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Clarifications
Clarification 1: Instruction includes the exploration and analysis of circular objects to examine the proportional relationship between circumference and diameter and arrive at an approximation of pi (π) as the constant of proportionality.Clarification 2: Solutions may be represented in terms of pi (π) or approximately.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Circumference
- Constant of Proportionality
- Diameter
- Pi (π)
- Proportional Relationship
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 6, students solved problems involving the perimeter and area of two-dimensional figures. In grade 7, students explore the proportional relationship between circumferences and diameters of circles and develop and learn a formula to solve circumference problems. In grade 8, students will learn and use the Pythagorean Theorem to find the distance between points in the coordinate plane, and this builds the foundation for the equation of a circle in high school geometry.- Instruction includes opportunities for students to see circular or cylindrical household objects of different sizes. Students will measure the diameter and the circumference of the circle in each object to the nearest tenth of a centimeter to arrive at an approximation of pi (π) as the constant of proportionality. Students can record the values in a table and plot the points on a coordinate plane to discover the pattern that arises (MTR.5.1). Students should complete multiple trials to best support their conclusions using both
radius and diameter.
Trial #1Trial #2
- Instruction emphasizes the relationship between radius and diameter so students will easily move between the equivalent forms of the circumference formula (MTR.3.1).
- Instruction includes student understanding that circumference of a circle is the same as perimeter of any other two-dimensional figure.
- Students are expected to know approximations of pi (, or 3.14).
Common Misconceptions or Errors
- Students may invert the terms radius and diameter. To address this misconception, review parts of a circle including radii, diameters and chords.
- Students may incorrectly believe pi is a variable, rather than a constant for every circle.
- Students may confuse circumference and area. To address this misconception, help students connect circumference as perimeter of a circle.
Strategies to Support Tiered Instruction
- Teacher provides opportunities for students to measure the radius and diameter of various circles and to explore and discuss the similarities and differences between radius and diameter.
- To clarify misconceptions between the relationship of the diameter and circumference, instruction includes solving for the constant of proportionality between a given diameter and circumference of a circle and discussing the patterns that arise. Teacher provides opportunities for students to solve for the circumference of a given circle in terms of pi before replacing the value of pi with an approximation to determine the estimated circumference.
- Teacher co-constructs a graphic organizer with students containing color-coded examples of circumference, area, diameter and radius.
Instructional Tasks
Instructional Task 1 (MTR.4.1, MTR.7.1)Amy and Eunice are participating in a bike-a-thon this weekend. Amy has 29-inch road bike wheels and Eunice has 26-inch mountain bike wheels, where the bike wheel measurements are based on their diameter.
- Part A. If they choose a bike-a-thon distance of 5 miles, whose bike wheels will need to do the fewest revolutions to reach the finish line?
- Part B. How many more revolutions will the other bike need to make to reach the finish line? Explain your reasoning.
Instructional Items
Instructional Item 1Determine the circumference of the following circles.
Instructional Item 2
When baking an apple pie, a strip of aluminum foil needs to be placed around the edge of the crust until the last 20 minutes of baking so that it will not burn. If using a 9-inch diameter pie pan, how long should the strip of foil be?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Experts
Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
Tutorials
STEM Lessons - Model Eliciting Activity
In this Model Eliciting Activities, MEA, students will calculate unit rate & circumference, compare & order decimals, convert metric units, and round decimals. Bubble Burst Corporation has developed some chewing gum prototypes and has requested the students to assist in the selection of which gum prototypes will be mass produced by using both quantitative and qualitative data to rank the prototypes for Bubble Burst Corporation.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
MFAS Formative Assessments
Students are asked to write the formula for the circumference of a circle, explain what each symbol represents, and label the variables on a diagram.
Students are asked to solve a problem involving the circumference of a circle.
Students are asked to explain the relationship between the circumference and diameter of a circle in terms of pi.
Original Student Tutorials Mathematics - Grades 6-8
Explore the origins of Pi as the ratio of Circumference to diameter of a circle. In this interactive tutorial you'll work with the circumference formula to determine the circumference of a circle and work backwards to determine the diameter and radius of a circle.
Learn to solve problems involving the circumference and area of circle-shaped pools in this interactive tutorial.
Student Resources
Original Student Tutorials
Explore the origins of Pi as the ratio of Circumference to diameter of a circle. In this interactive tutorial you'll work with the circumference formula to determine the circumference of a circle and work backwards to determine the diameter and radius of a circle.
Type: Original Student Tutorial
Learn to solve problems involving the circumference and area of circle-shaped pools in this interactive tutorial.
Type: Original Student Tutorial
Perspectives Video: Expert
<p>A math teacher describes the relationship between area and circumference and gives examples in nature.</p>
Type: Perspectives Video: Expert
Problem-Solving Tasks
The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.
Type: Problem-Solving Task
In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.
Type: Problem-Solving Task
In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.
Type: Problem-Solving Task
Tutorials
This video shows how the area and circumference relate to each other and how changing the radius of a circle affects the area and circumference.
Type: Tutorial
In this video, students are shown the parts of a circle and how the radius, diameter, circumference and Pi relate to each other.
Type: Tutorial
This video shows how to find the circumference, the distance around a circle, given the area.
Type: Tutorial
Parent Resources
Perspectives Video: Expert
<p>A math teacher describes the relationship between area and circumference and gives examples in nature.</p>
Type: Perspectives Video: Expert
Problem-Solving Tasks
The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.
Type: Problem-Solving Task
In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.
Type: Problem-Solving Task
In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.
Type: Problem-Solving Task
