EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

Clarifying Lessons

Grade 6: Exploring c/d = π

activity under revision

Lesson Overview

Students measure circular objects to collect data to investigate the relationship between the circumference of a circle and its diameter.

Mathematics Overview

Students use measurement procedures to collect data, organize the data in a table, graph the data on a coordinate graph, and use fractions and decimals to estimate the constant of proportionality, π, that describes the relationship between the circumference and diameter of the circle.

Set-up (to set the stage and motivate the students to participate)

  1. Students measure the perimeters of various objects around the room.
  2. Students discuss their findings. The teacher asks if students had difficulty measuring the perimeters of any objects or shapes (circles, for instance).
  3. Students brainstorm strategies for finding the "perimeter" of a circle (For example, use string, roll the object on a tape measure, etc.).
  4. The teacher informs students that the distance around a circle is called the circumference.
  5. Students brainstorm a list of circular objects (pizza, cookie, plate, etc.), and decide what would be the most fair way to divide a circle. For example, what is the most fair way to slice pizza? (Through the center)
  6. The teacher identifies the cut through the center of a circle as the diameter.
  7. In small groups, students cut a length of string approximately equal to the diameter of a circle found in the room. The teacher asks how many strings of this length it would take to go around the circumference of the circle. Students test their predictions and record their results. Each group uses string in this way to compare the diameter and circumference of several circles around the room.
  8. Students share and discuss their results with the whole class. The teacher emphasizes the consistency there seems to be in the relationship of circumference to diameter (i.e., the circumference of a circle seems to be a little longer than three of its diameters).
  9. Students work in pairs to use the rulers and tape measures to make more precise measurements (in both customary and metric units) to compute the ratio of circumference to diameter using a calculator. They organize their data in a table like the one below.

    table example

  10. Students graph the ordered pair using a classroom-sized coordinate graph. The coordinate graph will have diameter labeled along the horizontal axis and circumference along the vertical axis. Each group graphs a different coordinate pair of an object they measured.
  11. Students share and discuss the data in their charts and graph. The table should show the ratio they explored, c/d, actually has a constant value called pi that is a little greater than 3. The teacher gives its symbol, π.
  12. As a class, students discuss what patterns they see on the graph. For example, students might notice that the graph shows an increase that is constant. Students may not use this vocabulary; for example, students might say the graph looks like it's increasing by the same amount, it looks like a line can be formed if you were to join the points, etc. The teacher probes further to discuss that the amount of growth is the constant value π.

Teacher Notes (to personalize the lesson for your classroom)

Guiding Questions (to engage students in mathematical thinking during the lesson)

  • What is the perimeter of an object?
  • What is the circumference of a circle? The diameter?
  • How many diameters did you predict it would take to go all around the circle? How did you decide on your prediction?
  • Did it take more diameters to go around a larger circle? Or about the same number? Why do you think that happened?
  • What patterns are you seeing between the circumference and the diameter of a circle? How can you use fractions and decimals to describe it?
  • Does this relationship change or stay the same when you use a different unit to measure and compare the circumference and diameter of a circle? Why do you think that happens?
  • What relationship do you see in the graph?
  • What was graphed?
  • How can you describe the graph?
  • What relationship do you notice between the graph and the data in your table?

Teacher Notes (to personalize the lesson for your classroom)

Summary Questions (to direct students' attention to the key mathematics in the lesson)

To focus students' attention on the important relationship between the circumference of a circle and its diameter, ask questions such as

  • How is circumference of circle like the perimeter of a rectangle?
  • How is it different?
  • How does the circumference of any circle compare to its diameter?
  • What is the value that describes this proportional relationship called pi, or π? What is the numerical estimate of this constant of proportionality?
  • Does the size of the circle affect the relationship between the circumference and diameter? Why or why not?
  • Does the unit of measure affect the relationship between the circumference and diameter of a circle? Why or why not?
  • Why do you think this relationship exists? (It has to do with the fact that all circles are similar to all other circles, and similar figures have proportional relationships between corresponding parts.)
  • If you know the diameter of a circle, how can you figure out its circumference?
  • If you know the circumference of a circle, how can you figure out its diameter?
  • Based on the graph, would you be able to predict the circumference given the diameter? The diameter given the circumference?
  • Based on the graph, why do you think a relationship exists?

To highlight the use of ratios, fractions, and decimals to form equations that describe the relationship, ask questions such as

  • How did you use fractions as ratios to describe the proportional relationship you found between the circumference and the diameter of a circle?
  • How did you use decimals to describe the proportional relationship between the circumference and the diameter of a circle?
  • How can you use an equation to describe the relationship between the circumference and diameter of a circle? (c/d = π, so c = π x d).
  • How can you use this equation to determine the circumference if you know the diameter? To determine the diameter if you know the circumference?
  • Did you use a calculator in working on the problem? If so, how?

Teacher Notes (to personalize the lesson for your classroom)

Assessment Task(s) (to identify the mathematics students have learned in the lesson)

  • Students write in their journals about what they discovered about the circumference and diameter of a circle. The teacher should encourage the use of multiple representations, such as paragraphs, charts, drawings, equations, etc.
  • Given the measure of a circle's diameter, students predict the circumference; or given the measurement of a circle's circumference, students predict the diameter.

Teacher Notes (to personalize the lesson for your classroom)

Extension(s) (to lead students to connect the mathematics learned to other situations, both within and outside the classroom)

  • Students can investigate the various approximations used for π.
  • Students can graph the data from their table.

Students can explore the relationship between the perimeter of a square and the length of its diagonal (or the perimeter of a regular hexagon and the length of its diagonal, etc.).

Teacher Notes (to personalize the lesson for your classroom)