<p>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 measures. Students will:&nbsp;<br>- understand the relationship between perimeter and circumference.&nbsp;<br>- discover the relationship between the circumference and diameter of a circle.&nbsp;<br>- discover the relationship between a circle's radius and area.&nbsp;<br>estimate the area and circumference of a circle if radius or diameter is known.&nbsp;<br>- explain the difference between the area and circumference of a circle.&nbsp;<br>- use area and circumference to solve real-world situations.&nbsp;</p>

<p>- How may patterns be used to describe mathematical relationships?<br>- How may detecting repetition or regularity assist in solving problems more efficiently?&nbsp;<br>- How do spatial relationships, such as shape and dimension, help to create, construct, model, and represent real-world scenarios or solve problems?&nbsp;<br>- How may using geometric shape features help with mathematical reasoning and problem solving?&nbsp;<br>- How may geometric properties and theorems used to describe, model, and analyze problems?&nbsp;<br>&nbsp;</p>

<p>- Circumference: The distance around a circle. (C = 2π<i>r</i>)</p>

<p>- copies of Vocabulary Journal pages (M-7-4-1_Vocabulary Journal)<br>- 15–20 circular objects, cans, or other cylindrical objects for students to measure. (Everyday objects such as cans, containers, and tape rolls are best.)<br>- string cut into 14–18 inch pieces (one for each student or pair of students)<br>- 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)<br>- zipper 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)<br>- transparency and student copies of the Circles Lab Sheet (M-7-4-1_Circles Lab Sheet).<br>- one cylindrical container such as an oatmeal or coffee container for display<br>- chart paper and markers<br>- copies of the Lesson 1 Entrance Ticket (M-7-4-1_Lesson 1 Entrance Ticket and KEY).<br>- copies of the Switch sheet (M-7-4-1_The Switch and KEY).<br>- More on Parts of a Circle (M-7-4-1_More on Parts of a Circle and KEY), <i>optional</i><br>- copies of the Digits of pi (M-7-4-1_Digits of pi) for station activity</p>

<p>- The Lesson 1 Entrance Ticket (M-7-4-1_Lesson 1 Entrance Ticket and KEY) can be used to pre-assess students' knowledge levels.&nbsp;<br>- Students can be evaluated by using the More on Parts of a Circle activity sheet (M-7-4-1_More on Parts of a Circle and KEY).&nbsp;<br>- Assess students' grasp of the Circles Lab Sheet results (M-7-4-1_Circles Lab Sheet).<br>- If you want to use the Technology: Create Your Own Quiz Activity, students' understanding can be checked by trading and completing each other's quizzes.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement, Modeling<br>W: During a teacher demonstration using string and marker, students learn that a circle consists of multiple points equidistant from the center point. Students review the definition of perimeter and compare it to the circumference of a circle. This lesson will teach students a variety of measurements and calculations involving circles.<br>H: Hook students into the lesson by demonstrating how to measure the diameter and circumference of a circle and cylinder with string and a ruler. 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 the diameter.<br>E: Students explore and examining parts of a circle while looking for relationships among the parts. Students will compare radius-sized squares to circles of the same radius. They learn that slightly more than three radius squares are needed 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.&nbsp;<br>R: Student groups can revisit and develop their solutions throughout work time and after presenting to class. Furthermore, while any student or pair presents a solution, all other students in the class are encouraged to use the newly taught concepts to improve or add to their own work.&nbsp;<br>E: Students' knowledge is evaluated informally during work time and presentations of the circle problems. Each pair of students may complete a partner quiz to assess their levels of mastery.<br>T: Use the Extension suggestions to adapt the lesson to individual student needs. The Routine section includes strategies for reviewing lesson concepts throughout the school year. The Small Group section is appropriate for students who could benefit from additional practice, while the Expansion section is recommended for students who may be going beyond the standards. Additional activities are suggested for classroom stations, as well as the use of technology.<br>O: This lesson teaches students about the relationship between circumference, diameter, and area of a circle. Students learn by creating 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, discovering that the area of a circle is roughly three times (or π times) the area of a square whose edges are the same length as the radius of the circle. This is a lesson about discovering and connecting.<br>&nbsp;</p>

<p>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.&nbsp;<br><br><i>Optional:</i> For fun, have the class vote on the best freehand circle drawn on the board. Use this chance to review the properties of a circle.<br><br>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. Ask the class:&nbsp;<br><br><strong>"I scored 15 points on the board with my string and marker. What do you notice about these points?"</strong> (<i>They move around in a circle.</i>)<br><strong>"How many points would I have to make to actually form the full circle?"</strong> Demonstrate filling in several more points.<br><strong>"If I could add hundreds, thousands, or more, I'd eventually have a solid line that formed the outline of my circle. Can we use this concept to determine the definition of a circle?"</strong> Call on multiple students and come to class consensus on a definition. It will most likely be a combination of suggestions from several students. (<i>the set of all points, or an unlimited number of points, at the same distance from the center of the circle</i>)&nbsp;<br><br>Continue asking questions.&nbsp;<br><br><strong>"When you want to know the distance around a polygon or other straight-sided figure, what is it called?"</strong> (<i>perimeter</i>)&nbsp;<br><strong>"How do you find perimeter?"</strong> (<i>by adding the measurements of all straight sides</i>)<br><strong>"Can we use a ruler to find the perimeter (circumference) of a circle?"</strong> (<i>No, there are no straight sides.</i>)&nbsp;<br>Try showing with a ruler or having a student measure the circumference to show how difficult and inaccurate the measurement would be.&nbsp;<br><br>Tell students, <strong>"It is critical to be able to measure this distance because you will need to know the distance around circular objects to solve many real-world situations. Can anyone think of a time when the distance around a circle would be required?"</strong> (<i>fence distance around a round swimming pool, trim around a round window or project, edging quantity for a circular garden</i>)<br><br><strong>"What is a better way for us to measure the circumference?"</strong> Encourage a range of student suggestions.&nbsp;<br><br>End by saying, <strong>"In our next activity, we use string to make measurements for several different circles."</strong>&nbsp;<br><br>Hand out the Lesson 1 Entrance Ticket-Parts of a Circle (M-7-4-1_Lesson 1 Entrance Ticket and KEY). This is a review of vocabulary. Make sure to explain that a circle has 360 degrees.<br><br>Divide students into pairs. Display a collection of 15-20 circular or cylindrical objects. If possible, mark the center points of the circles and underline the radius or diameter of each object. Paper circles of various sizes can be used as a substitute. If paper circles are being 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.&nbsp;<br><br>Show how to find circumference by using a string, carefully outlining the distance around with the string, 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 are not careful to check that they are going through the center, their diameter measurements may be too small.&nbsp;<br><br>Give each pair of students a length of string, a ruler, and two copies of the Circles Lab Sheet (M-7-4-1_Circles Lab Sheet). Instruct each student pair to choose a circular 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 data 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.&nbsp;<br><br><strong>"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."</strong> While students are working, assist with any questions they may have.&nbsp;<br><br><strong>"Review the questions at the bottom of your lab sheet. Take a few minutes to identify similarities in your table columns and describe them as clearly as possible."</strong> Allow another 3-5 minutes.<br><br>1. Examine the circumference and diameter columns. Describe any patterns you see.<br>2. Consider the <i>C + d, C - d, C × d</i>, and <i>C ÷ d</i> columns. Describe any pattern(s) you observe.<br>3. Describe how these patterns can help you solve problems involving circles.<br>Read question 1 to the class, then say, <strong>"Turn to a partner and share the pattern(s) you found."</strong> 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 that dividing the circumference by 3 or a number somewhat greater than 3 yields a value close to to the diameter.<br><br>Repeat the process for question 2. Students should notice that dividing the circumference by the diameter of each circle yields a value slightly greater than 3. <strong>"You observed that the value is always slightly greater than 3. Can we narrow this down to a more exact value in tenths or hundreds?"</strong> Allow students to narrow it down to at least 3.1 or 3.2. It may be good to have each student calculate the mean for this column for his or her personal data and then compare the means for the whole class.&nbsp;<br><br><strong>"This value applies to all circles, no matter how small or how large. Pi is a special value that represents the ratio of circumference to diameter. Pi has a decimal value is infinite and never repeats. It is used in all calculations involving circles. When estimating measurements involving circles, it is sufficient to use the rounded value of 3. When we require a more accurate answer, we will still have to round pi. We usually round the value to 3.14. Using your discoveries, 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 centimeters. What is the circumference?"</strong> (<i>21 centimeters if students used 3; 21.98 centimeters if students used 3.14</i>).<br><br>This is a good opportunity to explore when an estimate is sufficient and when a more precise value is required.<br><br>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:<br><br>To calculate the distance around a circle (the circumference), multiply 3 or 3.14 times the diameter.&nbsp;<br>If you know the distance around a circular (circumference) of a circular object, divide it by 3 or 3.14 to calculate the distance across (the diameter).&nbsp;<br>If you want to find the radius, divide the diameter by 2.&nbsp;<br>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). 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.<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> Discuss the significance of understanding and using the right terms to express mathematical concepts accurately. During this lesson, students' vocabulary journals should include the following terms: <i>circle, circumference, diameter, pi,</i> and <i>radius</i>. Keep a supply of vocabulary journal pages on hand so that students can add them as needed (M-7-4-1_Vocabulary Journal). Highlight circumference and circular area examples from throughout 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 examples in each particular context. Explain the difference between identifying circumference with standard units and area of a circle with square units. Require students to use their labeling appropriately, both verbally and writing.&nbsp;<br><br><strong>Small Group: The Switch.</strong> Students who want further practice may be divided into small groups and given a copy of the Switch sheet (M-7-4-1_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 answers for 30 seconds. Choose a few partners to share their answers with the rest of the group. To demonstrate their partnership, the partners must place their names next to number 1. Then, have one person from each pair stand up. This person will switch places (and thus switch partners) with someone who is standing up. Students have two minutes with their new partners to answer number 2 (and write their new partners' names next to it). Have a few groups share their ideas openly. The person who did not move the previous time must move this time and switch with someone else standing (but students cannot have the same partner more than once). Then use this method to solve the remaining problems.<br><br>Throughout the lesson, ask students questions about critical thinking. For example, in question 2, ask, <strong>"Are there only three differences? What is the most significant difference?"</strong> When you reach number 5, ask about the properties of the number 1. <strong>"What is the most defining property of the number 1?"</strong> (<i>1×R=R</i>)&nbsp;<br><br>Keep students moving throughout the activity, and ensure that everyone has a chance to talk and answer a question. This activity could be turned into a game if it is appropriate for your class.&nbsp;<br><br>Students can use the equation A= π • <i>r²</i> to calculate the area of a circle and discuss volume.<br><br><strong>Expansion: Discovering Pi Activity.</strong> Students who have demonstrated skill with circle concepts should be challenged to find the next few digits of pi. Have each student trace 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 precisely as possible.<br>Students should calculate 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 to the 10 decimal place (or less if the decimal terminates). Finally, they will compare their value for pi to the first ten decimal digits (3.1415926535...) to see how accurate their calculations are. (M-7-4-1_Digits of pi contains more digits.)&nbsp;<br><br><strong>Technology: Create Your Own Quiz Activity.</strong> 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: <a href="http://www.calculatorsoup.com/calculators/geometry-plane/circle.php">http://www.calculatorsoup.com/calculators/geometry-plane/circle.php</a>&nbsp;<br><br>This website allows students to:<br>enter a radius to calculate both area and circumference.&nbsp;<br>enter the area to get both circumference and radius.&nbsp;<br>enter circumference to calculate both area and radius.&nbsp;<br>If time allows, have students or groups exchange and take each other's quizzes. Have the students or groups who 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 alternative possible methods of calculating the results.</p>