E XPLAINING A REA AND C IRCUMFERENCE OF A C IRCLE,LEARNING MAP INFORMATION. 7 G 4 Know the formulas for the area and circumference of a circle and solve problems give an informal derivation of. the relationship between the circumference and area of a circle. calculate the,area of a represent,explain explain polygon by equations. decomposing explain ratio with numbers,perimeter diameter it into and or. rectangles variables,calculate the construct,explain explain area of a simple. parallelogram equations to,circumference radius with the represent. formula problems,real world,explain pi equations,nonnegative. explain the,explain the,formula for,circumference,the area of a. use the use the,formula for formula for,the area of a circumference. circle to to solve,solve problems,explain the,relationship. circumference,and the area,of a circle, The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. Node Name Node Description, CALCULATE THE AREA OF A PARALLELOGRAM Calculate the area of a parallelogram using the formula A bh where b is. WITH THE FORMULA the base of the parallelogram and h is the height of the parallelogram. CALCULATE THE AREA OF A POLYGON BY Using words drawings models etc decompose a polygon including. DECOMPOSING IT INTO RECTANGLES special quadrilaterals into rectangles and or triangles and calculate the area. AND OR TRIANGLES of the polygon by summing the areas of the rectangles and triangles. CONSTRUCT SIMPLE EQUATIONS TO Construct simple equations one step or two step to represent real world. REPRESENT PROBLEMS or mathematical problems, Make known your understanding through words drawings concrete. EXPLAIN CIRCUMFERENCE, models etc that circumference is the perimeter of a circular area. Make known your understanding through words drawings concrete. EXPLAIN DIAMETER models etc that the diameter of a circle is any straight line that passes. through the center and touches the circle at each endpoint. Make known your understanding through words drawings manipulatives. EXPLAIN LENGTH etc that length is the distance along a path between two points on that. Make known your understanding through words drawings concrete. EXPLAIN PERIMETER models etc that perimeter is a path length that surrounds or encloses a. plane area, Make known your understanding through words drawings concrete. EXPLAIN PI models etc that pi is the ratio of the circumference of a circle to its. Make known your understanding through words drawings concrete. EXPLAIN RADIUS models etc that the radius of a circle is any straight line from the. circumference of the circle to the center, Make known your understanding through words drawings manipulatives. etc that a ratio represents a multiplicative comparison of two quantities or. EXPLAIN RATIO the joining of two quantities into a composed unit For example the ratio. of eyes to nose on a person is 2 1 because for every two eyes there is one. Make known your understanding through words drawings manipulatives. EXPLAIN THE CIRCUMFERENCE FORMULA, etc that the formula of the circumference of a circle is 2 pi r or pi d. EXPLAIN THE FORMULA FOR THE AREA OF A Make known your understanding through words drawings manipulatives. CIRCLE etc that the formula for the area of a circle is pi r 2. EXPLAIN THE RELATIONSHIP BETWEEN Make known your understanding through words drawings manipulatives. CIRCUMFERENCE AND THE AREA OF A CIRCLE etc of the relationship between circumference and area of a circle. REPRESENT EQUATIONS WITH NUMBERS Through writing or an appropriate assistive technology represent equations. AND OR VARIABLES with numbers variables or both,SOLVE REAL WORLD PROBLEMS USING. Use equations with nonnegative rational numbers to solve real world. EQUATIONS WITH NONNEGATIVE RATIONAL, USE THE FORMULA FOR THE AREA OF A Use the formula for area of a circle to solve real world or mathematical. CIRCLE TO SOLVE PROBLEMS problems, USE THE FORMULA FOR CIRCUMFERENCE TO Use the formula for circumference to solve real world or mathematical. SOLVE PROBLEMS problems, The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. ADDITIONAL NODES RELATED TO THIS UNIT OF INSTRUCTION. Node Name Node Description Related Node,Make known your understanding. through words drawings, EXPLAIN AREA manipulatives etc that area is a two Postrequisite of EXPLAIN LENGTH. dimensional quantity representing the,amount of space in a surface. The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. E XPLAINING A REA AND C IRCUMFERENCE OF A C IRCLE,TEACHER NOTES. This unit includes the following documents,u Learning Map Information. u Instructional Activity two lessons, u Instructional Activity Student Handout for Lesson 1 and Lesson 2. u Instructional Activity Supplement for Lesson 2,u Student Activity. u Student Activity Solution Guide, In this unit students will learn about the radius diameter circumference and area of a circle The learning. activities provide the context for students to notice properties and create mathematically accurate and. meaningful definitions for properties of and measurements in a circle Sinclair 2012 As recommended by. Van de Walle 2014 students will measure and explore relationships among radius diameter and. circumference which will lead them to discover that pi represents the ratio of the circumference of a circle to. the diameter Students need to develop an awareness that pi is an irrational value which must always be. approximated when making calculations Using a rounded value of 3 14 for pi means all values calculated. using the rounded value are approximations In addition reporting values where an exact value for pi is used. to calculate e g pi button on a calculator requires rounding and therefore those values are approximations. In the first lesson students will measure the radius diameter and circumference of several different circular. objects Series of guiding questions steer students toward understanding the relationship between the radius. and the diameter as well as the relationship between the diameter and the circumference of a circle By. examining circles of different sizes organizing the measurements in tables and computing the ratio of each. circle s circumference to its diameter students will develop an understanding of how the ratio. circumference diameter remains constant regardless of a circle s size By exploring the relationships among. the measurable features of circles students will develop a strong foundation for understanding two formulas. for circumference of a circle C d and C 2 r and an understanding of why both formulas are accurate. The second lesson addresses the area of a circle This lesson builds on students previous experiences with. area of polygons to establish a conceptual foundation for the formula for the area of a circle A key skill in. geometry is the ability to create attend to and learn how to work with imagery Sinclair 2012 Students will. use drawings of shapes to describe and analyze properties of shapes and to consider different ways to view. shapes The lesson uses imagery throughout beginning with an activity to guide the development of the. formula for area based on what students know about the circumference of a circle and the area of a. parallelogram An important conceptual prerequisite is understanding that decomposing and rearranging the. parts of a two dimensional shape preserves the area Building on this knowledge students will decompose. circles into sectors and rearrange the sectors into shapes that approximate parallelograms As the number of. sectors increases and the size of each sector decreases the resulting shapes look more like parallelograms. This activity promotes students use of imagery through opportunities to reimagine how a circle s area can be. The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. represented while preserving its original size By connecting the area of a circle to the area of a parallelogram. students can establish meaningful connections to their prior study of area of parallelograms in 6 G 1. The learning map section for this sequence of activities includes prerequisite knowledge of length perimeter. and the ability to represent and solve equations Building on an understanding of length students can learn to. explain diameter and radius An understanding of perimeter provides the foundation for students to explain. circumference Knowledge of circumference and diameter together permit students to explore relationships. between them to develop an understanding of pi as the ratio of the circumference of a circle to the diameter. Then students should be ready to explore and explain the formulas for area and circumference of circles use. the formulas to solve real world and mathematical problems and explain the relationship between. circumference and area, The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. REFERENCES, Sinclair N Pimm D Skelin M 2012 Geometry The big ideas and essential understandings Developing. essential understanding of geometry for teaching mathematics in grades 6 8 pp 7 68 Reston VA NCTM. Van de Walle J Bay Williams J Karp K Lovin L 2014 Exploring measurement concepts Teaching. student centered mathematics Developmentally appropriate instruction for grades 6 8 Volume I Second ed pp. 298 324 Upper Saddle River NJ Pearson Education Limited. The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. E XPLAINING A REA AND C IRCUMFERENCE OF A C IRCLE,INSTRUCTIONAL ACTIVITY. LEARNING GOAL, Students will develop an understanding of diameter radius circumference and pi and the relationships. among them and will solve problems using the circumference formula. PRIMARY ACTIVITY, Students will measure circular objects and create definitions to support their understanding of diameter. radius circumference and pi,OTHER VOCABULARY,Students will need to know. u Distance,u Perimeter, u Several circular objects of different sizes up to 10. u Rulers or tape measures with centimeter markings. u I NSTRUCTIONAL ACTIVITY S TUDENT H ANDOUT,IMPLEMENTATION. Begin the lesson by having students measure the circular objects available and record their results as described. in the I NSTRUCTIONAL ACTIVITY S TUDENT H ANDOUT The following table is an example row of. student work, The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. DISTANCE AROUND THE,DISTANCE FROM THE DISTANCE ACROSS THE. DISTANCE AROUND CIRCLE DISTANCE,OBJECT CENTER OF THE CIRCLE WIDEST PART OF THE. THE CIRCLE ACROSS THE WIDEST,TO THE EDGE CIRCLE,PART OF THE CIRCLE. about 3 218, small paper plate 9 centimeters 17 9 centimeters 57 6 centimeters. centimeters, Emphasize the importance of measuring in centimeters as accurately as possible for each object. Students should measure each object to the nearest tenth of a centimeter. Model the measurements students will make using a circular object Tape measures or string can be. used to measure the distance around the circle circumference and to measure straight distances. from the center of the circle to the edge radius or across the widest part of a circle diameter. Check the students work as they fill in the table to ensure their measurements are accurate. particularly in the column where students divide the circumference by the diameter to calculate. approximations of pi Note that the value in this column should be close to pi approximately. 3 14159 but will not be exact due to human error in calculation. Require students to answer Questions 1 7 to summarize their work in the table and look for. patterns in their data in pairs Students should write either d 2r or r d for Question 3 C d. for Question 6 and C 2 r for Question 7, Provide scaffolding when necessary using the following guiding questions As students describe and. show the diameter and radius of a circle they should articulate the fact that both the diameter and. radius pass through the center of the circle When students describe how to determine the distance. around an object it is important that they describe a process that measures the entire continuous. distance no gaps and does not measure any portion of the perimeter more than one time no. The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. GUIDING QUESTIONS,Elicit student thinking,u What can you tell me about this circle. u Can you describe different properties of this circle. Determine if the student can EXPLAIN DIAMETER, u What do you picture when you hear the word diameter. u Can you show me where the diameter of this circle is. u How would you describe the diameter of a circle in your own words. Determine if the student can EXPLAIN RADIUS, u What do you picture when you hear the word radius. u Can you show me where the radius of this circle is. u How would you describe the radius of a circle in your own words. Determine if the student can EXPLAIN PERIMETER,u What do we call the distance around a rectangle. u How can you determine the distance around a shape. u Can you show me the perimeter of this circle, The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. GUIDING QUESTIONS, Determine if the student can EXPLAIN CIRCUMFERENCE. u What shape are we referring to when we talk about circumference. u What is the difference between perimeter and circumference. u How would you describe the circumference of a circle in your own words. Determine if the student can EXPLAIN PI,u How did we determine the value of pi. u How would you describe pi, Determine if the student can EXPLAIN THE CIRCUMFERENCE FORMULA. u What is the circumference formula Is there another option for the circumference. u How do these formulas relate to your measurements. Once most students have completed Questions 1 7 review student answers as a class and ensure students. correct their work when needed Inform students that pi is approximately 3 14159 and that it is an irrational. number because it does not end or repeat Make sure students realize that operations involving pi result in an. approximate value because either pi or the result must be rounded Refer to the T EACHER N OTES for more. information regarding approximations and pi, Practice a few questions as a class where students are provided either radius diameter or. circumference and asked to find a different measurement Discuss what is given what students are. trying to find where each is located on the circle and which equation is most efficient based on the. situation Following are some example questions, u If the radius of a circle is 2 inches what is the diameter What is the circumference. u If the diameter of a circle is 10 centimeters what is the circumference What is the radius. u If the circumference of a circle is 25 feet what is the diameter What is the radius. u You are building a circular fence around your garden If the radius of garden is 3 feet how much. fencing do you need, NOTE Real world situations in this unit provide opportunities to ask. students to extend the measurement calculation and determine how much of. The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. the given material would need to be purchased In the preceding example. there is an opportunity to ask students how many sections of fence they. would have to purchase if it is sold in 6 foot sections This leads to a. discussion that in real world situations rounding up is necessary to ensure. there is enough material to complete the job, Require students to practice Questions 8 13 to find either circumference radius or diameter in real. world and mathematical problems,GUIDING QUESTIONS, Determine if the student can USE THE FORMULA FOR CIRCUMFERENCE TO SOLVE. u What information do you have about this circle, u What are you trying to determine Can you describe this measurement in. relation to the circle, u What equation or process is most efficient for answering this question. u How would you simplify solve this equation or carry out this process. Students should be required to show all work as they complete the I NSTRUCTIONAL ACTIVITY S TUDENT. H ANDOUT and write their answers in complete thorough well thought out sentences. At the end of the activity teachers should provide students with an example requiring them to use either C. d or C 2 r to solve a problem as an exit ticket for the day. The Dynamic Learning Maps including node names descriptions and connections have been developed by the Achievement and Assessment Institute. and are copyrighted by the University of Kansas Center for Research Learning map information is available for use by educators but may not be used. for commercial purposes including being shared without cost when bundled in a commercial transaction without written permission Dynamic Learning. Map nodes and connections may not be altered by anyone other than the staff of the Achievement and Assessment Institute at the University of Kansas. E XPLAINING A REA AND C IRCUMFERENCE OF A C IRCLE, For each object make the measurements described in the following table Use the string to help measure. when needed Be sure to measure in centimeters and be as precise as possible. DISTANCE AROUND,DISTANCE FROM THE DISTANCE ACROSS,DISTANCE AROUND THE CIRCLE DISTANCE. OBJECT CENTER OF THE CIRCLE THE WIDEST PART OF,THE CIRCLE ACROSS THE WIDEST. TO THE EDGE THE CIRCLE,PART OF THE CIRCLE,Copyright 2015 by The University of Kansas 1. 1 What do you notice about the distance across the widest part of the circle and the distance from the. center of the circle to the edge, 2 The distance across the widest part of the circle is called the diameter Write a definition of diameter in. your own words, 3 The distance from the center of the circle to a point on the edge is defined as the radius What is the. relationship between the length of the radius and the length of the diameter in each row of your. table Write an equation that relates the diameter d to the radius r. 4 What do you notice about the distance around the circle divided by the diameter of the circle These. numbers are in the last column of the table, 5 The ratio of the distance around the circle to the diameter of the circle is defined as pi Using the. values in one row of your table explain how the diameter the distance around the circle and pi are. related mathematically, 6 The distance around the circle is defined as the circumference Write an equation that relates the. circumference C to the diameter d, 7 Using what you know about the relationship between the radius of the circle and the diameter write. an equation that relates the circumference C to the radius r. Copyright 2015 by The University of Kansas 2, 8 The diameter of a circle is 3 feet What is the radius What is the circumference. 9 The radius of a circle is 4 5 centimeters What is the circumference What is the diameter. 10 The circumference of a circle is 15 7 yards What is the diameter What is the radius. 11 The diameter of a circular swimming pool is 20 feet What is the radius of the pool What is the. circumference, 12 The circumference of a pizza is 44 inches What is the approximate radius of the pizza. 13 Measuring around the outside of a circular fence you determine the length is 31 4 feet How far is it. across the widest part of the circle,Copyright 2015 by The University of Kansas 3.

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