Unit 3, Lesson 3: Exploring Circumference Lesson Goals Understand diameter and circumference of a circle. as the constant of proportionality in the relationship between the Recognize that the relationship between the diameter and circumference of a circle is proportional and the constant of proportionality is a little more than 3. Required Materials empty toilet paper roll measuring tapes cylindrical household items 3.1: Which Is Greater? (5 minutes) Setup: Students in groups of 2. Display the image for all to see. 1 minute of quiet think time, followed by small-group and whole-class discussion. Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 1
Student task statement Possible responses Clare wonders if the height of the toilet paper tube or the distance around the tube is greater. What information would she need in order to solve the problem? How could she find this out? Clare needs to measure the length and distance around the tube (for example, with a flexible tape measure). Anticipated misconceptions Students may not understand what is meant by the height of the tube because it can sit in two different ways. Point these students to the first picture of the tube, and ask them to identify the height as the tube is sitting in that picture. Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 2
3.2: Measuring Circumference and Diameter (25 minutes) Setup: Groups of 2 4. Distribute 3 circular objects and measuring tapes or string and rulers to each group. Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 3
Student task statement Your teacher will give you several circular objects. Possible responses 1. Answers vary. 1. Measure the diameter and the circumference of the circle in each object to the nearest tenth of a centimeter. Record your measurements in the table. object diameter (cm) circumference (cm) 2. Answers vary. The graph includes three points that are close to lying on a line that passes through. 3. Answers vary. The graph includes more points that are close to lying on a line that passes through. 2. Plot the diameter and circumference values from the table on the coordinate plane. What do you notice? 3. Plot the points from two other groups on the same coordinate plane. Do you see the same pattern that Anticipated misconceptions Students may try to measure the diameter without going across the widest part of the circle, or may struggle with measuring around the circumference. Mentally check that their measurements divide to get approximately 3 or compare with your own prepared table of data and prompt them to re-measure when their measurements are off by too much. If the circular object has a rim or lip, this Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 4
you noticed earlier? could help students keep the measuring tape in place while measuring the circumference. If students are struggling to see the proportional relationship, remind them of recent examples where they have seen similar graphs of proportional relationships. Ask them to estimate additional diameter-circumference pairs that would fit the pattern shown in the graph. Based on their graphs, do the values of the circumferences seem to relate to those of the diameters in a particular way? What seems to be that relationship? Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 5
Are you ready for more? Suppose you had another circular object with a diameter that is half as long as the diameter of your largest circle. What would its circumference be? Possible Responses Answers vary, depending on the size of the circular objects students measured. Possible solution: Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 6
3.3: Calculating Circumference and Diameter (10 minutes) Setup: Ask students to label pictures with measurements. Remind students of the constant of proportionality from previous activity. Quiet work time followed by whole-group discussion. Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 7
Student task statement Here are five circles. One measurement for each circle is given in the table. Possible responses See lesson plan for completed table. Anticipated misconceptions Some students may multiply the circumference by the constant of proportionality instead of dividing by it. Prompt them to consider whether the diameter can be longer than the circumference of a circle. Use the constant of proportionality estimated in the previous activity to complete the table. diameter (cm) circumference (cm) circle A 3 circle B 10 Some students may struggle to divide by 3.1 if that is the constant of proportionality decided on in the previous lesson. Ask these students if they could use an easier number as their constant, and allow them to divide by 3 instead. Then ask them how their answer would have changed if they divided by 3.1. circle C 24 circle D 18 circle E 1 Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 8
Are you ready for more? The circumference of the Earth is approximately 40,000 km. If you made a circle of wire around the globe, that is only 10 meters (0.01 km) longer than the circumference of the globe, could a flea, a mouse, or even a person creep under it? Possible Responses Yes! Each meter added to the diameter of a circle adds about 3.1 meters to the circumference of the circle. So if the circumference of the Earth is increased by 10 meters, this means that a little more than 3 meters have been added to the diameter. So there would be about 1.5 meters of distance between the rope and the Earth, making it easy for a flea, mouse, or person to go under the rope! Lesson Synthesis (5 minutes) What is the relationship between the diameter and the circumference of a circle? How does this relate to the number? 3.4: Identifying Circumference and Diameter (Cool-down, 5 minutes) Setup: None. Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 9
Student task statement Select all the pairs that could be reasonable approximations for the diameter and circumference of a circle. Explain your reasoning. Possible responses Only B, because is the only quotient that is a little more than 3. A. 5 meters and 22 meters. B. 19 inches and 60 inches. C. 33 centimeters and 80 centimeters. Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 10