Shake It With Pythagoras


Author:

Kelly DeNu

Unit Title:

Shake It With Pythagoras

Grade:

8

Subject:

Math

Estimated Duration:

6 Days – Block Scheduling 

Unit Activities:

Background Knowledge:

It would be helpful for students to know how to solve equations and know how to square and take the square root of a number.

Date:

July 2013

The Big Idea (including global relevance)

It is known that the Egyptians used a knotted rope as an aid to constructing right angles in their buildings. The rope had 12 evenly spaced knots, which could be formed into a 3-4-5 right triangle, thus giving an angle of exactly 90 degrees. This geometric history is still important today in the design of robust buildings enabling resistance to exceptional forces including earthquakes.  

The Essential Question
Can we create buildings using the Pythagorean Theorem that will be structurally sound enough to withstand an earthquake?  
Justification for Selection of Content

Often times students see the Pythagorean Theorem and can recite the formula by heart. However, a lot of times it is difficult for students to apply the formula that they have just recited. This lesson will enable students to derive the formula through investigation and allow them to see how it is used and appreciated in real life.

The Challenge

Create a model of a 3-story building that would be structurally sound to withstand an earthquake using the Pythagorean Theorem.  

The Hook

Show students a video titled, “When Not Knowing Math Can Cost You $15,000.” It is a clip from Who Wants to Be a Millionaire?  Show part of the video until the answer is revealed.  Teach activity one (students will learn about the Pythagorean Theorem) and then pick the video up after the lesson to reveal the answer.  This way they connect with the video, look for how to find the answer throughout the lesson, and become confident at the end of the lesson when they get the problem correct and the participant does not.  

Teacher's Guiding Questions
  • What do you want to know about Pythagoras?
  • Why do you think his theorem was important?
  • Do we use the theorem in present day? Has it been modified?
  • Would you be able to explain and make sense of a “proof?”
  • What is the Pythagorean Theorem and proofs?
  • Who was Pythagoras?  
  • Can we show a proof of the Pythagorean Theorem to explain the relationship of the sides of a 90 degree right triangle?
  • Can we use the Pythagorean Theorem to help determine missing side lengths when building a building?
  • Does a building’s base have to be 90 degrees?
  • Would a building’s height effect its stability?
  • Does the frequency of an earthquake have a correlation to the structure of the building and its stability?
  • Does cross bracing (using the Pythagorean Theorem) add stability to a building?

  • Is there a correlation between an earthquake’s frequency and how much a building moves?
ACS (Real world applications; career connections; societal impact)

A - The Pythagorean Theorem can help people measure a missing side length quickly when creating or designing flower gardens, retaining walls, housing additions, furniture placement, etc.

C - The Pythagorean Theorem can be used by carpenters and engineers when designing houses and structures.

S - The basis for understanding nearly all geometry, there is virtually no venue of modern society that is unaffected by it.  Example: Finding the distance between plates at a baseball game. 

Engineering Design Process (EDP)

  • Challenge - Create a building and prove that it contains a 90 degree angle by using the Pythagorean Theorem. The building should be structurally sound enough to withstand an earthquake.  
  • Research - Use websites to see how Pythagorean Theorem is used in construction. Also determine how the ancient Egyptians used this idea to make right angles. 
  • Brainstorm and Essential Questions – Look at building codes and the ancient Egyptian rope idea to figure out how they can use that to make a structure. 
  • Build and test for stability – Test using a shake table that would simulate an earthquake.
  • Redesign – Improve upon the initial design, perhaps using bracing to see if the structure is sounder & re-test. Research the effects of bracing in buildings. 
  • Final Conclusion – Videotape their findings and be able to explain the findings.  
Academic Standards 

Math:

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Science:

Earth & Space Sciences – Physical Earth -

The composition and properties of Earth’s interior are identified by the behavior of seismic waves. 

Unit Activities

Day 1: Activity 1

  1. Pre-test students on their prior Pythagorean Theorem Knowledge (Pre-test)
  2. Present students with the “hook” video on “When Not Knowing Math Can Cost You $15,000.”
  3. Are You Right Activity?  To show students the difference between right, acute, and obtuse angles.  They should be able to generate the Pythagorean Theorem from here.  (Activity 1, Worksheet A)
  4. Pythagorean Proof Example Activity- Handout: “Showing Proof” (Activity 2) – Students should be able to “see” how the side of a squared plus b squared will equal c squared.  (Activity 2, Worksheet A)
  5. Show a computer applet example of the Pythagorean Theorem Proof and a 3-D example with this water video
  6. Homework (Activity 2, Worksheet B)

Day 2: Researching the Pythagorean Theorem (Activity 3)

  1. Do a little Background overview and formal introduction of Pythagorean Theorem and use this slide show as a guide.
  2. Taco Cart Example
  3. Research – Complete a webquest on Pythagoras to discover his background and to understand at least one of his proofs.  Students should be able to recognize and explain one proof.  (Activity 3, Worksheet A)
  4. Triangles Triples Homework (Activity 3, Worksheet B)

Day 3:  Finding the Length of Line Segments on a Grid (Activity 4)

  1. Finding the length of a line segment demonstration
  2. Treasure map activity (Activity 4, Worksheet A)
  3. Puzzle Proof Activity (Activity 4, Worksheet B)

Day 4: Challenge and Brainstorm (Activity 5)

  1. Start the Engineering Design Process
  2. Challenge and Brainstorm (Activity 5, Worksheet A)
  3. Build and Test (Activity 6)

Day 5: Redesign and Test

  1. Redesign Using Braces or other researched approach to improve stability  (Activity 6, Worksheet A)
  2. Re-Test design.  Students should videotape results to use for presentation of their findings.

Day 6: Conclusion and Reflection

  1. Conclusion/Reflection (Activity 6, Worksheet B)
  2. Prepare for presentations
  3. Presentation using videos as explanations
  4. Rubric (Activity 6, Worksheet C) 

Where the CBL and EDP appear in the Unit

CBL - Activity 5

EDP - Activity 5 & 6

Misconceptions

A lot of times students don’t believe the concepts that are taught in Math can be applicable to the real world. This unit will let students apply their measuring skills and the importance of the Pythagorean Theorem. 

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument
Results: Evidence of Growth in Student Learning

Evidence of Growth in Student Learning
How to Make This a Hierarchical Unit

In order to make this a high school unit, teachers could have the students actually research one of the many proofs of Pythagoras and have the students demonstrate that proof with manipulatives. For the treasure map, the teacher could have students compare the Pythagorean Theorem to the distance formula and discuss the pros and cons. When approaching the engineering design process, the teacher could pull more Science into the unit by discussing earthquakes more in depth and applying logarithms to find the intensity of earthquakes. Students could also research different shapes of structures to see what would withstand the earthquakes the best.

Reflection
This was by far the most engaging lesson that I have ever created. The students were all very willing and excited to participate. The only shortcomings I had with this unit was dealing with students who were not getting along with their groups, and the time it took to implement the lesson. The students were so engaged that they wanted to keep on going with the project and extend the time allotted. Next time when I do this unit, I want to add a requirement to the rubric. I want the students to actually research a real life building and I would like them to use an actual scale factor and dilate the buildings down to model size. This would also tie in common core math standard G4 (scale factor and dilations).