What Happens When We Go Around a Circle?
Investigation Question(s)
• Is there a relationship (which can be developed into a mathematical formula) between the diameter, circumference, and radius of circles?
• Does the size of the circle make any difference in this relationship?
• Can we find a formula that can be the universal tool for explaining this relationship?
Investigation Goals
• What does represent?
• Why can we use certain formulas when finding specific geometric measurements?
MA State Frameworks
• 8.P.4 Create and use symbolic expressions and relate them to verbal, tabular, and graphical representations.
• 8.P.8 Explain and analyze—both quantitatively and qualitatively, using pictures, graphs, charts, or equations—how a change in one variable results in a change in another variable in functional relationships, e.g., C = d, A = r2 (A as a function of r), Arectangle = lw (Arectangle as a function of l and w).
• 8.P.9 Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate.
• 8.M.1 Select, convert (within the same system of measurement), and use appropriate units of measurement or scale.
• 8.M.3 Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.
• 8.D.1 Describe the characteristics and limitations of a data sample. Identify different ways of selecting a sample, e.g., convenience sampling, responses to a survey, random sampling.
Overview & Approximate Time
• This lesson will be part of a Geometry Unit.
• This lesson should take about 3-5 days to complete and analyze.
(This is based on hour-long blocks)
Data & Analysis Tools
• Image J
• Excel
Steps for Data Analysis
• Students will explore the world of circles in their everyday life. Using digital cameras we will search for circles in our surroundings. Each group will be responsible for finding at least 5 circles of various sizes. It is most important that students look for very large and small circles. The more evidence we have the more support we have for belief that there does exist a unique relationship between the radius of a circle and the circumference.
• Next students will open these pictures in Image J. Using the tools they will measure both the circumference and the diameter of the circles they are working on.
• Students will copy the information from Image J analysis and copy/paste the information into an Excel Spread sheet.
• The Excel spread sheet needs 3 columns-
o Column 1-circumference
o Column 2-diameter
o Column 3 formula to be applied. COL 1/COL 2
• Close attention will be focused on the results of COL 3.
• Results for each student will be placed on an overhead that will be created from an Excel spreadsheet.
•
Background Information
• Since fourth or fifth grade students have worked with the formulas for finding the circumference, diameter, radius, and area of circles. They have used the Greek symbol .
• Students have been told that is an irrational number. They are usually instructed to use either 3.14 or 22/7 for a value when evaluating problems using .
Sample Product
Student handouts for data collection & analysis (if provided); may be electronic
Investigation Question(s)
• Is there a relationship (which can be developed into a mathematical formula) between the diameter, circumference, and radius of circles?
• Does the size of the circle make any difference in this relationship?
• Can we find a formula that can be the universal tool for explaining this relationship?
Investigation Goals
• What does represent?
• Why can we use certain formulas when finding specific geometric measurements?
MA State Frameworks
• 8.P.4 Create and use symbolic expressions and relate them to verbal, tabular, and graphical representations.
• 8.P.8 Explain and analyze—both quantitatively and qualitatively, using pictures, graphs, charts, or equations—how a change in one variable results in a change in another variable in functional relationships, e.g., C = d, A = r2 (A as a function of r), Arectangle = lw (Arectangle as a function of l and w).
• 8.P.9 Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate.
• 8.M.1 Select, convert (within the same system of measurement), and use appropriate units of measurement or scale.
• 8.M.3 Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.
• 8.D.1 Describe the characteristics and limitations of a data sample. Identify different ways of selecting a sample, e.g., convenience sampling, responses to a survey, random sampling.
Overview & Approximate Time
• This lesson will be part of a Geometry Unit.
• This lesson should take about 3-5 days to complete and analyze.
(This is based on hour-long blocks)
Data & Analysis Tools
• Image J
• Excel
Steps for Data Analysis
• Students will explore the world of circles in their everyday life. Using digital cameras we will search for circles in our surroundings. Each group will be responsible for finding at least 5 circles of various sizes. It is most important that students look for very large and small circles. The more evidence we have the more support we have for belief that there does exist a unique relationship between the radius of a circle and the circumference.
• Next students will open these pictures in Image J. Using the tools they will measure both the circumference and the diameter of the circles they are working on.
• Students will copy the information from Image J analysis and copy/paste the information into an Excel Spread sheet.
• The Excel spread sheet needs 3 columns-
o Column 1-circumference
o Column 2-diameter
o Column 3 formula to be applied. COL 1/COL 2
• Close attention will be focused on the results of COL 3.
• Results for each student will be placed on an overhead that will be created from an Excel spreadsheet.
•
Background Information
• Since fourth or fifth grade students have worked with the formulas for finding the circumference, diameter, radius, and area of circles. They have used the Greek symbol .
• Students have been told that is an irrational number. They are usually instructed to use either 3.14 or 22/7 for a value when evaluating problems using .
Sample Product
Student handouts for data collection & analysis (if provided); may be electronic
555:1841
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