In this partner activity, students take turns sorting pictures of objects. The purpose of this activity is to build on students’ prior experience with circles and help them refine their definition of a circle. Monitor for different ways in which groups choose to categorize the objects, but especially for categories that distinguish between circles and non-circles.

Next, students compare the size of the circles to begin a discussion of what aspects of a circle can be measured. Finally, the teacher introduces the terms “diameter,” “center,” “radius,” and “circumference,” so students can identify these measurements in the pictures of the circular objects.

Because all of the objects pictured are three dimensional and circles are not, encourage students to focus on the circular (and non-circular) aspects of the objects. For example, the utility hole cover is actually a cylinder with a relatively short height, but the outline of the utility hole cover is circular. Some of the pictures could reasonably be placed into either category. For example, the outline of the pizza is not a complete circle, but the outline of a slice of pepperoni may be. The final categorization is not as important as students’ reasoning about what makes something a circle. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3).

The use of the words “smallest” and “largest” is purposely vague to encourage students to reason about the measurable things in a circle. As they work, listen for how students define “size,” the ways in which they determine the size, their estimation strategies, and any actual estimations. These will all be important in the whole-group discussion.

The goal of this discussion is to introduce terms that describe measurable aspects of circles. Invite students to share their reasoning and estimates about the relative sizes of the circles. Make sure that students articulate which aspect of the objects are circles (for example, the outline of the utility hole cover or the path of the yo-yo trick), because all of the objects are actually three dimensional.

Ask students to defend their order by estimating how big each circle is. Wait for the ambiguity of “what part of the circle are we measuring” to come up, or point out that different students are (likely) using different attributes when discussing the size of the circle. Challenge students to explain their answerswith more precision.

Next, introduce the terms “diameter,” “center,” “radius,” and “circumference” as they relate to the parts being measured. Some important points to cover include:

• Circles are one-dimensional figures that enclose a two-dimensional region.
• The size of a circle can be described using the length of its diameter—a segment with its endpoints on the circle that passes through the center. Any two diameters of the same circle have the same length. We will often use phrases like “What is the diameter of the circle?” to mean “What is the length of a diameter of the circle?”.
• The circumference is the length around the circle.
• A radius is a line segment from the center of a circle to any point on the circle. Its length is half of the diameter.

Direct students’ attention to the reference created using Collect and Display. Update the reference to include additional phrases such as “diameter,” “center,” “radius,” and “circumference.”

Consider asking students to identify pictures from the activity that draw attention to the radius, to the diameter, and to the circumference, and ask how they decided. Invite students to borrow language from the display as needed. For example:

• The radius is depicted in the wagon wheel, yo-yo trick, pivot irrigation, orange slice, dartboard, and airplane propeller.
• The diameter is depicted in the wagon wheel, dartboard, and grill.
• The circumference is depicted in all the circles but is especially prominent in the glow necklace and yo-yo trick.

Give students a minute to write down what each of the new terms means, using words or diagrams.

Finally, display the picture of the grill, utility hole cover, or bike wheel for all to see. Draw students’ attention to some of the other lines (chords) that are not a diameter or radius. Ask whether these lines depict the diameter or radius and why not.

The purpose of this activity is to continue developing the idea that we can measure different attributes of a circle and to practice using the terms “diameter,” “radius,” and “circumference.” Students reason about these attributes when three different-sized circles are described as “measuring 24 inches” and realize that the 24 inches must measure a different attribute of each of the circles. Describing specifically which part of a circle is being measured is an opportunity for students to attend to precision (MP6).

Students may think that all of the objects are the same size because they are focusing only on the “24 inches.” Ask students to describe each of the objects to make it clear they are not the same size.

The goal of this discussion is for students to describe the relationship between the radius and diameter of a circle.

Ask one or more students to share their choices for diameter, radius, or circumference as the measurement of the three circles. Prompt students to explain their reasoning until they come to an agreement.

Consider displaying this image of the bike wheel and the airplane propeller drawn to the same scale during the discussion.

Ask students:

• “What do you notice about the size of these two circles?” (The airplane propeller’s path is twice as wide as the bicycle wheel.)
• “What is the diameter of the airplane propeller? How do you know?” (48 inches, because .)
• “What is the radius of the bicycle wheel? How do you know?” (12 inches, because .)
• “How can we express the relationship between the radius and diameter of a circle with an equation?” ( or equivalent)
• “Is this relationship proportional? How do you know?” (Yes. The equation is of the form . The constant of proportionality is 2.)

The relationship between the circumference and the diameter will be addressed in future lessons.

The purpose of this activity is to reinforce students’ understanding of the terms “diameter,” “center,” and “radius” and also for students to practice using a compass. Students are asked to draw circles with a given radius or diameter, including one where they make the radius of the circle match another length they have already drawn.

Before using the compass, students first attempt to draw a circle freehand. Then, they recognize the compass as a strategic tool for drawing circles and transferring lengths (MP5). Students will apply this understanding in a later unit, when they construct a triangle given the lengths of its three sides.

If this is a student’s first time using a compass, direct instruction may be needed on how to use one. The circles that students draw may not be perfect, but as they gain more experience with a compass, they will improve. 

In the digital version of the activity, students use an applet to draw circles with a virtual compass and ruler. The applet allows students to see that a circle is made out of all the points that are the same distance from a given point. Use the digital version if physical compasses are not available. Additionally, the digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

The main goal for this discussion is to reinforce the relationship between radius and diameter. Ask selected students to share their strategies for drawing Circles C and D. Highlight the relationship that for diameter and radius after each. 

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response for Circle D by correcting errors, clarifying meaning, and adding details.

• Display this first draft:

  “Circle D should be bigger than Circle B because diameters are bigger than radii.”

  

  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.

• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, and then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

If time permits, display students’ recreations of the designs from the last question.

• Looking at the first design, ask “What is the same about all of these circles? What is different?” (They all have the same center, but different radii, diameters, and circumferences).
• Looking at the second, third, or fourth design, ask “What is the same about all of these circles? What is different?” (For the circles in each design, they all have the same radius, diameter, and circumference, but different centers.)