Lesson 5

In the previous lesson’s cool-down, students studied a similar situation involving a jet airplane rather than a rocket. In the cool-down students were given a scale diagram of the situation. Students may have used estimation strategies or they may have calculated ratios of side lengths and used the right triangle table to estimate a range of possible values for the launch angle. Through this warm-up and the synthesis, encourage students to think more about how calculating ratios and looking at the right triangle table can help them make a more accurate estimate.

Ensure that students have access to their right triangle table for this activity and the remainder of the lesson.

If students are struggling to make reasonable estimates for the angle measures, refer them to their
right triangle table.

Invite a student to share their method. (I used the right triangle table and compared the ratio of sides to the ratios in the table.) Remind students to use their right triangle table as reference throughout class.

In this activity, students estimate angle measures in right triangles by calculating the ratios of side lengths and using their right triangle table to associate those ratios with angle measures. In previous activities, students were estimating angle measures in a context. In this activity, they are estimating angle measures in right triangles with whole-number side lengths.

Monitor for:

• students calculating the ratios of side lengths and comparing them to ratios in the right triangle table
• students using complementary angles to find the second angle measure once the first has been established

Ask students what the Pythagorean Theorem says. (If a right triangle has legs \(a\) and \(b\) and hypotenuse \(c\), then \(a^2+b^2=c^2\).) Ask students what the converse of the Pythagorean Theorem says. (If a triangle has sides \(a\), \(b\) and \(c\) that satisfy \(a^2+b^2=c^2\), then it is a right triangle.)

Tell students, “That means a triangle with side lengths 7, 24, and 25 is a right triangle because \(7^2+24^2 =25^2\). A set of whole numbers that satisfies the converse of the Pythagorean Theorem is called a Pythagorean triple.”

If students are struggling to make reasonable estimates for the angle measures, refer them to their right triangle table. 

Invite students who used the right triangle table to find the angle measures by comparing and ordering the ratios of side lengths to explain their reasoning. 

Ask students if they repeated the process for the other acute angle. (No, once I know one acute angle in a right triangle the other one has to be complementary. Yes, if I check those ratios too I might be able to make a better estimate.)

This activity is the first time students are expected to use trigonometric ratios rather than a specific similar triangle to find unknown side lengths in right triangles.

Monitor for students who:

• use the information from the right triangle table as constants of proportionality and work with \(y=kx\) relationships
• use the information from the right triangle table as ratios and find a scale factor that scales them up to the size of the given triangle
• use the information from the right triangle table as values, and guess and check to find a side length that results in the right value

Select students to share each strategy, as different strategies will make sense to different students depending on how they conceptualize the relationships. Students only need one strategy, but it’s important for each student to have at least one strategy they can make sense of.

Display this image for all to see:

Ask students to think of at least one thing they notice and at least one thing they wonder.

Things students may notice:

• It’s a right triangle.
• One angle is 20 degrees.
• The other acute angle is 70 degrees.
• One side is 8.5 units long.
• Two of the sides are unknown.

Things students may wonder:

• Are they going to ask us to find the lengths of the missing sides?
• Is this triangle on the table?
• Do we need more information?

If using the right triangle table to find a missing side length does not come up during the conversation, ask students to discuss this idea.

Encourage students to draw a diagram to solve the problem if one is not provided.

Invite students to share their strategies. If a strategy is not used by any student it is not necessary to mention it.

Possible strategies:

• using the information from the right triangle table as constants of proportionality and working with \(y=kx\) relationships
• using the information from the right triangle table as ratios and finding a scale factor that scales them up to the size of the given triangle
• using the information from the right triangle table as values and guessing and checking to find a side length that results in the right value