<p>The students will be introduced to polygons. In this lesson, students will learn:&nbsp;<br>- measure with a ruler.&nbsp;<br>- cut a polygon shape using straight lines.&nbsp;<br>- understand and calculate basic formulas.&nbsp;<br>- find the interior angle sum of an n-gon.&nbsp;<br>- find the exterior angle sum of an n-gon.&nbsp;<br>- calculate the measure of an interior angle of an equiangular n-gon.&nbsp;<br>- calculate the exterior angle of an equiangular n-gon.&nbsp;<br>- calculate the missing angles of an n-gon.<br>&nbsp;</p>

<p>- What are the important aspects of polygons, and how can we utilize the characteristics of their sides and angles to make general conclusions about them?</p>

<p>- Exterior Angle: In a polygon, the angle between any side produced and the adjacent side not produced.&nbsp;</p><p>- Interior Angle: In a polygon, the angle between any two sides of the polygon, not produced, and lying within the polygon.</p><p>- Apothem: In a regular polygon, the line from the center of the polygon to the midpoint of one of the sides.</p><p>- Distinct Diagonals: The number of diagonals that can be drawn from one vertex of a polygon.&nbsp;</p>

<p>- copies of a full-face photo (should be a paper copy) of yourself (the teacher)</p><p>- scissors</p><p>- tape</p><p>- markers</p><p>- protractor</p><p>- ruler</p><p>- copies of Polygon Face (M-G-2-2_Polygon Face)</p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- shape sheets (M-G-2-2_Three- and Four-Sided Regular Polygons, M-G-2-2_Five- and Six-Sided Regular Polygons, M-G-2-2_Seven- and Eight-Sided Regular Polygons, and M-G-2-2_Nine- and Ten-Sided Regular Polygons)</span></p><p>- pictures of famous bridges</p><p>- balsa wood sticks (only if doing Activity 3)</p><p>- cardboard pieces (only if doing Activity 3)</p><p>- glue (only if doing Activity 3)</p><p>- twine/string (only if doing Activity 3)</p><p>- small paper cup (only if doing Activity 3)</p><p>- AA batteries (for weights) (only if doing Activity 3)</p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- copies of Bridge Diagrams (M-G-2-2_Bridge Diagrams) (only if doing Activity 3)&nbsp;</span></p><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);">- copies of Lesson 2 Exit Ticket (M-G-2-2_Lesson 2 Exit Ticket)</span></p>

<p>- Evaluate group presentations based on the accuracy of content, quality of group communication, and fair distribution of tasks.&nbsp;<br>- Journal replies might include subjective observations as well as factual or quantitative reporting.&nbsp;<br>- The Lesson 2 Exit Ticket (M-G-2-2_Lesson 2 Exit Ticket) encourages students to document particular polygon-related facts.&nbsp;<br>- The Bridge Activity should be evaluated using specific comparisons of predictions and outcomes.<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson covers the representation, properties, and basic measurements of polygons. They will be able distinguish between two-dimensional shapes that are and those that are not polygons, use the interior and exterior angles, and make appropriate classifications.&nbsp;<br>H: The Polygon Face activity visually demonstrates how polygons can be represented in a human face. Finding and combining the lines that form the polygons, as well as finding the diagonals, will help students develop the more general abilities needed to create meaningful polygonal representations.&nbsp;<br>E: After completing and reviewing the polygon face activity, students will create, write, and edit journal entries. Individual writing assignments will require students to explain their personal knowledge of what they have studied while also providing them with a useful tool for moving the lesson forward.&nbsp;<br>R: Students can learn about the relationships between a polygon's number of sides and the number of triangles it contains by subdividing it into triangles with diagonals. As students finish subdividing and counting triangles, they have a better understanding of whether adopting a predictable pattern to solve a problem is generalizable.&nbsp;<br>E: Students can identify the characteristics that distinguish stronger bridges from weaker ones by observing their failures under loads beyond their capability. Students can forecast structural strengths and weaknesses based on the design features of each bridge.&nbsp;<br>T: Review and emphasize proper use of new vocabulary. Ask students to keep a journal of their words. In addition to a definition, students should give examples, non-examples, and visuals to explain and expand their comprehension. When the class is working on other geometry concepts involving angles, have students calculate the interior/exterior angle measure sums of any polygons included in the lesson. They can also calculate the specific angles of the polygons.&nbsp;<br>O: This lesson teaches students about shapes with numerous sides, including triangles, decagons, and more. The lesson then discusses why exterior angles always equal 360 degrees and emphasizes intuitive thinking to develop the interior angle sum formula. This lesson takes students beyond simple memorization and rote robotics to gain a better grasp of how the angles of a regular polygon are related.<br>&nbsp;</p>

<p>This activity outlines how students can learn and practice some basic polygon properties. Students will recognize polygons based on their sides and label them as irregular or regular polygons. They'll use simple formulas to calculate perimeter, area, diagonals, and interior triangles.</p><p>Distribute a photo of your face or one from a magazine to each student. Instruct students to discover and draw as many polygons on the face as they can using their straightedge. You may need to demonstrate one or two before students comprehend the assignment. Ask them to look for polygons with various numbers of sides. You can show them the attached example (M-G-2-2_Polygon Face).</p><p>After 10 minutes, place one picture on the wall and ask if any of the students have a polygon they would want to share with the class. Ask him or her to approach the wall and draw a polygon on the face. This can be done at random or in sequence based on the number of sides. Invite students to come up and add their own polygon to the picture until everyone has had a chance to contribute.</p><p>As a class, encourage students to identify polygons with various sides, starting with triangles or trigons and ending with decagons. Students should be divided into groups and assigned to different polygons. Have the group cut its polygon from the larger photo and tape it to a blank sheet of paper.</p><p>Ask the group to identify if the polygon is convex or concave. Have the group sketch the polygon's diagonals. (Note that if the polygon is concave, some diagonals will be outside the shape.) Students can determine the number of diagonals for their polygon using this formula:</p><p><strong>number of distinct diagonals = \(n(n - 3) \over 2\)</strong></p><p><strong>(where </strong><i><strong>n </strong></i><strong>= number of sides or vertices)</strong></p><p>Students should apply their rules to measure the sides and perimeters of their polygons.</p><p>Allow a few students to present their findings in detail to the rest of the class, and then discuss whether they are correct. Compare and contrast the several polygons provided by the students.</p><p>Distribute shape sheet sets at random; each student should receive only one set (three- and four-sided, five- and six-sided, seven- and eight-sided, or nine- and ten-sided) (M-G-2-2_Three- and Four-Sided Regular Polygons, M-G-2-2_Five- and Six-Sided Regular Polygons, M-G-2-2_Seven- and Eight-Sided Regular Polygons, and M-G-2-2_Nine- and Ten-Sided Regular Polygons). Ask, <strong>"What do you notice about these shapes as compared to the shapes you were looking at before?"</strong> (They're all regular shapes. The previous shapes should have been irregular.)</p><p>Each shape is identified by its apothem and the length of one side. Ask students to find the perimeter of that polygon using simple measurements and/or the formula (written on the board):</p><p><strong>Perimeter =</strong><i><strong> ns</strong></i></p><p><strong>(where </strong><i><strong>n</strong></i><strong> = number of sides and </strong><i><strong>s</strong></i><strong> = length of each side)</strong></p><p>Ask them to break their polygons into triangles and connect each vertex to a single vertex. Point out that the number of triangles in a polygon can be calculated using the following formula (write on board):</p><p><strong>Number of triangles = </strong><i><strong>n – 2</strong></i></p><p><strong>(where </strong><i><strong>n</strong></i><strong> = the number of sides)</strong></p><p>Using the apothem and the length of one side, students may find the area of the regular polygon using the following formula (write on board):</p><p><strong>Area = \(AP \over 2\)</strong></p><p><strong>(where </strong><i><strong>A</strong></i><strong> = apothem and </strong><i><strong>P </strong></i><strong>= perimeter)</strong></p><p>Draw a regular hexagon and its apothem on the board, and ask them where the area formula came from. If they require prompting, draw the isosceles triangle, with the apothem as the altitude. Students should be able to see the six isosceles (or equilateral for the hexagon) partitions that comprise the total area. The perimeter of a regular polygon is the sum of the bases of its component triangles. Because the area of each triangle equals half the base multiplied by the height (apothem), the area of any regular polygon is the sum of the bases (perimeter) times the apothem (height), divided by half the number of sides.</p><p>Students are grouped based on the polygons they are using. Students should compare their work with their peers to see if they all came up with the same solution.</p><p>Have students write about what they've learned about polygons. The journal page should be turned in so you can track each student's development.</p><p><strong>Activity 2: Angles of Polygons</strong></p><p>Students will be introduced to the exterior and interior angle sums of polygons.</p><p><strong>"In this activity, we will look at the size of different polygons' internal and exterior angles, as well as the sum of those angles. We can obtain the angle measures of various polygons by knowing the sum of the interior or exterior angle measures."</strong></p><p>Before beginning this activity, use masking tape to draw a large pentagon (like to the one below) with exterior angles on the floor.</p><figure class="image"><img style="aspect-ratio:173/167;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_164.png" width="173" height="167"></figure><p>For this practice, have a student start at one intersection and walk along the tape lines, returning to the same point (the lines could be described as city streets or paths). Discuss with the students what this means.</p><p><strong>"How much did s/he turn in total?"</strong> (<i>540 degrees</i>)</p><p><strong>"What degree turn is it when you turn all the way around?"</strong> (<i>360 degrees</i>)</p><p><strong>"Which angles on the taped figure represent each turn?"</strong> (<i>exterior angles</i>)</p><p><strong>"What does that tell us about the sum of the exterior angles?"</strong> (<i>They equal 360 degrees.</i>)</p><p><strong>"If the pentagon was a different shape (sides are different lengths, interior angles are different sizes), would that change the sum of the exterior angles?"</strong> (<i>no</i>)</p><p><strong>"If it had a different number of sides, nine sides for example, would the sum of the exterior angles change?"</strong> (<i>No, you would still turn all the way around.</i>)</p><p>Draw the pentagon on the board and have students draw it on their paper. <strong>"We know that the sum of the exterior angles of a pentagon equals 360 degrees, but what about the interior angles?" </strong>Have each student connect two non-adjacent vertices of the pentagon before connecting the remaining two.</p><p><strong>"What is the sum of the interior angles of a triangle?"</strong> (<i>180 degrees</i>).</p><p><strong>"What shape do I get if I put two triangles together, leg to leg?"</strong> (<i>quadrilateral</i>)</p><p><strong>"What is the sum of the interior angles?" </strong>(<i>360 degrees</i>)</p><p>Have students repeat the process with three triangles to form a pentagon. Ask students to make a table of results. The table should include the number of sides (three to ten), the total of internal angles, the measure of each interior angle of an equiangular polygon, and the measure of each exterior angle. Start by filling in the number of sides and the sum of interior angles for the triangle-pentagon. Assist students as needed with filling out the table. Samples have been supplied below.</p><figure class="image"><img style="aspect-ratio:519/398;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_165.png" width="519" height="398"></figure><figure class="image"><img style="aspect-ratio:572/353;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_166.png" width="572" height="353"></figure><p><strong>"Do you notice a pattern in the sum of the interior angles?"</strong> (<i>It increases by 180 degrees each time.</i>)</p><p><strong>"What does 180 degrees have to do with each shape?"</strong> (<i>It is the sum of the angles of the triangle, and we add a triangle whenever we require another side.</i>)</p><p><strong>"If I had a regular pentagon, how could I find the measure of each angle?"</strong> (<i>Divide the total of the internal angles by 5. Remind students to record this answer in their tables.</i>)</p><p><strong>"What if I knew that two of the angles of a pentagon were each 75 degrees, and the other three angles were equal to each other? How would I calculate the measure of one of the other angles?"</strong> (<i>Potential solution: \(540-75-75 \over 3\)</i>)</p><p>Allow students to work independently or in groups to finish the table. Allow them to utilize additional triangles if they would like. Walk throughout the room, assisting those who require further practice. Check that the results are correct.</p><p><strong>"What if I needed to find the sum of the interior or exterior angles of a polygon with 25 sides?"</strong></p><p><strong>"What if I also needed to find the measure of each interior or each exterior angle of an equiangular polygon with 25 sides?"</strong></p><p><strong>"Would I want to make a chart, or is there a better way?"</strong> (<i>Find the rule</i>.)</p><p>Encourage students to work together in groups to create guidelines for:</p><p>The total number of interior angles in an <i>n</i>-gon.</p><p>The measurement of each interior angle of an <i>n</i>-gon.</p><p>The measurement of each exterior angle of an <i>n</i>-gon.</p><p>Have a few groups present their rules (and the process they used to establish them) to the class. Use the shape sheets to demonstrate this.</p><p>Examples of each could be:</p><p>1. (<i>n</i> – 2) × 180<br>2. \((n-2) × 180 \over n\)<br>3. 180 - \((n-2) × 180 \over n\)</p><p>The following activity is optional, but should be completed if time permits and the necessary supplies are available. If you decide not to do the following activity, distribute the Lesson 2 Exit Ticket (M-G-2-2_Lesson 2 Exit Ticket in the Resources folder).</p><p><strong>Activity 3: Bridges</strong></p><p>In this activity, students will learn about the practical application of polygons.</p><p>Students will examine the properties of squares, triangles, pentagons, and hexagons, as well as learn how to demonstrate which shape is the most stable when put under stress.</p><p>Begin by asking students what makes a good bridge. Good answers include solid materials, high-quality construction, and the utilization of strong design.</p><p>Ask about some common bridge structures. Show examples of famous bridges.</p><p>Ask students what shapes they think will build the strongest bridge: triangles, squares, or hexagons.</p><p>Divide students into six groups so that each group can work on one pattern at a time. Distribute the activity materials to each group, and hand out the bridge diagrams (M-G-2-2_Bridge Diagrams). Each group will be in charge of constructing a bridge according to one of the patterns. Assign one pattern to each group. Instruct students to use balsa sticks and glue to make the bridges as shown in the diagram.</p><p>Allow the bridges to dry overnight.</p><p>Before the following lesson, place cardboard tops on top of each bridge structure, then poke holes in the sides of a small paper cup and string twine through the holes. The other end of the string should be tied around one of the bridge tops (a cardboard piece), with the cup hanging down the middle of the bridge structure. This process will need to be repeated for each bridge as its solidity is tested. To test each bridge structure, place each bridge over a space between two desks, with the paper cup hanging down the middle. Add one battery (weight) to the cup at a time until the bridge collapses. Note the weight at which each bridge collapses.</p><p><strong>"The bridge with the most strength should be the bridge using the isosceles triangle with the one point of the triangle at the center of the top of the bridge."</strong></p><p>Following the deconstruction of the bridges, have each group examine the advantages and disadvantages of their assigned bridge structure. Have each group come up with one solution to strengthen their bridge.</p><p>Allow all groups to share their positives, negatives, and solutions to the rest of the class. Discuss the order in which the bridges failed (which collapsed under the least amount of weight and which collapsed under the most weight) as well as the possible or probable causes of this order. Discuss the principles of the shapes that made the bridges stronger or weaker. Discuss the relationship between the lengths of a triangle's sides and its angles, and how when one length changes, the angles must change as well. As a result of this principle, the rigidity of the triangle structure makes it the strongest and therefore the best load-bearing form. Discuss other potential causes of failure, such as building errors, material weaknesses, and so on.</p><p>Once the class discussion has ended, distribute the Lesson 2 Exit Ticket (M-G-2-2_Lesson 2 Exit Ticket). (Note: This is the same Exit Ticket that the class would use if you choose not to complete Activity 3.)</p><p><strong>Extension:</strong></p><p>Students can look for patterns in interior/exterior angle sums when the polygon is not concave.</p><p><strong>Small Groups: </strong>Groups can be adjusted larger to help students who need more practice, or smaller to allow proficient students to participate more.</p><p>Students who may require additional learning opportunities can be given simpler shapes (such as squares and triangles) in each of the activities so that they can understand the ideas with less effort. While students are forming groups to discuss their findings, you can check with students who may require additional practice to ensure that they understand everything.</p>