Pythagorean Theorem


Author:

Tyler Styons

Unit Title:

Pythagorean Theorem

Grade:

8th

Subject:

Pythagorean Theorem

Estimated Duration:

8, 50 minute class periods

Unit Activities:

Acitivity 1: Introduction

Activity 2: Introduction

Activity 3: Application Shell

Activity 4: Challenege

Background Knowledge: 

  1. Students should have a basic understanding of what makes up a triangle
  2. Students should be able to solve for an unknown variable in a given equation
  3. Students should understand what is means to square a number
  4. Students should know what a square is
  5. Students should be able to find the area of a square

Date:

July 2014

The Big Idea (including global relevance)

Cell phones, radio, and television are all staples in most individual’s lives. By the time a student arrives at school, or an adult arrives at work, they have most likely encountered all three of these forms of communication in some fashion. Both cell phones and radio (television to some extent) are able to function based on their ability to communicate with an external tower. We see radio and cell phone towers often times while we are driving, but we fail to understand the importance of them being there in the first place. The most important topic to consider is this: without these towers in place, we would not be able to use such things as radio or cell phones which have become important pieces in our everyday lives. 

The Essential Question

How are cell/radio towers built in order to withstand any adverse weather or conditions?

Justification for Selection of Content

When examining student’s in my class data from the previous year, student scores have indicated that Pythagorean Theorem was an area of weakness. According to the end of the year benchmark testing, approximately 75% of my students performed in the “Basic” level on questions that involved Pythagorean Theorem. While the pre and post tests will assess student ability levels on this particular topic, data will also be collected from the end of the year benchmark along with OAA testing data. The pre/post tests will be given in tandem with the lesson while the OAA testing data will be reported in June of 2015.

The Challenge

Students will be constructing model cell phone towers from dowel rods and fishing line (guy lines). The cell phone towers will need to be able to withstand a fan blowing on it to simulate real world weather conditions. 

The Hook

When presenting the Hook to the students, they should first be asked if they have ever seen a cell/radio tower before. Connections could be made to the radio tower that is in close proximity to the school. If some students are still not making this connection, pictures could be shown of what these towers look like. Before showing the following video, ask if these structures look dangerous? What do you think people have to think about when they are making these structures?

Show the following clip: http://fox2now.com/2013/03/06/cell-tower-collapse-could-have-been-prevented/

Ask the students what the man said specifically that could have prevented the fall. (The tower needed to be tethered down in some way) Students should then be asked if they know any systems that are used to tie something down. At this point the instructor could bring up an example of a tent that uses stakes to tie down a tent.

At this point in the lesson show this video to the students so they have the proper understanding of what the function of a guy line is and what it looks like on the structure itself. https://www.youtube.com/watch?v=GP2rfNNgh48

This is what happens when things do not work properly: https://www.youtube.com/watch?v=yN_KU4lmgO0

Some key points to note when presenting this hook is to focus on the vocabulary of “guy line” along with its purpose. Students should now have a visual of what this structure looks like and have a beginning understand of how they might use a structure similar to this in order to solve their challenge. 

Teacher's Guiding Questions
  1. What are vocabulary words that we can use to describe a right triangle?
  2. How can we apply what we know about squares to develop a relationship between the sides of a right triangle?
  3. What is the Pythagorean Theorem?
  4. How can we use the Pythagorean Theorem to find the missing side length in a triangle?
  5. When can we use the Pythagorean Theorem to answer problems in the real world?
  6. What are the obstacles that engineers have when constructing a tower?
  7. How are cell phone towers connected to the ground?
  8. What is a “guy line”?
  9. What is a way we can determine the length of a guy line?
  10. How could we determine where a guy line needs to be attached on a cell phone tower?
ACS (Real world applications; career connections; societal impact)

Real World Applications:

Cellular and radio communications are highly used in modern society. These types of communication would not be possible without the constructions of towers that allow certain devices to work. These towers must be able to withstand a variety of adverse conditions including wind, rain, and snow. 

Career Connections:

Those in the civil engineering field are involved heavily with zoning rules and regulations which factor heavily in the process of constructing a cellular or radio tower. There is also a field of engineering referred to as broadcast engineering that is directly involved with the construction of these towers. The building of the tower will be addressed when the students construct a model tower in the final challenge.

Societal Impact:

Cellular and radio communications are relied upon when delivering a variety of messages in our society. These types of communication are not only used in interpersonal relationships, but are also invaluable tools in the business sector. Students through their everyday lives have a sense of the importance of this topic.

Engineering Design Process (EDP)
  1. Identify and Define: Students will identify the problem when they receive their individual height/ geographical requirement that they have to meet.
  2. Gather Information: Students will research how cell phone towers are built and the purpose of guy lines.
  3. Identify Alternatives: Each group member will be required to design blueprint of what the ramp should look like.
  4. Select Best Solution to Try: Each group must discuss the pros and cons of each design and decide which one will best meet the challenge.
  5. Implement Solution: Students will create a final design drawing that includes labeled dimensions and materials.  They will then construct the prototype.
  6. Test and Evaluate: Each group will test their solution. Then they will describe the test and results.  They will also explain whether the design was effective and provide reasons.
  7. Communicate: Each group will give a short presentation on their tower and how it performed when it was tested against the adverse wind conditions.  
Unit Academic Standard

CCSS.MATH.CONTENT.8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Unit Activities

Unit 3: Cellular and radio communication – Design a model tower that has the ability to withstand adverse weather conditions.

Lesson 1: Exploring the Pythagorean Theorem and its ability to solve problems (2, 50 minute class periods)

Lesson 1 introduces the Pythagorean Theorem at the most basic of levels and gives students the opportunity to explore the proof on the Pythagorean Theorem in a concrete manner before then extending that knowledge with practice of this theorem to solve situational problems. Throughout this lesson students are getting the basic skill of applying the Pythagorean Theorem however it should be noted that the instructor should continually refer to the challenge that was issued at the beginning of the unit. The students should be able to visualize the guy lines being a right angle. This allows for the students to see the math application as it applies to their challenge.

Activity 1:

Introduction of the Big Idea, Generating the Essential Question, Challenge, Hook (see    description in the previous section of this document, and guiding questions.  

Activity 2:

Explaining the proof of the Pythagorean Theorem in a hands on fashion (building the proof with square tiles). Before the instructor delves straight into this lesson, it might be helpful to draw on the board a model tower with guy lines. Show the students what shape this looks like. The students should be able to conclude that it is a right triangle.    

Lesson 2: Synthesizing knowledge of Pythagorean Theorem into an effective tower design (6, 50 minute class periods) 

After their work in Lesson 1, students now have a baseline understanding of what the Pythagorean Theorem is and the mechanics of how to use the theorem to solve real world problems. In this lesson, students now move more to the application of this topic and the ultimate challenge of building an effective tower.

Activity 1:                                                        

In this activity students will have to become “certified” in using the Pythagorean Theorem in real world problems. Students will be responsible for turning in three separate artifacts to demonstrate their knowledge. Certifications that the students must complete will have varying levels of difficultly to meet the varying needs of students.

Activity 2:

Design a tower that can withstand adverse weather conditions.

Where the CBL and EDP appear in the Unit

Challenge based learning appears in Lesson 1, Activity 1 when the students are presented with a challenge. Of building a cell/radio tower. Both CBL and EDP appear in Lesson 2, Activity 2 when the students are constructing their cell phone tower. The key piece is that the students should arrive at the solution to add guy lines to their tower without the instructor giving this to the students. 

Misconceptions

Students may think that you can add two legs of a triangle to get the hypotenuse

Students may confuse the legs with the hypotenuse

Students may not be able to identify which variable corresponds with the appropriate side of the triangle

Additional Resources

YouTube (See links embedded in the document)

http://teacherweb.com/GA/MarionCountyMiddleHigh/Welch8thmath/Pythagorean-Word-Problems3.pdf

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument
How to Make This a Hierarchical Unit

CCSS.MATH.CONTENT.HSG.SRT.D.11

(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

When doing this problem with high school students, instead of providing students with a set length for their guy lines, instructors could provide angle measurements for the guy lines which would result in students having to use the Law of Sines and the Law of Cosines to find the unknown measurements.