<p>Students in fifth grade should comprehend angle measures of 90, 180, 270, and 360 degrees. This will be necessary to comprehend rotational symmetry. Students will learn about rotational symmetry through experimentation and discovery. Students will:&nbsp;<br>- identify figures that have rotational symmetry.&nbsp;<br>- explain why a figure does or does not have rotational symmetry.&nbsp;<br>- understand rotations at 90, 180, 270, and 360 degrees.&nbsp;<br>- rotate a geometric figure on a coordinate plane.&nbsp;<br>- determine whether polygons (equilateral triangle, square, parallelogram, rhombus, rectangle, hexagon, pentagon) and other figures (heart, clover leaf, etc.) have rotational symmetry when rotated in the center.&nbsp;<br>- estimate the degree of rotation when the rotational symmetry is not a multiple of 90 degrees.&nbsp;<br>- create designs with rotational symmetry.&nbsp;</p>

<p>- What strategies might we use to verify symmetry and congruency?&nbsp;<br>- What strategies may we take to continue a sequence?</p>

<p>- Angle of Rotation: The number of degrees that something is rotated about a fixed point.&nbsp;<br>- Rotational Symmetry: A figure is said to have rotational symmetry if it can be rotated less than a full turn (360 degrees) about a fixed point interior to the figure and the rotated image matches, or coincides, with the original figure.&nbsp;<br>- Vertex: A point where lines, rays, sides of a polygon, or edges of a polyhedron meet (corner).</p>

<p>- student copies of Vocabulary Journal pages (M-5-7-1_Vocabulary Journal)&nbsp;<br>- student copies of Rotational Record sheets (M-5-7-2_Rotational Record, M-5-7-2_Pattern Cutouts 3, and M-5-7-2_Rotational Record KEY)&nbsp;<br>- 4 large congruent trapezoids and 4 large congruent parallelograms cut from colored construction paper&nbsp;<br>- a piece of cardboard (about 12 x 12 inches) for each student (this does not need to be purchased; it can be cut from boxes, as print or designs on it do not matter)&nbsp;<br>- 2 copies of Coordinate Axes sheets for each student (M-5-7-2_Coordinate Axes)&nbsp;<br>- 1 rectangle or triangle cutout for each student (M-5-7-2_Push Pin Figures)&nbsp;<br>- a push pin for each student&nbsp;<br>- regular and irregular polygon paper cutouts&nbsp;<br>- student copies of Lesson 2 Exit Ticket (M-5-7-2_Lesson 2 Exit Ticket and KEY)&nbsp;<br>- copies of snowflake station directions (M-5-7-2_Snowflake Station)&nbsp;<br>- scissors&nbsp;<br>- markers, crayons, or colored pencils&nbsp;<br>- 9 x 12-inch colored construction paper or 8.5 x 11 colored printer paper</p>

<p>- To determine students' comfort level with concepts, use the random reporter approach for student observations during the introduction activity.&nbsp;<br>- Observations during the Recording Rotations Activity will allow for any necessary clarifications and corrections.&nbsp;<br>- The evaluation of the Initials Rotation Project should help in determining which students are prepared to work beyond the standards and which may benefit from additional instruction.&nbsp;<br>- Use the Lesson 2 Exit Ticket (M-5-7-2_Lesson 2 Exit Ticket and KEY) for additional examination.&nbsp;</p>

<p>Scaffolding, Modeling, and Explicit Instruction&nbsp;<br>W: Show four rotations in four different colors to the class. Tell the class about a new type of transformation called a rotation.&nbsp;<br>H: Encourage students to practice rotations by tracing shapes at 90°, 180°, 270°, and 360° counter-clockwise. Define and show rotational symmetry.&nbsp;<br>E: Have students determine if additional shapes have rotational symmetry. Distribute Rotational Record sheets (M-5-7-2_Rotational Record) so that students can test and record their findings. They can also work on the Initials Rotation project, which they will use their initials to build a design and rotating it around the origin of a set of axes.&nbsp;<br>R: Monitor students' rotation projects. Ask students to demonstrate certain rotations and summarize their findings. Review or explain as needed.&nbsp;<br>E: Have students complete the Lesson 2 Exit Ticket to evaluate if additional practice is needed.&nbsp;<br>T: Provide magazines with picture of items with rotational symmetry for students to explore. Students can categorize the images and identify the symmetry in each item they see. Have students design kaleidoscopes or quilts exhibiting rotational and reflective symmetry patterns.&nbsp;<br>O: In this lesson, students learn rotational symmetry through hands-on crafts.&nbsp;</p>

<p><strong>Prior to class, use colorful paper to cut out four congruent shapes such as parallelograms or trapezoids. Arrange them on the front board or wall, with the original on or centered over 0° and the other figure rotated 90°, 180°, and 270°. If the classroom is set up to project images from your computer, you can make and display a rotating design with </strong><a href="http://www.mathsisfun.com/geometry/symmetry-artist.html"><strong>http://www.mathsisfun.com/geometry/symmetry-artist.html</strong></a><strong>.</strong><br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_46.png" width="379" height="684"><br><br>Ask students to write three observations about the arrangement of the shape on a piece of paper. Make a random selection of students to share their observations. Take note of the trapezoid example, where one side of the original (red) figure is on the axis line of 0°. The remaining figures are rotated, not reflected. The same side of each figure is resting against the axis lines at 90°, 180°, and 270°. Each point that touches an axis is the same distance from the origin in each rotation. If the red figure is considered the original figure, moving counter clockwise is the 90° rotation (green figure), the 180° rotation (blue figure), and the 270° rotation (yellow figure). It is also interesting to note the pattern formed by the trapezoids on the inside white space surrounding the origin.&nbsp;<br><br>The parallelogram layout centers each figure on the degree markings rather than resting on them. It should be observed that, in addition to being rotated, these figures appear to have been reflected (flipped) both horizontally (green to yellow) and vertically (red to blue).&nbsp;<br><br><strong>"In today's lesson, we'll be working on designs made by rotating figures. The angle of rotation is the number of degrees by which we rotate each figure. We will analyze and create designs that have symmetry, similar to our reflections in Lesson 1. This will be another type of symmetry, called rotational symmetry."</strong><br><br><strong>"We will begin by rotating the figures at your desk. Each of you will be given a piece of cardboard, two coordinate axes, a geometric figure, and a push pin. To safeguard your desk top, you should only push your pin into the paper when the cardboard is below it."</strong><br><br>Distribute one piece of cardboard, two Coordinate Axes sheets (M-5-7-2_Coordinate Axes), one geometric shape (M-5-7-2_Push Pin Figures), and a push pin to each student. Instruct students to put one sheet of axes paper down on the cardboard. Bring tape to help students secure the page to the cardboard.&nbsp;<br><br><i>Note to teacher:</i> You may want to lightly tape one axis page to the front and back of the cardboard pieces to save time during the activity,&nbsp;<br><br>Students will use paper models on the coordinate axes to investigate what happens when shapes are rotated. Students will use a rectangle or triangle cutout. The shape should be pinned down, but it should also be able to rotate around the origin.<br><br><strong>"Now we're ready to start. Take your geometric figure and set one corner at the origin (center) of your coordinate axes, and one edge on top of the x-axis at 0°, as shown on the board with my trapezoid example. Push your pin carefully onto the same vertex of your geometric shape, as close to the origin as possible." </strong>Walk around and assist students.&nbsp;<br><br><strong>"Trace your shape with a pencil. Now, while keeping your pin in place, rotate your figure 90° counterclockwise (to the left). Make sure to line it up in the same position relative to the axis as when it was first place. When you're sure that it's in the correct place, trace it. Raise your hand if you need support.”</strong> Visually verify that each student accurately traced the rotation. Then repeat the process for 180°, 270°, and 360°.&nbsp;<br><br><strong>"What did you notice when you rotated 360°?"</strong> (<i>the figure overlapped exactly with the original traced figure</i>)&nbsp;<br><br><strong>"Let's try something else. Take your pin and figure off of the cardboard, then flip it over. We'll be using your second Coordinate Axis Sheet. Pin the vertex of your figure to the origin (center), but with the figure centered over the x-axis at 0◦, as in my parallelogram example."</strong> Repeat the steps taken on the first side.<br><br><strong>"The designs you just made have rotational symmetry. If a design or figure has rotational symmetry, when rotated counterclockwise around a fixed interior point, it will completely coincide (overlap) with the original at least once before being rotated 360°. The number of degrees it is rotated to get to the first overlap is called the degree of rotation. What is our degree of rotation?"</strong> (<i>90°</i>)&nbsp;<br><br><strong>"In one full turn (360°), how many times would our design be able to rotate and rest (overlap) entirely on the original design?"</strong> (<i>4</i>)<br><br>To help students grasp the action, show them an equilateral triangle that has been rotated with a push pin in its center rather than a vertex. Make note that when rotated about its own center, there are only three positions, whereas when rotated about the origin, there may be four. Consider inserting a push pin into the center (origin) of any of the student designs and rotating the entire design. Show this rotation movement using one of the student's designs, or have a student show it on his or her own axes by rotating the figure with the push pin in the center.<br><br><strong>Recording Rotations Activity</strong><br><br><strong>"With your partner, you will determine whether some additional shapes have rotational symmetry and the degree of rotation for those that do."</strong>&nbsp;<br><br>Hand out the Rotational Symmetry record sheet (M-5-7-2_Rotational Record), a bag of leftover pattern cutouts from Lesson 1 (M-5-7-1_Pattern Cutouts 1, M-5-7-1_Pattern Cutouts 2), and Pattern Cutouts 3 (M-5-7-2_Pattern Cutouts 3). Allow students to explore with the cutouts by using cardboard and a push pin. If students don't know the degree of rotation (other than 90°, 180°, and 270°), encourage them to figure out how to calculate or estimate a logical value. Distribute among the groups to help explain misconceptions. Have the teacher key available so that students can check their own work or to check it for them (M-5-7-2_Rotational Record KEY). Once students have completed the Recording Rotations activity, instruct them to begin the Initials Rotation Project. If the initials are not completed during class, this could be assigned as a take-home project.<br><br><strong>Initials Rotation Project (in class or take home):</strong><br><br>Give each student another Coordinate Axis paper (M-5-7-2_Coordinate Axes). Students will write their initials in capital letters and place a symbol on each side of the initials on the x-axis at 0°. The letters should follow a creative pattern or design. The symbols can be the same or different from those used in the Lesson 1 Name Card activity. The symbols should represent something about the students or something they enjoy, such as a flower or a basketball. Instruct students to rotate their initial designs 90°, 180°, and 270°. Provide markers, crayons, or colored pencils. Students should design the outline in pencil first, then add the design color once their initials have been correctly rotated. Allow students about 20 to 30 minutes to work. If additional time is required, this task can be completed at home.<br><br>Keep an eye on students while they work on the Recording Rotations and Initials Rotation assignments. Encourage students to discuss their ideas and results. Have students show specific rotations and ask for their observations with their project elements. Make suggestions or ask clarifying questions to help students who have misconceptions. Encourage students to make adjustments as needed. Within each pair or small group, ask at least one student to summarize the definition of rotational symmetry and lead a brief discussion with the group. Assess whether students can distinguish between reflectional (mirror) and rotational symmetry.&nbsp;<br><br>At the end of the class, each student should fill out the Exit Ticket (M-5-7-2_Lesson 2 Exit Ticket and KEY). Use the results to decide whether the additional instructional strategies recommended below are appropriate for individual students.<br><br><strong>Extension:</strong><br><br>Use the options below to personalize this lesson to meet the needs of your students' throughout the unit and the year.<br><br><strong>Routine:</strong> Discuss how important it is to comprehend and use the appropriate vocabulary words while communicating mathematical ideas. During this lesson, students should record the following terms in their Vocabulary Journals (M-5-7-1_Vocabulary Journal): angle of rotation, coordinate axes, edge, origin, rotational symmetry, vertex. Keep a supply of Vocabulary Journal pages on hand so that students can add them as needed. Bring up examples of rotation and rotational symmetry from throughout the school year. Ask students to bring up any examples they see (signs, hubcaps, flowers, etc.). Create a special place in the classroom, such as a bulletin board, to display pictures of symmetry that students have discovered and shared with the class. If the example is rotational, post and label the type of symmetry as well as the degree of rotation.<br><br><strong>Small Group:</strong> Rotational Practice<br><br>Extra practice is required for those who are struggling with the idea of rotation. Work directly with these students in a small group while other students complete stations or the extension activity. Give students a Coordinate Axis sheet (M-5-7-2_Coordinate Axes). Also provide a variety of patterns blocks or pattern cutouts (M-5-7-1_Pattern Cutouts 1, M-5-7-1_Pattern Cutouts 2, or M-5-7-2_Pattern Cutouts 3). Begin by asking students to place one figure at 0°. Demonstrate a 90° rotation on a coordinate grid, then allow students to practice under supervision and guidance. Continue to practice by rotating the individual cutout 180°, 270°, and 360°. As students gain proficiency with these rotations, they can progress to a figure made by combining two pattern blocks or cutouts. If students are having difficulties manipulating the pattern pieces during the rotation, tape the pattern pieces together to start. Move on to more complex designs when time allows.&nbsp;<br><br><strong>Station 1: Snowflake Symmetry</strong><br><br>Provide square sheets of paper which are cut into 8 x 8 inches, 9 x 12 colorful sheets of paper, glue sticks, and scissors. Post a directions page at the station (M-5-7-2_Snowflake Station). Ask students to glue their snowflake to a colorful sheet of paper, leaving space to write below it. Direct students to write on the symmetry in their snowflake. A variation is, instead of writing it down, students can describe the symmetry verbally to you, to a small group, or to the full class. Students should hang their unique creations at the station or another specified location.<br><br><strong>Station 2: Hubcap and Sign Symmetry</strong><br><br>Offer a range of magazines, scissors, glue sticks, and lined paper. Auto magazines and advertisements are good choices because of the types of pictures students are trying to find. Students should cut out 3-5 pictures of signs and/or hubcaps with rotational symmetry or numerous forms of symmetry. If it is difficult to find enough pictures for each student, have them work in pairs. Another alternative is to title each page with a different topic based on the magazines available (for example, fashion, food, home décor, auto, brand logos, sports, and so on). Ask students to cut out and glue examples to their papers. Below each image, ask students to describe the symmetry used in the design. If time allows, invite students to build their own hubcap design or new traffic signs using rotational symmetry.&nbsp;<br><br><strong>Expansion: Kaleidoscope Symmetry Project</strong><br><br>Many wallpaper, floor covering, fabric, quilt square, and kaleidoscope designs use patterns with both reflective and rotational symmetry (M-5-7-1_Pattern Cutouts 1, M-5-7-1_Pattern Cutouts 2, and M-5-7-2_Pattern Cutouts 3). Show several examples, such as those below. Ask students to use pattern blocks or pattern cutouts to create a kaleidoscope pattern with both rotational and reflective symmetry. Suggest that students start by creating one triangular pattern (60°, 90°, or 120° center angle) and then reflecting and rotating it to create the full design. Provide a coordinate grid (M-5-7-2_Coordinate Axes) on which students can arrange their pattern blocks. You might want to help by drawing one triangular outline on the grid paper before making duplicates, so that students are given the first part of the design. Once the design is finished, have students trace the pieces and color it.<br><br>Kaleidoscope examples:<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_47.png" width="471" height="309"><br><br>Quilt square examples:<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_48.png" width="407" height="157"><br><br><strong>This lesson was designed to be exploratory in nature. Hands-on projects with student summaries at the end of each activity help students remember the ideas and allow the instructor to evaluate understanding. The lesson began with a rotation demonstration, in which students made observations about the characteristics of rotated figures. Students were encouraged to rotate their own models in two different ways. The lesson scaffolds on the reflective symmetry examined in Lesson 1 by using similar vocabulary and similar activities, as well as comparing the types of symmetry covered in both lessons. In the expansion at the end of the lesson, students are required to combine rotational and reflective symmetry in the same design. Lesson 3 will continue the scaffolding and introduce translational transformations.</strong></p>