Area and Circumference
of Circles
Learning Objectives
23
Materials
Geometry
Lesson
•Student Guided Practice
Book (pages 160–166)
• Know the formulas for the area and circumference
of a circle and use them to solve problems.
• Math Fluency Game Sets
Mathematical Practices and Processes
• Reason abstractly and quantitatively.
• Model with mathematics.
• Look for and make use of structure.
• Digital Math
Fluency Games
• Centimeter Graph Paper
(filename: centgraph.pdf)
•Rulers
(filename: ruler.pdf)
Progress Monitoring
• sticky notes
The Student Guided Practice Book pages below can be used
to formally and informally assess student understanding of
the concepts.
• chart paper
•markers
•scissors
• circular objects
•string
Name: _____________________________________________________________________
Date: _________________________
Lesson
23
Serious Circles
Directions: Read the word problem. Answer the questions.
Date: _________________________
___________________________
____________________________
Name: ______________
23
the word problem.
Directions: Read
The Parent
in the shape of a circle.
your work.
playground? Show
The diameter is 40
Explain how you found
yards.
Area = _______________________________________________________________________
the answer.
nce of
2 April wants to sew ribbon around the outside of the rug. What is the circumference of the rug?
_____
Show your work.
the playground?
circumference of
the circle?
Circumference = ______________________________________________________________
Explanation:
the answer.
3 At the park there
is a merry-go-r
_____________________________________________________________________________
ound with a radius
What is the circumfere
nce?
_____________________________________________________________________________
________
of 6 feet.
A 18.84 ft.
B 37.68 ft.
C 75.36 ft.
D 113.04 ft.
_________
__________________
_________
__________________
Explanation:
_____________________________________________________________________________
_____
__________________
__________________
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3 Explain your reasoning for how to find the area and circumference of a circle.
_____
__________________
__________________
__________________
__________________
Directions: Write
an equation. Solve
the problem.
_____________________________________________________________________________
_____________________________________________________________________________
how to find the
your reasoning for
nce of a circle.
area and circumfere
______________
Student
21208—Focused Mathematics—
161
Guided Practice Book
162
Name: ____________________________________
_________________________________
Lesso n
_______
__________________
Date: _________________________
Find
Refocus
_
________________________
question.
Answer each
2 Find a circle.
of the circle,
circumference
What is the
and a ruler?
of the circle,
using string
circumference
2
___________
What is the
____________
and a ruler?
____________
using string
___________
of the circle,
____________
circumference
What is the
____________
?
of the circle,
using the formula
circumference
___________
What is the
______
?
______
____________
using the formula
___________
___________
______
______
______
______
____________
____________
the two
___________
ces between
____________
Are there differenIf so, explain why.
____________
?
the two
measurements
3
ces between
___________
Are there differenIf so, explain why.
____________
?
____________
measurements
___________
___________
____________
____________
____________
____________
___________
___________
______
______
______
______
____________
____________
using
___________
circle,
______
the
of
______
area
What is the
____________
using
graph paper?
area of the circle,
___________
What is the
____________
____________
graph paper?
4
using
___________
____________
area of the circle,
What is the
____________
using
the formula?
area of the circle,
___________
What is the
____________
______
ula?
______
the form
___________
___________
____________
____________
____________
____________
the two
___________
ces between
____________
Are there differenIf so, explain why.
____________
?
the two
measurements
ces between
___________
Are there differenIf so, explain why.
____________
?
____________
measurements
___________
___________
____________
____________
____________
____________
164
___________
___________
____________
____________
____________
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Date: _________________________
Mini Golf
21208—Focused Mathematics—
Student
Circumference: ____________
Unpack the Problem
______________________
Area: ________________________
23Less
on
23
1
21208—Focused
Practice Book
dent Guided
Mathematics—Stu
________
Solution
2
Draw a
visual repr
________
esentatio
________
n to expla
in your
10 in.
______________________
Area: ________________________
Look Back and Explain
_____________
____________
Circumference: ____________
12 in.
______________________
Area: ________________________
© Teacher Created Materials
dent Guided Practice Book
21208—Focused Mathematics—Stu
21208—Focused Mathematics—Stu
dent Guided Practice Book
© Teacher Created Materials
166
163
Materials
© Teacher Created
© Teacher Created Materials
165
21208—Fo
cused Mathe
matics—St
udent Guide
d
Practice
Book
© Teach
er Create
d Mater
ials
__
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__
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________
answer.
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Circumference: ____________
__
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e.
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of a circl
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mference
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and circu
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________
ng the area
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n
een findi
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Date: ______
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Reflect
io
rence betw
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Circumference: ____________
______________________
© Teacher Created
Materials
____________
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the diffe
________
________
Find a circle.
Area: ________________________
Guided Practice Book
Name:
Explain
________
Make a Plan
Directions:
6 in.
______
Lesson
A new miniature golf course
is being constructed in
town. One of
the holes will be placed
in the exact center of a
circle. The hole is
8 feet from the edge of
the circle. The owners
are trying to find
the cost to cover the entire
circle with artificial turf.
The cost is
$3.50 per square foot. What
is the cost of the turf?
area of each circle.
the circumference and
4 in.
Math in the
Real World
Independent Practice
23Directions:
1
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__________________
_________________________________
Name: ____________________________________
23
Date:
__________________
Solution: _________
_________
Materials
© Teacher Created
Lesson
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__________________
__________________
Name: _________
Equation: _________
_________
14 in.
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__________________
21208—Focused Mathematics—Student Guided Practice Book
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© Teacher Created Materials
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4 What is the area of
the circle?
_____________________________________________________________________________
______________
__________________
__________________
_________
__________________
1
A 14.13 ft.
B 28.26 ft.
C 56.52 ft.
D 63.585 ft.
9 ft.
Explain how you found
Teacher
Background
2 What is the
Show your work. Explain how you found the answer.
_____
__________________
__________________
__________________
__________________
2 What is the circumfere
A 3.14 sq. in.
B 15.7 sq. in.
C 31.4 sq. in.
D 78.5 sq. in.
5 in.
_____________________________________________________________________________
__________________
__________________
__________________
__________________
160
the circle?
Explanation:
_________________
Date: _________________________
Check
each problem.
1 What is the area of
_____________________________________________________________________________
Explanation:
3 Explain
Quick
Directions: Solve
__________________
__________________
_________
Area = _________
Circumference =
Name: ______________
____________________________
___________________________
23
1 What is the area of the rug? Show your work. Explain how you found the answer.
Answer the questions.
a playground
Association made
the
1 What is the area of
Lesson
April wanted a circle rug to cover the majority of her room. She determined the radius of the
circle is 5 feet long.
struction
Playground Con
Lesson
________
__
__
In previous grades, students
have learned how to find the
perimeter and area of polygons,
such as rectangles and triangles.
They have also learned to find
the area of complex figures by
decomposing them into other
shapes. Now students
apply that knowledge
to finding the
circumference and
area of a circle by
using formulas.
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
237
Area and Circumference
of Circles (cont.)
Lesson
23
Warm-Up
min.
1.On the board, draw a number of shapes (rectangle, squares, and triangles). Include
dimensions for each shape. Label each shape with a letter for easy identification.
2.Place students into groups of two to three. Provide each group with the same number of
sticky notes as the shapes on the board. For example, if there are six shapes, each group
should get six sticky notes.
3.Have groups find the perimeter and area of each shape. Have them label the name of
the shape on a sticky note, and include its calculated area and perimeter.
4.When everyone has finished, have students place the sticky notes next to the correct
shapes. When all of the notes are on the board, have students explain how they found
the area and perimeter of each shape and name the formulas they used.
Language and Vocabulary
min.
1.Prior to the lesson, create a poster titled Circles. Draw a circle in the middle of
the poster.
2.Write the following vocabulary terms on the board:
pi (π)
circumference
radius
diameter
3.On chart paper, draw a point in the middle of the circle and draw a line that extends
from the center to the edge of the circle. Tell students that the line represents the
radius of the circle. Label the radius on the chart. Explain that the diameter is a line
that starts at one side of a circle, extends to the other side of a circle, and crosses through
the center. Draw and label the diameter.
4.Tell students that the circumference of a circle is the distance around the circle. Ask,
“How can we show what circumference is on the chart?” Students may suggest drawing
an arrow around the outside of the circle. Tell students that pi (π) is approximately equal
to 3.14. Explain that this decimal number keeps going indefinitely. On the chart write
π = 3.14.
5. Keep the poster up for students to refer to throughout the lesson.
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21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
© Teacher Created Materials
Area and Circumference
of Circles (cont.)
Whole-Group Lesson
Lesson
23
min.
Focus
1.The following lesson will address this focus question:
How do you find the area and circumference of a circle?
2.You may wish to write the focus question on the board and read it aloud to students.
Explain that you will revisit the focus question at the end of the lesson.
I Do
1.Say, “Today we are going to find both the circumference and area of circles.” Write
the following problem on the board as you read it aloud: Bob wants to know the
circumference of the circular walking path in the park. He knows the diameter is
10 feet. What is the circumference?
2.Ask, “What is circumference?” (It is the distance around a circle; it is the perimeter
of the circle.) “What strategy can we use to solve this problem?” If no one suggests
it, draw a sketch on the board. Say, “We know the diameter, the distance across the
circle, is 10 feet.” Draw a sketch and label the diameter. Say, “We want to know the
distance around the circle.”
10 ft.
3.Say, “To find the circumference, we multiply π by the diameter. We know that
π is approximately 3.14.” On the board, write C = πd = 10 × 3.14. Say, “The
circumference is 31.4 feet.”
4.Ask, “What is the radius of this circle?” (Since the radius is half the diameter, the
radius is 5 feet.) Say, “We can also use the formula C = 2πr to find the circumference.
We multiply 2 times π times the radius.” Write the following example on the board:
C = 2πr = 2 × 3.14 × 5 = 31.4.
Language Support
Emphasize that the circumference of a circle is the distance around
it, or the perimeter. The area is the number of square units that
cover the circle. To help students make the distinction, have
them draw a circle and an arrow going around the circle labeled
circumference. Then, have students draw square units on the
surface of the circle and label these area.
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239
Area and Circumference
of Circles (cont.)
Lesson
23
Whole-Group Lesson
I Do
(cont.)
(cont.)
5.Ask, “How do you find the area of a rectangle?” Have a student write this formula
on the board: Area = base × height. Next to it, write the formula for the area of
a circle, Area = πr². Say, “Even though it doesn’t look like it, finding the area of a
circle is basically the same as finding the area of a rectangle.” Pass out paper and
scissors to each pair of students. In addition, provide circular objects for students to
trace.
6.Tell students to trace and cut out a circle. Then, have them fold the circle so it has
eight equal sections. Have students identify the center point, diameter, and radius
on the circle. Next, have students cut out the eight sections and place them as
shown to make a parallelogram. Tell students to cut the section at the end in half
and move it to the other end of the parallelogram, making a rectangle as shown.
πr
r
7.Ask, “What is the height of our decomposed circle called?” (the radius) Ask, “What
is the base?” Students should recognize that the top and bottom of the rectangle is
made up of the outside edges of the circle, the circumference. So, the base is half
of the circumference, or 12 (2πr). On the board, write the following equations in a
vertical column so students can follow along and understand where each number is
coming from.
Area = base × height
Area =
1
2
(2πr) × r
Area = πr × r
Area = πr²
8.Say, “So the area of the circle is made up of the same formula as we use for the area
of a rectangle.”
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21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
© Teacher Created Materials
Area and Circumference
of Circles (cont.)
Whole-Group Lesson
We Do
Lesson
23
(cont.)
1.Refer students to the Playground Construction activity sheet (Student Guided
Practice Book, page 160). Say, “Let’s look at another problem together.” Write
the problem on the board as you read it aloud: The Parent Association made a
playground in the shape of a circle. The diameter is 40 yards.
2.Ask, “How can we find the area of this playground?” Suggest that students make
a sketch of the playground and label the diameter. Students should explain that
the formula for a circle is basically the same formula for a rectangle, base × height.
Write the formula on the board: Area = πr².
3.Ask, “Do we know the radius?” (If the diameter is 40 yards, then the radius is
half that measurement, or 20 yards.) Guide students to insert the radius into the
formula. Write the following on the board: A = πr² = 3.14 × (20)² = 3.14 × (20 × 20)
= 3.14 × 400 = 1,256. Say, “The area of the playground is 1,256 square yards.” Have
students write an explanation about how they found the solution on their activity
sheet.
4.Refer students to Question 2. Ask, “How do we find the circumference of the
playground?” (Finding the circumference is the same as finding the perimeter.)
Write the formula on the board: C = πd. Guide students to insert the diameter into
the formula. Write the following on the board: C = πd = 3.14 × 40 = 125.6. Say,
“The circumference of the playground is 125.6 yards.” Have students write an
explanation about how they found the solution on their activity sheet.
5.Have students explain their reasoning in Question 3. To help students explain their
reasoning, provide them with the following sentence frames:
• The circumference of a circle means _____.
• I found the circumference by _____.
• The area of a circle means _____.
• I found the area by _____.
© Teacher Created Materials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
241
Lesson
23
Area and Circumference
of Circles (cont.)
Whole-Group Lesson
(cont.)
You Do1.Refer students to the Serious Circles activity sheet (Student Guided Practice Book,
page 161). Provide the sentence frames from Step 5 of the We Do section to help
students explain their reasoning.
2.Have students share their equations and reasoning. If students have difficulty
explaining their reasoning, remind them to use the sentence frames and
vocabulary terms.
Closing the Whole-Group Lesson
Revisit the focus question for the lesson: How do you find the area and circumference of a
circle? Have students explain the difference between finding the circumference and area
of a circle. Ask students to recall the formulas used for each. Students should indicate
that the circumference of a circle is the distance around the circle, whereas the area is the
number of square units that cover the surface. The formula used for the perimeter of a
circle is C = πd or C = 2πr and the formula for area is A = πr².
Progress Monitoring
min.
1.Have students complete the Quick Check activity sheet (Student Guided Practice
Book, page 162) to gauge student progress toward mastery of the Learning Objectives.
2.Based on the results of the Quick Check activity sheet and your observations during
the lesson, identify students who may benefit from additional instruction in the
Learning Objectives. These students will be placed into a small group for reteaching.
See instructions on the following page.
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Area and Circumference
of Circles (cont.)
Differentiated Instruction
Lesson
23
min.
Gather students for reteaching. The remaining students will complete the Independent
Practice activity sheet (Student Guided Practice Book, page 164) to reinforce their learning
and then play the Math Fluency Games.
Refocus
PPT
Revisit the focus question for the lesson: How do you find the area and circumference of a
circle? Students who struggled to understand how to find the area and circumference of
a circle may benefit from additional instruction with concrete models. Provide students
with circular objects, string, rulers, and centemeter graph paper. To help students find
the circumference of a circle, have them cut a piece of string that is the same length as the
distance around the circle. Have them measure the string with a ruler. Then, ask students to
measure the circle’s diameter and use the formula to calculate its circumference. Have them
compare the actual measurement to the calculated measurement.
To help them find the area of a circle, have students trace a circle onto graph paper. Ask them
to count the square units that the circle covers, estimating for partial square units. Then, have
them find the radius in square units and calculate the area using the formula. Prompt students
to compare the actual measurement to the calculated measurement. Discuss whether the
two measurements were close, and if so why or why not. Finally, support students as they
complete Question 1 on the Refocus activity sheet (Student Guided Practice Book, page 163),
and then have them solve Question 2 independently.
Math Fluency Games
Math Fluency Game Sets
Digital Math Fluency Games
Extend Learning
Ask students how they might find the radius or diameter if they know the
circumference or the area of the circle. Demonstrate using the formulas to find
the unknown measurements. Then, have students complete the Lesson 23 Extend
Learning Task (filename: extendtask23.pdf).
© Teacher Created Materials
21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
243
Area and Circumference
of Circles (cont.)
Lesson
23
Math in the Real World
min.
1.Refer students to the Math in the Real World: Mini Golf task (Student Guided Practice
Book, page 165). Have a student read the task aloud. Tell students to explain or
summarize the task to their partner. Have a few students share their summaries.
2.Ask students to think about what information they will need to solve the task and what
the task is asking them to do. Then, have them share with a partner. Ask a few students
to share out. Students should indicate that they know the hole in the center is 8 feet
from the edge of the circle and that the cost per square foot for artificial turf is $3. 50.
They need to find the cost of the turf. Have students work in groups of two or three to
complete the task.
3.As students are working, circulate and ask focusing, assessing, and advancing questions:
•
What information do you know?
• What are you trying to find out?
• How can you use concrete models to solve the problem?
Sentence Frames for Explaining Reasoning
• I found the area of the circle by _____.
• I found the cost of the turf by _____.
4.Observe how students are solving the task, and choose a few groups who solved the task
in different ways to share their solutions and reasoning. Try to have the solutions move
from concrete representations to more abstract representations. For example, have
students share solutions with a visual representation (drawing) and then the symbolic
representation (equation). Make sure students explain their reasoning as they share
solutions. Students should have found the area of the circle to find the number of square
units that will cover the surface. Their work should include: A =πr² = (3.14)(8)2 = 200.96
sq. feet. Students should have found the cost of the turf by multiplying the area of the
circle by the cost per square foot. Their work should include: 200.96 × 3.5 = 703.36.
The cost to cover the surface of the circle is $703.36.
5.As groups are sharing their solution paths, reasoning, and strategies, ask questions:
• How is this strategy similar to one that we have seen in a previous task?
• Who can restate _____’s strategy/solution path/reasoning?
• Which solution path makes the most sense to you? Why?
Lesson Reflection
min.
Have students summarize their learning about finding the area and circumference of a
circle, and provide feedback on any questions they still have about the content on the
Reflection activity sheet (Student Guided Practice Book, page 166).
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21199—Focused Mathematics Intervention Level 7—Teacher’s Guide
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