RATIO AND PROPORTION

Properties of Circles
(Teacher Copy)

### Activity 1:  Finding p

Students measure circular objects to discover that the ratio of the circumference to the radius of a circle is p.

To find relationships between the radius, diameter, and circumference of a circle.

Introduction:
Ask the students if they can tell or show you the meaning of the terms radius, diameter, and circumference (see the
Reference Copy).  Can they think of when they might want to know the radius, diameter, or circumference of a circle?  Discuss why these concepts are important when buying automobile tires.  (Tire sizes are based on tire radius and width.  You might want to have students examine tire ads and discuss the various sizes that are available for a given car.)

Part A

 Object All objects diameter Answers will vary radius Answers will vary circumference Answers will vary diameter divided by radius » 2 for all entries circumference divided by radius » 6.2 for all entries circumference divided by diameter » 3.1 for all entries

Summary Questions

1. Students should say something to the effect that the size of the circumference increases as the radius (diameter) increases.
2. Students should notice that the answers in the last three rows of the table are the same across the row.
3. Yes,  you should get the same ratios for any circle.
4. The calculated circumferences should be close to the measured circumferences in the table.
5. Students should report that calculated circumferences are close to measured circumferences in the table.
Closing Discussion
• When you divide the circumference by the radius of any circle, will you always get about 3.1?  Yes, you will always get an approximation of p.
• If you measured the radius, diameter, and circumference in inches instead of centimeters, would it change the results of the experiment?  (If time permits, you may want to try this out!)  No, the ratio is a constant.
• Why do you get some numbers that are a little more or a little less than 3.1?  Accuracy of measurement will affect the calculations.

### Activity  2:  Graphing Relationships

Students will create a graph of the diameter in relation to the circumference of a circle.  It is assumed that students have completed Activity 1 and have the student copy available.  As an alternative to graph paper, have students use computer graphing software to make plots.

To find relationships between the radius, diameter, and circumference of a circle.

Introduction:
Ask students how they could  make a graphic representation of the numbers they found in Activity 1.   [Review with students how to plot pairs of points on an x-y graph.]

The graphs drawn by students should be similar to the graph in the Student Copy.  The line drawn should have approximately an equal number of points on each side of the line.  Numbering on the graphs should be in equal increments.

Part C: The answer to a-d is p.  Student answers should be about 3.1 but may vary due to measurement error.

Summary Questions

1. Students should say something to the effect that plotted measurements of other circles would appear on or near the line.
2. The ratio of circumference to diameter is a constant; it is a linear relationship.

Closing Discussion
• Why did we draw one straight line on the graph rather than "connect the dots"?  There is a linear relationship between diameter and circumference of a circle; the points plotted are only approximations of this linear relationship.
• Is there a relationship between p and the slope of a line?  The slope of the line is p.