Dr Everett McCoy The Learning Circle You have a round lid and a stick Break the stick into two unequal lengths Place the sticks into the lid with the ends at the rim so that they cross Place a mark at the crossing point on each ID: 655148
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Slide1
The Learning Circle
Knowledge & Formulas
Dr. Everett McCoySlide2
The Learning Circle
You have a round lid and a stick.
Break the stick into two unequal lengths.
Place the sticks into the lid with the endsat the rim so that they cross.Place a mark at the crossing point on each
stick. (
refer to the diagram)
Label each measurement as a, b, c, and d.
How long is it from each end of each stick to the mark?
What is the relationship between these four lengths?
[Try combining them using multiplication]
How long is each?Slide3
The Learning Circle
Counting and Cardinality K.CC
Geometry
K.G, 1.G, 2.GMeasurement and Data 4.MDGeometry 7.GReasoning with Equations and Inequalities A-REI
Congruence G-CO
Circles G-CExpressing Geometric Properties with
Equations G-GPE
Geometric Measurement and Dimension G-GMD
Trigonometric Functions
F-TF
Common Core StandardsSlide4
The Learning Circle
In
this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. Students develop informal arguments justifying common formulas for circumference, area, and volume of geometric objects, especially those related to circles.
Unit 3: Circles and VolumeSlide5
The Learning Circle
Discovery
4
6
8
Find
x
.
x
ASlide6
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
vertical angles are
B
D
C
E
x
ASlide7
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
m
D
= ½
m
AB
The measure of an inscribed angle is half the subtended arc
B
D
C
E
x
ASlide8
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
m
D
= ½
m
AB
m
C
= ½
m
AB
The measure of an inscribed angle is half the subtended arc
B
D
C
E
x
ASlide9
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
D
C
Two inscribed angles subtending the same arc are congruent
B
D
C
E
x
ASlide10
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
D
C
A B
If two corresponding angles of two triangles are congruent, so are the third angles.
B
D
C
E
x
ASlide11
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
D
C
A B
AED
~
BEC
AAA similarity
B
D
C
E
x
ASlide12
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
D
C
A B
AED
~
BEC
x
/6 = 4/8
ratios of corresponding parts of similar triangles are constant
B
D
C
E
x
ASlide13
The Learning Circle
Discovery
4
6
A
8
Find
x
.
AED
BEC
D
C
A B
AED
~
BEC
x
/6 = 4/8
and
x
= 3
B
D
C
E
x
ASlide14
The Learning Circle
Refinement
4
6
8
Find
x
.
Since
a
/
b
=
c
/
d
or
ad
=
bc
then
8
x
= 24
or
x
= 3
x
ASlide15
The Learning Circle
Use
4
6
8
Find
x
.
x
A
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide16
The Learning Circle
Use
4
6
8
Find
x
.
8
x
=
6
4
x
A
(circle-to-intersection)
(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide17
The Learning Circle
Use
4
6
8
Find
x
.
8
x
=
6
4
x
A
(circle-to-intersection)
(intersection-to-circle)
=
(
circle-to-intersection)(intersection-to-circle)Slide18
The Learning Circle
Use
4
6
8
Find
x
.
8
x =
6
4
x
A
(circle-to-intersection)(intersection-to-circle)
=
(
circle-to-intersection)(intersection-to-circle)Slide19
The Learning Circle
Use
4
6
8
Find
x
.
8
x =
6
4
x
A
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)
(intersection-to-circle)Slide20
The Learning Circle
Use
4
6
8
Find
x
.
8
x =
6
4
x
A
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)
(intersection-to-circle)Slide21
The Learning Circle
Use
4
6
8
Find
x
.
8
x =
6
4
or
x
= 3
x
A
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide22
The Learning Circle
Use
4
6
8
Find
x
.
8
x =
6
4
or
x
= 3
x
(OX)(XO)=(OX)(XO)
ASlide23
The Learning Circle
You have a round lid and a stick.
Break the stick into two unequal lengths.
Place the sticks into the lid with the endsat the rim so that they cross.Place a mark at the crossing point on each
stick. (
refer to the diagram)
Label each measurement as a, b, c, and d.
How long is it from each end of each stick to the mark?
What is the relationship between these four lengths?
[Try combining them using multiplication]
How long is each?
Calculate results based on our discussion.
Are your results different? Why or why not?Slide24
The Learning Circle
You have a round lid and a stick.
Arrange the sticks so that they now
intersect at one end outside of the lid.The sticks touch the lid at one end and crossthe lid at another point.
(refer to the diagram)
How long is it from each point where a stick meets the lid to the intersection?
What is the relationship between the lengths outside and the total lengths?
[Try combining them using multiplication,… again]
How long is each?Slide25
The Learning Circle
Discovery
4
6
A
8
Find
x
.
B
D
C
E
x
BSlide26
The Learning Circle
Discovery
4
6
A
8
Find
x
.
A B
inscribed angles subtending the same arc are congruent
B
D
C
E
x
BSlide27
The Learning Circle
Discovery
4
6
A
8
Find
x
.
A B
E E
an angle is congruent to itself
B
D
C
E
x
BSlide28
The Learning Circle
Discovery
4
6
A
8
Find
x
.
A B
E E
ACE BDE
If two corresponding angles of two triangles are congruent, so are the third angles.
B
D
C
E
x
BSlide29
The Learning Circle
Discovery
4
6
A
8
Find
x
.
A B
E E
ACE BDE
AEC ~
BED
AAA similarity
B
D
C
E
x
BSlide30
The Learning Circle
Discovery
4
6
A
8
Find
x
.
A B
E E
ACE BDE
AEC ~
BED
ratios of corresponding parts of similar triangles are constant
B
D
C
E
x
BSlide31
The Learning Circle
Discovery
4
6
A
8
Find
x
.
A B
E E
ACE BDE
AEC ~
BED
and
112 = 4
x
+ 16
x
= 24
B
D
C
E
x
BSlide32
The Learning Circle
Use
4
6
A
8
Find
x
.
B
D
C
E
x
B
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide33
The Learning Circle
Use
4
6
A
8
B
D
C
E
x
B
(circle-to-intersection)
(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)
4
(8 +
x
) = 4(6 + 4)
Find
x
.Slide34
The Learning Circle
Use
4
6
A
8
Find
x
.
B
D
C
E
x
B
(circle-to-intersection)
(intersection-to-circle)
=
(
circle-to-intersection)(intersection-to-circle)
4(4 +
x
)
= 4(6 + 4)Slide35
The Learning Circle
Use
4
6
A
8
Find
x
.
B
D
C
E
x
B
(circle-to-intersection)(intersection-to-circle)
=
(
circle-to-intersection)(intersection-to-circle)
8(8 +
x
) =
4(6 + 4)Slide36
The Learning Circle
Use
4
6
A
8
Find
x
.
B
D
C
E
x
B
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)
(intersection-to-circle)
4(4 +
x
) = 8
(6 + 4)Slide37
The Learning Circle
Use
4
6
A
8
Find
x
.
B
D
C
E
x
B
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)
(intersection-to-circle)
4(4 +
x
) = 8(6 + 8)Slide38
The Learning Circle
Use
4
6
A
8
Find
x
.
B
D
C
E
x
B
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)
4(4 +
x
) = 8(6 + 8)
16 + 4
x
= 112
.
.
4
x
= 96
.
or
.
x
= 24
.Slide39
The Learning Circle
You have a round lid and a stick.
Arrange the sticks so that they now
intersect at one end outside of the lid.The sticks touch the lid at one end and crossthe lid at another point.
(refer to the diagram)
How long is it from each point where a stick meets the lid to the intersection?
What is the relationship between the lengths outside and the total lengths?
[Try combining them using multiplication,… again]
How long is each?
Calculate results based on our discussion.
Are your results different? Why or why not?Slide40
The Learning Circle
You have a round lid and a stick.
Rearrange the sticks so that one of them
is now tangent to the lid.The other stick touches the lid at one endand crosses the lid at another point.
(refer to the diagram)
How long is it from each point where the stick meets the lid to the intersection?
What is the relationship between the length outside and total length of the one stick to the length of the tangent?
[Try a variation of the method we have been using]
How long is each?Slide41
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
.Slide42
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
A CBD
m
CBD
=
1
/
2
CB =
m
ASlide43
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
A CBD
D D
an angle is congruent to itselfSlide44
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
A CBD
D D
DCB ABD
If two corresponding angles of two triangles are congruent, so are the third angles.Slide45
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
A CBD
D D
DCB ABD
ABD ~
BCD
AAA similaritySlide46
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
A CBD
D D
DCB ABD
ABD ~
BCD
ratios of corresponding parts of similar triangles are constantSlide47
The Learning Circle
Discovery
4
A
6
Find
x
.
C
B
D
x
C
A CBD
D D
DCB ABD
ABD ~
BCD
or
4(
x
+ 4) = 6
2
x
= 5Slide48
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
.
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide49
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
4
(4 +
x
) = 6
2
(circle-to-intersection)
(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide50
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
4(4 +
x
)
= 6
2
(circle-to-intersection)
(intersection-to-circle)
=
(
circle-to-intersection)(intersection-to-circle)Slide51
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
4(4 +
x
) =
6
2
(circle-to-intersection)(intersection-to-circle)
=
(
circle-to-intersection)(intersection-to-circle)Slide52
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
4(4 +
x
) = 6
2
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)
(intersection-to-circle)Slide53
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
4(4 +
x
) = 6
2
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)
(intersection-to-circle)Slide54
The Learning Circle
Use
4
A
6
Find
x
.
C
B
D
x
C
4(4 +
x
) = 6
2
or
16 + 4
x
= 36
4
x
= 20
x
= 5
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide55
The Learning Circle
Use
x
A
6
Find
x
.
C
B
D
5
C
2
x
(5 +
x
) = 6
2
or
x
2
+ 5
x
– 36 = 0
x
= –9, 4
since x>
0,
x
= 4
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide56
The Learning Circle
Use
4
A
x
Find
x
.
C
B
D
5
C
3
4(5 + 4) =
x
2
or
x
2
= 36
x
= ±6
since x>
0,
x
= 6
(circle-to-intersection)(intersection-to-circle) =
(
circle-to-intersection)(intersection-to-circle)Slide57
The Learning Circle
You have a round lid and a stick.
Rearrange the sticks so that one of them
is now tangent to the lid.The other stick touches the lid at one endand crosses the lid at another point.
(refer to the diagram)
How long is it from each point where the stick meets the lid to the intersection?
What is the relationship between the length outside and total length of the one stick to the length of the tangent?
[Try a variation of the method we have been using]
How long is each?
Calculate results based on our discussion.
Are your results different? Why or why not?Slide58
The Learning Circle
Because he is not allowed inside, to determine the diameter of Stonehenge, a researcher stretches a rope 120ft long from the edge of Stonehenge along a line through the center. His assistant stretches a rope 240ft long from the side of and tangent to the circle to a point where the ropes meet. What is the diameter of the Stonehenge circle?
What Can We Do?
12
0
24
0Slide59
The Learning Circle
Because he is not allowed inside, to determine the diameter of Stonehenge, a researcher stretches a rope 120ft long from the edge of Stonehenge along a line through the center. His assistant stretches a rope 240ft long from the side of and tangent to the circle to a point where the ropes meet. What is the diameter of the Stonehenge circle?
What Can We Do?
12
0
24
0
r
r
r + 120
r
2
+ 240
2
= (r+120)
2
240
2
= 240r + 120
2
r = 180
Pythagoras’ TheoremSlide60
The Learning Circle
Because he is not allowed inside, to determine the diameter of Stonehenge, a researcher stretches a rope 120ft long from the edge of Stonehenge along a line through the center. His assistant stretches a rope 240ft long from the side of and tangent to the circle to a point where the ropes meet. What is the diameter of the Stonehenge circle?
What Can We Do?
12
0
24
0
d
120(d + 120) = 240
2
120d
+
120
2
= 240
2
120d = 240
2
– 120
2
120d = (
240–120)(240+120)
120d = (120)(360)
d = 360 so r
=
180
Secant-Tangent IntersectionSlide61
The Learning Circle
Because he is not allowed inside, to determine the diameter of Stonehenge, a researcher stretches a rope 120ft long from the edge of Stonehenge along a line through the center. His assistant stretches a rope 240ft long from the side of and tangent to the circle to a point where the ropes meet. What is the diameter of the Stonehenge circle?
Why Can We Do It?
d
t
r
r
r + d
r
2
+ t
2
= (r + d)
2
r
2
+ t
2
= r
2
+2rd + d
2
t
2
= 2rd + d
2
r = (t
2
–d
2
)/2d
r = (
t+d
)(t–d)/2d
r =
t+d
.
t–d
2 d
r = (average)(relative error to diameter line)Slide62
The Learning Circle
Applications
75
110
x
80
A farmer would like to find the distance between two trees on opposite sides of a lake. Using the map, estimate how the farmer would do that.Slide63
The Learning Circle
Applications
75
110
x
80
A farmer would like to find the distance between two trees on opposite sides of a lake. Using the map, estimate how the farmer would do that.Slide64
The Learning Circle
You have a round container, your sticks, and string.
Find the diameter of the container.
Guess at a line through the center of thecontainer. You may place a hole in thecontainer if that helps.
What if you can’t measure through the center?We know that two tangent segments that meet
are equal in length. Can you devise aninstrument that would help achieve a
better measurement of the diameter
without having to put a hole in the container?
How long is each?Slide65
The Learning Circle
Applications
x
An astronaut is in a craft 28 miles above the Earth’s equator. The diameter of the Earth is approximately 7972 miles. If the craft flies in a straight line, how far is it to the horizon?Slide66
The Learning Circle
Applications
An offset cam wheel made of two eccentric circles is broken and needs to be replaced. The cam wheel must be specially manufactured. You must determine the diameter of the larger wheel from the fragment you have.Slide67
The Learning Circle
Applications
An offset cam wheel made of two eccentric circles is broken and needs to be replaced. The cam wheel must be specially manufactured. You must determine the diameter of the larger wheel from the fragment you have.Slide68
The Learning Circle
Applications
You have a fragment of ancient pottery that was roughly circular. Determine the size of the pottery.Slide69
The Learning Circle
Applications
You have a fragment of ancient pottery that was roughly circular. Determine the size of the pottery.Slide70
The Learning Circle
Applications
You have a fragment of ancient pottery that was roughly circular. Determine the size of the pottery.
If the pottery is a tapered vessel, can you devise a plan to more efficiently determine
where
in the vessel the fragment goes?Slide71
The Learning Circle
Thank You