America's Finest City Flag
Did you know the City of San Diego has an official city flag? (pictured right)
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According to the 5 Principles of Flag Design, it's... well...
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What do you think? Does it inspire civic pride? What symbolism would you use to represent our city?
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In this project, students used ratios, proportion, similarity to design a new flag for the city of San Diego. They researched San Diego's history, geography, and culture, and how shape and proportion are used in flag symbology. Students also used trigonometry to find the height of the large flagpole in Liberty Station, where their design might fly one day!
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Essential Questions:​​
How are ratios, proportions, similarity, and trigonometry used in design and displays of flags?
Students use a laser rangefinder to measure the linear distance and the angle of elevation to the top of the flagpole.
Student flag design. The blue represents the Pacific Ocean and the Coronado Bridge; the gold references the Spanish flag and San Diego's history as a Spanish colony, as well as the 1915 World's Fair at Balboa Park; the green references the Mexican flag and the city's historical status as a Mexican city between 1821 and 1848; and the purple represents the jacaranda, the city's much-beloved official flower. The sun represents the sunny climate and beaches that attract 29 million visitors every year.
Portion of a student's final unit problem poster.
Student flag design. The sun represents the sun and weather in San Diego, while the center smaller circle with the four colors represents the Kumeyaay people; these colors are on the Kumeyaay emblem that can be seen in a mural in Chicano park.
Students sort pictures and definitions to review vocabulary around geometric transformations.
PBL Design Notes
This project ran throughout our unit on Similarity and Trigonometry. In the first phase of the project, students explored similarity, ratio, and proportion in various flag constructions and developed criteria for two figures, especially triangles, to be similar. As a collaboration with our Humanities class, students brainstormed in groups, researched in pairs, and took a field trip to the San Diego History Center to identify which themes they wanted to feature in their flag, and then puzzled through the geometric representation of their ideas in flag form. Designs underwent multiple phases of peer critique before a final presentation and class vote for the best flag design. The flag with the most votes was printed 3 ft x 5 ft and hung in the classroom for the rest of the semester. In the second phase of the project, students used learned about trigonometric ratios from the lens of flagpoles and flag designs, and they used trigonometry to find the height of the big flagpole near our school.
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If I were to do this project again, I would want to raise the stakes with a public exhibition and a proposal to the City of San Diego to adopt the new flag. I would also curate more resources for students and scaffold the research portion of the project--many of the flags were very generic "blue for ocean, yellow for sand" designs that could have applied to any beach city. For the Unit Final Problem that capped off the project, where students found the height of the flagpole, I would consider using different methods and instruments. In the first run of the project, the class shared a laser rangefinder which output the angle of elevation, the linear distance to the top of the flagpole, and the horizontal distance to the flagpole. For a future iteration, I would consider having the students build clinometers, then use the clinometer with a tape measure to find the height. I would also consider having one student stand next to the flag pole and another student stand at a distance with a ruler to find the height that way. In their final unit problem poster, students could discuss the results of the different methods and which they think is the most accurate/practical.
Content Standards
HSG-SRT.A.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain
using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
HSG-SRT.A.3
Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.
HSG-SRT.A.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
HSG-SRT.A.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions
of trigonometric ratios for acute angles.
HSG-SRT.A.7
Explain and use the relationship between the sine and cosine of complementary angles.
HSG-SRT.A.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.