Area and Circumference of Circles


 Austin McLaughlin
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1 Teacher Created Materials Focused Mathematics Student Guided Practice Book of Circles Learning Objectives Geometry Know the formulas for the area and circumference of a circle and use them to solve problems. Mathematical Practices and Processes Reason abstractly and quantitatively. Model with mathematics. Look for and make use of structure. Progress Monitoring The Student Guided Practice Book pages below can be used to formally and informally assess student understanding of the concepts. Playground Construction Directions: Read the word problem. Answer the questions. The Parent Association made a playground in the shape of a circle. The diameter is 40 yards. 1What is the area of the playground? Show your work. Explain how you found the answer. Area = Explanation: 2What is the circumference of the playground? Show your work. Explain how you found the answer. 160 Circumference = Explanation: 3Explain your reasoning for how to find the area and circumference of a circle Focused Mathematics Student Guided Practice Book Teacher Created Materials Directions: Answer each question. Refocus 1Find a circle. 2Find a circle. What is the circumference of the circle, What is the circumference of the circle, using string and a ruler? using string and a ruler? What is the circumference of the circle, What is the circumference of the circle, using the formula? using the formula? Are there differences between the two Are there differences between the two measurements? If so, explain why. measurements? If so, explain why. What is the area of the circle, using What is the area of the circle, using graph paper? graph paper? What is the area of the circle, using What is the area of the circle, using the formula? the formula? Are there differences between the two Are there differences between the two measurements? If so, explain why. measurements? If so, explain why. Teacher Created Materials Focused Mathematics Student Guided Practice Book 163 Independent Practice Directions: Find the circumference and area of each circle in. 4 in. 10 in. 12 in. Serious Circles Directions: Read the word problem. Answer the questions. April wanted a circle rug to cover the majority of her room. She determined the radius of the circle is 5 feet long. 1 What is the area of the rug? Show your work. Explain how you found the answer. Area = Explanation: 2 April wants to sew ribbon around the outside of the rug. What is the circumference of the rug? Show your work. Explain how you found the answer. Circumference = Explanation: 3 Explain your reasoning for how to find the area and circumference of a circle. Circumference: Area: Circumference: Area: Circumference: Area: Circumference: Area: Focused Mathematics Student Guided Practice Book Teacher Created Materials Math in the Real World Unpack the Problem Solution Look Back and Explain Mini Golf A new miniature golf course is being constructed in town. One of the holes will be placed in the exact center of a circle. The hole is 8 feet from the edge of the circle. The owners are trying to find the cost to cover the entire circle with artificial turf. The cost is $3.50 per square foot. What is the cost of the turf? Make a Plan Teacher Created Materials Focused Mathematics Student Guided Practice Book 166 Directions: Solve each problem. 1What is the area of the circle? 2What is the circumference of the circle? Quick Check 9 ft. 5 in. 3At the park there is a merrygoround with a radius of 6 feet. What is the circumference? Directions: Write an equation. Solve the problem. 4What is the area of the circle? 14 in. A 3.14 sq. in. B 15.7 sq. in. C 31.4 sq. in. D 78.5 sq. in. A ft. B ft. C ft. D ft. A ft. B ft. C ft. D ft. Equation: Solution: Focused Mathematics Student Guided Practice Book Teacher Created Materials 165 Reflection 1Explain the difference between finding the area and circumference of a circle. 2Draw a visual representation to explain your answer Focused Mathematics Student Guided Practice Book Teacher Created Materials Materials Student Guided Practice Book (pages ) Math Fluency Game Sets Digital Math Fluency Games Centimeter Graph Paper (filename: centgraph.pdf) Rulers (filename: ruler.pdf) sticky notes chart paper markers scissors circular objects string Teacher Background In previous grades, students have learned how to find the perimeter and area of polygons, such as rectangles and triangles. They have also learned to find the area of complex figures by decomposing them into other shapes. Now students apply that knowledge to finding the circumference and area of a circle by using formulas. Teacher Created Materials Focused Mathematics Intervention Level 7 Teacher s Guide 7
2 WarmUp 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 WholeGroup 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 WholeGroup (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 WholeGroup (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 WholeGroup (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 WholeGroup 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
8 Math in the Real World 1. Refer students to the Math in the Real World: Mini Golf task (Student Guided Practice Book, page 165). Have a student read the task aloud. Tell students to explain or summarize the task to their partner. Have a few students share their summaries. 2. Ask students to think about what information they will need to solve the task and what the task is asking them to do. Then, have them share with a partner. Ask a few students to share out. Students should indicate that they know the hole in the center is 8 feet from the edge of the circle and that the cost per square foot for artificial turf is $ They need to find the cost of the turf. Have students work in groups of two or three to complete the task. 3. As students are working, circulate and ask focusing, assessing, and advancing questions: What information do you know? What are you trying to find out? How can you use concrete models to solve the problem? Sentence Frames for Explaining Reasoning I found the area of the circle by. I found the cost of the turf by. 4. Observe how students are solving the task, and choose a few groups who solved the task in different ways to share their solutions and reasoning. Try to have the solutions move from concrete representations to more abstract representations. For example, have students share solutions with a visual representation (drawing) and then the symbolic representation (equation). Make sure students explain their reasoning as they share solutions. Students should have found the area of the circle to find the number of square units that will cover the surface. Their work should include: A =πr² = (3.14)(8) 2 = sq. feet. Students should have found the cost of the turf by multiplying the area of the circle by the cost per square foot. Their work should include: = The cost to cover the surface of the circle is $ As groups are sharing their solution paths, reasoning, and strategies, ask questions: How is this strategy similar to one that we have seen in a previous task? Who can restate s strategy/solution path/reasoning? Which solution path makes the most sense to you? Why? Reflection Have students summarize their learning about finding the area and circumference of a circle, and provide feedback on any questions they still have about the content on the Reflection activity sheet (Student Guided Practice Book, page 166) Focused Mathematics Intervention Level 7 Teacher s Guide Teacher Created Materials
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