br> <font face="Arial, Helvetica, sans-serif" size="7"><font size="5"><b>Exploring Snowflake Shapes </b></font></font></div> </td> <td width="29"></td> <td width="17"></td> </tr> <tr> <td height="85" colspan="4" valign="top"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="1178" height="68" valign="top"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="1187" height="85" valign="top"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="1200" height="90" valign="top"><br> <font face="Arial, Helvetica, sans-serif" size="4"><b>Objectives:</b> Students learn about the properties of snowflakes through reading a NASA press release. By viewing a NASA eClips video segment, students discover how snowflakes form and how their shapes relate to their history. Students then explore the geometric shapes of snowflakes by viewing them as polygons and calculating their areas using the area properties of right triangles. <p></font></p> </td> <td width="3"></td> </tr> </tbody> </table> </td> <td width="25"></td> </tr> </tbody> </table> </td> </tr> </tbody> </table> </td> <td></td> </tr> </tbody> </table> </td> </tr> <tr> <td colspan="5" valign="top" rowspan="4"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="545" height="426" valign="top"><img src="SnowflakeNOAAsmall.jpg" height="540" width="540"></td> <td width="9"></td> </tr> </tbody> </table> </td> <td width="12" height="105"></td> <td width="10"></td> <td width="10"></td> <td valign="top" colspan="6" bgcolor="#000000"><img src="HMLogo.jpg" height="75" width="203"> </td> <td width="16"></td> <td valign="top" colspan="4" bgcolor="#00CCFF"> <p align="center"><font face="Arial, Helvetica, sans-serif" size="4"><b>Mathematics Skill or Topic Area: </b></font></p> <p align="center"><b><font face="Arial, Helvetica, sans-serif" size="4">Area of Polygons</font></b></p> </td> <td width="17"></td> <td width="30"></td> </tr> <tr> <td height="211"></td> <td></td> <td></td> <td valign="top" colspan="11"> <p><font face="Arial, Helvetica, sans-serif" size="4" color="#333333"><b>Next Gen Science Framework: </b></font><font face="Arial, Helvetica, sans-serif" size="2">PS1.A: Structure and Properties of Matter; PS1.B: Chemical Reactions; Patterns </font></p> <p><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"><b>Common Core ELA for Science:</b></font><font face="Arial, Helvetica, sans-serif" size="2"> RST.6-8.2. Determine the central ideas or conclusions of a text; provide an accurate summary of the text distinct from prior knowledge or opinions. RST.6-8.8. Distinguish among facts, reasoned judgment based on research findings, and speculation in a text. RST.6-8.9. Compare and contrast the information gained from experiments, simulations, video, or multimedia sources with that gained from reading a text on the same topic. </font> </p> <p><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"><b>Common Core Math Standard:</b> </font><font face="Arial, Helvetica, sans-serif" size="2">CCSS.Math.Content.6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. CCSS.Math.Content.6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices. Apply these techniques in the context of solving real-world and mathematical problems. </font></p> </td> <td valign="top"></td> <td valign="top"></td> </tr> <tr> <td height="9"></td> <td></td> <td></td> <td valign="top" colspan="3" rowspan="2"><img src="EclipsLOGO.gif" height="106" width="138"></td> <td width="12"></td> <td width="13"></td> <td width="49"></td> <td></td> <td></td> <td width="38"></td> <td width="96"></td> <td width="87"></td> <td valign="top"></td> <td valign="top"></td> </tr> <tr> <td height="115"></td> <td></td> <td></td> <td></td> <td colspan="8" valign="top"> <p><font face="Arial, Helvetica, sans-serif" size="3"><b><font size="4">Video Engagement: <font size="3"></font></font></b><font size="3"><i><b>Snowflakes - </b></i><font size="2">Snowflakes are actually a history of what was happening in the atmosphere as the snowflake formed and fell to the ground. Find out how you can help NASA scientists study different types of snowflakes to learn about the atmosphere and changes in climate. (4 minutes) <a href="http://www.nasa.gov/audience/foreducators/nasaeclips/search.html?terms=%22our%20world%3A%20snowflakes%22&category=0000&disp=grid"> <font color="#000000"><b>View Program</b> </font></a></font></font></p> </font></td> <td valign="top"></td> </tr> <tr> <td height="9" width="9"></td> <td width="100"></td> <td></td> <td></td> <td width="105"></td> <td></td> <td></td> <td></td> <td width="12"></td> <td width="89"></td> <td width="37"></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td rowspan="10"></td> <td valign="top" rowspan="8" colspan="4"> <table width="100%" border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td width="530" height="689" valign="top"> <p><font face="Arial, Helvetica, sans-serif" size="4"><b><font size="5">Engage your students with a press release:</font></b> </font></p> <p align="center"><font face="Arial, Helvetica, sans-serif" size="4"><i><u>Olympic Ice is Different in a 'Frozen Light' </u></i></font></p> <p><font face="Arial, Helvetica, sans-serif" size="4"></p> <p>Scientists and spectators at the Winter Olympics are all seeing the same ice and snow, but one scientist can see exactly how different it can be for each sport, by using light. </p> <p>Snow and ice can look like long tubes or tiny ping-pong balls. NASA scientist Peter Wasilewski is using &quot;polarized&quot; light to see if snow and ice is the right texture for different Olympic sports, and also creating beautiful colored pictures of the snow and ice. It's important to see the shapes of the snow and ice because in Olympic sports, they have to be different for each sport, whether skiing or snowboarding, figure skating or hockey. The shape of snow for example, can be tiny snow flakes or snow balls. Typically, snow flakes, like the one shown above (NOAA photo) are not good for skiers, and can slow them down. By looking at the snow with polarized light, Wasilewski can see what type of snow is on the ground and if its right for skiing. </p> <p>&quot;Ice is different for each Olympic sport,&quot; Wasilewski said. &quot;The ice is softer for figure skaters than it is for hockey players. Figure skaters need to dig in with their toe picks for jumps. Ice hockey players want the hard ice that makes the ice fast and easier to skate on. With a microscopic look at the ice using the spectrum, I'm able to see how the ice differs.&quot; </p> <p>Wasilewski realized that color and shape of ice and snow changes depending on the thickness and direction of the ice. He got started by just playing around, making ice crystals in his refrigerator and photographing them. From there, the colorful art of snow and ice he calls &quot;Frizion&quot; was born. </font></p> <p><font face="Arial, Helvetica, sans-serif" size="4">Press release date line - February 26, 2006 [<a href="http://www.nasa.gov/vision/earth/lookingatearth/olympic_ice.html"> Click Here</a> ]</p> <p>For more information about the artwork by Peter Wasilewski: [<a href="http://www.frizion.com/about.html">Click Here</a>] </font> <p></p> <p></p> </td> <td width="15"></td> </tr> <tr> <td height="9"></td> <td></td> </tr> </tbody> </table> </td> <td rowspan="10"></td> <td rowspan="10"></td> <td height="108"></td> <td></td> <td colspan="3" rowspan="2" valign="top"><img src="SpaceMathICONSM.jpg" height="124" width="124"></td> <td></td> <td colspan="7" valign="top" bgcolor="#CCCCCC"> <p align="center"><font face="Arial, Helvetica, sans-serif" color="#000000" size="4"><b><font size="5">Explore math connections with </font></b></font><font face="Arial, Helvetica, sans-serif" size="5"></p> <p align="center"><b><i>SpaceMath@NASA</i></b></font></p> </td> <td></td> </tr> <tr> <td height="16"></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td height="9"></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td height="500"></td> <td></td> <td colspan="11" valign="top" bgcolor="#CCCCCC"> <p><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"><b>Problem I - </b></font><font face="Arial, Helvetica, sans-serif" color="#000000" size="4"><u>Graphing a Snowflake Using Symmetry</u> </font><font face="Arial, Helvetica, sans-serif" size="4">- <font size="3">Students plot the coordinate points for a snowflake in the First Quadrant and connect the points with line segments. Then students use reflection symmetry to 'mirror' these points into the other three Cartesian quadrants. [<a href="6Mod11Prob1.pdf">Open PDF </a>]</font> </p> <p><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"><b>Problem II - </b></font><font face="Arial, Helvetica, sans-serif" color="#000000" size="4"><u>The Surface Area of a Snowflake</u> </font><font face="Arial, Helvetica, sans-serif" size="4">- <font size="3">Students use the properties of right triangles, specifically the area formula, to determine the area of a snowflake pattern. <font size="3">[<a href="6Mod11Prob2.pdf">Open PDF </a>] </font> </p> <p></p> <p></p> <p></p> <p><font face="Arial, Helvetica, sans-serif" color="#000000" size="4"><b><font size="5">Explain your thinking:</font></b> </font></p> <p><font face="Arial, Helvetica, sans-serif" size="4"><font size="3"><font size="4"><b>Write your own problem</b></font><b><font size="4"> </font></b>- Using information found in the Math Connection problems, the press release or the video program, create your own math problem. Explain why you set the problem up this way, and how you might find its answer. </font></font></p> <p></p> <p></p> <p></p> <p><font face="Arial, Helvetica, sans-serif" color="#000000" size="4"><b><font size="5">Evaluate your understanding:</font> </b></font></p> <p><font face="Arial, Helvetica, sans-serif" size="4"><b>Challenge Problem</b> - <font size="3"> Snowflakes come in many shapes and sizes. In this problem, students explore how the surface area of a snowflake is related to how fast it grows. [<a href="6Mod11Prob3.pdf">Open PDF </a>] </font></p> <p>&nbsp;</p> <p><font size="5">For additional problems and resources relating to snow research, visit the NASA, History of Winter website at:</font></p> <p> <center> <a href="http://education.gsfc.nasa.gov/how/"><font size="5">http://education.gsfc.nasa.gov/how/</font></a> </center> <br> <center><img src="HOWlogo.jpg" width="222" height="113"> </center></p> </font></font></font></font> <td></td> </tr> </tbody> </table> </body> </html>