Unit Five Organizer: “CIRCLES AND GRAPHS” (4 weeks) |
OVERVIEW: This unit, which falls at the end of the second nine weeks, should be viewed as a “bridge“ between units taught in the first semester and the units to come. It addresses the geometry of the circle which will eventually be taught in the fifth grade GPS but is not currently taught in the fifth grade QCC. Students will: • Understand the relationship between the circumference and the diameter of a circle; • Find the radius, diameter, circumference and/or the area of a circle given appropriate information; • Organize data in grouped frequency tables; • Create circle graphs to display data; • Operate with fractions, decimals and percents to answer questions related to graphs; • Evaluate, in context, algebraic expressions, including those with exponents; and • Solve algebraic equations related to circles. Students will discover Pi (π) and use the equation C/d = π to find the circumference, the radius and the diameter of a circle. They will derive the formula for the area of a circle by cutting a circle into equal sectors and noticing that the more sectors into which the circle is cut, the closer the area formed approximates the area of a rectangle. They will organize available data into a grouped frequency table and represent that data with a circle graph. To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources. |
ENDURING UNDERSTANDINGS: • The ratio of the circumference to the diameter of any circle is a constant approximately equal to 3.14. • Formulas can be used to help us find missing measurements of figures. • The area of a circle can be approximated using the area of a rectangle. • Fractions, decimals and percents help us solve problems and make sense of data. • Some data sets are best displayed using circle graphs. |
ESSENTIAL QUESTIONS: • What is the relationship between the circumference and the diameter of a circle? • How can we determine the formula for the area of a circle? • When should I use a circle graph? • How do circle graphs help me compare different groups? • How can fractions, decimals and percents help me answer questions related to data? |
STANDARDS ADDRESSED IN THIS UNIT |
Mathematical standards are interwoven and
should be addressed throughout the year in as many different units and
activities as possible in order to emphasize the natural connections that exist among mathematical topics. KEY STANDARDS: M6D1. Students will pose questions, collect data, represent and analyze the data, and interpret results. b. Using data, construct frequency distributions, frequency tables, and graphs. c. Choose appropriate graphs to be consistent with the nature of the data (categorical or numerical). Graphs should include pictographs, histograms, bar graphs, line graphs, circle graphs, and line plots. d. Use tables and graphs to examine variation that occurs within a group and variation that occurs between groups. e. Relate the data analysis to the context of the questions posed. M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems. d. Add and subtract fractions and mixed numbers with unlike denominators. e. Multiply and divide fractions and mixed numbers. f. Use fractions, decimals, and percents interchangeably. g. Solve problems involving fractions, decimals, and percents. M6M2. Students will use appropriate units of measure for finding length, perimeter, area and volume and will express each quantity using the appropriate unit. a. Measure length to the nearest half, fourth, eighth and sixteenth of an inch. b. Select and use units of appropriate size and type to measure length, perimeter, area and volume. M6A1. Students will understand the concept of ratio and use it to represent quantitative relationships. M6A3. Students will evaluate algebraic expressions, including those with exponents, and solve simple one-step equations using each of the four basic operations. RELATED STANDARDS: M6P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M6P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. M6P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. M6P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics.
M6P5. Students will represent mathematics in multiple ways. |
CONCEPTS TO MAINTAIN: It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. • the number of degrees in a complete revolution • measuring angles using a protractor • drawing angles of a given degree |
EVIDENCE OF LEARNING: By the conclusion of this unit, students should be able to demonstrate the following competencies: • Use the formula C = πd or C = 2πr to find the missing lengths on a circle; • Find the area of a circle given the radius or the diameter; • Organize data into grouped frequency tables; • Display data using a circle graph; • Evaluate algebraic expressions, including those with exponents; and • Solve one step algebraic equations related to the formula for the circumference of a circle. The following task represents the level of depth, rigor, and complexity expected of all 6^{th} grade students. This task or a task of similar depth and rigor should be used to demonstrate evidence of learning. Culminating Activity: “Data and Circle Graphs” Students will organize, represent and analyze a set of given data. |
STRATEGIES FOR TEACHING AND LEARNING: • Students should be actively engaged in developing their own understanding; • Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words; • Appropriate manipulatives and technology should be used to enhance student learning; and • Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection. |