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The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary.
Many more GaDOE approved instructional plans are available by using the Search Standards feature located on GeorgiaStandards.Org.
Georgia Performance Standards Framework for Mathematics – Grade 6
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.
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Kathy Cox, State Superintendent of Schools
Unit 5 Organizer y CIRCLES AND GRAPHS
September 20, 2006 y Page 1 of 16
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Georgia Performance Standards Framework for Mathematics – Grade 6
•
•
•
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.
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Kathy Cox, State Superintendent of Schools
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Georgia Performance Standards Framework for Mathematics – Grade 6
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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 5 Organizer y CIRCLES AND GRAPHS
September 20, 2006 y Page 3 of 16
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Georgia Performance Standards Framework for Mathematics – Grade 6
M6P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
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
SELECTED TERMS AND SYMBOLS:
You may visit www.intermath-uga.gatech.edu and click on dictionary to see definitions and specific examples of terms and symbols
used in the sixth grade GPS.
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;
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 5 Organizer y CIRCLES AND GRAPHS
September 20, 2006 y Page 4 of 16
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Georgia Performance Standards Framework for Mathematics – Grade 6
•
•
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 6th 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.
TASKS:
The collection of the following tasks represents the level of depth, rigor and complexity expected of all sixth grade students to
demonstrate evidence of learning.
• Discovering Pi
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Unit 5 Organizer y CIRCLES AND GRAPHS
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Georgia Performance Standards Framework for Mathematics – Grade 6
Discovering Pi
Today you will be working in groups to take and record measurements from circles of varying sizes. You will measure the
circumference and the diameter of each circle and then examine the ratio (length of the circumference/length of the
diameter) of these two lengths. The measurements should be recorded in the table below. In the third column, express the
ratio, represented by C/d, as a fraction and then as a decimal. You may use your calculator to calculate the decimal value.
Round each ratio in decimal form to the nearest hundredth.
What do you notice about the ratio of the length of the circumference to the length of the diameter? Make a conjecture about
this ratio. Compare your results with the results of other groups. Do the findings of the other groups support your
conjecture?
Description of object
Length of circumference
Length of diameter
C/d
*********************************************************************************
Discovering Pi
Discussion, Suggestions, Possible Solutions
Make sure that you provide students with circles of varying sizes, including some that are relatively small and some that are
relatively large. Suggestions include jewelry (rings, bracelets), spools of thread, flower pots, tires, hula hoops, round tables,
etc.
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Unit 5 Organizer y CIRCLES AND GRAPHS
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Georgia Performance Standards Framework for Mathematics – Grade 6
You may want to begin this task with a mini-lesson on rounding since students are asked to round to the nearest hundredth.
Allow students to decide on the units they should use to measure the lengths on each circle. After the groups begin, check to
be sure that students understand that they must use the same units of measure to measure the circumference and the
diameter of a given circle.
In a closing discussion, be sure to clear up any issues or misconceptions related to measurement that still exist. Guide
students to the idea that the ratio C/d is around 3.14. End the lesson by telling students some form of the following. “You
have discovered one of the most important ratios in mathematics. It is so important that it was given a special name by the
Greeks. That name is Pi and is represented by the symbol, π. Later on, when you study irrational numbers, you will learn
that the actual value of Pi is a decimal value that is non-terminating and non-repeating. It goes on forever! For our work
this year, we will use the approximation 3.14 for π.”
• Using the Equation C/d = π
Using the Equation C/d = π
a. In the equation C/d = π, C and d are called variables. π is referred to as a constant. Explain why.
3 cm
Use this circle to help you answer parts b-d.
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b. Look at the given circle. What is its radius? What is its diameter? Explain how you know.
c. Use the equation you have discovered, C/d = π and what you have learned about solving equations to find the
circumference of this circle.
d. We can write the equation C/d =π as C = πd. Explain why.
e. We say the equation C =πd is “solved for C in terms of d”. In other words the equation gives us the length of the
circumference of a circle if we know its diameter. We can also find the diameter if we know the circumference.
Solve the equation for d. What operation did you use? Why?
f. Suppose the circumference of a circle is 24 inches. What is its diameter to the nearest tenth of an inch?
g. Use what you know about the relationship between the diameter and the radius of a circle to write an equation for the
circumference in terms of the radius. Explain your thinking.
h. Practice what you have learned. Find the indicated length for each circle below. Round each of your answers to the
nearest tenth of a unit.
A circle has a radius of 5 inches. Find its circumference.
A circle has a circumference of 21 feet. Find its diameter.
A circle has a radius of ¾ yards. Find its circumference.
A circle has a diameter of 12.8 inches. Find its circumference.
*********************************************************************************
Using the Equation C/d = π
Discussion, Suggestions, Possible Solutions
An excellent mini-lesson for this task could involve the conventions for multiplying and dividing numbers and letters. Be sure
that students know that 4n means 4 times n and could also be written as 4·n. Ask students why they think we might not want
to write 4xn in algebra. Conventions that should be mentioned include:
• 1·n, 1xn, 1n are usually written as n. Ask students why they think this might be true.
• mn means m times n
• n/m means n divided by m and can also be written n÷m.
• n/n = 1. Ask students why they think this might be true.
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Unit 5 Organizer y CIRCLES AND GRAPHS
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Georgia Performance Standards Framework for Mathematics – Grade 6
At some point students will also need to discuss the difference between calculating values using the approximation 3.14 for π
and calculating values using the π button on their calculators.
• Deriving the Area of a Circle
Deriving the Area of a Circle
Today you will work in groups of four. You will be given four paper circles all the same size. You are to cut your circles
into equal sectors. One circle should be cut into 6 sectors, one into 8 sectors, one into 12 sectors and one into 18 sectors. Use
your sectors to help you do the following:
a. Estimate the area of the circle.
b. Explain in words how you think you might find the area of the circle. Justify your thinking.
c. Use your work in parts a and b to explain how you could find the area of any circle.
d. Suppose the radius of a circle is represented by the variable r. Use the work that you have done so far to write a
formula for the area of a circle.
*********************************************************************************
Deriving the Area of a Circle
Discussion, Suggestions, Possible Solutions
Give each group of students four paper circles. Circles with a diameter in the neighborhood of 8 inches work well. Ask
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students how they might find the center of the circle. (Folding works well.) Once the center has been marked, students
may divide their circles into equal sectors by folding and/or using a protractor. This task is good preparation for the
work on circle graphs that is to follow. Let students struggle with this activity. The diagram below shows how students
may arrange their sectors to approximate the area of the circle. They should grasp the idea that the more sectors they
have, the closer the figure approximates a rectangle. As they work through the task, students should be able to
generalize their work to the formula for the area of a circle.
• Circles and Sectors
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Georgia Performance Standards Framework for Mathematics – Grade 6
Circles and Sectors
The shaded regions above are called sectors of the circle.
1. Estimate the degree measure of each sector. Explain your thinking.
2. Using your estimates, determine the fractional part of the circle represented by each sector. Show how you know.
Use your fractions to be sure that your estimates account for the whole circle?
3. Using your estimates, determine the percent of the circle represented by each sector. How can you use your percents
to be sure that your estimates account for the whole circle?
4. If the radius of this circle were 4 inches, what would its area be? Show your calculations.
5. Using your estimates, give the area of each sector of the circle. Show how you might check your work.
*********************************************************************************
Circles and Sectors
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Discussion, Suggestions, Possible Solutions
The purpose of this task is to help students get ready to draw accurate circle graphs. Before beginning the tasks, students
should know that there are 360° in a complete revolution, how to use a protractor, and how to calculate the area of a circle.
Possible solution:
1. The number of degrees in each sector should be estimated, moving clockwise from North, at around 90 degrees, 30
degrees, 90 degrees and 150 degrees. Students should see the angle in the top, right quadrant and “bottom” angle as
about 90 degrees and estimate the other two based on these estimates.
2. Students should use ratios as part-to-whole relationships in representing the sectors as fractional parts of the circle:
90 1
30
1
150 5
of the circle,
of the circle. Students should know that the sum
= of the circle,
=
=
360 4
360 12
360 12
1 1
5 1 3
1
5
3 12
of their fractions should be very close to 1. + + + = + + + =
=1
4 12 12 4 12 12 12 12 12
3. Students should be able to convert from fractions to percents and know that the sum of their percents should be near
100% since they are estimating-exactly 100% if they made sure that the sum of their degrees is 360 degrees.
1
1
5
= 25%, = 8. 3%, = 41. 6%.
25% + 25% + 41. 6% + 8. 3% = 100% .
4
12
12
This is a good place to discuss what happens when we add repeating decimals. Ask students what they get when they
add
1
5
and
. They can readily see that they get
12
12
6
1
or which is 50%. Ask what this might suggest about
12
2
49. 9% . What does that suggest about . 9 ?
4. A = πr2=16 π≈50.24 square inches
5. .25(50.24) = 12.6 square inches, .0833(50.24) = 4.2 square inches, ,4166…(50.24) = 20.9 square inches.
2(12.6) + 4.2 + 20.9 = 50.3 square inches. A = πr2=16 Π≈50.24 square inches.
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Georgia Performance Standards Framework for Mathematics – Grade 6
• Data and Circle Graphs
This culminating task represents the level of depth and rigor and complexity expected of all 6th grade students to demonstrate
evidence of learning.
UNIT FIVE TASK: “Data and Circle Graphs”
Forty middle school students were asked how many CDs they own. The responses were as follows:
6, 2, 38, 27, 22, 36, 11, 43, 26, 19, 61, 0, 30, 16, 23, 38, 17, 20, 25, 29, 28, 19, 16, 24, 29, 45, 16, 20, 19, 8, 12, 27, 27,
28, 3, 13, 21, 20, 22, 27
a. Organize the data into a grouped frequency table with no more than 6 classes.
b. Represent you data using a circle graph. Show and explain all of your calculations. Be sure that your sectors are
drawn accurately and that your graph is attractive and easy to read.
c. If this data were representative of your class, how many students would fall into each of the sectors on your graph?
Explain your thinking and show how you know.
If this data were representative of all students in your school, how many students would fall into each of the sectors on
your graph? Explain your thinking and show how you know.
Standards Addressed in this Task
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
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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.
Concepts/Skills to Maintain
•
•
•
The number of degrees in a complete revolution
Measuring angles using a protractor
Drawing angles of a given degree
Suggestions for Classroom Use
While this task may serve as a summative assessment, it may also be used for teaching and learning. It is important that all
elements of the task be addressed throughout the learning process so that students understand what is expected of them.
This task, unlike those in other units, does not cover all of the standards addressed in the unit. Although circles will be
addressed again in unit 8, it is important to be sure that students understand the relationships and calculations addressed in
this unit.
•
•
•
•
Peer Review
Display for parent night
Place in portfolio
Photographs
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Unit 5 Organizer y CIRCLES AND GRAPHS
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Discussion, Suggestions and Possible Solutions
a. Students may use different classes to group their data. One reasonable grouped frequency table is presented below.
Number of CDs Owned
0-9
10 - 19
20 - 29
30 - 39
40 - 49
Over 50
Frequency
5
10
18
4
2
1
b. Calculations for circle graph (based on data grouped as shown in the above table).
• 5/40 = 1/8 = .125 = 12.5%; number of degrees in sector = .125 x 360 = 45 degrees or 1/8 x 360 = 45 degrees
a. 10/40 = ¼ = .25 = 25%; number of degrees in sector = .25 x 360 = ¼ x 360 = 90 degrees
b. 18/40 = 9/20 = .45 = 45%; number of degrees in sector = .45 x 360 = 9/20 x 360 = 162 degrees
c. 4/40 = 1/10 = .10 = 10%; number of degrees in circle = ,1 x 360 = 1/10 x 360 = 36 degrees
d. 2/40 = 1/20 = .05 = 5%; number of degrees in sector = 18 degrees
e. 1/40 = .025 = 2.5 %; number of degrees in sector = 9 degrees
Encourage students to use methods easiest for them when computing. Help them utilize the simple proportional
relationships that exist here. (i.e. 2/40 is one half of 4/40 so one half of 36 degrees is 18 degrees, etc. Ask questions that
prompt them to check their work by adding percents to obtain 100%, fractions to obtain 1 whole and/or degrees to
obtain 360 degrees.
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Georgia Performance Standards Framework for Mathematics – Grade 6
Graphs should be carefully labeled, neat and accurate.
Number of CDs Owned
5
6
Number of CDs Owned
1
4
2
3
1→ 0 – 9
2→ 10 – 19
3→ 20 – 29
4→ 30 – 39
5→ 40 – 49
6→ Over 50
c.- d. Answers will vary for parts c and d. Students may need to do some research to determine the number of students in
their school.
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