In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, grouped by the angle measure of one acute angle). For now, call this the “Right Triangle Table” so students remember that these ratios apply only to right triangles. Building these ratios from the similar triangles that they are based on helps students make sense of and build connections between similar triangles and trigonometry (MP1).

Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, doing this provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles that have an acute angle that matches the angle listed in the table will be similar and thus have the same ratios.

In the digital version of the activity, students use a spreadsheet to organize data about a triangle’s angles and side lengths collected using an applet. The applet allows students to quickly create four different right triangles to record measurements from. The digital version may be helpful for expediting the creation of the triangles and performing the quotient calculations with accuracy.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language and then restate their observation with more precise language in order to communicate more clearly. Do not use the word “trigonometry” or any of the names of the ratios yet.

The purpose of this discussion is to elicit the ideas and questions that students have about the Right Triangle Table. 

Display the class table, and distribute the blackline master with the completed table. Students will need their completed Right Triangle Table for the next several lessons. Suggest that students tape it into their workbook or staple it to their reference chart. They will need to reference it only during this unit.

Display the table and the prompts, “What do you notice? What do you wonder?” Ask students to think of at least one thing they notice and at least one thing they wonder about the tables generated in this activity. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice and wonder with their partner.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the table. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement. 

Things students may notice:

• The angles are increasing by multiples of ten.
• The values in the “opposite leg hypotenuse” column are increasing.
• The values in the first two quotient columns are all less than 1.
• The first two quotient columns are the same numbers but in reverse orders.

Things students may wonder:

• Will any triangle have these quotients?
• Do all complementary angles have the same ratios?
• What happens if the angle is in between these?
• What happens if the angle is larger than 90 degrees?

If the idea that this table describes all right triangles with the same angles does not come up during the conversation, ask students to discuss this idea.

In the previous activity, students noticed that if right triangles have the same angle measures, they have the same ratios of their side lengths. Students may also have noticed patterns in the table—for example, when two angles are complementary, the ratios in the first two columns are the same but in reverse order. 

In this activity, students look for patterns in the Right Triangle Table, including patterns that let them make predictions about other data points.

If students are struggling to make the connection between the 35-degree and 55-degree angles, prompt them to draw a right triangle with a 35-degree angle.

The goal of this activity is to continue to introduce students to patterns in the Right Triangle Table. Students need not resolve or master concepts such as the relationships between complementary angles at this point.

Invite a student who used the angles measuring 30 and 40 degrees to estimate the 35-degree angles to share their method as a conjecture. “Let’s test that hypothesis using other angles on the chart.” (On this table, if one angle measure is between two other angle measures, then the values of the associated ratios will be between the values of the ratios associated with the larger and smaller angles.)

Invite a student who used the complementary relationship between 55 degrees and 35 degrees to reason about the ratios of the adjacent side to the hypotenuse or the opposite side to the hypotenuse to share their method as a conjecture. “Let’s test that hypothesis using other angles on the chart.” ( , and for the 20-degree and 70-degree angles, the values of the ratios in the first two columns are swapped, so it seems as if the same pattern would work here.) Remind students that we define complementary angles as two angles whose measures add up to 90 degrees.

Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” in the image shown here, then the leg you are touching is the adjacent leg and the leg you are not touching is the opposite leg. The hypotenuse is always the longest side.