LESSON 6 Overview | Solve Area and Circumference Problems Involving Circles STANDARDS FOR MATHEMATICAL Objectives Vocabulary PRACTICE (SMP) Content Objectives Math Vocabulary SMP 1, 2, 3, 4, 5, and 6 are integrated into the • Derive formulas for circumference and center (of a circle) a point that is the same Try-Discuss-Connect routine.* area of a circle. distance from every point on the circle. This lesson provides additional support for: Understand that a circle’s diameter and • circle a two-dimensional shape in which 4 Model with mathematics. its circumference have a proportional relationship; the constant of every point is the same distance from the 7 Look for and make use of structure. proportionality is called pi (p). center. * See page 1q to learn how every lesson includes • Use proportional reasoning and the circumference the distance around the these SMP. formula for circumference of a circle to outside of a circle. It can be thought of as solve problems involving radius, the perimeter of the circle. circumference, and diameter. diameter a line segment that goes • Use the formula for area of a circle to solve area problems. through the center of a circle and has endpoints on the circle. Also, the distance Language Objectives across a circle through the center. • Explain the relationship between pi (p) in a circle, the quotient ​​ circumference ​​ . ···········diameter circumference, diameter, radius, and area 22 Common approximations are 3.14 and ​​ . using the lesson vocabulary. ··7 • Connect strategies for determining the proportional relationship between radius (of a circle) a line segment from the diameter and circumference by analyzing center of a circle to any point on the circle. models and describing how the Also, the distance from the center to any strategies are alike and different. point on a circle. • Interpret and solve word problems about circles using circumference and Review the following key terms. area formulas. area the amount of space inside a closed two-dimensional figure. Area is measured in square units such as square centimeters. Prior Knowledge unit rate the numerical part of a rate. For • Know that the perimeter is the distance example, the rate 3 miles per hour has a around a two-dimensional figure. unit rate of 3. For the ratio a : b, the unit rate a • Know that the area of a two-dimensional is the quotient . b figure is the amount of space enclosed ·· inside it and that area is measured in square units. Academic Vocabulary • Know that a proportional relationship is approximate almost exact. one in which ratios of quantities all share the same constant of proportionality, or unit rate. Learning Progression Earlier in Grade 7, students learned In this lesson, students apply their In Grade 8, students will apply their about ratios, rates, scale models, and knowledge to recognize that the understanding of proportional proportional relationships. They learned circumference of a circle is proportional relationships to identify the properties that the measurements of a geometric to its diameter and that pi is the of linear functions. They will extend figure can be proportional to one constant of proportionality. They their knowledge of circles to find the another and what makes one figure a identify the area of a circle as a property volumes of spheres, cylinders, and scale copy of another. that is related to its circumference and cones. radius but that is not proportional to either of them. 97a LESSON 6 Solve Area and Circumference Problems Involving Circles ©Curriculum Associates, LLC Copying is not permitted. LESSON 6 Overview Pacing Guide MATERIALS DIFFERENTIATION Items marked with are available on the Teacher Toolbox. SESSION 1 Explore Circumference of a Circle (35–50 min) • Start (5 min) Math Toolkit grid paper, string PREPARE Interactive Tutorial • Try It (5–10 min ) RETEACH or REINFORCE Visual Model • Discuss It (10–15 min) Materials For display: meter stick, sticky notes • Connect It (10–15 min) Presentation Slides • Close: Exit Ticket (5 min) Additional Practice (pages 101–102) SESSION 2 Develop Using the Relationship Between a Circle’s Circumference and Diameter (45–60 min) • Start (5 min) Math Toolkit compasses, flexible RETEACH or REINFORCE Hands-On Activity • Try It (10–15 min ) tape measures, rulers, string Materials For each pair: at least 3 different-sized • Discuss It (10–15 min) circular objects, ruler, string • Connect It (15–20 min) Presentation Slides REINFORCE Fluency & Skills Practice • Close: Exit Ticket (5 min) EXTEND Deepen Understanding Additional Practice (pages 107–108) SESSION 3 Develop Finding the Area of a Circle (45–60 min) • Start (5 min) Math Toolkit grid paper, RETEACH or REINFORCE Hands-On Activity • Try It (10–15 min ) tracing paper Materials For each pair: ruler, scissors, tape • Discuss It (10–15 min) Activity Sheet Grid Paper • Connect It (15–20 min) Presentation Slides REINFORCE Fluency & Skills Practice • Close: Exit Ticket (5 min) EXTEND Deepen Understanding Additional Practice (pages 113–114) SESSION 4 Refine Solving Circumference and Area Problems Involving Circles (45–60 min) • Start (5 min) Math Toolkit Have items from RETEACH Hands-On Activity • Monitor & Guide (15–20 min ) previous sessions available Materials For each small group: compass, • Group & Differentiate (20–30 min) for students. Activity Sheet Grid Paper • Close: Exit Ticket (5 min) REINFORCE Problems 4–10 Presentation Slides EXTEND Challenge PERSONALIZE Lesson 6 Quiz or RETEACH Tools for Instruction Digital Comprehension Check REINFORCE Math Center Activity EXTEND Enrichment Activity ©Curriculum Associates, LLC Copying is not permitted. LESSON 6 Solve Area and Circumference Problems Involving Circles 97b LESSON 6 Overview | Solve Area and Circumference Problems Involving Circles Connect to Culture ➤ Use these activities to connect with and leverage the diverse backgrounds and experiences of all students. Engage students in sharing what they know about contexts before you add the information given here. SESSION 1 Try It When you fly over the Great Plains in an airplane, you can see green sections of irrigated crops surrounded by dry grasslands. Soon after its invention in the 1940s, the center pivot irrigation system became very popular because it delivered water easily across a wide area. In 1976, Scientific American published the opinion that the invention was the most significant mechanical improvement to agriculture since the tractor. Ask students to speculate about how farms were watered before this system was invented. SESSION 2 Apply It Problem 9 The word gong comes from the language of Java, an island in southeast Asia. Gongs date back thousands of years in regions across Asia, and they remain in use today in many cultures. A gong is generally circular and flat, like a plate, and made of metal. Like bells and chimes, a gong is a type of percussion instrument, meaning it is played by being struck. The sound of a gong is often used as an effect in television and radio programs, often to signify the end of an event. Ask students to name and discuss other examples of circular shapes in music, such as drum faces, cymbals, tambourines, and the rim of the bell of a horn. SESSION 3 Apply It Problem 8 Ask students to describe their experiences playing darts or similar games, such as lawn darts or ring toss. Games of darts are played in basements, club halls, recreation centers, and many places where people gather to relax. However, darts is also played as a serious sport. Professional darts is played according to a long list of rules and regulations. For example, the dartboard must be centered at a height of 5 feet, 8 inches above the ground, and the horizontal distance from the board to the toe line must be 7 feet, 9.25 inches. DIFFERENTIATION | ENGLISH LANGUAGE LEARNERS SESSION 4 Apply It Problem 3 Paving stones, or pavers, can be made of natural stone, brick, or concrete. Each has different properties, including cost, strength, durability, and resistance to stains. All provide a hard, relatively flat surface for outdoor use. Patios can take on a wide variety of colors and designs based on the pavers used to make them. Ask students to discuss the patios or outdoor walkways with designs made from pavers or bricks that they have seen. 97c LESSON 6 Solve Area and Circumference Problems Involving Circles ©Curriculum Associates, LLC Copying is not permitted. LESSON 6 Overview Connect to Family and Community ➤ After the Explore session, have students use the Family Letter to let their families know what they are learning and to encourage family involvement. LESSON 6 | SOLVE AREA AND CIRCUMFERENCE PROBLEMS INVOLVING CIRCLES LESSON Activity Thinking About Circles Around You 6 ➤ Do this activity together to investigate circles in the real world. Have you ever wondered why runners start at diff erent spots Dear Family, Circles Involving Problems Circumference and Area Solve on a track instead of lining up together? This is called a This week your student is learning about circumference and area of circles. staggered start. Circumference is the distance around a circle, or its Since tracks have semicircles (or half circles) on the ends, if all perimeter. Area is the amount of space inside a circle. the runners lined up together, runners in outer lanes would The radius, r, is the distance from the center of a circle to need to run farther than runners in inner lanes. Why? Because any point on the circle. The diameter, d, is the distance diameter, d the circumference of the outer lane is greater than the from one side of the circle to the other, passing through circumference of the inner lane. the center of the circle. center Staggering starting positions ensures that all runners run the radius, r A special relationship exists between the circumference of same distance! a circle and its diameter.