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Saturday 2: Geometry. Ted Coe, September 2014. cc-by-. sa. 3.0 . unported. unless otherwise noted. Warm-up: Geometric Fractions. If = . . What is 1?. How can you use this to show that . ID: 230282

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The Arizona Mathematics Partnership:

Saturday 2: Geometry

Ted Coe, September 2014

cc-by-

sa

3.0

unported

unless otherwise noted

Slide2Warm-up: Geometric Fractions

Slide3If = . What is 1?How can you use this to show that

Check for Synthesis:

3

Source:

Slide4Geometric Fractions

Slide5THE Rules of Engagement

Speak

meaningfully

— what you say should carry meaning;

Exhibit

intellectual integrity

— base your conjectures on a logical foundation; don’t pretend to understand when you don’t;

Strive

to make sense

— persist in making sense of problems and your colleagues’ thinking.

Respect

the learning process of your colleagues

— allow them the opportunity to think, reflect and construct. When assisting your colleagues, pose questions to better understand their constructed meanings. We ask that you refrain from simply telling your colleagues how to do a particular task.

Marilyn

Carlson, Arizona State University

Slide6Define

Square

Triangle

Angle

Slide7Quadrilaterals

Slide8Quadrilaterals

Slide9The RED broomstick is three feet long

The YELLOW broomstick is four feet longThe GREEN broomstick is six feet long

The Broomsticks

Slide1010

Source

: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

Slide1111

Source

: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

Slide1212

Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

Slide13Slide14

Perimeter

What is “it”?

I

s

the perimeter a measurement

?

…or

is “it” something we

can

measure?

Slide15Perimeter

Is

perimeter a one-dimensional, two-dimensional, or three-dimensional thing

?

Does this room have a perimeter?

Slide16What do we mean when we talk about “measurement”?

Measurement

Slide17How about this?

Determine the attribute you want to measureFind something else with the same attribute. Use it as the measuring unit.Compare the two: multiplicatively.

Measurement

Slide18So.... how do we measure circumference?

Circumference

Slide19Tennis Balls

Slide20Circumference

If I double the RADIUS of a circle what happens to the circumference?

Slide21Slide22

The circumference is three and a bit times as large as the diameter.

http://tedcoe.com/math/circumference

Slide23What is an angle?

Angles

Slide24Using objects at your table measure the angle

Angles

Slide25CCSS, Grade 4, p.31

Slide26Slide27

Slide28

Measure the length of

s. Choose your unit of measure carefully.

Measure the angle. Choose your unit carefully.

s

Slide29Define: Area

Slide30Area: Grade 3 CCSS

Slide31Slide32

What about the kite?

Slide33Slide34

Slide35

Slide36

Area of whole square is 4r^2

Area of red square is 2r^2

Area of circle is…

Slide37Cut out a right triangle from a 3x5 card – try to make sure that one leg is noticeably larger than the other.

What strategies could you use to create this?

a

b

c

Slide38Lay down your triangle on construction paper

. Match my orientation with the right angle leaning right.

Draw squares off each of the three sides.

Measure the

areas of these squares.

Slide39Let’s try something crazy…

I came across an interesting diagram and I want to walk you through the design.

See: A

.

Bogomolny

,

Pythagorean Theorem and its many proofs

from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/pythagoras/index.shtml

, Accessed 12 September 2014

Slide40See: A

.

Bogomolny

,

Pythagorean Theorem and its many proofs

from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/pythagoras/index.shtml

, Accessed 12 September 2014

Slide41See: A

.

Bogomolny

,

Pythagorean Theorem and its many proofs

from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/pythagoras/index.shtml

, Accessed 12 September 2014

Slide42Perpendicular

See: A

.

Bogomolny

,

Pythagorean Theorem and its many proofs

from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/pythagoras/index.shtml

, Accessed 12 September 2014

Slide43Perpendicular

See: A

.

Bogomolny

,

Pythagorean Theorem and its many proofs

from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/pythagoras/index.shtml

, Accessed 12 September 2014

Slide441

2

3

4

5

See: A

.

Bogomolny

,

Pythagorean Theorem and its many proofs

from Interactive Mathematics Miscellany and Puzzles

http://www.cut-the-knot.org/pythagoras/index.shtml

, Accessed 12 September 2014

Slide45Slide46

If the Pythagorean Theorem is true

AND

If you have constructed and cut correctly

THEN

You should be able to show that the sum of the area of the smaller squares equals the area of the larger square.

Slide47Is this a proof?

Slide48Slide49

Area of blue square:

c

2

a

b

Area of whole (red) square:

(

a + b

)(

a + b

)

b

a

Area of green triangle:

ab

OR

4*

+

c

2

c

This means that:

(

a + b

)(

a + b

) =

2ab + c2

a2 + ab + ab + b2 = 2ab + c2

a2 + 2ab + b2 = 2ab + c2

a2 + b2 = c2

Slide50CCSS: Grade 8

Slide51CCSS: HS Geometry

Slide52http://www.cut-the-knot.org/pythagoras/index.shtml

Slide53Slide54

Find the dimensions of the rectangle

Find the area of the rectangle

Find a rectangle somewhere in the room

similar to the shaded triangle

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