Ted Coe, Scottsdale Community College, 2012. Some materials were also created or refined as part of the development of the “Math 5: Geometry” curriculum for Arizona State University’s Teaching Foundations Project.

*AMP 2012-2013:

Saturday #1

*The 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

The Plot...

*Define

SquareTriangleAngle

*Quadrilaterals

*Quadrilaterals

The Broomsticks

The RED broomstick is three feet long

The YELLOW broomstick is four feet long

The GREEN broomstick is six feet long

The Broomsticks

The Willis tower (formerly the Sears tower) is 1730 feet high. The Burj Khalifa (formerly Burj Dubai) is 2717 feet high.

The Burj is ______________ times as

large as the Willis tower.

The Willis tower is _____________times as

large as the Burj

The Burj is _____________ percent the

size of the Willis tower.

The Willis tower is _____________ percent

the size of the Burj.

*Perimeter

*Is perimeter a one-dimensional, two-dimensional, or three-dimensional thing?

*Does this room have a perimeter?

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

*Measurement

How about this?

•Determine the attribute you want to measure•Find something else with the same attribute. Use it as the measuring unit.•Compare the two: multiplicatively.*Measurement

*What is a circle?

*What is circumference?

*Tennis Balls

The circumference is about how many times as large as the diameter?

The diameter is about how many times as large as the circumference?

•Using objects at your table measure the angle

*Angles

What is the length of “d”? You may choose the unit.

Define: Area

Area has been defined* as the following:

“a two dimensional space measured by the number of non-overlapping unit squares or parts of unit squares that can fit into the space”

Discuss...*State of Arizona 2008 Standards Glossary

*What about the kite?

Area of whole square is 4r^2

Area of red square is 2r^2

Area of circle is…

*Pizza

• $10 for 14” large

• $12 for 16” x-large

• Papa John's claim: 30% more!

*How many

Rotations?

Cut 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?

ab

c

Lay down your triangle on construction paper.

Draw squares off each of the three sides.

Estimate the areas of these squares.

X

Y

ZDraw the following parallel and perpendicular lines:

X: Along the right side of the hypotenuse’s square

Y: Perpendicular to line “X” passing through the corner of the opposite square

Z: Perpendicular to line “Y” passing through the intersection of the square and line “Y”.

If the Pythagorean Theorem is true AND

If you have constructed and cut correctly

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

Is this a proof?

Area of blue square =

a

b

Area of whole (red) square =

(𝑎+𝑏)(𝑎+𝑏)b

a

Area of one green triangle =

OR

c

This means that:

𝑎2+𝑎𝑏+𝑎𝑏+𝑏2=2𝑎𝑏+𝑐2

𝑎2+2𝑎𝑏+𝑏2=2𝑎𝑏+𝑐2

𝑎2+𝑏2=𝑐2

a

a

b

b

cc

c