Geometry

According to PSSM, students should be exposed to dynamic  geometric representations and be able to use them to solve problems. More specifically, students should be able to analyze characteristics and properties of two-and –three dimensional shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational systems; and students should be able to use visualization, spatial reasoning and geometric modeling to solve problems.

Help geometry come alive using the following interactives:

#1
 * Geometry Evaluations: **

 **//An interactive lesson in finding the Circumference of a circle //** || **Geometry: ** Solve Problems involving Circumference of Circles || * || //Rationale //: This applet is designed to allow students to investigate the circumference of circles and apply that knowledge to unknown circles. Students interact within the applet in multiple ways; there are multiple choice questions as well as fill in the blank questions and students receive immediate feedback along with explanations as they are problem solving. There is also a built in calculator to help students follow the procedures of calculating circumferences of circles. || Instrumental understanding Relational understanding || * || //Rationale //<span style="font-family: 'Times New Roman','serif';">: The focus of this activity is to support students in their instrumental understanding of circumference, and then guides them to the relational aspect of radius and diameter to circumference and how and why pi is important in this process. At the end of the applet, students are directed to think about how circles were constructed before the use of technology and then asked to apply that information to the invention of gears.Students are given a formula and asked to find how many rotations a small gear has to make for one rotation of a large gear. The problem is put into the context of mechanical engineering and students are supported by website links they can check out to help them investigate the problem. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">practice <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">direct instruction/explanation <span style="color: #000000; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in; text-indent: -0.25in;"> · <span style="color: black; font-family: 'Times New Roman','serif';">learning through exploration  || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Students practice performing calculations given the formula of the circumference of a circle and finding and identifying the radius and diameter. There are thorough explanations of each that students may access as they are working through each problem in the “Say What?” section found in the menu. They learn through exploration in the “So What?”, “Dig Deeper” and “Talk About It” sections described above. Students are asked to apply the information they have learned and reflect about their learning process at the end of the activity. ||
 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> **//<span style="font-family: 'Times New Roman','serif'; font-size: 18pt;">[|Finding Your Way Around Circles] //**
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Which standard? **
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|analyze characteristics] and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;"> [|use visualization], spatial reasoning, and geometric modeling to solve problems. || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet focuses on the relationship between pi and the diameter (and radius) of a circle to its circumference. Students are given step-by step explanations of each part of the formula and work through several examples with circles to derive the formula C= Once they have the formula, they use interactive tools to find radius, diameter and circumference of various circles. When working through the applet, students are able to use reasoning and modeling to solve problems focused around the circumference of circles.  ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What mathematical content is being learned (or intended to be learned)? ** <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in; text-indent: -0.25in;">Identify Radius, Diameter and Circumference
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Is the focus on instrumental or relational understanding? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Response //<span style="font-family: 'Times New Roman','serif';">: Technology provides a visualization tool that allows students to virtually “see” how radius and diameter affect the circumference of various circles. Within the investigation, students can click on a virtual ruler to measure diameters and use the calculator to calculate results. This automates the formula and allows students to simplify the big idea to within the realm of the investigation. This technology represents knowledge and thinking in several different ways and allows students to discover information on their own by providing links to internet resources. Students also communicate and collaborate with a partner at the end of the activity to complete practice problems and a reflection. The interactive is very easy to navigate and the layout is almost intuitive for students to complete. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What instructional function(s) does the resource serve? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What kinds of representations of the mathematics are used? **

<span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">symbolic <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">graphical <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">visual/spatial <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">concrete or real-world objects <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">dynamic || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet allows students to view mathematics represented in a variety of ways. Symbolically, math is represented through numbers and symbols that students can manipulate throughout the activity. Graphically, students see circles drawn in the four quadrants and use that information to determine either the radius or diameter of the circle given. This interactive is very visual as the user can see the host mouse actually “saw” off the line for diameter to be exactly at the two points on the edge of the circle and again the mouse uses a saw to represent the radius as half of diameter. Real world objects are used during the course of the investigation, culminating in rotating gears at the end. This applet is extremely dynamic as there is movement and sound all the way through the investigations. ||

<span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> **//<span style="color: #000000; font-family: 'Times New Roman','serif'; font-size: 20pt;">An interactive lesson in geometric transformations //** || **<span style="color: green; font-family: 'Times New Roman','serif';">Geometry: **
 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> //<span style="font-family: 'Times New Roman','serif'; font-size: 24pt;">[|Bathroom Tiles] //
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Which standard? **
 * <span style="color: windowtext; font-family: 'Times New Roman','serif'; text-decoration: none; textunderline: none;">[|specify locations] <span style="color: #000000; font-family: 'Times New Roman','serif';">and describe spatial relationships using coordinate geometry and other representational systems;
 * <span style="color: windowtext; font-family: 'Times New Roman','serif'; text-decoration: none; textunderline: none;">[|apply transformations] <span style="color: #000000; font-family: 'Times New Roman','serif';">and use symmetry to analyze mathematical situations;
 * <span style="color: windowtext; font-family: 'Times New Roman','serif'; text-decoration: none; textunderline: none;">[|use visualization] <span style="color: #000000; font-family: 'Times New Roman','serif';">, spatial reasoning, and geometric modeling to solve problems. || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet focuses on relationships between rotational geometry and the coordinate plane. Students are required to identify specific degrees of rotation a portion of a tile must undergo to get from the original point to its image. Students also need to identify equations of lines in order to reflect given objects. Students apply each mathematical transformation in no particular order within the applet. Throughout the investigation, students need to reason through their answers and use geometric models to visualize how each transformation will be produced.

<span style="font-family: 'Times New Roman','serif';">These standards coincide with the 6th grade GLCE’s for geometry in Michigan. Our benchmarks <span style="color: black; font-family: 'Times New Roman','serif';">state that students should understand the basic rigid motions in the plane (reflections, rotations, translations), relate these to congruence, and apply them to solve problems. Students should also understand and use simple compositions of basic rigid transformations, e.g., a translation followed by a reflection. || <span style="color: #000000; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in; text-indent: -0.25in;"> · <span style="font-family: 'Times New Roman','serif';">Geometric Transformations || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet is designed to allow students to use an interactive game to perform transformations using reflections, rotations and translations. Students also need to use this information to apply these transformations in the coordinate plane. Students identify equations of lines and determine slope in order to find lines of reflection or points of rotation. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Instrumental understanding <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Relational understanding || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: The focus of this interactive activity is to promote relational understanding between geometric transformations and the coordinate plane. Students use the idea of the coordinate plane to transform geometric figures within the four quadrants of the coordinate plane. If students need a refresher, however, rules and procedures for performing transformations and ideas about the coordinate plane are summarized in the <span style="font-family: 'Times New Roman','serif';">[|“Key Ideas”] section. There are also [|“Tips”] to give the user hints on how to solve a given transformation. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What mathematical content is being learned (or intended to be learned)? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Is the focus on instrumental or relational understanding? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Response //<span style="font-family: 'Times New Roman','serif';">: Technology provides a visualization tool that allows students to virtually “see” the movement of each transformation. It also gives students practice in defining rotations by degrees, reflections through lines and translations through movement. This technology also transforms the learners experience to a quick and time-saving image that is automated and much more accurate than free hand drawing. Students see the facilitated movement from the original object to its image. This technology also allows students to represent their mathematical thinking through the interactive interface and precise measurements of rotations and reflections. Students cannot simply guess to arrive at the answer.

<span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">This technology also supports the role of teacher by providing different levels to enhance differentiated instruction. Each level requires a different challenge to the student all surrounding the basic principles of geometric transformations. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">practice <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">direct instruction/explanation <span style="color: #000000; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in; text-indent: -0.25in;"> · <span style="color: black; font-family: 'Times New Roman','serif';">learning through exploration  || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Students practice performing transformations and use what they know about the coordinate plane to reason through their actions. Direct instruction is provided about each of the geometric transformations as well as some basic information about the coordinate plane (see above). Students learn throughout the activity by exploring different transformations and their applications in geometry. Students begin to realize patterns within the coordinate plane that help them determine degrees of rotation or lines of reflection. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What instructional function(s) does the resource serve? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What kinds of representations of the mathematics are used? **

<span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">symbolic <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">graphical <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">visual/spatial <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">concrete or real-world objects <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">dynamic || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet represents mathematics in a variety of ways. Symbolically, the interactive shows numerals that can be manipulated by the user and geometric shapes that are transformed. Graphically, math is represented through the use of the coordinate plane. Students are able to visualize 2-dimentional objects in motion as they are reflected, rotated, and translated within the coordinate plane. Each shape is a representation of bathroom tile designs, which students are both familiar with and can imagine geometric transformations happening to. All of these transformations are dynamic as students are able to see perpetual movement from the original object to its image. Sound is also apparent within this interactive and a distinct sound is heard differentiating between correct and incorrect answers. ||


 * <span style="color: #000080; font-family: 'Comic Sans MS',cursive; font-size: 130%; margin: 0in 0in 10pt;">Geometry Annotated Links: **


 * **Screen Shot** ||  **Name and Description**  ||
 * [[image:Angle_Sums.jpg width="365" height="247" align="center" link="http://illuminations.nctm.org/ActivityDetail.aspx?ID=9"]] || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt;"> In [|Angle Sums], students <span style="color: #000000; font-family: 'Times New Roman','serif';">explore the sum of angles property for different polygons. Students choose a polygon, and are then able to drag vertices, thus changing the shape of the polygon. The applet continuously updates angle measurements as well as the sum of the angles. Students are able to see that even though the angles change, the sum remains the same. By observing how changing the angles affects the sum of the angles, students are able to discover that all triangles have an angle sum of 180°, quadrilaterals have an angle sum of 360°, etc. In addition, for triangles and quadrilaterals, students can watch a short animated clip that shows the angles being put together to create a straight angle and circle, respectively.  ||
 * [[image:Surface_Area_and_Volume.jpg width="384" height="264" align="center" link="http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/"]] || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt;"> In [|Surface Area and Volume], <span style="color: #000000; font-family: 'Times New Roman','serif'; font-size: 12pt;">students manipulate three-dimensional rectangular prisms, or triangular prisms to experiment with surface area and volume. Students practice skills at estimating surface area and volume and calculate volume to see how surface area changes. There is a slider where students can manipulate the width, height and depth of the prism. From there, they can either practice calculating the surface area and volume, or the program can calculate it for them so students can look at trends within the changes in the surface area and volume. Once they have explored the relationship between surface area and volume in rectangular prisms, they can take what they have noticed and try and apply it to the triangular prism and develop an understanding between the surface area and volume of that shape. All that is given to the students is the calculations- it is up to the students to determine the relationship between the data and how it changes within the two shapes.  ||