# Area and Circumference of Circles

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2 Warm-Up 1. On the board, draw a number of shapes (rectangle, squares, and triangles). Include dimensions for each shape. Label each shape with a letter for easy identification. 2. Place students into groups of two to three. Provide each group with the same number of sticky notes as the shapes on the board. For example, if there are six shapes, each group should get six sticky notes. 3. Have groups find the perimeter and area of each shape. Have them label the name of the shape on a sticky note, and include its calculated area and perimeter. 4. When everyone has finished, have students place the sticky notes next to the correct shapes. When all of the notes are on the board, have students explain how they found the area and perimeter of each shape and name the formulas they used. Language and Vocabulary 1. Prior to the lesson, create a poster titled Circles. Draw a circle in the middle of the poster. 2. Write the following vocabulary terms on the board: pi (π) circumference radius diameter 3. On chart paper, draw a point in the middle of the circle and draw a line that extends from the center to the edge of the circle. Tell students that the line represents the radius of the circle. Label the radius on the chart. Explain that the diameter is a line that starts at one side of a circle, extends to the other side of a circle, and crosses through the center. Draw and label the diameter. 4. Tell students that the circumference of a circle is the distance around the circle. Ask, How can we show what circumference is on the chart? Students may suggest drawing an arrow around the outside of the circle. Tell students that pi (π) is approximately equal to Explain that this decimal number keeps going indefinitely. On the chart write π = Keep the poster up for students to refer to throughout the lesson Focused Mathematics Intervention Level 7 Teacher s Guide Teacher Created Materials

3 Whole-Group Focus 1. The following lesson will address this focus question: How do you find the area and circumference of a circle? 2. You may wish to write the focus question on the board and read it aloud to students. Explain that you will revisit the focus question at the end of the lesson. I Do 1. Say, Today we are going to find both the circumference and area of circles. Write the following problem on the board as you read it aloud: Bob wants to know the circumference of the circular walking path in the park. He knows the diameter is 10 feet. What is the circumference? 2. Ask, What is circumference? (It is the distance around a circle; it is the perimeter of the circle.) What strategy can we use to solve this problem? If no one suggests it, draw a sketch on the board. Say, We know the diameter, the distance across the circle, is 10 feet. Draw a sketch and label the diameter. Say, We want to know the distance around the circle. 10 ft. 3. Say, To find the circumference, we multiply π by the diameter. We know that π is approximately On the board, write C = πd = Say, The circumference is 31.4 feet. 4. Ask, What is the radius of this circle? (Since the radius is half the diameter, the radius is 5 feet.) Say, We can also use the formula C = 2πr to find the circumference. We multiply 2 times π times the radius. Write the following example on the board: C = 2πr = = Language Support Emphasize that the circumference of a circle is the distance around it, or the perimeter. The area is the number of square units that cover the circle. To help students make the distinction, have them draw a circle and an arrow going around the circle labeled circumference. Then, have students draw square units on the surface of the circle and label these area. Teacher Created Materials Focused Mathematics Intervention Level 7 Teacher s Guide 9

4 Whole-Group (cont.) I Do (cont.) 5. Ask, How do you find the area of a rectangle? Have a student write this formula on the board: Area = base height. Next to it, write the formula for the area of a circle, Area = πr². Say, Even though it doesn t look like it, finding the area of a circle is basically the same as finding the area of a rectangle. Pass out paper and scissors to each pair of students. In addition, provide circular objects for students to trace. 6. Tell students to trace and cut out a circle. Then, have them fold the circle so it has eight equal sections. Have students identify the center point, diameter, and radius on the circle. Next, have students cut out the eight sections and place them as shown to make a parallelogram. Tell students to cut the section at the end in half and move it to the other end of the parallelogram, making a rectangle as shown. πr r 7. Ask, What is the height of our decomposed circle called? (the radius) Ask, What is the base? Students should recognize that the top and bottom of the rectangle is made up of the outside edges of the circle, the circumference. So, the base is half of the circumference, or 1 2 (2πr). On the board, write the following equations in a vertical column so students can follow along and understand where each number is coming from. Area = base height Area = 1 2 (2πr) r Area = πr r Area = πr² 8. Say, So the area of the circle is made up of the same formula as we use for the area of a rectangle Focused Mathematics Intervention Level 7 Teacher s Guide Teacher Created Materials

5 Whole-Group (cont.) We Do 1. Refer students to the Playground Construction activity sheet (Student Guided Practice Book, page 160). Say, Let s look at another problem together. Write the problem on the board as you read it aloud: The Parent Association made a playground in the shape of a circle. The diameter is 40 yards. 2. Ask, How can we find the area of this playground? Suggest that students make a sketch of the playground and label the diameter. Students should explain that the formula for a circle is basically the same formula for a rectangle, base height. Write the formula on the board: Area = πr². 3. Ask, Do we know the radius? (If the diameter is 40 yards, then the radius is half that measurement, or 20 yards.) Guide students to insert the radius into the formula. Write the following on the board: A = πr² = 3.14 (20)² = 3.14 (20 20) = = 1,256. Say, The area of the playground is 1,256 square yards. Have students write an explanation about how they found the solution on their activity sheet. 4. Refer students to Question 2. Ask, How do we find the circumference of the playground? (Finding the circumference is the same as finding the perimeter.) Write the formula on the board: C = πd. Guide students to insert the diameter into the formula. Write the following on the board: C = πd = = Say, The circumference of the playground is yards. Have students write an explanation about how they found the solution on their activity sheet. 5. Have students explain their reasoning in Question 3. To help students explain their reasoning, provide them with the following sentence frames: The circumference of a circle means. I found the circumference by. The area of a circle means. I found the area by. Teacher Created Materials Focused Mathematics Intervention Level 7 Teacher s Guide 241

6 Whole-Group (cont.) You Do 1. Refer students to the Serious Circles activity sheet (Student Guided Practice Book, page 161). Provide the sentence frames from Step 5 of the We Do section to help students explain their reasoning. 2. Have students share their equations and reasoning. If students have difficulty explaining their reasoning, remind them to use the sentence frames and vocabulary terms. Closing the Whole-Group Revisit the focus question for the lesson: How do you find the area and circumference of a circle? Have students explain the difference between finding the circumference and area of a circle. Ask students to recall the formulas used for each. Students should indicate that the circumference of a circle is the distance around the circle, whereas the area is the number of square units that cover the surface. The formula used for the perimeter of a circle is C = πd or C = 2πr and the formula for area is A = πr². Progress Monitoring 1. Have students complete the Quick Check activity sheet (Student Guided Practice Book, page 162) to gauge student progress toward mastery of the Learning Objectives. 2. Based on the results of the Quick Check activity sheet and your observations during the lesson, identify students who may benefit from additional instruction in the Learning Objectives. These students will be placed into a small group for reteaching. See instructions on the following page Focused Mathematics Intervention Level 7 Teacher s Guide Teacher Created Materials

7 Differentiated Instruction Gather students for reteaching. The remaining students will complete the Independent Practice activity sheet (Student Guided Practice Book, page 164) to reinforce their learning and then play the Math Fluency Games. Refocus PPT Revisit the focus question for the lesson: How do you find the area and circumference of a circle? Students who struggled to understand how to find the area and circumference of a circle may benefit from additional instruction with concrete models. Provide students with circular objects, string, rulers, and centemeter graph paper. To help students find the circumference of a circle, have them cut a piece of string that is the same length as the distance around the circle. Have them measure the string with a ruler. Then, ask students to measure the circle s diameter and use the formula to calculate its circumference. Have them compare the actual measurement to the calculated measurement. To help them find the area of a circle, have students trace a circle onto graph paper. Ask them to count the square units that the circle covers, estimating for partial square units. Then, have them find the radius in square units and calculate the area using the formula. Prompt students to compare the actual measurement to the calculated measurement. Discuss whether the two measurements were close, and if so why or why not. Finally, support students as they complete Question 1 on the Refocus activity sheet (Student Guided Practice Book, page 163), and then have them solve Question 2 independently. Math Fluency Games Math Fluency Game Sets Digital Math Fluency Games Extend Learning Ask students how they might find the radius or diameter if they know the circumference or the area of the circle. Demonstrate using the formulas to find the unknown measurements. Then, have students complete the Extend Learning Task (filename: extendtask.pdf). Teacher Created Materials Focused Mathematics Intervention Level 7 Teacher s Guide 243

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