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The Learning Circle Knowledge & Formulas The Learning Circle Knowledge & Formulas

The Learning Circle Knowledge & Formulas - PowerPoint Presentation

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The Learning Circle Knowledge & Formulas - PPT Presentation

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

intersection circle find learning circle intersection learning find lid discovery long stick diameter point stonehenge bec tangent aed determine 120 lengths angles

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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