Umbrella Frame


Author:

Thomas Haas

Unit Title:

Umbrella Frame

Grade:

9-10

Subject:

Geometry

Estimated Duration:

8-9 (45 min) classes

Unit Activities:

Activity 1: Constructing congruent lines and angles 

Activity 2: Constructing triangles 

Activity 3: Testing each shortcut 

Activity 4: Build an Umbrella 

Background Knowledge:

Students must have a background knowledge of the interior sum of angles in a triangle. They must also have a general knowledge of how a compass works. 

Date:

July 2013

The Big Idea (including global relevance)

Whether being outdoors is a desire or a must, it is always nice to have some kind of portable protection from the elements when they turn harsh. Some form of the umbrella has been dated back to Egyptian days or arguably earlier. While the early umbrellas were designed for sun, they have been modified to give protection from the sun, rain, and even wind in some cases. 

The Essential Question
What kind of durable, portable protection could be built to protect people from the sun and rain?
Justification for Selection of Content

Students have a rough time working with the theorems and postulates that prove triangles congruent. This unit is built to help students see the real life applications in which congruent triangles are prevalent. Students will be pre-tested to assess their knowledge of congruent triangles as well as their ability to construct congruent triangles with only a compass and straight edge. After the constructions, students will be formatively assessed on constructing triangles from given angles and sides. 

The Challenge
Design and build a fully functional umbrella to meet the following requirements:
  • Shade the greatest area
  • Protect from heavy rain
  • Withstand high speed wind

The Hook
Show a video on how useful it is to have portable protection from the elements. Fans at a Rangers game get hit hard by rain and wind. 
Teacher's Guiding Questions
  1. How is the umbrella frame constructed?
  2. What do you notice about the frame of the umbrella that would connect to our learning of congruent triangles?
  3. What are the constraints for the challenge?
  4. How many stretchers should be used to support the canopy?
  5. What is the maximum angle at which the ribs should form with the shaft when the umbrella is fully opened?
  6. How can you show on your design that the umbrella is supported by congruent triangles?
  7. How can you maximize the shaded area?
  8. How can you prevent the umbrella from folding under wind/rain?
  9. How can you keep the umbrella from flipping up when high wind comes underneath it?

ACS (Real world applications; career connections; societal impact)

A- This unit is relevant to the development and design of the umbrella. Students will need to realize how the umbrella opens by forming congruent triangles with the stretchers and the shaft. This is addressed in Activity 4.

C- By learning this unit, students will have options opened in architecture, surveying, astronomers, civil engineering, and mechanical engineers to name a few. Understanding congruent triangles plays a role in some part of each of the previous careers listed. This is addressed in all of Activity 3 & 4.

S- Umbrellas are still being redesigned today to fit more needs and to become more durable against heavy rains or high winds. An example of this is the stealth umbrella which was created to withstand high winds without flipping inside out. This is addressed in Activity 4.

Engineering Design Process (EDP)
The EDP is found in Lesson 2, Activity 2 when the students research umbrellas and start designing them to appeal to consumers with the ideal of making a large profit margin. The idea is to develop an umbrella that will stand out to consumers and be cost efficient in making it. All students will have the same material available to be used for the frame and covering. Students will take their research and brainstorm ideas to design their umbrella with the constraints of supplies and cost. A price sheet will be given to them so they can calculate the cost of their umbrella. Students will then design and build their umbrella. They will run tests to make sure it can withstand rain and heavy winds. They will be allowed time to redesign to ensure that their product is the most cost efficient as well as durable against the elements and desirable to the consumer. A final product along with design sheets of the umbrella in four different positions (closed, one-third open, two-thirds open, and fully opened) will be submitted. 
Academic Standards 

Activity 1:

G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Activity 2:

G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Activity 3:

G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Activity 4:

G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects

G-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost)

Unit Activities

Construction

Students will learn how to construct congruent line segments and angles through the use of a straight edge and compass. They will also learn to verify two segments/angles as congruent or not by checking the construction of the pair of segments/angles. They will then combine the construction of angles and segments to construct congruent triangles.

Activity 1: Constructing congruent lines and angles

  • Rules for congruent lines (3.1.1a)
  • Rules for congruent angles (3.1.1b)

Activity 2: Constructing triangles

  • Construct congruent triangles from given triangles (3.1.2a)
  • Define triangles as congruent or not based on measured parts (3.1.2b)

Congruent Triangles

Students will use their learned skills in construction to test triangle postulates in order to discover shortcuts to proving two triangles congruent. They will be split into groups and everyone in the group will individually construct a triangle with given information onto wax paper. They will then compare each of their triangles to everyone else in the group to make an argument of the truth behind the postulate they were studying. After studying triangles, they will apply their knowledge to congruent triangles seen in real life through designing and constructing an umbrella. The frame of the umbrella has congruent triangles with the shaft and stretchers. Students will build an umbrella that should be functional, cost efficient, and desirable to the Cincinnati market.

Activity 3: Testing each shortcut

  • Study SSS and AAA as possible postulates (3.1.3a)
    • Homework (3.1.3b)
  • Study SAS and SSA as possible postulates(3.1.3c)
    • Homework (3.1.3d)
  • Study ASA and SAA as possible postulates (3.1.3e)
    • Homework (3.1.3f)

Activity 4: Build an Umbrella

  • Challenge expectations (3.1.4a)
  • Material sheet (3.1.4b)
  • Design template (3.1.4c)

Where the CBL and EDP appear in the Unit

CBL appears in Activity 3 when the students are using their methods of construction to check different shortcuts to discovering congruent triangles.

EDP appears in Activity 4 when students define their challenge of building an umbrella with constraints of materials and making it appealing to buyers as well as being able to function against the elements. 

Misconceptions

It is a common misconception that SSA works to be a correct postulate, but it does not always make congruent triangles. It might be necessary to show students this video if they have this misconception.

Students may also have a hard time understanding the word ‘inclusion’ when it comes to noticing the difference between SAS and SSA. While SAS works to prove congruence, the included angle makes the difference to ensure congruent triangles. It is not advised to allow students to make right triangles to prove any of these cases as right triangles have their own special case to prove congruence. 

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

The chart shows the growth of students’ content knowledge as a result of completing this unit.  Progress was significant for all students

How to Make This a Hierarchical Unit

Based on the Academic Standards identified earlier in the document:

  • What Academic Standards would be used if this unit were taught at a lower grade level?
    • Students in the 8th grade begin to learn about congruent figures through transformations.
  • Describe what activities might accompany the lower grade level unit.
    • Students could learn to construct congruent lines and angles and test their theories of two congruent figures in the plane by reconstructing the same angle and segment. This is done in Activities 1 and 2.