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The Arizona Mathematics
Partnership:
Saturday 3: Geometry
Ted Coe, January 31, 2015
cc-by-sa 4.0 unported unless otherwise noted
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
Perimeter
What is “it”?
Is the perimeter a measurement?
…or is “it” something we can measure?
Measurement
What do we mean when we talk about
“measurement”?
•Using objects at your table measure the angle
Angles
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
Warm-up: Geometric Fractions
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
Is this a proof?
1
a
Area of one green triangle = 2 𝑎𝑏
b
Area of blue square = 𝑐 2
a
c
Area of whole (red) square =
c
b
(𝑎 + 𝑏)(𝑎 + 𝑏)
OR
1
4 ∙ 𝑎𝑏 + 𝑐 2
2
This means that:
c
c
a
b
𝑎 + 𝑏 𝑎 + 𝑏 = 2𝑎𝑏 + 𝑐 2
𝑎2 + 𝑎𝑏 + 𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐 2
b
a
𝑎2 + 2𝑎𝑏 + 𝑏2 = 2𝑎𝑏 + 𝑐 2
𝑎2 + 𝑏2 = 𝑐 2
Indiana (1896)
House Bill 296, Section 2:
“…that the ratio of the diameter and circumference is as fivefourths to four;”
What is the mathematical value they are proposing for Pi?
From http://www.agecon.purdue.edu/crd/Localgov/Second%20Level%20pages/indiana_pi_bill.htm
5
Illustration: 4 ÷
4
 Find the dimensions
of the rectangle
 Find the area of the
rectangle
 Find a rectangle
somewhere in the
room similar to the
shaded triangle
When
we say
two figures are
similar we
mean…
Answer on your own. Share.
What is a scale factor?
Teaching Geometry According to the Common Core Standards, H. Wu Revised: April 15,
2012. Grade 7 notes, p.49:
Working with similar figures
“Similar means same shape different
size.”
“All rectangles are the same shape.
They are all rectangles!”
“Therefore all rectangles are similar.”
CCSS: Grade 2 (p.17)
CCSS: Grade 7 (p.46)
CCSS: Grade 7 (p.46)
From the Progressions
ime.math.arizona.edu/progressions
CCSS: Grade 8 (p.56)
Teaching Geometry According in Grade 8 and High School According to the Common
Core Standards, H. Wu Revised: October 16, 2013, p.45
http://math.berkeley.edu/~wu/CCSS-Geometry.pdf
CCSS: Grade 8 (p.54)
From the progressions documents
Source: http://commoncoretools.me/wp-content/uploads/2013/07/ccss_progression_functions_2013_07_02.pdf p.5
CCSS: HS Geometry (p.74)
CCSS: HS Geometry (p.74)
CCSS: HS Geometry (p.74)
CCSS: Grade 8 (p.56)
CCSS: HS Geometry (p.77)
Pythagorean Theorem?