This is the first Which Three Go Together? routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together? Why do they go together?”

Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepens their awareness of connections across representations.

This Warm-up prompts students to compare four triangles. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and make sure the reasons given are correct.

During the discussion, prompt students to explain the meaning of any vocabulary they use, such as “right,” “equilateral,” or “scalene,” and to clarify their reasoning as needed. Consider asking:

• “How do you know ?”
• “What do you mean by ?”
• “Can you say that in another way?”

To continue investigating relationships in triangles, in this activity students create a table or revisit a table from a previous lesson and add information about angles. Students adjust an angle between two segments with given lengths to see patterns and generalize (MP8) how the side opposite the angle is affected. The goal of this activity is for students to determine that, in a triangle, larger angles are opposite longer sides and conversely, longer sides are opposite larger angles.

In the digital version of the activity, students use an applet to adjust an angle between two segments to see how the opposite side is affected. The applet allows students to more clearly see the side opposite the angle being adjusted and play with different options more dynamically. The digital version may be preferable if it is available and students would benefit from visualizing the connection.

If students don't have individual access, displaying the applet for all to see would be helpful during the Activity Synthesis.

This activity gives students another chance to convince themselves that, in a triangle, larger angles are opposite longer sides and longer sides are opposite larger angles. Starting with a common right triangle, students will review Pythagorean Theorem. Then, by adjusting one side of the right triangle, students will explore what happens to angles when two sides of a triangle stay constant and a third changes length.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).