Lesson 7

Students are asked to calculate side lengths of a right triangle. They can apply their new understanding from the previous lesson and use trigonometry.

Remind students that \(\sin(20)\) is shorthand for “the length of the side opposite the 20 degree angle divided by the length of the hypotenuse for any right triangle with an acute angle of 20 degrees.” So when they ask the calculator to display \(\sin(20)\), they are asking the calculator for the ratio that is constant for all the triangles similar to this one by the Angle-Angle Triangle Similarity Theorem.

This info gap activity gives students an opportunity to determine and request the information needed to calculate side lengths of right triangles using trigonometry.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Tell students they will continue to use trigonometry to solve for side lengths of right triangles. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What information did you need to ask for?” (What letters and where they go. An acute angle and a side length.)
• “What could you figure out once you knew angle \(C\) was the right angle?” (The legs of the right triangle both have the letter \(C\) and the hypotenuse uses the other two letters)

Highlight for students that drawing a diagram and labeling it with the information provided is an important strategy. It’s too easy to forget something or mix up information without a clear diagram to work from.

Students continue to use trigonometry to calculate side lengths of right triangles. In this case they apply that skill to a real world context and engage in some error analysis during the synthesis.

Consider showing where Dubai and Philadelphia are on a map and defining masonry.

If students struggle with the city hall question prompt then to draw a diagram with the information they know. 

Display this image of the Philadelphia City Hall:

“The exact heights are 829.8 meters for the Burj Khalifa and 548 feet for City Hall. Why don’t those numbers match your calculations?” (Rounding or measurement error. The difference between \(\tan(23)\) and \(\tan(23.2)\) makes a big difference when you work with large numbers.) "Is it reasonable to assume you are accurate to the nearest tenth in this case?" (No, the provided measurements were rounded to the nearest ten meters or hundred feet so our usual rounding scheme doesn't apply.)