Transcript The Arizona Mathematics Partnership: Saturday 2: Geometry Ted Coe, September 2014 cc-by-sa 3.0 unported unless otherwise noted.

The Arizona Mathematics
Partnership:
Saturday 2: Geometry
Ted Coe, September 2014
cc-by-sa 3.0 unported unless otherwise noted
Warm-up: Geometric Fractions
Check for Synthesis:
If
𝟐
𝟑
= . What is 1?
How can you use this to show that
𝟏
𝟐
𝟑
=
𝟑
𝟐
?
Geometric Fractions
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 Rules of Engagement
Square
Triangle
Angle
Define
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
What is “it”?
Is the perimeter a measurement?
…or is “it” something we can measure?
Perimeter
Is perimeter a one-dimensional, twodimensional, or three-dimensional
thing?
Does this room have a perimeter?
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
•So.... how do we measure circumference?
Circumference
Tennis Balls
Circumference
If I double the RADIUS of a circle what
happens to the circumference?
The circumference is three and a bit times as large as the diameter.
http://tedcoe.com/math/circumference
•What is an angle?
Angles
•Using objects at your table measure the angle
Angles
CCSS, Grade 4, p.31
s
Measure the length of s. Choose your unit of measure carefully.
Measure the angle. Choose your unit carefully.
Define: Area
Area: Grade 3 CCSS
What about the kite?
Area of whole
square is 4r^2
Area of red square
is 2r^2
Area of circle is…
Cut out a right triangle from a 3x5 card – try to
make sure that one leg is noticeably larger than the
other.
b
a
c
What strategies could you use to create this?
Lay 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.
Let’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
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
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
Perpendicular
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
Perpendicular
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
1
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
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.
Is this a proof?
1
Area of green triangle: 2ab
Area of blue square: c2
b
Area of whole (red) square:
(a + b)(a + b)
OR
4*
a
1
𝑎𝑏
2
+ c2
This means that:
(a + b)(a + b) = 2ab + c2
c
a2 + ab + ab + b2 = 2ab + c2
b
a
a2 + 2ab + b2 = 2ab + c2
a2 + b2 = c2
CCSS: Grade 8
CCSS: HS Geometry
http://www.cut-the-knot.org/pythagoras/index.shtml
 Find the dimensions
of the rectangle
 Find the area of the
rectangle
 Find a rectangle
somewhere in the
room similar to the
shaded triangle