Topic:  

Properties of Circles

Software:  

Microsoft Excel

Software rationale:

Here is the rationale for the use of technology, according to the National Council of Teachers of Mathematics:

Technology has changed the way in which mathematics is used and has led to the creation of both new and expanded fields of mathematical study. Thus, the technology is driving change in the content of mathematics and the way it is learned and accessed. A vital aspect of such change is a teacher’s ability to select and use appropriate instructional technology to develop, enhance, and extend students’ understanding and application of mathematics. It is essential that teachers continue to explore the impact of instructional technology and the perspectives it provides on an expanding array of mathematics concepts, skills, and applications.

This lesson plan employs the spreadsheet (Excel) as a convenient tool for graphing C vs. D and A vs D² to discover that the quotients C/D and A/D² have the same value for all circles.

Class time:  

1 class.

Level:  

Grade 8

Objectives:

Students will be able to:

• Make accurate measurement estimations based on an understanding of formulas and results of investigation.
• Measure the radius, diameter, and circumference of a circle using concrete materials.
• Recognize that there is a constant relationship between the radius, diameter, and circumference of a circle.
• Develop the formula for finding the circumference and formula for finding the area of a circle.
• Draw a circle given its area and or circumference.
• Define radius, diameter, and circumference.

Computer functions:

Students need to be familiar with Excel’s graph or chart menu and Excel’s calculator.

Problem:  

What is the relationship between the  circumference and diameter of a circle?

Procedure:  

Group students in pairs and provide each pair with five cans of different diameters, a tape measure which reads in millimeters, a copy of the worksheet and access to a Microsoft Excel (spreadsheet).

Discuss briefly the terms: diameter, radius, circumference and surface area. Ask the students to complete the following worksheet, ensuring that each produces their own table and graph. Students must use a spreadsheet when completing exercise 4 to construct the table and scatter plot.

Worksheet:

1. What geometric solid resembles the shape of a can? Explain in a sentence or draw a diagram to show what is meant by each of these measures of a can.
    

   
    
    
    

   a) its diameter b) its circumference c) its surface area d) its volume
    
    
    
    
   
    

2. Describe how you would measure the diameter, circumference, and height of a can.
    

   
    
    
    
    
    
    
    

3. class="maintext">a) Without measuring, estimate which is greater, the circumference of a can or its height.
    

   
   
    
    
    
    
    
    

   b) Estimate how many times bigger the circumference is compared to the diameter.
    
    
    
    
    
    

4. a) Measure and record the diameter, circumference, and height of each can. Check your estimates in exercise 3. Were you surprised? Explain.

When all the students have finished, ask which is bigger - the height or circumference of a can. Remind them that the circumference of a circle is about three times its diameter. Ask them to estimate the corresponding circumferences to different diameters. Then ask them to estimate the corresponding diameters to different circumferences. After reinforcing the approximate 3 to 1 relationship between circumference and diameter, formalize it by introducing p and explaining that it’s just a symbol that stands for the exact ratio.

Assessment:

This activity should be used as a group project. In addition to evaluating the worksheet, I will circulate during the investigation. This time provides me with an opportunity to observe their ability to follow directions and to work in groups.

Learners will be given 5 percent towards the project mark if they show acceptable group skills. The students understanding of the concepts will graded according to the following rubric. A Level 1 achievement will earn the students a D grade, Level 2 will earn the student a C to B grade, Level 3 will earn the student B⁺ to A grade, Level 4 will earn the student an A⁺.

Concepts	Level 1	Level 2	Level 3	Level 4
Understanding of the Relationship between the circumference of a circle and its diameter, and that C/D is the same for all circles.	The response to exercise 4 includes fewer than three of the following elements: A completed table showing C/D between 3.0 and 3.3. A formula of the from C = p D or C » 3.14D. An appropriate graph comparing C and D for several cans. Correct application of the formula relating C and D.	The response to exercise 4 includes at least three of the following elements: A completed table showing C/D between 3.0 and 3.3. A formula of the from C = p D or C » 3.14D. An appropriate graph comparing C and D for several cans Correct application of the formula relating C and D.	The response to exercise 4 includes all of the following elements: A completed table showing C/D between 3.0 and 3.3. A formula of the from C = p D or C » 3.14D. An appropriate graph comparing C and D for several cans Correct application of the formula relating C and D.	In addition to Level 3: The graph comparing C and D for several cans is a scatter plot or line graph.