<p>In this lesson, students investigate the properties of triangles through three different activities. It begins with a review of the basic concepts of triangles. After students acquire a solid understanding of the general elements, they will investigate particular triangle segments and their points of concurrency. The lesson also covers triangle congruency and the process involved in establishing that two triangles are congruent. At the end of this lesson, students will be able to recognize aspects of a triangle, such as the names of various types of triangles and the properties of triangle angles. During this lesson, students will:<br>- think creatively to uncover triangles in nontraditional places.&nbsp;<br>- draw a stick figure.&nbsp;<br>- draw a triangle accurately, as described in the lesson.&nbsp;<br>- use folding techniques to find special segments and their points of intersection on any type of triangle.&nbsp;<br>- determine the center of gravity of a triangle.&nbsp;<br>- create a circumscribed circle.&nbsp;<br>- draw an inscribed circle.&nbsp;<br>- to draw an equilateral triangle, use a straightedge, a compass, or a protractor.&nbsp;<br>- find the median of a side of a triangle.&nbsp;<br>- find the centroid of a triangle by taking the medians of its sides.&nbsp;<br>- determine the congruence of two triangles using SAS, SSS, AAS, and ASA.</p>

<p>- What are the key features of triangles and other polygons?</p>

<p>- Centroid: In a triangle, the point of intersection of the lines joining the median of each side and the opposite vertex.&nbsp;<br>- Center of Gravity: In geometry, the point about which the body is at equilibrium; if the object is of constant density, it is also the centroid and also known as the center of mass.<br>- Circumcenter: In a triangle, the point of intersection of the perpendicular bisectors of each side.<br>- Circumscribed Polygon: A polygon that has each of its sides tangent to the same circle.<br>- Incenter: In a triangle, the point of intersection of the three internal angle bisectors.<br>- Inscribed Polygon: A polygon, each of whose vertices lies on a given circle.<br>- Orthocenter: In a triangle, the intersection point of the lines adjoining an altitude of the triangle with its opposite vertex.</p>

<p>- paper<br>- straightedge<br>- protractor<br>- pencil<br>- ruler<br>- colored marker<br>- patty paper, four pieces per student (Patty papers are squares that are waxed on one side and are used to separate hamburgers before they are cooked.) Note: for most satisfactory results, use high-quality paper and only use pencils.<br>- compass<br>- copies of Dragon’s Eye Triangle (M-G-2-1_Dragon's Eye Triangle)<br>- copies of Triangle Visual Activity (preferably color) (M-G-2-1_Triangle Visual Activity Resource)<br>- copies of Points of Intersection in Triangles (M-G-2-1_Points of Intersection in Triangles)<br>- copies of three triangle types (M-G-2-1_Equilateral Triangle, M-G-2-1_Isosceles Triangle, and M-G-2-1_Scalene Triangle)<br>- copies of Lesson 1 Exit Ticket and KEY (key for teacher reference) (M-G-2-1_Lesson 1 Exit Ticket and KEY)<br>&nbsp;</p>

<p>- Observation during group activity is an effective approach to assess how correctly students measure and record.&nbsp;<br>- Consider how well each group shared the tasks, communicated within the group, and utilized each member's efforts during the Triangle Pose Activity group presentation.&nbsp;<br>- Lesson 1 Exit Ticket and KEY (M-G-2-1_Lesson 1 Exit Ticket and KEY) evaluates students' understanding of triangle categorization and congruence.<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson teaches students how to recognize and make representations of various triangles, identify them by their properties, detect real-world representations of triangles, and find the triangle concurrencies such as incenter, circumcenter, orthocenter, and centroid.&nbsp;<br>H: In Activity 1, students will use some lines from their bodies in various poses to portray several types of triangles. This kinesthetic activity requires an additional level of thought. To complete the triangle representations correctly, students must move, evaluate, and reposition themselves to comply to the proper alignments as prompted by their partners.&nbsp;<br>E: In the second part of Activity 1, students measure and draw triangles based on their body lines. Students will use their knowledge of angle measurement and triangle classification to the measuring and drafting processes. They will compare their results with the triangle inequality theorem to other students' drawings and measurements and the drawings and measurements of other students.&nbsp;<br>R: The properties that distinguish acute, equilateral, obtuse, scalene, right, and isosceles triangles from each other are based on their side lengths and angle measures. Each category is generalizable, and students must examine all characteristics when making classifications.&nbsp;<br>E: The exit ticket for Lesson 1 checks students' mastery of triangle classification and congruence requirements.&nbsp;<br>T: Small groups are used during this lesson. Students who need more practice with the particular sections of this lesson can discuss the principles with you while the rest of the class works independently. If paired learning is the most successful method for your students, some sections could be completed in groups.&nbsp;<br>O: Working individually and in groups allows students to study and explore new concepts independently before sharing their findings with groups (and finally the full class in the line drawing presentations).<br>&nbsp;</p>

<p><strong>Activity 1: Triangle Pose</strong></p><p>In this activity, students will learn about different types of triangles that they come across in their daily lives.</p><p>This lesson is intended as an introduction to triangles, particularly for students who are not readily inclined to mathematics. It uses tactile and creative sensibility to connect students to mathematical concepts.</p><p>Begin by asking students to provide instances of basic triangles in their daily lives. Road signs, pine trees, a piece of pie, bridge supports, and so on are all possible options.</p><p>Inform students that triangles can be discovered in the way we move and use our bodies. Models, dancers, and pop performers create angles with their movements. Show them an example of a triangle made by placing your hand on your hip, or use the examples from the Triangle Visual Activity (M-G-2-1_Triangle Visual Activity Resource).</p><p>For this exercise, students will compete in a posing competition against their friends. The idea is to develop a pose that produces as many triangles as possible.</p><p>Ask students to form matching-gender partners, "armed" with a pencil and a few sheets of paper. One student will be responsible for drawing the pose, while the other will create poses on the floor and/or the wall. Both students must participate in the triangle identification. The more complex the poses and the more creative the students are, the more triangles they will find.</p><p>Allow students 10 minutes to find their pose. After 10 minutes, students must have a stick figure drawing of the pose, as well as all triangles sketched over their stick-figure drawing using their colored marker.</p><p>After students have finished, have them present their pose to the class, displaying the drawing and distinguishing the many types of triangles (obtuse, acute, or right; scalene, isosceles, or equilateral).</p><p>The team with the most triangles in a single pose using the floor and/or the wall will win the competition.</p><p>Students should choose one triangle from their team's pose and sketch it to fill the paper, then measure the sides and angles of the triangle.</p><p>Compare the measurements and angles to students around, and explain how different poses utilizing the same appendages can result in such different triangles.</p><p>Ask students to calculate the sum of the angles in their triangle. Point out, <strong>"This will always equal 180 degrees."</strong></p><p>Have students add the two short sides of their triangle. Say, <strong>"This will be larger than the longest side of the triangle."</strong></p><p><strong>The Triangle Inequality Theorem asserts that the length of one side must be smaller than the total of the other two sides but greater than the difference between the sides.</strong></p><p>Return to the poses chosen by the students. Have them re-identify the triangle they chose. Instruct them to shorten or lengthen one side of the triangle. Resketch the pose and triangle. Remeasure the lengths of the sides and recalculate the angles of the triangle. Compare the results with the original triangle. Take note of the angle variations caused by changing the length of one side of the triangle.</p><p>Discuss in class the various properties of scalene, isosceles, equilateral, and right triangles. Encourage students to connect the attributes to what they noticed throughout the posing exercises.</p><p><strong>Alternate Activity: </strong>This activity is designed to require a lot of physical activity. The following are adjustments to reduce movement.</p><p>Depending on the general tone or attitude of the classroom, the competition part of this lesson can be removed without compromising the activity's goal.</p><p>If the class is reluctant to participate in movement, the posing section of the activity can be replaced with searching for objects in the room and identifying triangles within those objects. For examples, see the Triangle Visual Activity (M-G-2-1_Triangle Visual Activity Resource). Another option for the less mobile classroom is to look for poses in a magazine and identify triangles in those poses.</p><p><strong>Activity 2: Triangle Concurrency</strong></p><p>Note: Prior and frequent practice with patty paper will lead to better results in this lesson. Students' fine-motor abilities vary, and some will require further assistance and support. Student assistants with experience are really useful.</p><p>In this activity, students will learn about the special segments of a triangle and their points of concurrency. Distribute copies of Points of Intersection in Triangles (M-G-2-1_Points of Intersection in Triangles) for the students to take notes with.</p><p><strong>"In this activity, we'll look at some of the distinctive properties of triangles. We shall focus on the special segments of a triangle (perpendicular bisectors, medians, angle bisectors, and altitudes) and their points of congruence."</strong></p><p>Begin by distributing four sheets of patty paper to each student. Meanwhile, ask students what they know about triangles and what makes them unique. Consider asking a student to make a list of these characteristics on the board. Answers may include three sides, 180 degrees, the strongest shape, acute, obtuse, right, and so on.</p><p><strong>"We are going to learn some new and unique features of triangles."</strong></p><p>Draw an acute triangle on the board, then instruct students to design an acute triangle on one piece of patty paper that is similar to the example on the board (triangle should be large and roughly scalene). Students should use their straightedge and pencils. Keep protractors on hand so they may double-check their measurements. It is crucial that the triangle is acute.</p><p>Have students trace their triangle on the remaining three pieces of patty paper.</p><p>Demonstrate the following and have students do the same with their own pieces of patty paper.</p><p><strong>"First, find one altitude by folding the patty paper. Fold through the top vertex, keeping the bottom edge aligned with itself. Use a straightedge to draw along the fold. Repeat for the remaining two sides. Label this triangle </strong><i><strong>Altitudes.</strong></i><strong>"</strong></p><p>Ask students what they notice about the three altitudes. Explain that the three altitudes are <i>concurrent</i> and name the common point as the <i>orthocenter</i>.</p><p><strong>"Second, find one angle bisector by folding another piece of patty paper. Fold through a vertex, aligning the edges on either side of it. Use a straightedge to draw along the fold. Repeat for the remaining two vertices. Label this triangle </strong><i><strong>Angle Bisectors</strong></i><strong>."</strong></p><p><strong>"What do you notice about the three angle bisectors?"</strong> Pay attention to the use of vocabulary terms such as <i>concurrent or point </i>of<i> concurrency</i>. <strong>"Label the point </strong><i><strong>incenter</strong></i><strong>."</strong></p><p><strong>"What do you notice about the location of the incenter?"</strong> If necessary, instruct them to consider the distance from the incenter to the edge of the triangle. Assume that the incenter is equidistant from both sides of the triangle; thus, it is the center of the inscribed circle. Draw the circle with a compass.</p><p>Third, show students how to identify one perpendicular bisector on the third piece of patty paper. Fold by aligning the endpoints of the bottom edge. Use a straightedge to draw along the fold. Repeat for the remaining two sides. <strong>"Label this triangle </strong><i><strong>Perpendicular Bisectors.</strong></i><strong>"</strong></p><p>Again, ask students what they see about the three perpendicular bisectors. Students should notice that these are concurrent, with the point of concurrency equidistant from the triangle's vertices.<strong> "Label the point </strong><i><strong>circumcenter</strong></i><strong>. The circumcenter, which is equidistant from the vertices, is the center of the circumscribed circle."</strong> Draw the circle with a compass.</p><p>Finally, have students illustrate how to find one median using the last piece of patty paper. To find the midpoint of the bottom edge, line up a fold like the perpendicular bisector. Mark that point. Fold the patty paper through the midpoint and opposite vertex. Use a straightedge to draw along the fold. Repeat for the remaining two sides. <strong>"Label this triangle as </strong><i><strong>Medians</strong></i><strong>."</strong></p><p>Alternative: Use pre-cut triangles, measure to the nearest 0.1 cm, identify the midpoint after measuring, and draw the medians.</p><p>Encourage students to notice that the three medians are likewise concurrent. Ask them to label the point centroid.</p><p>Cut out the median triangle from heavier-weight paper. Show how to balance this triangle on the tip of a pencil, pen, or compass point. Ask students the name of the balancing point (centroid). <strong>"What does that say about the median of a triangle?"</strong> Encourage students to understand that the median divides a triangle in half, just as the median of a list of numbers does. <strong>"The two smaller triangles formed by dividing the main triangle will have the same area. They will share the same base (half of the median split) and altitude. As a result, both triangles will be identical."</strong></p><p><strong>Students should use the eraser on the end of a pencil to balance the cut triangles.</strong></p><p>Divide the class into groups of four students. Provide each group with a certain sort of triangle (M-G-2-1_Equilateral Triangle, M-G-2-1_Isosceles Triangle, and M-G-2-1_Scalene Triangle) and four pieces of patty paper. Each group should draw a triangle of their given type on the first piece of patty paper and trace it on the other three.</p><p>Each student in the group should select one of the four special segments (altitude, angle bisector, perpendicular bisector, or median) and find the point of concurrency of those segments on his or her triangle using the same folding method used in the acute triangle lesson.</p><p>Allow students to share with their group the points of concurrency they discovered. Encourage them to consider the following question:</p><p><strong>"How does the type of triangle affect or change the location of each point of concurrency?"</strong></p><p>Groups should take note of the following:</p><p>The altitudes/orthocenter and circumcenter lie outside the obtuse triangle.</p><p>The orthocenter is located at the right angle of a right triangle.</p><p>A right triangle's circumcenter is located at the midpoint of its longest side.</p><p>Incenter is always the center of the inscribed circle, and it is always inside the circle.</p><p>The circumcenter is always the center of a circumscribed circle.</p><p>The orthocenter, incenter, circumcenter, and centroid are all the same point in an equilateral triangle.</p><p>While students are discussing their results, move about the classroom and provide leading questions to groups who may require extra practice.</p><p>After students have explored their findings in groups, each group should report their findings to the class. Encourage other groups to ask questions to the group presenting. Have the students make a list of their discoveries to publish in the classroom.</p><p><strong>Activity 3: Dragon’s Eye</strong></p><p>Show this on an overhead projector with clear transparency. Mark and fold it to guide students. Practice before giving a demonstration.&nbsp;<br>Students should have past and frequent experience using a ruler, compass, and straightedge.</p><p><strong>"In this activity, we will look at the congruency of triangles and ways to prove that two triangles are congruent. We will also practice determining the centroid of a triangle and examine the unique qualities of equilateral triangles."</strong></p><p><strong>"The Dragon's Eye is an ancient Germanic symbol that represents strength and protection. It's an equilateral triangle divided into three identical triangles."</strong></p><p><strong>"On a piece of paper draw an equilateral triangle using your straightedge."</strong></p><p>Students can accomplish this in one of two ways:</p><p>Using a Protractor and a Straightedge</p><p>Using a Compass and a Straightedge</p><p><strong>"What will the angles of the triangle be?"</strong> (<i>60 degrees</i>)</p><p><strong>"What are the respective lengths of the sides of the triangle?"</strong> (<i>They can be any length as long as they're all the same length.</i>)</p><p><strong>"Make sure that this length is determined and written next to the triangle."</strong></p><p>Divide the classroom into four groups.</p><p>1. Instruct Group 1 to find the triangle's centroid.</p><p>2. Instruct Group 2 to find the triangle's orthocenter.</p><p>3. Instruct Group 3 to find the triangle's circumcenter.</p><p>4. Instruct Group 4 to find the triangle's center.</p><p>The task can be completed via measurements or construction, however measurement is preferred in this circumstance.</p><p><strong>"What do you notice about all of these points?"</strong> Compare the groups. (<i>They all have the same point.</i>)</p><p><strong>"Draw a line segment from the center of the triangle to each vertex. You will now have three isosceles triangles that divide the equilateral triangle."</strong></p><p>Instruct students in the previously established groups to apply congruency rules to determine whether the three triangles are equal. Each group should use a different method.</p><p>Group 1: SAS (Side, Angle, Side)</p><p>Group 2: SSS (Side-Side-Side)</p><p>Group 3: ASA (Angle-Side-Angle)</p><p>Group 4: AAS (Angle-Angle-Side)</p><p>Present and discuss each group's findings with the class. Proofs do not require formal treatment, but they should be addressed.</p><p>Distribute one Dragon's Eye Triangle example (M-G-2-1_Dragon's Eye Triangle) each student. Students should use their ruler and straightedge to find the triangle's centroid.</p><p><strong>"The line that connects the center of the triangle to each vertex divides the larger triangle into three smaller ones. Use the four previously used rules (SAS, SSS, ASA, and AAS) to determine whether or not these triangles are congruent."</strong></p><p><strong>"Report your findings to your group. Explain why the triangles are or are not congruent."</strong> Ask groups to share their findings with the rest of the class. <strong>"Which triangles are congruent and how do you know they are or are not congruent?" </strong>Make sure students use precise angle and side measurements to describe their findings.</p><p>Before leaving class, have students fill out the Lesson 1 Exit Ticket (M-G-2-1_Lesson 1 Exit Ticket and KEY).</p><p>Students can collect these tickets as they leave. You can use the exit tickets to determine which students understand congruence and the properties of equilateral triangles better than others.</p><p><strong>Extension:</strong></p><p><strong>Technology:</strong> If Geometer's Sketchpad is unavailable, a useful resource for reinforcing the centroid, circumcenter, orthocenter, and incenter constructs is provided at <a href="https://www.mathopenref.com/triangle.html">https://www.mathopenref.com/triangle.html</a>. This website has an interactive component that allows students to immediately learn how changes in just one aspect of a triangle can affect other properties of the same triangle.</p><p>Using the website (or Sketchpad), draw a triangle with altitudes, angle bisectors, perpendicular bisectors, and medians marked (use a different color for each kind of line). Instruct students to determine the orthocenter, incenter, circumcenter, and centroid. Then instruct students to move a point on the triangle to make it equilateral, isosceles, obtuse, and right. Ask students to describe, verbally or in writing, what occurs to the orthocenter, incenter, circumcenter, and centroid when the type of triangle changes. This task could be utilized for review, reinforcing students' learning, or as a make-up activity for students who missed the group lesson.</p><p><strong>Routine:</strong> As a warm-up activity, distribute squares of paper and have students fold them in half diagonally to form a right triangle. Have students measure the triangle's sides and angles. Fold to shorten one side of the triangle (not the hypotenuse) by one inch. Remeasure the angles and sides, and analyze what has changed about the triangle.</p><p><strong>Expansion:&nbsp;</strong><br><br><strong>Part 1:</strong> On the divided equilateral triangle, bisect one of the smaller scalene triangles from the center of the longest side to the opposite vertex. You'll now have two smaller right triangles. Prove that these triangles are congruent using the RHS (Right-angle-Hypotenuse-Side) rule: RHS triangles are congruent because the angles opposite congruent sides are congruent, making them congruent according to SAS or ASA.</p><p>Students can further study if the special segments are concurrent in various shapes, starting with special quadrilaterals (rhombus, square, rectangle, trapezoid, and parallelogram). They might also look at whether any points of concurrency have retained their value. (Incenter refers to the center of an inscribed circle, etc.). Add references to relevant facts that relate triangle concurrencies for each type of triangle. For example, the incenter represents places that are equidistant from all three sides.</p><p><strong>Part 2: </strong>Which triangles (equilateral, isosceles, right, and isosceles right) have the same points of concurrency for the incenter? Circumcenter? Orthocenter? Centroid? (equilateral: all)</p><p>These activities were created to provide students with the tools and practice they need to remember the concepts of triangle congruency, as well as a hands-on method to identifying distinctive segments of a triangle. Students work with a variety of triangles to determine when their generalizations can be extended to other types of triangles and when their conclusions must be modified for different types of triangles.</p>