This is the first Notice and Wonder activity in the course. Students are shown a geometric construction composed of seven circles and asked, “What do you notice? What do you wonder?”

Students are given a few minutes to write down what they notice and wonder about the construction and then time to share their thoughts. Their responses are recorded for all to see. Often, the goal is to elicit observations and curiosities about a mathematical idea students are about to explore. Pondering the two open questions allows students to build interest about and gain entry into an upcoming task.

The purpose of this Warm-up is to elicit the idea that many shapes can emerge within the regular hexagon construction, which will be useful when students reason about other shapes and construct equilateral triangles in a later activity. While students may notice and wonder many things about this image, the different polygons that students could draw using the given image is the important discussion point. (See following activities in this lesson for examples of these polygons.)

This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If connecting intersection points to make polygons does not come up during the conversation, ask students to discuss this idea.

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 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).