Lesson 4

This is the first Notice and Wonder activity in the course. Students are shown a geometric construction composed of seven circles. The prompt to students is: “What do you notice? What do you wonder?”. Students are given a few minutes to write down things they notice and things they wonder. After students have had a chance to write down their responses, ask several students to share things they noticed and things they wondered; record these for all to see. Often, the goal is to steer the conversation to wondering about something mathematical that the class is about to focus on. The purpose is to make a mathematical task accessible to all students with these two approachable questions. By thinking about them and responding, students gain entry into the context and might have their curiosity piqued.

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, naming the polygons that students could draw is the important discussion point.

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).

Display the image for all to see.  Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

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. 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, rhombi, parallelograms, rectangles, isosceles trapezoids, regular hexagons, and others. Also monitor for students whose conjectures only involve claims about distance. These are the most appropriate conjectures to justify during discussion, because all the circles have the same radius.

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.

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?”

If need be based on the pre-unit diagnostic, use the various shapes students generated to review vocabulary. 

• “What is it called when a triangle has two congruent sides?” (Isosceles triangle.)
• “What is the quadrilateral that has two parallel sides called? And what if two of its sides are congruent?” (Trapezoid. Isosceles trapezoid.)
• “What is the quadrilateral with two pairs of parallel sides called? What if all 4 of its sides are congruent?” (Parallelogram. Rhombus.)

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).

Encourage students to construct the equilateral triangles using different methods rather than repeating the same method twice.

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.

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 not mentioned by students, conjecture that, in some of these constructions, the entire construction remains the same even when rotated \(\frac13\) 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. This plants ideas about symmetry that will be explored in later lessons.