In this lesson, students will investigate the relationship between the circumference and area of a circle. Students will use a number of strategies to measure radius, diameter, circumference, and area while learning about the relationships between these measurements. Students will:
- understand the relationship between perimeter and circumference.
- discover the relationship between the circumference and diameter of a circle.
- discover the relationship between a circle's radius and area.
- estimate the area and circumference of a circle if radius or diameter is known.
- explain the difference between the area and circumference of a circle.
- use area and circumference to solve real-world problems.
- How can we use the relationship between area and volume to draw, construct, model, and represent real-world situations, as well as solve problems of area and volume?
- Circle: The set of all points equidistant from a given point, called the center.
- Circumference: Distance around a circle.
- Diameter: A segment with both endpoints on the circle (chord), which also extends through the center of the circle.
- pi: The ratio of circumference to diameter for any circle, often approximated to be 3.14 (the symbol representing pi is π).
- Radius: A segment from the center of a circle to the edge.
- student copies of Vocabulary Journal pages (M-6-2-1_Vocabulary Journal)
- 15–20 circular objects, cans, or other cylindrical objects for students to measure (everyday objects such as cans, containers, and tape rolls are best)
- string cut into 14–18 inch pieces (one for each student or pair of students)
- 4-, 6-, and 8-centimeter squares cut from centimeter grid paper (M-6-2-1_Centimeter Grid Paper); need four to five of the same size per student or pair of students
- single circles (radius 4, 6, and 8 centimeters) drawn on centimeter grid paper, (one per student or pair with the same radius as the set of squares provided)
- resealable bags to store centimeter squares and circles (one circle with four to five squares with side lengths equal to the radius of the circle in each bag)
- transparency and student copies of the Circles Lab Sheet (M-6-2-3_Circles Lab Sheet)
- a cylinder container such as an oatmeal or coffee container for display
- chart paper and markers
- copies of the Lesson 3 Entrance Ticket–Parts of a Circle (M-6-2-3_Lesson 3 Entrance Ticket and KEY)
- copies of the Switch sheet (M-6-2-3_The Switch and KEY)
- More on Parts of a Circle (M-6-2-3_More on Parts of a Circle and KEY), optional
- copies of the Digits of pi for station activity (M-6-2-3_Digits of pi)
- Informally assess student comprehension by observing them during the Investigating Circumference and Investigating Area exercises.
- Use the Partner Problems and presentations to assess student understanding.
- At the end of the lesson, use the Partner Quiz to formally evaluate student understanding.
Scaffolding, Active Engagement, Modeling and Explicit Instruction
W: Use a string and a marker to show that a circle is numerous points equidistant from a center point. Students review the definition of perimeter and compare it to the circumference of a circle.
H: Using string and a ruler, demonstrate how to measure the diameter and circumference of a circle and cylinder. Discuss how to use and compare these values. Students use string to determine the diameter and circumference of various circular objects. Using these measurements, students determine that circumference is π times greater than diameter.
E: Students compare radius-sized squares to circles of the same radius to identify the relationships between the parts of a circle. They learn that it takes slightly more than three radius squares to cover the two-dimensional surface (area) of the circle. This discovery leads to the area formula for a circle. Students use this formula to make various real-world area estimations and calculations, some of which are presented to the class.
R: Student groups are given the opportunity to change their solutions during work time and after presenting problems to the class. Furthermore, while any student or pair presents a solution, all other students in the class are encouraged to use new ideas learned to modify or supplement their own work.
E: The teacher evaluates students' understanding informally during work time and during problem presentations. Each pair of students takes a partner quiz to assess their levels of understanding.
T: Use the extension suggestions to personalize the lesson to individual student needs. The small-group activity is appropriate for students who could benefit from additional practice, and the expansion is recommended for students who may be going beyond the standards. Additional exercises are suggested for classroom stations, as well as the use of technology.
O: This lesson teaches students about the relationship between circumference, diameter, and area of a circle. Students learn from a table of values that the circumference of a circle is approximately three times the diameter, and pi is approximately 3.14. Students then investigate the relationship between the area of a circle and radius squares. They discover that the area of a circle is roughly three times (or π times) the area of a square which edges are the same length as the radius of the circle.
As students enter the room, have them each draw a circle on the whiteboard. When class begins, ask students to consider the circles on the board. Ask them to describe the similarities and differences between the circles.
Optional: For fun, have the class vote on the best free-hand circle drawn on the board. Use this chance to review the properties of a circle.
Cut a piece of yarn or string about 12 inches long. Add a marker to one end. Holding the loose end of the string in the middle of the board (or a sheet of chart paper), extend the marker end straight out and mark a point on the board. Rotate the marker while keeping the string taut, and make roughly 15 points spaced out along the perimeter of the circle you'll eventually fill in. Ask the class,
"I have made 15 points on the board with my string and marker. What do you notice about these points?" (they move around in a circle)
"How many points do I need to form the full circle?" Demonstrate filling in several more points.
"If I could add hundreds, thousands, or more, I'd eventually have a solid line to form the outline of my circle. Can we use this concept to determine the definition of a circle?" Call on multiple students and reach a class consensus on a definition. It will most likely be a combination of suggestions from a few different students. (the set of all points, or an unlimited number of points, the same distance from the center of the circle)
Continue asking questions,
"When you want to know the distance around a polygon or other straight-sided figure, what is it called?" (perimeter)
"How do you find perimeter?" (by adding the measurements of all the straight sides)
"Can we use a ruler to find the perimeter (circumference) of a circle?" (No, there are no straight sides.)
Try showing with a ruler or having a student measure the circumference using a ruler to show how difficult and inaccurate the measurement would be.
Tell students, "It is important to be able to measure this distance because you will need to know the distance between circular objects to solve many real-world situations. Can anyone think of a time when the distance around a circle would be required?" (fence distance around a round swimming pool, trim around a round window or project, amount of edging for a circular garden)
"What would be a better way for us to measure the circumference?" Encourage various student suggestions.
End by saying, "In our next activity, we use string to make measurements for several different circles."
Give students the Lesson 3 Entrance Ticket–Parts of a Circle (M-6-2-3_Lesson 3 Entrance Ticket and KEY). This is a review of vocabulary. Make sure to explain the fact that a circle has 360 degrees.
Divide students into pairs. Display a collection of 15-20 circular or cylindrical objects. If possible, mark the center points of the circles and highlight the radius or diameter of each object. A variety of circle sizes cut out of paper can be substitute. If paper circles are used, a fold line along the symmetry line highlights the diameter. This substitution works best with eight to ten different sizes, each on a different color of paper.
Show how to find circumference by using a string, carefully outlining the distance around it, and measuring it using a ruler. Ask students to explain how they would go about measuring the diameter of a circle or the circular base of a cylinder. If they are unsure, demonstrate this using only a ruler and no string, and then again with a string. Remind students that the diameter must pass through the center of the circle and all the way across the circle. A common inaccuracy is measuring a smaller chord or only the radius. If students do not ensure that they are passing through the center, their diameter measurements may be too small.
Give each pair of students a length of string, a ruler, and two copies of the Circles Lab Sheet (M-6-2-3_Circles Lab Sheet). Instruct each student pair to choose one round or cylindrical object from the collection. They must measure the diameter and circumference of each circle or the base of a cylinder and write the results on their lab sheet in the appropriate columns. Direct students to measure in centimeters and round to the nearest tenth. Students should record the information on their own sheet. Walk around the room to help with any problems that arise and to ensure student accuracy with string placement and ruler measurements. After students have completed the first round of measurements, have them switch their object for another. Repeat the measuring procedures until the table's eight rows are filled with various objects or circles. Ask students to return the strings, rulers, circles, and cylinders.
"Now that you have eight sets of measurements, you will do some calculations on the data. This step requires you to work on your own lab sheet. I will allow you around 10 minutes to complete your calculations. Examine the headings for the next four columns of your table. You will add, subtract, multiply, and divide the circumference and diameters of each object you measure. You could use a calculator for this. When your calculations are complete, please turn your paper over so that I can know when everyone is finished." While students are working, assist with any questions they may have.
"Review the questions at the bottom of your lab sheet. Take a few minutes to look for patterns in your table columns and describe them as precisely as possible." Allow another 3-5 minutes.
1. Examine the circumference and diameter columns. Describe any pattern(s) you see.
2. Consider the C + d, C - d, C × d, and C ÷ d columns. Describe any pattern(s) you observe.
3. Describe how these patterns can help you solve problems involving circles.
Read question 1 to the class, then say, "Turn to a partner and share the pattern(s) you found." After a minute or two, invite a few students to share their observations with the class. Students should notice that the circumference of each circle is approximately three times larger than the diameter, or if you divide the circumference by 3 or a number slightly greater than 3, you will get a value close to to the diameter.
Repeat the process for question 2. Students should notice that dividing the circumference by the diameter of each circle gets a result slightly greater than 3. "You have discovered that the value is always slightly greater than 3. Can we narrow this down to a more exact value in tenths or hundreds?" Allow students to narrow it down to at least 3.1 or 3.2. It may be helpful to have each student calculate the mean for this column for his or her own data and then compare the means for the whole class.
"This value applies to all circles, no matter how small or how large. The ratio of circumference and diameter is always a special value called pi. Pi's decimal value is infinite and never repeats. It is used in all calculations involving circles. When estimating measurements involving circles, simply use the rounded number of 3. When we require a more accurate answer, we will still have to round pi. We normally round the figure to 3.14. Using your findings, we can calculate the circumference of any circle using the formula C = 3.14 × diameter. Let me offer you an example problem. Raise your hand as soon as you know the answer. I have a circle with a diameter of 7 cm. What is the circumference?" (21 centimeters if students used 3; 21.98 centimeters if students used 3.14).
This is a great opportunity to explore when an estimate is sufficient and when a more accurate value is required.
Repeat the process for Question 3. Allow students to share their responses. Clarify any misconceptions that arise. If these conclusions are not mentioned, introduce them.
To calculate the distance around a circle (the circumference), multiply 3 or 3.14 times the diameter.
If you know the circumference of a circular object, divide it by 3 or 3.14 to calculate the diameter. If you want to find the radius, divide the diameter by 2.
If further practice is needed, give out to students More on Parts of a Circle activity sheet (M-6-2-3_More on Parts of a Circle and KEY). It discusses line naming conventions before asking students to use this convention to name various areas of the circle. It consists of chord, radius, diameter, center, and circumference.
Extension:
Routine: Discuss the significance of understanding and using correct words to express mathematical concepts accurately. During this lesson, students should record the following terms in their vocabulary journals: circle, circumference, diameter, pi, radius. Keep a supply of vocabulary journal pages on hand so that students can add them as needed (M-6-2-1_Vocabulary Journal). Point out circumference and circular area examples from the school year. When studying ratios and proportions, use the circumference and diameter ratios. Ask students to bring in circular objects that may be measured and compared, as well as other examples of circles they come across outside of class, such as circle graphs. Discuss the use and meaning of such instances in each specific context. Explain the difference between identifying circumference with standard units and area of a circle with square units. Require students to mark their comments appropriately, both verbally and writing.
Small Group: The Switch. Students continue the lesson exercise by comparing circumference and area, then applying that logic to the volume of a cylinder. Give each student a copy of the Switch sheet (M-6-2-3_The Switch and KEY). You'll need a timer. Tell students they have 1.5 minutes to finish question 1. When the time is up, have them pick a partner (just one, unless there are an odd number of students, in which case only one group of three is permitted) and compare their responses for 30 seconds. Pick a few partners to share their responses with the class. The partners must write their names next to number one to indicate that they were partners. Then, have one person from each pair stand up. This person will switch places (and thus switching partners) with someone who is standing up. Students with new partners have 2 minutes to answer number 2 (and write their new partners' names next to it). Have a few groups share their ideas aloud. Then the person who did not move the previous time must move this time and switch with someone else standing (although students may not have the same partner more than once). Then use this method to solve the remaining problems.
Throughout the lesson, ask students questions about critical thinking. For example, in number two, ask them if there are only three differences. Which is the most important difference? When you reach to number five, ask about the features of number one. What is the most distinguishing characteristic of the number one?
(1 x R = R)
Keep students moving throughout the activity, and make sure that everyone has a chance to speak and answer questions. This activity could be turned into a game if it is appropriate for your classroom.
Students use A= π • r² to calculate the area of a circle and then use it to talk about volume.
Introduce the concept of volume in a cylinder. Hold up a tuna can and describe it as a cylinder with a height of one (you will need four tuna cans). "Now that you know how to calculate the area of a circle, consider how you could determine how much tuna would fit inside this canister. Does anyone know what this measurement is called?" (volume)
Afterwards ask, "What units of measure do you think we would use to label volume?" (cubic, because we're measuring a three-dimensional space)
Then place another can of tuna on top and ask, "What happens to the volume when another can of tuna is added? What happens if we 'double' the height of the tuna cans? Or adjust the height in any way?" Another way to get students thinking about this is to imagine flat rectangles stacked on top of each other. The area of the base times the height (number of rectangles stacked up) is similar to the volume of the tuna cans - the area of the circle times the height (number of circles in the stack).
Expansion: Discovering Pi Activity. Students who have shown proficiency with circle concepts should be challenged to find the next few digits of pi. Have each student draw three circles on chart paper or use three circular objects from the lesson. Ask students to use string and a ruler to measure the circumference and diameter ( in centimeters to the nearest tenth) as carefully as possible.
Students should calculate the mean value for each of their three circumferences and diameters. On paper, students should divide the mean circumference by the mean diameter to the 10 decimal place. Finally, they will compare their value of pi to the first ten decimal digits (3.1415926535…) to see how accurate their calculations are (see M-6-2-3_Digits of pi) for additional digits.)
Technology: Create Your Own Quiz Activity. Students can work individually or in groups. Provide paper, pencils, markers, and computer access. Students create three real-world problems that involve the area and circumference of circles. Students submit a quiz with diagrams, using their three questions, and supplying an answer key. To assist with making initial calculations or checking calculations, allow students to use:
http://www.calculatorsoup.com/calculators/geometry-plane/circle.php
This website helps students to:
enter a radius to calculate both area and circumference.
enter area to get the circumference and radius.
enter circumference to calculate both area and radius.
If time allows, have students or groups exchange and take each other's quizzes. Have the students or groups that created each quiz grade the quiz, using the answer key they created. After students have exchanged and scored one other's quizzes, allow them to discuss their reasoning, errors, and different possible methods of computing the results.
