• Which shape is used the most?
• What do you know about this shape?

Triangles are the most widely used shape in construction and engineering fields. In this unit we will discover properties of triangles, medians, altitudes, angle bisectors, perpendicular bisectors, mid segments, and special relationships among segments of triangles.

Students can use skills learned in this unit to:

• Use properties of special segments in triangles
• Use triangle inequalities to determine what triangles are possible
• Extend methods for justifying and proving relationships
• To solve problems in science classes and all future math courses.

Standards (NCTM and CT)

NCTM Standards

• Apply proper techniques to find measures
• Use representations to communicate mathematical ideas
• Analyze properties of 2-D shapes 

State Standards

• Geometric constructions, either with drawing tools or software, can be used to investigate relationships among plane geometric figures.
• Geometric relationships may be verified and proved using synthetic, coordinate, and transformation.

Unit Objectives

• Use properties of medians and altitudes of triangles
• Use properties of perpendicular bisectors and angle bisectors of triangles
• Use properties of angle bisectors to identify equal distances, such as the lengths of beams in roof.

Materials

Computers with LAN School software, Cabri Jr., geometers sketchpad, smart view, smart board, textbook, graphing calculator, notebook, metric rulers, scissors, glue, cardboard, graphing paper, colored pencils.

Introduction

In this research you will find out why properties of triangles are widely used in engineering and construction fields. Around 4000 years ago, the Babylonians and the Chinese were aware of the fact that a triangle with the sides of 3, 4 and 5 must be a right triangle. They used this concept to construct right angles and designed a right triangle by dividing a long string into twelve equal parts, such that one side of the triangle is three, the second side is four and the third side is five sections long.  Around 2500 BC, the Megalithic monuments in Egypt and Northern Europe comprised of right triangles with integer sides.  Pythagoras (569–475 BC), used algebraic methods to construct Pythagorean triples. In 400 BC, Plato established a method for finding Pythagorean triples which blended both algebra and geometry. Around 300 BC, in the Euclid’s Elements, the oldest existing axiomatic proof of the theorem is presented.

Assignment

• Each group of students will research properties of triangles and find out why triangles are widely used in construction, especially when building bridges.  You will prepare a presentation to share your findings with other groups. The groups then will build a bridge using different triangles and their properties.
• Use questions below to guide you in your quest for information needed to complete your projects.
• At the conclusion of this project, each group will prepare a list with recommendations for the engineering and construction companies. The final proposal with properties of triangles will include only main characteristics of triangles used in construction fields.
•  This is a group project and will be graded accordingly. Refer to the attached rubric to see how you will be graded.

1. Researching the answers to the first group of questions will give you the background information and help you identify the properties of triangles.

• What is the relationship between segments formed by medians and altitudes of a triangle?
• What is the relationship between segments formed by the medians of a triangle?
• How does the length of the segment from a vertex to the center compare with the length of the segment from the center to the midpoint of the opposite side?
• At what point do the medians of a triangle intersect? How do you know that? Explain?
• Based on your observation what can you conclude the point of concurrency?

1. Use the next group of questions to help you discover what are mid-segments, centroid, orthocenter, and circumcenter of a triangle. 

• How do you find the centroid, orthocenter, and circumcenter of a triangle and what are the relationships among these segments in a triangle?
• What is the relationship of the mid-segment of a triangle and the opposite (third) side of a triangle?
• Classify triangles based on their sides and their angles? What are properties of special right triangles?
• What are some things you want to avoid when building your bridge?
• Discuss how your bridge will be able to support weight?

Resources

 The following resources can be used to investigate the questions posed above. These resources are meant to serve as a guide for your research, and you are encouraged to explore further Web sites on your own. You do not have to limit your research for this project on answering only the questions posed above.

www.pbs.org/wgbh/nova/bridge

http://mathforum.org/alejandre/frisbie/bridge.html

http://42explore.com/bridge.htm

http://southvalleyms.net/Resources/building_bridges.htm

http://www.analyzemath.com/Geometry/triangles.html

Conclusion

You will learn why and how triangles are used in engendering and construction projects. You will discover also when triangles are used for the first time. At the end of this activity you will make a list with your findings emphasizing the special relationships that exist among segment of the triangles. Then the class will summarize main ideas and conclusions and make suggestions to the engineering and construction companies about special triangles and their properties.  

Webquest Assessment Rubric (maximum score: 100)

Resources:

Internet resources:

http://www.buzzle.com/articles/history-of-pythagorean-theorem.html

Other resources:

Mathematics Teacher, Vol. 102, No. 9, May 2009, Designing a Webquest, page 667.

1. Engage: Warm up exercises –   (5 minutes)
   • Students work to answer the questions and justify their answers.
   • Questions for students: What shapes are used to construct this bridge? What do you know about these shapes? What else do you see?

The teacher asks students to share their answers with the class. Raises questions to uncover what students know and think about properties of triangles and where these properties are used. Draw out from students responses to connect with the new lesson.

Exploration 1

Students: During the exploration stage students will find the relationship between segments formed by medians and altitudes of a triangle. They will complete the following activity and answer questions. The concept and properties of segments of triangles are widely used in a verity of engineering and construction fields. Students will discover the relationships using simple drawings and than use Geometers sketchpad or Cabri Jr. to test their findings. Students will share with the class what they have found in their groups. Students need to do some research to support their conclusions. When they complete this task they will prepare a list with recommendations for the engineering and construction companies. (20 min)

Teacher: Before starting the exploration activity, the teacher makes sure that students are in groups and ready to start this activity. When students start the activity, the teacher observes each group and asks questions to students to redirect the student’s investigation when necessary.  

What is the relationship between segments formed by the medians of a triangle?

                                                              i.      Find the balance point of a triangle.

  ii.      Construct the medians of a triangle.

Use your observations to answer these questions and complete these exercises:

1. Copy and complete the tale. Measure in millimeters.

How does the length of the segment from a vertex to origin P compare with the length of the segment from the origin P to the midpoint of the opposite side?

1. How does the length of the segment from a vertex to origin P compare with the length of the segment from the vertex to the midpoint of the opposite side? 

3. Explain/Discuss: (10 min) the teacher starts explaining this lesson by asking students the following question as a starting point of the lesson: 

What can you conclude about the line that contains a vertex and the balance point?

1.  
   • At what point do the medians of a triangle intersect? How do you know that? Explain?
   • Based on your observation what can you conclude the point of concurrency?
   • Students will solve simple exercises.

4. Evaluate/Closing the lesson (10 min)

1. Students work with a partner to solve exercises that the teacher puts on the smart board. High-achieving students will be differentiated-challenged based on their ability to complete their work.
2. The teacher answers to the questions that students might have and help students that need help. Also, the teacher takes notes about what students need more reinforcement and what they have mastered.

• Have students summarize the major points of the lesson and answer the Essential Question: How do you find the centroid of a triangle?

 Engage: Interdisciplinary Application (5 min)

Teacher asks students: what do you know about the segments on the triangle park?

Students answer questions and calculate the segments that are unknown.

Explore 2: How are the points of concurrency in a triangle? (20 min) 

Students will use geometry drawing software to investigate concurrency. They work using with a partner. They take turns when drawing and taking notes.

Teacher: Before starting the exploration activity, the teacher makes sure that students are in groups and ready to start this activity. When students start the activity, the teacher observes each group and asks questions to students to redirect the student’s investigation when necessary.  

1. Draw the perpendicular bisectors of a triangle

Draw the medians of a triangle

 Draw the altitudes of a triangle

Explain: 10 min

Teacher: Before the explanation starts the teachers ask students to explain their key findings. What kinds of triangles are formed? What special triangles have all three segments perpendicular to the base?

Evaluate/Closing the lesson (10 min)

1. Students work with a partner to an activity involving the concept of concurrency. Students will be differentiated-challenged based on their ability to complete their work.

• The teacher helps students to apply the new concept that they learned and take notes about their progress.  
• Have students summarize the major points of the lesson.

1. Explore: (20 min)

Students with a partner to complete the activity and answer the following questions:

• Draw a triangle and find the midpoint of each side.
• Draw and measure the segment joining the midpoints of these sides
• Compare the length of this segment to the length of the third side of the triangle
• Compare the slope of this segment and the third side of the triangle
• Complete this work for all three sides of the triangle.

Teacher: Introduces to students a sample of a mid-segment. Ask students to explore the special relationship between the mid-segment and the third side of the triangle. When students start working in their groups, the teacher interacts with students by asking questions and helping students to complete this task as necessary.

1. Explain/Discuss: (10 min) Teacher starts the explanation/discussion of this concept by asking students to share with the class what they found during their exploration stage. The teacher uses students input to explain the special relationship between the mid-segment and the third side of the triangle. 
2. Evaluate: (10 min) Students work with a partner to solve five exercises that teacher post on the smart board. The teacher walks around the room to check students work and help them to apply the new concept on these exercises when necessary.
3. Summarize the main points of the lesson and assign homework. (5 min) 

Extend/Elaborate/Evaluate (25 min)

• Students work in groups to solve problems that include properties of triangles learned in this unit. Students that have mastered these concepts will solve more challenging and multistep problems that requires critical thinking. This is the time for the teacher to reinforce and re-teach properties of triangles to students that might need further explanations. 
• Teacher observes and takes notes about students’ performance, asks questions, and help students as necessary.

Note: the teacher makes adjustments as needed in this stage to help students that might need extra help.

 Research (20 min)

• Students research properties of triangles. In their research students have to find when these properties are used for the first time, and how the design of the object has changed-improved over time. There is a list with questions regarding properties of triangles that students have to answer.
• Teacher assigns to each group an object that students have to research, build, and present to the class. Object to be considered: bridge, house with triangular roof, bird house, and triangular park. Students can build an object of their choice if the properties of triangles learned in this unit apply to that object.   

Note: the teacher makes adjustments as needed in the next lesson if students need more time to complete their research.

Engage:

• Students will be engaged in a discussion about the properties of triangles used in the object that each group has to build.
• Teacher asks students ask questions about special relationships of segment of triangles.

Students work in groups and start building their objects. Each group has to record dimensions of the shapes used to build the object. They have to explain to class what special triangles they have used in their objects. Also, the have to explain the relationships that exist among segment of triangles used.

Teacher: asks each group to explain how they have used their knowledge to construct their objects.