Download The Amazing Circle

The Amazing Circle
Grade Levels 3-5
CAMT Houston, Texas
Dates: July 18- 21, 2012
Session Presenter: Lynn Carr
Email: [email protected]
Book Source
All Around Circles Shapes, Solids and More
Amazing Circle #1,#2,#3,#4 The Amazing Circle
Circle Circus What's Next Volume 1
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Download the CAMT- CARR conference packet
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Exploring the Amazing Circle
Distribute the student page, Amazing Circle Number 1.
1
Fold the Amazing
Circle in half and
crease.
a. What is this new shape called? [Semi-circular region]
b. What is its straight edge called? [Diameter]
c. In your own words, define a diameter. [the longest straight line segment with
endpoints on the circle, a line segment through the center of the circle, etc.]
d. What is the curved edge of this new shape called? [Semi-circle]
e. How many semi-circles do you see? [Two]
Relationship
f. What part of the circular region lies within a semi-circular region? [One-half]
Exploration
g. Can you find a way to locate the center of the circle? [Fold a different semicircular region and crease. The center of the circle is where the two creases
intersect.]
Student Record
h. Have the students label their Amazing Circles as shown in the diagram. Instruct
them to color or shade one of the semi-circular regions. Once completed, tell
them to tape their Amazing Circle to their student page.
TEACHER GUIDELINES
Identifications
semi-circular region
1 of circular region
2
A
AB is a diameter
1 of circular region
2
semi-circular region
AMAZING CIRCLE, Vol. 1
B
tape to student
sheet at this
position
4
© 2005 AIMS Education Foundation
Name
Class
Amazing Circle Number
Date
1
Cut out an Amazing Circle.
Fold in half and crease.
Identifications
a. What is this new shape called?
b. What is the straight edge called?
c. In your own words, define a diameter.
d. What is the curved edge of this new
shape called?
e. How many semi-circles do you see?
Relationship
f. What part of the circular region lies within
a semi-circular region?
Exploration
ta
pe
he
re
g. How can you locate the center of
the circle?
h. Tape your Amazing Circle to the outlined circular region.
AMAZING CIRCLE, Vol. 1
5
© 2005 AIMS Education Foundation
Exploring the Amazing Circle
Distribute the student page, Amazing Circle Number 2.
2
a
Fold the Amazing Circle in half (a).
Open the circle and fold a second,
different semi-circle. Open the circle.
Mark the point where the diameters
intersect.
a. What can you say about the two diameters? [They intersect at the center of
the circle, bisect each other, are congruent, create four radii, form four central
angles, create four sectors, etc.]
b. What is the line segment equal to one-half a diameter called? [Radius ]
c. The plural of radius is radii. How many radii do you see? [Four]
d. The diameters cut the circle into parts. What are they called? [Arcs. In this
instance they are shorter than a semi-circle and are therefore called minor arcs.
Those greater than a semi-circle are called major arcs.]
e. The two diameters created four regions. What are these called? [Sectors]
f. What makes up the boundary of a sector? [Two radii and an arc]
g. How many pairs of congruent sectors are there? [Two]
h. How many non-overlapping angles have been formed at the center? [Four.
These are known as central angles.]
Relationships and Measurements
i. How do the shapes and sizes of opposite sectors compare? [They are congruent:
same size, same shape.]
j. How does the measure of opposite angles compare? [The angles are congruent.]
k. What is the sum of any two adjacent central angles? [180o or a straight angle]
l. What shape do any two adjacent sectors form? [Semi-circular region]
m. What is the sum of any two adjacent arcs? [Semi-circle]
ADBE major arc
Student Record
DB minor arc
n. Have the students label their
Amazing Circles as shown
in the diagram. Instruct
them to color or shade one
pair of opposite sectors.
Once completed, have them
tape their Amazing Circle to
their page.
1 + 2 = 180˚
4
AMAZING CIRCLE, Vol. 1
TEACHER GUIDELINES
Identifications
6
tape to student
sheet at this
position
© 2005 AIMS Education Foundation
Name
Class
Amazing Circle Number
a
Date
2
Cut out an Amazing Circle. Fold the circle in
half (a). Open the circle and fold a second,
different semi-circle. Open the circle. Mark
the point where the diameters intersect.
Identifications
b. What is the line segment equal to
one-half a diameter called?
a. What can you say about the two
diameters?
c. The plural of radius is radii. How
many radii do you see?
d. The diameters cut the circle into
parts. What are those parts called?
e. The two diameters created four
regions. What are these called?
f.
g. How many pairs of of congruent
sectors are there?
What makes up the boundary of a
sector?
h. How many non-overlapping angles
have been formed at the center?
he
re
i.
Relationships and Measurements
How do the shapes and sizes of
j. How does the measure of opposite
opposite sectors compare?
angles compare?
l.
ta
pe
k. What is the sum of any two adjacent
central angles?
What shape do any two adjacent
sectors form?
m. What is the sum of any two adjacent
arcs?
n. Tape your Amazing Circle to the outlined circular region.
AMAZING CIRCLE, Vol. 1
7
© 2005 AIMS Education Foundation
Exploring the Amazing Circle
Distribute the student page, Amazing Circle Number 3.
b
a
3
Fold an Amazing Circle in
half (a). Fold a second (b)
and then a third semi-circular
region.
a. What can you say about the three diameters? [They all pass through the center
of the circle, are congruent, create six sectors, six radii, six minor arcs, six
central angles, etc.]
b. What name is given to any line segment that passes through the center of the
circle and has its endpoints on the circle. [Every such line segment must be a
diameter.]
Relationships and Measurements
c.
d.
e.
f.
g.
What shape do any three consecutive sectors form? [Semi-circular region]
What do any three consecutive arcs form? [Semi-circle]
What is the sum of any three consecutive central angles? [Straight angle, 180˚]
How do opposite sectors compare? [They are congruent]
How many pairs of opposite angles are there? [Three if the required condition is
that they do not overlap. Students might label them and then use combinations
of two angles and their opposite two angles to further explore this question.]
h. How does the angle measure of opposite angles compare? [They are congruent.]
TEACHER GUIDELINES
Identifications
Student Record
i. Have the students label their
Amazing Circles as shown in the
diagram. Instruct them to color
or shade opposite sectors. Once
completed, have them tape their
Amazing Circle to their page.
AMAZING CIRCLE, Vol. 1
tape to student
sheet at this
position
8
© 2005 AIMS Education Foundation
Name
Class
Amazing Circle Number
b
Date
3
Cut out an Amazing Circle. Fold the
circle in half (a). Open the circle and
fold a second (b) and then a third
different semi-circular region.
a
Identifications
b. What name is given to any line
a. What can you say about the three diameters?
segment that passes through
the center of the circle and has
its endpoints on the circle?
Relationships and Measurements
c. What shape do any three consecutive
sectors form?
d. What do any three consecutive
arcs form?
e. What is the sum of any three
consecutive central angles?
f. How do opposite sectors compare?
h. How does the angle measure of
opposite angles compare?
ta
pe
he
re
g. How many pairs of opposite angles
are there?
i. Tape your Amazing Circle to the outlined circular region.
AMAZING CIRCLE, Vol. 1
9
© 2005 AIMS Education Foundation
Exploring the Amazing Circle
Distribute the student page, Amazing Circle Number 4.
a
4
b
Identifications
a. What is the straight line segment formed by the fold called? [A chord. All
creases with endpoints on the circle are called chords of the circle. A diameter
is a chord that passes through the center of the circle.]
b. What constitutes the boundary of this new figure? [A chord and a major arc.]
c. What constitutes the boundary of the part that has been folded? [A chord and
a minor arc.]
Student Record
d. Have the students label their Amazing Circles as shown in the diagram. Instruct
them to color or shade the shape bounded by the chord and the major arc.
Once completed, have them tape their Amazing Circle to their page.
TEACHER GUIDELINES
Fold the Amazing Circle in half
and crease (a). Fold a second
and different diameter to locate
the center of the circle. Open
the circle (b). Bring one point on
the circle to the center and fold,
creating a sharp crease.
tape to student
sheet at this
position
AMAZING CIRCLE, Vol. 1
10
© 2005 AIMS Education Foundation
Name
Class
Amazing Circle Number
Date
4
b
Cut out an Amazing Circle. Fold two
diameters (a and b) to locate the
center of the circle. Bring one point
on the circle to the center and fold,
creating a sharp crease.
a
Identifications
a. What is the straight line segment formed by the fold called?
b. What constitutes the boundary of this
new figure?
ta
pe
he
re
c. What constitutes the boundary of the
folded over part?
d. Tape your Amazing Circle to the outlined circular region.
AMAZING CIRCLE, Vol. 1
11
© 2005 AIMS Education Foundation
TM
Thank you for your purchase!
Please be sure to save a copy of this document to your local computer.
This activity is copyrighted by the AIMS Education Foundation. All rights reserved. No part of this
work may be reproduced or transmitted in any form or by any means—except as noted below.
•
A person or school purchasing this AIMS activity is hereby granted permission to make up to
200 copies of any portion of it, provided these copies will be used for educational purposes and
only at one school site.
•
For a workshop or conference session, presenters may make one copy of any portion of a
purchased activity for each participant, with a limit of five activities or up to one-third of a
book, whichever is less.
•
All copies must bear the AIMS Education Foundation copyright information.
•
Modifications to AIMS pages (e.g., separating page elements for use on an interactive white
board) are permitted only within the classroom or school for which they were purchased, or by
presenters at conferences or workshops. Interactive white board files may not be uploaded to
any third-party website or otherwise distributed. AIMS artwork and content may not be used on
non-AIMS materials.
AIMS users may purchase unlimited duplication rights for making more than 200 copies, for use at
more than one school site, or for use on the Internet. Contact Duplication Rights or visit the AIMS
website for complete details.
P.O. Box 8120, Fresno, CA 93747
www.aimsedu.org • [email protected] • 1.888.733.2467
Topic
Circumference
Background Information
Circumference is the distance around a circle. It
is equivalent to the perimeter of a polygon, but it has
its own name. Because circles are round, directly
measuring circumference can be challenging. Many
methods can be employed, and some work better than
others, depending on the circle being measured. One
method is to use a flexible measuring tape to directly
measure the circle. This method works well when
finding the circumference of jars or cans. Another
method is to use a piece of string to go around the
circle, then straighten the string and measure it using
a ruler or meter stick. This works well with circles
drawn on paper or paper plates. A third method that
can be used when the circle is made from a flexible
material (such as a rubber band or a loop of yarn) is
to collapse the circle into a straight line, measure its
length, and double it.
Key Question
What is the circumference of a circle and how can it
be measured?
Learning Goals
Students will:
• learn that circumference is the distance around a
circle, and
• practice various techniques for measuring the
circumference of a circle.
Guiding Document
NCTM Standards 2000*
• Describe attributes and parts of two- and threedimensional shapes
• Relate ideas in geometry to ideas in number and
measurement
• Understand such attributes as length, area,
weight, volume, and size of angle and select
the appropriate type of unit for measuring each
attribute
• Understand how to measure using nonstandard
and standard units
• Use tools to measure
Management
1. If you do not have a wind-up meter tape, you will
need to tape several metric measuring tapes end
to end so that you have 25-30 meters.
2. Gather a variety of circular or cylindrical objects
like rubber bands, paper plates, grouping circles,
cans, jars, lids, etc., in as many different sizes as
possible to give multiple opportunities for finding
circumference.
3. You need a large, open area either indoors or
outdoors that will allow your entire class to stand in
a line, holding hands, with their arms outstretched
as far as they can.
4. For Part Two, set up a table with the string, meter
sticks, and metric measuring tapes. Have enough
of each so that several students can be using them
at the same time.
Math
Geometry
circles
circumference
Measurement
Integrated Processes
Observing
Comparing and contrasting
Relating
Applying
Procedure
Part One
1. Ask students what geometry terms they have heard
associated with circles. Record their responses
on the board. If no on mentions it, write the word
circumference.
2. Tell students that today they will be learning about
circumference. Ask if anyone knows what the
circumference of a circle is.
3. Discuss that the circumference is the distance
around a circle. It’s the same as the perimeter of a
square, rectangle, triangle, or other polygon, it just
has a unique name that only applies to circles.
Materials
Wind-up meter tape, 30 m
String
Meter sticks
Metric measuring tapes
Assorted objects (see Management 2)
Student page
SHAPES, SOLIDS, AND MORE
181
© 2009 AIMS Education Foundation
4. Use the end of a can to trace a circle on the board.
Challenge students to think about how they could
measure the circumference of the circle you
just drew.
5. After listening to some of their suggestions, tell
students that they are going to try a few different
ways to measure circumference. Take the class to
the open area you have selected. Instruct them to
stand in a circle holding hands. Have the students
spread apart so that the circle is as large as it can
be without letting go of their hands. (Students’
arms should all be fully extended.)
6. Ask students how they could measure the distance
around the circle formed by their bodies—the
circle’s circumference. Show them the wind-up
meter tape.
7. Try all of the students’ ideas that are feasible
and will result in an accurate measure of the
circumference, being sure to include the following
three methods:
a. Use the meter tape to measure all the way around
the circle. Since the tape is flexible, it can be
used to directly measure the circumference.
b. Turn the circumference into a straight line.
Select two adjacent students to let go of their
hands and begin to walk apart so that the
circle becomes a straight line. Have students
continue to keep their arms fully extended so
that the distance remains the same. Use the
wind-up meter tape to measure the length from
the first student’s outstretched arm to the last
student’s outstretched arm.
c. Have the circle “fold in half.” Direct the students
to walk toward each other until they are making
a line, two people deep while still holding hands
and keeping their arms outstretched. Measure
the distance from one end of the line to the
other, then double that distance to find the
circumference.
8. Have students return to the classroom. Discuss the
different methods used to find the circumference
and how they compared.
4. Instruct students to trade objects and to repeat
the process using a different method of finding
the circumference. (Remind them to return the
measuring tools to the table once they are finished
so that others can use them.)
5. Repeat until all students have used at least three
different methods for finding the circumference.
6. Discuss which method students prefer and why.
Connecting Learning
Part One
1. What is the circumference of a circle? [the distance
around the circle]
2. What are some ways that circumference can be
measured?
3. How did the different ways we measured the
circumference of our “student circle” compare?
Which do you think was easiest? …most difficult?
Part Two
1. What methods did you use to find the
circumferences of your circular objects?
2. How did the circumferences of your circles
compare? [the bigger the circle, the larger the
circumference]
3. Which method for finding the circumference did
you like best? Why?
4. Do you think one method is more accurate than
another? Justify your response.
5. What method would you use to find the
circumference of a rubber band? Why? [It is
difficult to measure around a rubber band with
either string or a tape measure, so folding it in half
and the doubling the length is likely the easiest
way to find the circumference.]
6. When would you need to know the circumference
of a circle?
*
Reprinted with permission from Principles and Standards for
School Mathematics, 2000 by the National Council of Teachers
of Mathematics. All rights reserved.
Part Two
1. Explain that students are now going to have the
opportunity to apply the same methods you used
to find the circumference of their circle to their
own circles.
2. Distribute a circular object and the student page
to each student. Show students the table with
the measuring tools that are available. Invite
them to decide on one method for finding the
circumference of the circle and to select the
appropriate measuring tool(s) for this method.
3. Have them record their measurements on the
student page along with the method used to find
the circumference.
SHAPES, SOLIDS, AND MORE
182
© 2009 AIMS Education Foundation
Key Question
What is the circumference
of a circle and how can it
be measured?
Learning Goals
• learn that circumference is the
distance around a circle, and
• practice various techniques for
measuring the circumference
of a circle.
SHAPES, SOLIDS, AND MORE
183
© 2009 AIMS Education Foundation
Describe each circle. Find its circumference.
Tell how you found the circumference. Use a
different way each time.
My circle: _________________________ Circumference: __________
How I measured the circumference: ___________________________
______________________________________________________________
My circle: _________________________ Circumference: __________
How I measured the circumference: ___________________________
______________________________________________________________
My circle: _________________________ Circumference: __________
How I measured the circumference: ___________________________
______________________________________________________________
My circle: _________________________ Circumference: __________
How I measured the circumference: ___________________________
______________________________________________________________
SHAPES, SOLIDS, AND MORE
184
© 2009 AIMS Education Foundation
Connecting Learning
Part One
1. What is the circumference
of a circle?
2. What are some ways that
circumference can be measured?
3. How did the different ways we
measured the circumference of
our “student circle” compare?
Which do you think was easiest?
…most difficult?
Part Two
1. What methods did you use to
find the circumferences of your
circular objects?
SHAPES, SOLIDS, AND MORE
185
© 2009 AIMS Education Foundation
Connecting Learning
2. How did the circumferences
of your circles compare?
3. Which method for finding the
circumference did you like best?
Why?
4. Do you think one method is more
accurate than another? Justify
your response.
5. What method would you use to
find the circumference of a rubber
band? Why?
6. When would you need to know the
circumference of a circle?
SHAPES, SOLIDS, AND MORE
186
© 2009 AIMS Education Foundation
TM
Thank you for your purchase!
Please be sure to save a copy of this document to your local computer.
This activity is copyrighted by the AIMS Education Foundation. All rights reserved. No part of this
work may be reproduced or transmitted in any form or by any means—except as noted below.
•
A person or school purchasing this AIMS activity is hereby granted permission to make up to
200 copies of any portion of it, provided these copies will be used for educational purposes and
only at one school site.
•
For a workshop or conference session, presenters may make one copy of any portion of a
purchased activity for each participant, with a limit of five activities or up to one-third of a
book, whichever is less.
•
All copies must bear the AIMS Education Foundation copyright information.
•
Modifications to AIMS pages (e.g., separating page elements for use on an interactive white
board) are permitted only within the classroom or school for which they were purchased, or by
presenters at conferences or workshops. Interactive white board files may not be uploaded to
any third-party website or otherwise distributed. AIMS artwork and content may not be used on
non-AIMS materials.
AIMS users may purchase unlimited duplication rights for making more than 200 copies, for use at
more than one school site, or for use on the Internet. Contact Duplication Rights or visit the AIMS
website for complete details.
P.O. Box 8120, Fresno, CA 93747
www.aimsedu.org • [email protected] • 1.888.733.2467
Topic
Geometry: properties and language
Integrated Processes
Observing
Comparing and contrasting
Relating
Key Question
How can a circle be transformed into another shape?
Materials
Scissors
Rulers, optional
Protractors, optional
Set of 3-D models, for reference (optional)
Focus
Students will turn a paper circle into a truncated
tetrahedron by a series of folding steps. They will
observe the shapes made along the way—comparing
the length and parallel/perpendicular relationships of
sides, comparing angles, and comparing areas in terms
of fractional parts.
Background Information
Geometry, with all its appeal to the visual and
tactile senses, is also a language. The level of success
in acquiring this language is heightened by using
interesting contexts and encountering concepts
repeatedly and over time. Amazing Circle, The
Beginnings provides a way to determine progress
in acquiring this geometric understanding and
language.
The big ideas of geometr y are shape and
dimension. Attention is directed to shape with each
fold that is made. Dimension comes into play as the
two-dimensional circle is transformed into a threedimensional truncated tetrahedron.
Other geometric concepts of importance in this
activity are parallel/perpendicular relationships and
congruence. Lines on a flat surface (in the same
plane) that never intersect are called parallel lines.
Perpendicular lines intersect at right angles. Geometric
figures are congruent or equal if they have the same
size and shape. Two line segments are congruent if
they have the same length. Two angles are congruent
if they have the same measure.
Guiding Documents
Project 2061 Benchmarks
• Mathematical ideas can be represented concretely,
graphically, and symbolically.
• Many objects can be described in ter ms of
simple plane figures and solids. Shapes can be
compared in terms of concepts such as parallel
and perpendicular, congruence and similarity, and
symmetry. Symmetry can be found by reflection,
turns, or slides.
• Areas of irregular shapes can be found by dividing
them into squares and triangles.
NCTM Standards 2000*
• Identify, compare, and analyze attributes of twoand three-dimensional geometric shapes and
develop vocabulary to describe the attributes
• Investigate, describe, and reason about the results of
subdividing, combining, and transforming shapes
• Explore congruence and similarity
• Use geometric models to solve problems in other
areas of mathematics, such as number and
measurement
Management
This activity can be used as a culmination and
review of geometric topics and language: 2-D shapes
(both polygons and circles); one 3-D shape (a
tetrahedron); the concepts of parallel, perpendicular,
and congruence; angle measure; and area relationships
expressed as fractions.
Though the spontaneous and fun-loving aspect of
the activity may be tempered, it can also be used as a
more formal assessment. Students could record their
answers to the questions on another piece of paper. It
would be helpful to generate a word bank as a class
to fend off frustrations with spelling, etc.
Math
Geometry and spatial sense
2-D and 3-D
terminology
Measurement, formal or informal
length
angle
area (see Numeration)
Numeration
fractions as an expression of area relationships
1
© 2005 AIMS Education Foundation
Procedure
1. Ask, “Have you ever tried origami, Japanese
paper folding?” Explain that a square piece of
paper is the beginning of most origami creations;
but today, students will use a circle to do paper
folding.
2. Give each student the circle and the two direction
pages. Have students cut out the circle.
3. Proceed through the steps on the direction pages
together. At each step, address the questions
appropriate for your students and have them
respond orally.
4. Enjoy the process and results while assessing the
geometric knowledge and language the class has
gained.
6.
7.
Discussion (on direction pages)
1. Fold the circle in half.
• What is the new shape called? [semi-circle]
• What is the fold called? [diameter, line
segment]
• How does the area of the semi-circle compare
to the area of the whole circle? [The semi-circle
has one-half the area of the circle.]
2. Open the circle. Fold a second diameter. Mark
the point where the diameters intersect.
• Describe what you see. [Examples: The two
diameters are equal length. Four radii extend
from the center of the circle to the edge.]
• How do opposite angles compare? [They are
the same or congruent.]
• How do you know? [One angle can be put over
the top of another to directly compare. They
can be measured with a protractor.]
• Where will any two diameters always intersect?
[in the center of the circle]
3. Open the circle and fold the edge to the center.
• This fold is called a chord. It starts and ends
on the circle but it does not pass through the
center.
4. Make a second fold to the center so the two
chords meet at one end.
• What is the measure of the angle formed by the
two chords? [60°]
5. Make a third fold to the center. It should meet
the ends of the other two chords.
• What is this shape called? [triangle]
• Describe everything you notice about this
triangle. [It has equal (congruent) sides and
equal (congruent) angles. Each angle is 60°
(acute), adding to a total of 180°.]
• What kind of triangle is this? [equilateral
triangle]
8.
9.
2
Find the midpoint of one side. Make a fold
through this midpoint and the opposite vertex.
• What kind of triangle is this? [right triangle]
• What are the angle measures? [30°, 60°, and
90°] What is their sum? [180°]
• How does the area of the right triangle compare
to that of the equilateral triangle? [It is one-half
the area of the equilateral triangle.]
Open back to the equilateral triangle. Bring one
vertex to the opposite midpoint and crease.
• What is this shape called? [trapezoid, more
specifically an isosceles trapezoid]
• Describe its sides and angles. [Two opposite
sides are parallel but one is twice as long as
the other. The other opposite sides are equal
(congruent) but not parallel. The two upper
angles are congruent, obtuse, and 120°. The two
lower angles are congruent, acute, and 60°.]
• How many triangles fill the shape? [3]
• How does the area of one of these triangles
compare to the area of the trapezoid? [Its area
is one-third the area of the trapezoid.]
• How does the area of the trapezoid compare
with the large equilateral triangle? [The area is
three-fourths the area of the large triangle.]
Fold the second and third vertices to the same
midpoint.
• What is this shape called? [rectangle]
• How does the height of this rectangle compare
with the height of the original triangle? [It is
one-half as high.]
• Notice that the vertices of the triangle are all
together. What is the sum of these angles? [The
sum is 180° and forms a straight angle.]
Open to the trapezoid. Fold one of the outside
triangles over the center triangle.
• What shape is formed? [rhombus, also part of
a larger group called parallelograms]
• How do the sides and angles compare? [The
sides are equal (congruent) and opposite angles
are equal (congruent).]
• How does its area compare to the area of the
trapezoid? [Its area is two-thirds that of the
trapezoid.]
© 2005 AIMS Education Foundation
10. Fold the remaining triangle over the center
triangle.
• What is this shape called? [equilateral
triangle]
• How does its area compare with that of the
rhombus? [It is one-half as large.]
11. Open to the original triangle. Fold the sides up
into a three-dimensional pyramid.
• What is another name for this shape?
[tetrahedron]
12. Open to the original triangle. Fold one vertex to
the center of the circle.
• What shape is formed? [trapezoid, more
specifically an isosceles trapezoid]
13. Fold another vertex to the center.
• What is this shape called? [pentagon]
• How do its sides compare? [Two sides are the
same length and three others are shorter, but
equal lengths. ]
• How do its angles compare? [One of the five
angles is different from the others. Or there are
four obtuse angles and one acute angle. Or four
angles are 120° and one angle is 60°.]
14. Fold the third vertex to the center.
• What is the name of this shape? [hexagon,
regular hexagon]
• What can you say about its sides and angles?
[The sides are equal (congruent) and so are
the angles. The angles are all obtuse and
measure 120°.]
• How many small triangles do you see? [6 (unless
students start counting all the smaller fold-line
triangles)]
• How does the area of the hexagon compare with
that of the original triangle? [It is six-ninths or
two-thirds as large.]
15. Open up to the original triangle. Tuck one of
the small triangles at one vertex into the small
triangle at another vertex. Tuck the remaining
triangle underneath.
• To which three-dimensional model is this
shape related? [the tetrahedron or triangular
pyramid
(Explain that this is called a truncated
tetrahedron. Truncated means it has a part that
has been cut off.)]
*
Reprinted with permission from Principles and Standards for
School Mathematics, 2000 by the National Council of Teachers
of Mathematics. All rights reserved.
3
© 2005 AIMS Education Foundation
Key Question
How can a circle
be transformed into
another shape?
Learning Goals
• turn a paper circle into a truncated
tetrahedron by a series of folding steps; and
• observe the shapes made along the
way—comparing the length and parallel/
perpendicular relationships of sides,
comparing angles, and comparing areas in
terms of fractional parts.
4
© 2005 AIMS Education Foundation
Duplicate as many circles as required.
Cut along the outside edge of the Amazing Circle so that the dark
line of the circle is saved.
5
© 2005 AIMS Education Foundation
1.
Fold the circle in half.
What is the new shape called? What is the fold called?
How does the area of the semi-circle compare to the area of the whole circle?
2.
3.
Open the circle and fold the edge to the center.
This fold is called a chord. It starts and ends on the circle
but it does not pass through the center.
4.
5.
6.
Open the circle. Fold a second diameter.
Mark the point where the diameters intersect.
Describe what you see.
How do opposite angles compare? How do you know?
Make a second fold to the center so the
two chords meet at one end.
What is the measure of the angle formed by the two chords?
Make a third fold to the center. It should meet the
ends of the other two chords.
What is this shape called?
Describe everything you notice about this triangle.
What kind of triangle is this?
Find the midpoint of one side. Make a fold
through this midpoint and the opposite vertex.
What kind of triangle is this?
What are the angle measures? What is their sum?
How does the area of the right triangle compare
to that of the equilateral triangle?
7.
6
Open back to the equilateral triangle. Bring
one vertex to the opposite midpoint and crease.
What is this shape called?
Describe its sides and angles.
How many triangles ll the shape?
How does the area of one of these triangles
compare to the area of the trapezoid?
How does the area of the trapezoid
compare with the large equilateral triangle?
© 2005 AIMS Education Foundation
8.
Fold the second and third vertices to the same midpoint.
What is this shape called?
How does the height of this rectangle compare with
the height of the original triangle?
Notice that the vertices of the triangle are all together.
What is the sum of these angles?
9.
Open to the trapezoid. Fold one of the
outside triangles over the center triangle.
What shape is formed?
How do the sides and angles compare?
How does its area compare to the area of the trapezoid?
10. Fold the remaining triangle over the center triangle.
What is this shape called?
How does its area compare with that of the rhombus?
11. Open to the original triangle. Fold the sides up
into a three-dimensional pyramid.
What is another name for this shape?
12. Open to the original triangle. Fold one vertex
to the center of the circle.
What shape is formed?
13. Fold another vertex to the center.
What is this shape called?
How do its sides compare?
How do its angles compare?
14. Fold the third vertex to the center.
What is the name of this shape?
What can you say about its sides and angles?
How many small triangles do you see?
How does the area of the hexagon compare
with that of the original triangle?
15. Open up to the original triangle. Tuck one of the small
triangles at one vertex into the small triangle at another
vertex. Tuck the remaining triangle underneath.
To which three-dimensional model is this shape related?
7
© 2005 AIMS Education Foundation
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CIRCLE CIRCUS
Find the maximum number of interior regions
created by 10 intersecting circles.
First determine the maximum number of interior
regions possible when fewer circles intersect.
Look for the pattern that develops and use it
to complete the table.
1 Circle
Maximum
Number
of Maximum
Number of
Number of
Number
of
Circles
Circles
Regions
Regions
2 Circles
3 Circles
4 Circles
1
1
2
3
3
7
4
5
6
7
8
20
5 Circles
WHAT’S NEXT? VOL. 1
n
34
© 2004 AIMS EDUCATION FOUNDATION
SUM WILL, SUM WON’T
(pg.32)
CIRCLE CIRCUS
(pg.34)
Numberof
Number
ofDice
Dice
Smallest
Smallest
Possible
Possible
Sum
Sum
Largest
Largest
Possible
Possible
Sum
Sum
Sum(s)
With
Sum(s) With
Greatest
Greatest
Chance of
Chance
of
Occurring
Occuring
Occurring
2
2
12
3
3
4
4
5
5
6
Maximum
Number
of Maximum
Number of
Number of
of
Circles
Circles
Regions
Regions
1
1
2
3
7
3
7
18
10 or 11
4
13
24
14
5
21
30
17 or 18
6
31
6
36
21
7
43
7
7
42
24 or 25
8
57
8
8
48
28
9
9
54
31 or 32
20
20
120
70
40
40
240
140
100
100
600
350
n (even)
n
6n
7n
2
n (odd)
n
6n
7n-1 7n+1
2 or 2
20
301
n
nn --n+1
n+1
2
The first set of differences for the numbers in the second
column is 2, 4, 6, 8, 10, etc. This pattern allows students
to continue the table indefinitely. The horizontal solution
is n(n-1) +1 = n2 - n + 1. To help students, suggest they
subtract one from each number in the second column,
obtaining 0, 2, 6, 12, 20, etc.
TWENTY-ONE CONNECT
(pg.35)
THE END OF THE WORLD
(pg.33)
Number of
Number
Disks
Disks
Minimum
Minimum
Number
Number of
Moves
Moves
1
1
2
3
3
7
4
15
5
31
64
2264-1- 1
n
22nn-1
-1
64
Extra Challenge Solution:
11
Approximately 5.85 x 10 = 585,000,000,000 years.
Students will need a scientific calculator to solve the
extra challenge.
WHAT’S NEXT? VOL. 1
56
Students should count
the number of segments
for circles with 5 points
and 6 points. To be sure
all possible segments are drawn, have students examine
how many originate from each point. This should always
be one fewer than the number of points on the circle.
Note that the differences for the numbers in the second
column are 1, 2, 3, 4, 5, etc. This discovery allows students to continue the table indefinitely. If students are
familiar with triangular numbers, they will recognize the
numbers in the second column as triangular numbers.
© 2004 AIMS EDUCATION FOUNDATION