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 images. 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 goes 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 ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any geometric terminology they use, such as “round,” “corners,” “circular,” or “symmetric,” 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?”

Focus students’ attention on the concept of complete rotational symmetry in three dimensions. If it does not come up during the discussion, introduce the idea. Invite students to envision a vertical axis of rotation, or a line around which things rotate in three dimensions, through the center of each object. The cylindrical block, and the conical hut can rotate through any number of degrees using this axis and look identical to the original orientation. The puzzle cube must rotate a multiple of 90 degrees in order to look identical to its original positioning. If it were stacked as a cylinder, the spring toy would have an axis of rotation with symmetry for any angle of rotation. As pictured, a circle could be rotated 180 degrees to trace out the shape of the spring, but not rotated a full 360 degrees.

Math Community

After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.

Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.

After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.

In this activity, students explore solids of rotation in a hands-on way. As students visualize the connection between two- and three-dimensional objects, they are thinking abstractly and quantitatively (MP2).

In the digital version of the activity, students use an applet to check their predictions for the solids that result from rotating figures around different axes. The applet allows students to observe solids get traced out and to see the solids from different perspectives. Use the digital version if students would benefit from using a dynamic visual to see the solids traced out by a shape.

If students don’t have individual access, projecting the applet would be helpful during the Activity Synthesis.

The purpose of this activity is for students to practice visualizing the two-dimensional figure that was rotated to make a three-dimensional shape.

In this activity, students draw the two-dimensional figure that, when rotated around the given axis of rotation, would produce the given three-dimensional solid. Monitor for students who draw a silhouette of the entire figure rather than just one half of it to highlight during discussion.

In the digital version of the activity, students use an applet to create their own three-dimensional solids. The applet allows students to view a created solid from different perspectives before drawing its original two-dimensional figure. If time allows, the digital version may be preferable because students create the three-dimensional shapes themselves.