• Observation during group work is a good way to evaluate how accurately students are measuring and recording.

• In the group presentation for Triangle Pose Activity, consider how well each group shared the tasks, communicated within the group, and made use of the efforts of each member.

• Lesson 1 Exit Ticket (M-G-2-1_Lesson 1 Exit Ticket.doc) assesses students’ knowledge of triangle classification and triangle congruence.

Activity 1: Triangle Pose

In this activity, students will discover the presence of different types of triangles in their daily life.

This lesson is designed as an introduction to triangles particularly for students who are not readily inclined to mathematics. It appeals to kinesthetic and artistic sensibilities to connect students to the mathematic principles.

Begin by asking students for examples of basic triangles in their daily lives. Possible ideas might be road signs, pine trees, a piece of pie, bridge supports, etc.

Tell students that triangles can also be found in how we move and use our bodies. Models, dancers, and pop artists create angles with the movements they make. Show them an example of the triangle formed by putting your hand on your hip or the examples provided in the Triangle Visual Activity (M-G-2-1_Triangle Visual Activity Resource.doc). [IS.6 - All Students]

For this activity students will be challenged to a posing competition against their peers. The goal is to create a pose that will yield as many triangles as possible.

Ask students to get into matching-gender pairs “armed” with a pencil and a few pieces of paper. One student will be responsible for drawing the pose, and the other student will be responsible for creating poses using the floor and/or the wall. Both students must be involved in the triangle identification. The more complicated the poses, and the more creative the students are, the more triangles they will discover.

Give students 10 minutes to find their pose. At the end of 10 minutes they must have a stick figure drawing of the pose as well as all triangles sketched over their stick-figure drawing using their colored marker.

After students are finished, have them present their pose to the class, showing the drawing as well as identifying the types of triangles (obtuse, acute, or right; scalene, isosceles, or equilateral).

The winner of the competition will be the team that can find the most triangles in one single pose using the floor and/or the wall.

Have students choose one triangle from their team’s pose and sketch it to fill the paper, measure the sides of the triangle, and measure the angles of that triangle.

Compare the measurements and angles to students nearby, discussing how the different poses using the same appendages can create such different triangles.

Have students find the sum of the angles of their triangle. Point out, “This will always equal 180 degrees.”

Have students add together the two short sides of their triangle. Say, “This will be larger than the longest side of the triangle.”

The Triangle Inequality Theorem states that the length of a given side must be less than the sum of the other two sides, but greater than the difference between the sides.

Go back to the poses chosen by students. Have them re-identify the triangle that they have chosen. Instruct them to shorten or lengthen one side of that triangle. Resketch that pose and triangle. Remeasure the length of the sides and recalculate the angles of that triangle. Compare the results to the original triangle. Note the changes in the angle due to the change in length of one side of the triangle.

Discuss as a class the different properties of scalene, isosceles, equilateral, and right triangles. Have students connect the properties to what they noticed with the posing exercises.

Alternate Activity: This activity is intended to involve a lot of physical activity. The following are modifications to include less movement.

Depending on the general tone or atmosphere of the classroom, the competition aspect of this lesson can be omitted without sacrificing the intent of the activity.

Also, if the class is reluctant to participate in movement, the posing portion of the activity can be substituted with searching for objects in the room and finding triangles within those objects. See examples in the Triangle Visual Activity (M-G-2-1_Triangle Visual Activity Resource.doc). Another option for the less-mobile classroom would be to find poses in a magazine and locate triangles in those poses.

Activity 2: Triangle Concurrency

Note: Prior and frequent practice with patty paper will result in more successful outcomes for this lesson. Students’ fine-motor skills vary and some will need extra help and support. Student helpers with experience are very useful.

In this activity, students will be introduced to the special segments of a triangle and their points of concurrency. Distribute copies of Points of Intersection in Triangles for the students to use as notes (M-G-2-1_Points of Intersection in Triangles.doc).

“In this activity, we will explore some of the unique properties of triangles. Specifically, we will look at the special segments of a triangle (perpendicular bisectors, medians, angle bisectors, and altitudes) and their points of concurrency.” [IS.7 - All Students]

Begin by giving four sheets of patty paper to each student. Meanwhile, ask students what they know about triangles and what makes triangles special. Consider having a student make a list of these traits on the board. Answers may include things like: three sides, 180 degrees, strongest shape, acute, obtuse, right, etc.

“We are going to learn some new and unique features of triangles.”

Draw an acute triangle on the board and then have students draw an acute triangle on one piece of patty paper similar to the example on the board (triangle should be large and roughly scalene). Students should use their straightedge and pencils. Have protractors available so they can check their measurements. It is important that the triangle is acute.

Have students trace their triangle onto the other three pieces of patty paper.

Demonstrate the following, having students do the same using their own pieces of patty paper:

“First, find one altitude by folding the patty paper. Fold through the top vertex, with the bottom edge aligned with itself. Use the straightedge to draw along the fold. Repeat for the other two sides. Label this triangle Altitudes.”

Ask students what they notice about the three altitudes. Explain that the three altitudes are concurrent and label their common point orthocenter.

“Second, find one angle bisector, again by folding a different piece of patty paper. Fold through a vertex, lining up the edges on either side of the vertex. Use the straightedge to draw along the fold. Repeat for the other two vertices. Label this triangle Angle Bisectors.”

“What do you notice about the three angle bisectors?” Listen for the use of the vocabulary words concurrent or point of concurrency. “Label the point incenter.”

“What do you notice about the location of the incenter?” Guide students to consider the distance from the incenter to the edge of the triangle if needed. Conclude that the incenter is equidistant from each side of the triangle; therefore, it is the center of the inscribed circle. Draw that circle with a compass.

Third, guide students to find one perpendicular bisector on the third piece of patty paper. Fold by lining up the endpoints of the bottom edge. Use the straightedge to draw along the fold. Repeat for the other two sides. “Label this triangle Perpendicular Bisectors.”

Again ask students what they notice about the three perpendicular bisectors. Students should observe that these are concurrent and that the point of concurrency is equidistant from the vertices of the triangle. “Label the point circumcenter. Since the circumcenter is equidistant from the vertices, it is the center of the circumscribed circle.” Draw that circle with a compass.

Last, have students demonstrate how to find one median on the last piece of patty paper. Line up a fold in the same way as the perpendicular bisector to find the midpoint of the bottom edge. Mark that point. Fold the patty paper through the midpoint and the opposite vertex. Use the straightedge to draw along the fold. Repeat for the other two sides. “Label this triangle as Medians.”

Alternative: Use pre-cut triangles, measure to nearest 0.1 cm, find midpoint after measuring, and draw the medians.

Lead students to observe that the three medians are also concurrent. Have them label the point centroid.

Cut out the median triangle (on heavier-weight paper). Show how this triangle can be balanced on the tip of a pencil/pen/compass point. Ask students the name of the balancing point (centroid). “What does that say about the median of a triangle?” Lead students to realize that the median divides a triangle in half, similar to the median of a list of numbers. “The two smaller triangles that the large triangle is divided into will have the same area. They will have the same base (half of the side the median split), and they share an altitude. Therefore, will be identical for both triangles.”

Have students use the eraser on the end of a pencil to balance the cut triangle.

Arrange the class into groups of four students. Give each group a specific type of triangle (M-G-2-1_Equilateral Triangle.doc, M-G-2-1_Isosceles Triangle.doc, and M-G-2-1_Scalene Triangle.doc) and four pieces of patty paper. Have each group draw a triangle of their assigned type on the first piece of patty paper and trace the same triangle on the remaining three pieces.

Each student in the group should choose one of the four special segments (altitude, angle bisector, perpendicular bisector, or median) and find the point of concurrency of those segments on his/her triangle, using the same folding process that was used as a class on the acute triangles.

Give students time to share with their group the points of concurrency they found. Encourage them to discuss the following question:

“How does the type of triangle affect or change the location of each point of concurrency?”

Some things groups should notice:

• Altitudes/orthocenter and circumcenter are outside the obtuse triangle.

• Orthocenter is at the right angle of a right triangle.

• Circumcenter is at the midpoint of the longest side of a right triangle.

• Incenter is always the center of the inscribed circle and is always inside the circle.

• Circumcenter is always the center of the circumscribed circle.

• Orthocenter, incenter, circumcenter, centroid are the same point for an equilateral triangle.

As students are discussing their results, move about the classroom, posing leading questions to groups that might need additional practice.

After students have discussed their findings within their groups, have each group present their findings to the class. Encourage other groups to ask questions of the group presenting. Have the class create a list of their findings to post in the classroom.

Activity 3: Dragon’s Eye [IS.8 - All Students]

Demonstrate this on an overhead projector with a clear transparency. Mark and fold it to lead students. Practice before demonstrating.
Students should have prior and frequent practice with a ruler, compass, and straightedge. [IS.9 - All Students]

“In this activity we will explore the congruency of triangles and ways to prove that two triangles are congruent. We will also practice finding the centroid of a triangle and look at the special properties of equilateral triangles.”

“The Dragon’s Eye is an ancient Germanic symbol that stands for strength and protection. It is an equilateral triangle divided into three identical triangles.”

“On a piece of paper draw an equilateral triangle using your straightedge.”

Students can do this in one of two ways:

• using a protractor and a straightedge

• using a compass and a straightedge

“What will the angles of the triangle be?” (60 degrees)

“What are the respective lengths of the sides of the triangle?” (They can be any length as long as they are all the same length.)

“Make sure that this length is determined and written next to the triangle.”

Divide the class into four groups.

1. Instruct Group 1 to find the centroid of the triangle.

2. Instruct Group 2 to find the orthocenter of the triangle.

3. Instruct Group 3 to find the circumcenter of the triangle.

4. Instruct Group 4 to find the incenter of the triangle.

The task can be done by using measurements or construction, though measurement is preferred in this case.

“What do you notice about all of these points?” Compare the groups. (They are all the same point.)

“Draw a line segment connecting the center of the triangle with each vertex. You will now have three isosceles triangles dividing the equilateral triangle.”

In the same groups previously established, instruct students to use the rules of congruency to determine whether the three triangles are equal to each other. Each group should use a different method.

Group 1: SAS (Side-Angle-Side)

Group 2: SSS (Side-Side-Side)

Group 3: ASA (Angle-Side-Angle)

Group 4: AAS (Angle-Angle-Side)

Present and discuss the findings of each group with the class. Proofs need not be formally treated, but should be addressed.

Distribute Dragon’s Eye Triangle samples (M-G-2-1_Dragon's Eye Triangle.doc), one to each student. Have students find the centroid of the triangle using their ruler and straightedge.

“The line connecting the center of the triangle to each vertex divides the larger triangle into three small ones. Prove that these triangles are or are not congruent using the four rules previously used (SAS, SSS, ASA, AAS).”

“Present your findings to your group. Discuss why the triangles are or are not congruent.” Ask groups to present their findings to the rest of the class. “Which triangles are congruent and how do you know they are or are not congruent?” Make sure students are using specific angle and side measurements to describe their findings.

Before leaving class, have students complete the Lesson 1 Exit Ticket (M-G-2-1_Lesson 1 Exit Ticket.doc).

These tickets can be collected as students leave. You can review student performance on the exit tickets to gain a sense of which students have a greater or lesser grasp of congruence and the properties of equilateral triangles.

Extension:

• Technology: If Geometer’s Sketchpad is not available, a good resource for reinforcing the centroid, circumcenter, orthocenter, incenter constructs can be found at https://www.mathopenref.com/triangle.html. This Web site offers an interactive component so students can quickly see how changes in just one aspect of a triangle can affect different properties of that same triangle.

  Using the Web site (or the Sketchpad), create a triangle with altitudes, angle bisectors, perpendicular bisectors, and medians marked (use a different color for each kind of line). Ask students to identify the orthocenter, incenter, circumcenter, and centroid. Then have students move a point on the triangle to make it equilateral, then isosceles, obtuse, and right. Have students describe, in verbal or written form, what happens to the orthocenter, incenter, circumcenter, and centroid each time the type of triangle changes. This activity could be used as review, to reinforce students’ understanding, or as a make-up activity for students who miss the group lesson.

• Routine: As a warm-up activity, pass out squares of paper and have students fold them in half diagonally to create a right triangle. Have students measure the sides and determine the angles of the triangle. By folding, shorten one side of the triangle (not the hypotenuse) by one inch. Remeasure the angles and sides and discuss what changed about the triangle.

• Expansion: Part 1: On the divided equilateral triangle, bisect one of the smaller scalene triangles from the center of the longest side to the opposite vertex. You will 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 angles opposite congruent sides are congruent, which makes them congruent by SAS or ASA.

  You can also have students investigate whether the special segments remain concurrent for other shapes, beginning with special quadrilaterals (rhombus, square, rectangle, trapezoid, and parallelogram). They can also investigate whether any points of concurrency retain their significance. (Incenter is the center of the inscribed circle, etc.) Add references to relevant facts that relate triangle concurrencies to each type of triangle. For example, the incenter shows points equidistant from all three sides.

• Expansion: Part 2: For which triangles (equilateral, isosceles, right, isosceles right) are the points of concurrency the same for the incenter? Circumcenter? Orthocenter? Centroid? (equilateral: all)

• These activities were designed to give students the tools and practice to retain the principles of congruency of triangles and also to give them a hands-on approach to finding special segments of a triangle. Students practice with many types of triangles so they can see when their generalizations can be applied to other types of triangles and when their conclusions need to be altered for other types of triangles.