Similar Triangles and the Flatiron Building

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Objective

SWBAT apply similar triangles to a real-world situation as they complete a Performance Task.

Big Idea

The Flatiron Building located in New York City is one of the most iconic buildings in the country. Its unique triangular shape makes it a great subject for mathematical investigations.

Do Now

5 minutes

To begin the lesson I show the video, Similar Triangles and the Flatiron Building. After it finishes, I hand out the worksheet “Similar Triangles and the Flatiron Building Worksheet.” Students are instructed to read the description about the Flatiron Building and highlight key information. This is a short passage, which should only take students at most 5 minutes to read. 

When students finish reading, we discuss the passage using the following questions: 

  • How tall is the Flatiron Building? 
  • What is the measure of the smallest angle?  
  • How wide is the 22nd St side? 
  • Why was the Flatiron’s base built as a triangle?  
  • What is meant by the statement, “Although from the ground the base appears to be an isosceles triangle, the base of the Flatiron building is actually a right triangle?”

When viewing the building from the ground, it seems that two sides of the Flatiron building are congruent, however, they are not. Additionally, the building has one right angle.

Mini-Lesson

10 minutes

There are two options for today's mini-lesson depending on the confidence of my students. If most of students are confident in how to set up and use proportions to prove triangles are similar, they may not need much more than an introduction to the task. These students should be able to figure out what they need to do (i.e., draw a triangle over the picture of the roof and measure the sides). They can then set up a proportion using the information in the passage and use it to find the actual lengths of sides of the buildings.

Some students will be able to use the worksheet Version A, but may need some guidance in setting up the task. I instruct these students to draw a triangle over the picture of the roof of the Flatiron building and to label the vertices A, B, and C. It is helpful to draw the triangle using a pink, green or blue highlighter on top of the “Ariel View of Flatiron Building.” I have my students extend the 5th Ave side to make an angle at the corner instead of the slight curve. There is opportunity to discuss how this affects results later in the lesson.

In Version B of the worksheet, students are given a triangle on their picture and are asked questions to guide them more. Instruct them to label the vertices of the angles A, B and C. 

Activity

25 minutes

Students begin today's Activity by measuring the angles of the triangles they drew on the picture of the roof of the Flatiron Building. Angle measurements may differ due to the differences in the triangles the students drew. Most students should get 90o, ~25o, and ~65o.

Using Angle-Angle-Similarity Postulate, students can verify that the triangular roof in the picture is similar to the actual building as stated in the passage. Students then measure the length of the sides of their triangles. It is important to remind students to use centimeters in order to get a more precise measurement.  

Some students will draw and label a triangle to represent the actual roof of the Flatiron Building, however other students may be able to set up a proportion without drawing a second triangle. For those students who draw a second triangle, make sure to remind them to label the sides of the triangle with the information given in the passage. It is also helpful to label the vertices with letters and the sides of the triangle with the name of the street it is representing. The shorter leg is 22nd St, the longer leg is 5th Ave, and the hypotenuse is Broadway.  The only known actual length is that of the 22nd St side, which is 87 feet. Students should use this information to set up proportions in order to find the length of the Broadway side and the 5th Ave side of the Flatiron. 

As students are working, I circulate around the classroom, offering guidance when necessary. Some students may need more help than others.  I have the students explain the directions to each other and figure out the necessary steps together.

When students complete the task, check their answers. In the actual Flatiron Building, the Broadway side is 190 ft wide and the 5th Ave side is 173 ft wide. At this point, several things can be done. 

  • Students can redraw their triangles on the picture to find measurements, which will give a more accurate measurement for the actual lengths.
  • Students can calculate the relative error for their lengths. They can then write an explanation about how accurate their measures are.
  • Students can help other students who may not have finished. 

 

Summary

5 minutes

During the summary, I ask a few students for their lengths of the actual building. We discuss the question,

What factors may affect the accuracy of the calculated lengths?  

I hope to draw out “human error” as a response and bring in points such as:

  • The triangle that students drew in the Mini-Lesson is not truly accurate for the shape of the roof.
  • Student measurements may not have been precise.  
  • The picture may not have been a true representation of the roof of the Flatiron Building. 

At the end of the discussion, I ask one or two students to explain how they used similar triangles to calculate the actual lengths of the roof of the Flatiron Building.