The purpose of this activity is to get students thinking about what other shapes are possible within the regular hexagon construction. This leads into the next activity about constructing an equilateral triangle.

Identify students who find various shapes to share during discussion. For example, right triangles, equilateral triangles of various sizes, rhombuses, parallelograms, rectangles, isosceles trapezoids, regular hexagons, and others. Also monitor for students whose conjectures only involve claims about distance. These are likely to be conjectures students can justify during the Activity Synthesis, using the fact that all the circles have the same radius.

The purpose of this discussion is to illustrate that many conjectures, even if they appear obvious, are difficult to justify with only the information that all the circles have the same radius. Display several student responses for all to see to emphasize the vast possibilities this simple construction allows.

Here are some questions for discussion:

• “Are all the circles the same size?” (Yes. Connecting the centers of any pair of adjacent circles shows they have the same radius. Comparing successive pairs shows that they all have the same length of radius.)
• “For which of these conjectures can we give a convincing explanation as to why it’s true?”
• “For which of these conjectures do we not know enough about yet to explain?”

The purpose of this activity is for students to compare the possible ways of constructing equilateral triangles. Look for student work that illustrates a variety of methods and sizes of triangles.

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

If students get stuck, ask them to refer to the Warm-up to see whether they can spot equilateral triangles. Then encourage them to do the compass and straightedge moves required to make those equilateral triangles.

The purpose of this discussion is to practice informal justification. Display several student responses for all to see. Ask students to explain their methods for producing equilateral triangles and justify how they know each triangle is equilateral. If no student uses tracing paper, remind students both compasses and tracing paper can be used to compare lengths. For example, the entire triangle remains the same even when rotated of a full turn (120 degrees) around the center. This means that each side can be rotated onto the other sides, and each angle can be rotated onto the other angles.

Some students may focus only on polygons whose vertices are marked on the diagram. Prompt them to look for other points of intersection that could be used as vertices.