As students enter the room, have them each draw a circle on the board. When class begins, ask students to consider the circles on the board. Ask them to state some similarities and differences between the circles.
Optional: For fun, you can have the class vote to determine the best freehand circle drawn on the board. Use this opportunity to reinforce the properties of a circle.
Cut a piece of yarn or string about 12 inches long. Attach a marker to one end. Holding the loose end of the string in the middle of the board (or piece of chart paper), stretch the marker end straight out and make a point on the board with the marker. Rotate the marker around, keeping the string taut, and make about 15 points spaced out along the perimeter of the circle you eventually fill in. Ask the class:
- “I have made 15 points on the board using my string and marker. What do you notice about the points?” (They go around in a circle.)
- “How many points would I have to make to actually form the full circle?” Demonstrate filling in several more points.
“If I could add hundreds, thousands, or more, I would eventually make a solid line to form the outline of my circle. Can we use this idea to come up with the definition of a circle?” Call on several students and come to class consensus on a definition. It will likely be a combination of ideas from a few different students. (the set of all points, or an infinite number of points, the same distance from the center point 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 demonstrating with a ruler, or have 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 are going to need to know the distance around circular objects to answer many real-life problems. Can any of you think of a time the distance around a circle would be needed?” (distance around a round swimming pool for a fence, trim around a circular window or project, amount of edging for a circular garden)
“What would be a better way for us to measure the circumference?” Encourage a variety of student suggestions.
Conclude by stating, “In our next activity, we use string to make measurements for several different circles.”
Give students the Lesson 1 Entrance Ticket–Parts of a Circle (M-7-4-1_Lesson 1 Entrance Ticket and KEY.docx). This is a review of the vocabulary. Be sure to introduce the fact that a circle has 360 degrees.
Place students into pairs. Display a set of 15–20 circular or cylindrical objects. If possible, mark the center points of the circles and highlight the radius or diameter on each object. A variety of circle sizes cut out of paper can be substituted. A fold line at the line of symmetry highlights the diameter if paper circles are being used. If using this substitution, eight to ten different sizes with each size on a different color of paper would work best.
Demonstrate finding circumference by using a string, carefully outlining the distance around with the string and measuring the string with a ruler. Ask students to explain how they could measure the diameter of a circle or the circular base of a cylinder. If they are not sure, demonstrate this as well using just a ruler without the string, and a second time using a string. Remind students that the diameter must go through the center of the circle and all the way across the circle. A common error is to measure a smaller chord or just the radius. If students are not careful to check that they are going through the center, their diameter measurements may be too small.
Provide each pair of students with a piece of string, a ruler and two copies of the Circles Lab Sheet (M-7-4-1_Circles Lab Sheet.docx). Instruct student pairs to select one circular or cylindrical object from the collection. They need to measure the diameter and circumference of each circle or the base of a cylinder and record the measurements on their lab sheet in the appropriate columns. Direct students to measure in centimeters, rounded to the nearest tenth. Students should record the data on their own sheet. Walk about the room to assist with difficulties that arise and to monitor student accuracy with string placement and ruler measurements. When students complete the first set of measurements, have them exchange their object for another. Repeat the measurement steps until all eight rows in the table are filled with different objects or circles. Have students return the strings, rulers, circles, and cylinders.
“Now that you have eight sets of measurements, you will do some calculations with the data. You will work individually on your own lab sheet for this step. I will give you about 10 minutes to do your calculations. Look at the headings on the next four columns of your table. You will add, subtract, multiply, and divide the circumference and diameters for each object you measured. You may use a calculator for this. Turn your paper over when your calculations are complete so I will know when everyone is finished.” While students work, assist with questions students may have.
“Look at 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 clearly as you can.” Allow another 3 to 5 minutes.
- Study the circumference and diameter columns. Describe any pattern(s) you see.
- Look at the C + d, C − d, C × d, and C ÷ d columns. Describe any pattern(s) you notice.
- Describe how these patterns may be able to help you solve problems involving circles.
Read question 1, above, to the class and then say, “Turn to a partner and share the pattern(s) you found.” After a minute or two, allow several students to share their observations with the class. Students should notice that the circumference for each circle is about three times larger than the diameter, or if you divide the circumference by 3 or a number slightly greater than 3, you will get close to the diameter.
Repeat this process for question 2. Students should notice that when you divide the circumference by the diameter for each circle you get a value slightly greater than 3. “You have discovered that the value is always slightly greater than 3. Can we narrow this down to a more specific value to the tenths or hundredths?” Let students narrow it down at least to 3.1 or 3.2. It may be helpful to have each student find the mean for this column for his/her own data and compare the means for the whole class.
“This value is actually the same for all circles no matter how small or how large. The ratio of circumference and diameter is always a special value called pi. Pi has a decimal value that never ends and never repeats. It is used in all calculations regarding circles. When estimating measurements involving circles it is sufficient to use the rounded value of 3. When we need a more accurate answer we will still need to round pi. We usually round to the value 3.14. Based on your discovery, we can find the circumference of any circle by using the formula C = 3.14 × diameter. Let me give you a sample problem. Raise your hand as soon as you have the answer. I have a circle with a diameter of 7 centimeters. What is its circumference?” (21 centimeters if students used 3, or 21.98 centimeters if students used 3.14)
This is a good opportunity to discuss when an estimate is acceptable and when a more accurate value is needed.
Repeat the process for question 3. Allow students to share responses. Clarify misconceptions as they arise. If these conclusions are not mentioned, introduce them:
- If you need the distance around a circle (the circumference), you can multiply 3 or 3.14 times the diameter.
- If you know the distance around a circular object, you can divide that by 3 or 3.14 to find the distance across (the diameter).
- You can also divide the diameter by 2, if you need to find the radius.
Students may be given the More on Parts of a Circle activity sheet (M-7-4-1_More on Parts of a Circle and KEY.docx). It touches on naming conventions for lines and then asks students to identify different parts of the circle using this naming convention. It includes chord, radius, diameter, center, and circumference.
Extension:
- Routine: Discuss the importance of understanding and using the correct words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ vocabulary journals: circle, circumference, diameter, pi, and radius. Keep a supply of vocabulary journal pages on hand so students can add pages as needed (M-7-4-1_Vocabulary Journal.docx). Point out instances of circumference and circular area examples seen throughout the school year. Use the circumference and diameter ratio when studying ratio and proportional reasoning. Ask students to bring in circular objects that can be measured and compared and other examples of circles that they see outside of class, such as circle graphs. Discuss the use and meaning of such examples in each particular context. Distinguish the difference in labeling circumference with standard units and area of a circle with square units. Require students to use appropriate labeling in both verbal and written responses.
- Small Group: The Switch. Students who would benefit from additional practice may be pulled into small groups and given a copy of the Switch sheet (M-7-4-1_The Switch and KEY.docx). You will need a timer. Tell students that they have 1.5 minutes to complete question 1. When the time is up, have them find a partner (only one, unless there is an uneven number of students, and then only allow one group of three) and compare their answers for 30 seconds. Pick a few partners to share what they answered with the rest of the group. The partners need to put their names next to number 1 to show that they were partners. Then have one person in each pair stand up. This person is to switch places (thus switching partners) with someone else who is standing up. With their new partners students have 2 minutes to answer number 2 (and put their new partners’ names next to number 2). Have a few groups give their ideas aloud. Then the one who didn’t move last time needs to move this time and switch with someone else standing (but students are not to have the same partner more than once). Then do the rest of the problems this way.
Ask students critical thinking questions throughout. For example in number 2, ask, “Are there only three differences? Which is the most important difference?” When you get to number 5, ask about the properties of the number 1. “What is the most defining property of the number 1?” (1×R=R)
Keep students moving throughout the activity, and be sure that everyone has an opportunity to talk and answer a question. This activity could turn into a game if appropriate for your class.
Using A= π • r2, have students find the area of a circle and then use it to talk about volume.
- Expansion: Discovering Pi Activity. For students who have shown proficiency with the circle concepts, challenge them to find the next several digits of pi. Have each student trace three circles onto chart paper or use three circular objects from the lesson. Ask students to measure the circumference and diameter (in centimeters to the nearest tenth) as carefully as they can, using string and a ruler.
Have students find the mean value for their three circumference values, and the mean value for their three diameters. On paper, students should divide the mean circumference by the mean diameter, dividing out to 10 decimal places (or less if the decimal terminates). Lastly, they will compare their value for pi to the first ten decimal digits of pi (3.1415926535…) to see how accurate their calculations are. (See M-7-4-1_Digits of pi.docx for additional digits.)
- Technology: Create Your Own Quiz Activity. Students may work individually or in groups. Provide paper, pencils, markers, and computer access. Students write three real-world problems involving the area and circumference of circles. Students turn in a quiz including diagrams, using their three questions and providing 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 Web site allows students to:
- enter radius to get both area and circumference.
- enter area to get both circumference and radius.
- enter circumference to get both area and radius.
If time permits, have students or groups exchange and take each other’s quizzes. Have students or groups who wrote each quiz grade the quiz, using the answer key they created. Provide students time to engage in a discussion about the reasoning, errors, and other possible ways of calculating the solutions after they have exchanged and graded one another’s quizzes.
