Lesson 1.6 Drawing Circles with a Compass

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Objective
1
To provide practice using a compass.
materials
Teaching the Lesson
Key Activities
Students draw circles with a compass and construct a square inscribed in a circle
by folding paper.
Key Concepts and Skills
•
•
•
•
Use a compass to measure distance. [Measurement and Reference Frames Goal 1]
Use a compass to draw circles. [Geometry Goal 2]
Construct an inscribed square. [Geometry Goal 2]
Verify that the sides of a square are the same length. [Geometry Goal 2]
Math Journal 1, pp. 14 and 15
Study Link 1 5
slate; compass; paper (colored, if
available); straightedge; cardboard;
scissors (optional)
Key Vocabulary
compass • circle • center (of a circle) • inscribed square
2
Ongoing Learning & Practice
Students play Polygon Pair-Up to practice identifying properties of polygons.
Students practice and maintain skills through Math Boxes and Study Link activities.
Ongoing Assessment: Recognizing Student Achievement Use journal page 16.
[Geometry Goal 1]
materials
Math Journal 1, p. 16
Student Reference Book, p. 258
Study Link Master (Math Masters,
p. 23)
Game Masters (Math Masters,
pp. 496 and 497)
scissors
See Advance Preparation
3
Differentiation Options
ENRICHMENT
Students solve an inscribed square puzzle.
materials
EXTRA PRACTICE
Students use a compass to create
circle designs.
Teaching Master (Math Masters,
p. 24)
straightedge; compass; scissors;
glue; colored paper (optional)
Ed Emberley’s Picture Pie: A Circle
Drawing Book
See Advance Preparation
Additional Information
Advance Preparation In Part 2, students cut apart the Polygon Deck and Property Deck from
Math Masters, pages 496 and 497. Consider copying the cards on cardstock for students and
making overhead transparency cards for demonstrations.
For the optional Extra Practice activity in Part 3, obtain the book Ed Emberley’s Picture Pie:
A Circle Drawing Book by Ed Emberly (Little Brown, 1984).
Technology
Assessment Management System
Math Boxes, Problem 3
See the iTLG.
Lesson 1 6
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Getting Started
Mental Math and Reflexes
Math Message
Write numbers such as those listed below on the board.
Your job is to draw a large circle on the playground.
How will you do it? Discuss the problem with a partner.
Record your ideas on a half-sheet of paper.
2,510
9,246
3,082
7,682,041
4,502,639
67,314,851
32,756
172,908
530,175
Study Link 1 5 Follow-Up
For each number, ask questions such as the following:
Have small groups of students compare answers to
Problems 1–3. Then ask students to pose their own
riddles to the group.
• What is the value of the digit x?
• Which digit is in the thousands place?
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Encourage students to share ideas. One method is to anchor one
end of a rope to the ground, attach a piece of chalk to the other
end, pull the rope taut, and then rotate it around the anchor to
draw the circle. Let students judge for themselves the relative
effectiveness of the methods they suggest. You might have them
try out their methods on the playground.
Student Page
Date
LESSON
16
䉬
One way of drawing a circle without a compass
Time
An Inscribed Square
Follow the directions below to make a square
that you will tape on the next page.
Step 1 Use your compass to draw a circle on
a sheet of colored paper. The circle
should be small enough to fit on the
next page. Cut out the circle.
Step 2 With your pencil, make a dot in the
center of the circle, where the hole is,
on both the front and the back.
Step 3 Fold the circle in half. Make sure that
the edges match and that the fold line
passes through the center. Be sure to
make sharp creases.
Step 4 Fold the folded circle in half again so
that the edges match.
Step 5 Unfold your circle. The folds should
pass through the center of the circle
and form 4 right angles.
Step 6 Using a straightedge, connect the
endpoints of the folds at the edge of
the circle to make a square. Cut out
the square.
116
Tell students that in this lesson they will learn how to use a
compass to draw circles. Students who have never used a
compass will need plenty of practice with the most basic
construction—that of a circle—before attempting more
difficult constructions.
Drawing Circles with
a Compass
Demonstrate two methods for drawing circles with a compass.
(These directions are for right-handed students.) Students should
draw on top of cardboard or several sheets of paper to prevent
damage to the desk or tabletop and to keep the anchor from
slipping as the pencil is rotated. For either method, the compass
should be held at the top, not by its arms. (See margin on page 49.)
14
Math Journal 1, p. 14
48
WHOLE-CLASS
ACTIVITY
Unit 1 Naming and Constructing Geometric Figures
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Method 1:
Students should experiment with both methods and use the one
with which they feel more comfortable. Have them draw circles
of various sizes.
Remind students of the following:
As with polygons, the interior of a circle is not part of the circle.
The point located at the anchor of the compass is called the
center of the circle; the center is not part of the circle.
All points on a circle are the same distance from the center of
the circle.
Constructing an Inscribed Square
INDEPENDENT
ACTIVITY
(Math Journal 1, pp. 14 and 15)
Construction of an inscribed square (a square whose vertices all
lie on a circle) relies on paper folding. Demonstrate the construction
while students follow your directions, or have students follow the
directions on their own.
1. Press the anchor of the compass firmly onto the paper.
(Some teachers find it helpful
to tape the paper to the work
surface.)
2. Rotate the pencil point of the
compass around the anchor,
keeping the paper fixed in
place.
3. If the pencil is rotated clockwise, start with the pencil close
to where the 5 would be located on a clock face. (If rotating
counterclockwise, start near
the 7.)
Adjusting the Activity
Have students construct an inscribed regular octagon. One way to do
this is to draw and cut out a circle and fold it into eighths. The fold lines meet the
circle in eight equally spaced points. These points become the vertices of the
octagon. If students need a hint, you might tell them to start as though they were
making a square using the method on journal page 14.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Method 1
Method 2:
Rotate a single sheet of paper,
keeping the anchor and pencil
point fixed in place. This method
is especially useful when drawing
smaller circles.
Student Page
Date
LESSON
16
䉬
Time
An Inscribed Square
continued
Now use your compass to find out whether the 4 sides of your square are about the
same length.
Place the anchor on one endpoint of a side and the pencil point on the other endpoint
of the side. Then, without changing the compass opening, try to place the anchor and
pencil point on the endpoints of each of the other sides.
Method 2
If the sides of your square are about the same length, tape the square in the space
below. If not, follow the directions on page 14 again. Tape your best square in the
space below.
NOTE These directions are for a traditional,
two-arm compass. Other compasses are
available, including ruler-type compasses.
These methods work with both types of
compasses.
15
Math Journal 1, p. 15
Lesson 1 6
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2 Ongoing Learning & Practice
Playing Polygon Pair-Up
PARTNER
ACTIVITY
(Student Reference Book, p. 258; Math Masters,
pp. 496 and 497)
Students play Polygon Pair-Up to practice identifying properties of
polygons. Consider playing a game or two against the class on the
overhead projector to help students learn the rules.
Adjusting the Activity
Polygon Cards from Math Masters, page 496
Use these game variations as appropriate:
Variation 1:
Use only the Property Cards.
Players take turns drawing cards and tracing a shape from their Geometry
Template that matches the property described on the card.
The first time a shape from the Geometry Template is used, a player earns
3 points for a correct match. If the same shape is traced again, players earn
only 1 point.
All sides are the
same length.
All angles are right
angles.
When time runs out, the player with the highest score wins.
Variation 2:
Place all the Polygon Cards faceup. Place the Property Cards facedown
between the players.
Players take turns drawing a Property Card and searching the Polygon Cards
to find a match.
Property Cards from Math Masters, page 497
The player with the most pairs wins.
A U D I T O R Y
1 6
䉬
a. 9 ⫺ 5 ⫽
b. 11 ⫺ 2 ⫽
c.
d.
7
8
3
B and C
⫽ 14 ⫺ 7
A
⫽ 12 ⫺ 4
e. 13 ⫺ 6 ⫽
f.
polygons?
⫽ 12 ⫺ 9
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夹
V I S U A L
INDEPENDENT
ACTIVITY
Writing/Reasoning Have students write a response
to the following: Explain why the shapes you chose in
Problem 2 are not polygons. Sample answer: B has curved
sides, and A does not have sides that connect end to end.
C
B
7
3. Draw a quadrangle with only 1 right angle.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 1-8. The skill in Problem 6
previews Unit 2 content.
2. Which of the shape(s) below are NOT
4
9
T A C T I L E
(Math Journal 1, p. 16)
Math Boxes
1. Subtract mentally.
Time
LESSON
K I N E S T H E T I C
Math Boxes 1 6
Student Page
Date
4. Circle the convex polygon(s).
Draw in the right angle symbol.
Sample answers:
Ongoing Assessment:
Recognizing Student Achievement
How do you know it is a right angle?
It is a square corner.
Use Math Boxes, Problem 3 to assess students’ understanding of right angles.
Students are making adequate progress if they are able to draw an appropriate
quadrangle. Some students may be able to explain how they know the angle is
a right angle.
6. In the numeral 42,318, the 2 stands
Draw point T on it.
for 2,000.
Sample answer:
HA
T
a. The 1 stands for
៮៬?
What is another name for HA
b. The 8 stands for
៮៬
HT
91
c. The 4 stands for
d. The 3 stands for
10 .
8
.
40,000 .
300 .
[Geometry Goal 1]
4
16
Math Journal 1, p. 16
50
97
93 99
5. Draw and label ray HA.
Math Boxes
Problem 3
Unit 1 Naming and Constructing Geometric Figures
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Study Link Master
Name
Study Link 1 6
INDEPENDENT
ACTIVITY
Date
STUDY LINK
Time
Properties of Geometric Figures
1 6
䉬
(Math Masters, p. 23)
96–100
Home Connection Students match geometric figures
and properties.
A
B
C
D
E
F
G
H
I
Write the letter or letters that match each statement.
3 Differentiation Options
ENRICHMENT
Solving an Inscribed-Square
INDEPENDENT
ACTIVITY
1.
These are polygons.
2.
These are regular polygons.
3.
These are quadrangles.
4.
These are concave.
5.
These are NOT parallelograms.
A, B, C, E, F, G, and I
B and C
C, E, F, and I
A
A, B, D, F, G, H, and I
6.
These do NOT have any right
angles or angles whose
measures are larger than a
right angle.
D, G, and H
15–30 Min
Puzzle
Try This
Take a paper clip and two pencils. Create a homemade compass. You may not
bend or break the paper clip. How many different size circles can you make with it?
7.
2
Practice
(Math Masters, p. 24)
30 50 8.
60
11.
To apply students’ understanding of inscribing polygons within
circles, have them solve an inscribed-square puzzle. Polygon 1 is
said to be inscribed in Polygon 2 if all of the vertices of Polygon 1
are on Polygon 2.
80
9.
80 20
12.
100
10.
250 140 120 70
13.
460 230 40 60 50
390
230
Math Masters, p. 23
Have students describe the squares they drew. Encourage words
like center of circle and inscribed.
EXTRA PRACTICE
Creating Circle Designs
INDEPENDENT
ACTIVITY
30+ Min
Art Link To provide practice using a compass to draw
circles, have students create circle designs based on the
ones in Ed Emberley’s Picture Pie: A Circle Drawing
Book by Ed Emberley (Little Brown, 1984). Circles that
have been constructed with a compass and cut into halves,
fourths, or eighths are the basis for the artwork in the book. The
circles can be constructed on paper of various colors and can be
used to form elaborate designs.
Teaching Master
Name
Date
LESSON
1 6
䉬
Time
A Crowded-Points Puzzle
Nine points are crowded together in a large, square room.
The points do not like crowds.
1.
2.
Use a straightedge to draw 2 squares so that each point will have
a room of its own.
Explain what you did to solve this puzzle.
䉬 Describe the squares you drew using vocabulary words you learned in class.
䉬 Tell how you know that the 2 polygons you drew are squares.
Sample answer: I drew 2 squares, one
inside the other. They are inscribed squares
because their vertices touch the other
squares. Both shapes are squares because
they have equal sides and right angles.
Math Masters, p. 24
Lesson 1 6
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