Overview. Essential Questions. Grade 7 Mathematics, Quarter 3, Unit 3.3 Area and Circumference of Circles. Number of instruction days: 3 5


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1 Area and Circumference of Circles Number of instruction days: 3 5 Overview Content to Be Learned Develop an understanding of the formulas for the area and circumference of a circle. Explore the relationship between circumference and area of a circle to develop an understanding of how the area of a circle is found. Understand the relationship among area, circumference, radius, and diameter in circles. Solve reallife mathematical problems involving circumference and area of a circle. Mathematical Practices to Be Integrated 5 Use appropriate tools strategically. Demonstrate familiarity with tools (i.e., protractors, rulers) appropriate to grade level. Make sound decisions about when these tools might be helpful, recognize the insight to be gained from them and their limitations. Detect possible errors by strategically using estimation and other mathematical knowledge. 6 Attend to precision. Use clear definitions and reasoning in discussion with others. Specify units of measure. Calculate accurately and efficiently. Express numerical answers with a degree of precision appropriate for the problem context. 7 Look for and make use of structure. Look closely to discern patterns or structure. See complicated things as single objects or as being composed of several objects. Essential Questions How can you determine the area of a circle? How can you determine the circumference of a circle? What is the relationship among circumference, radius, and diameter in circles? Given the area of a circle, how do you find the circumference? Given the circumference of a circle, how do you find the area? Providence Public Schools D63
2 Area and Circumference of Circles (3 5 days) Standards Common Core State Standards for Mathematical Content Geometry 7.G Solve reallife and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Common Core State Standards for Mathematical Practice 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several D64 Providence Public Schools
3 Area and Circumference of Circles (3 5 days) Grade 7 Mathematics, Quarter 3, Unit 3.3 objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In Grade 6, students solve a variety of problems involving angle measure, perimeter, area, surface area, and volume. Current Learning Students learn the formulas for area and circumference of a circle. They solve realworld and mathematical problems involving area, volume, and surface area of two and threedimensional objects. Future Learning Students will know the formulas for the volume of cones, cylinders, and spheres. They will use these formulas to solve realworld and mathematical problems. Additional Findings Students understand relationships among the angles, side lengths, perimeters, and areas and volumes of similar objects. (Curriculum Focal Points, p. 37) An understanding of the concepts of perimeter, area, and volume is initiated in lower grades and extended and deepened in grades 68. Whenever possible, students should develop formulas and procedures meaningfully through investigation rather than memorize them. Even formulas that are difficult to justify rigorously in the middle grades, such as that for the area of a circle, should be treated in ways that help students develop an intuitive sense of their reasonableness. (Principles and Standards for School Mathematics, p. 244) Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. Providence Public Schools D65
4 Area and Circumference of Circles (3 5 days) The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Know and apply the formulas for the area and circumference of a circle to solve mathematical and real world problems. Demonstrate an understanding of the relationship among area, circumference, radius, and diameter in circles. Find the area of a circle given the circumference. Find the circumference of a circle given the area of a circle. Learning Objectives Students will be able to: Instruction Measure the circumference of circles and develop a formula for finding circumference of any circle. Determine a formula for finding the area of a circle. Use formulas for the area and circumference of a circle to solve mathematical and real world problems. Determine the relationship between a circle s area and its circumference. Demonstrate knowledge of area and circumference of circles. Resources Connected Mathematics 2, Pearson/Prentice Hall, 2008: Covering and Surrounding Investigation 5: Measuring Irregular Shapes and Circles Problem 5.2: Surrounding a Circle, Student Book (pages 7273) Problem 5.3: Pricing Pizza, Student Book (pages 7475) Problem 5.4: Squaring a Circle, Student Book (pages 7677) Teacher s Guide Implementing and Teaching Guide Teaching Transparencies Assessment Resource Book Additional Practice and Skills Workbook Strategies for English Language Learners Special Needs Handbook Parent Guide Prentice Hall Teacher Station Software D66 Providence Public Schools
5 Area and Circumference of Circles (3 5 days) Grade 7 Mathematics, Quarter 3, Unit 3.3 Exam View Software (Students can enter webcodes) Connected Mathematics 2, Pearson/Prentice Hall, 2011: Common Core Additional Investigations Grade 7 CC Investigation 4: Geometry Topics; Problem 4.2 Investigations are located in the Worksheets tab Common Core Investigations Teacher s Guide Implementing a Common Core Curriculum Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010) Available with the Algebra resources Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials Centimeter grid paper (2 to 3 sheets per student), student notebooks, rulers, calculators, string, tape measure, scissors, several circular objects, glue, construction paper, geometric nets, clear tape or glue sticks, scissors, tennis balls or other spheres, student handouts Instructional Considerations Key Vocabulary diameter circumference radius (radii) pi ( ) Planning for Effective Instructional Design and Delivery Reinforced vocabulary from previous grades or units: area, center, circle. Living word walls assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, both new and reinforced, and have pictures, examples, and/or diagrams to accompany the definitions. Teachers should review the Mathematics of the Unit found on page 3 of all CMP2 teacher editions. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the CMP2 resource. In Covering and Surrounding Problem 5.2 uses a cues, questions, and advance organizers strategy to introduce new knowledge. Page 72 in the student book has an expository advance organizer that uses a pictographic representation to introduce key vocabulary for this lesson. Have students copy and label this picture in their notes. For Problem 5.2, have students measure many different sizes of circles. If time allows, students would enjoy finding a way to measure the circles on a basketball court. This would generate interest and allow them to study the relationship between radius, diameter, and circumference for large and small circles. This work will help students lay the foundation for an understanding of the number pi. Providence Public Schools D67
6 Area and Circumference of Circles (3 5 days) The CC Investigation 4: Geometry Topics Problem 4.2 should be done after the problems in Covering and Surrounding because Problem 4.2 only focuses on application of the area and circumference formulas. A common error students make is to use the diameter instead of the radius when finding the area of a circle. Check in with them as they work through the problems to monitor and correct this common error. Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions should be included in the unit assessment. CMP2 has online resources that may be helpful in planning for all units of study. Visit and sign on to SuccessNet. You will find the Common Core Additional Investigations and Common Core Investigations Teacher s Guide under the worksheet tab. Notes D68 Providence Public Schools
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