The Arizona Mathematics Partnership: Saturday 2: Geometry Ted
Coe, September 2014 cc-by-sa 3.0 unported unless otherwise
noted
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Warm-up: Geometric Fractions
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Check for Synthesis: 3 SOURCE:
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Geometric Fractions
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THE Rules of Engagement Speak meaningfully what you say should
carry meaning; Exhibit intellectual integrity base your conjectures
on a logical foundation; dont pretend to understand when you dont;
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
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Define Square Triangle Angle
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Quadrilaterals
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The RED broomstick is three feet long The YELLOW broomstick is
four feet long The GREEN broomstick is six feet long The
Broomsticks
Perimeter What is it? Is the perimeter a measurement? or is it
something we can measure?
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Perimeter Is perimeter a one-dimensional, two- dimensional, or
three-dimensional thing? Does this room have a perimeter?
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What do we mean when we talk about measurement?
Measurement
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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
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So.... how do we measure circumference? Circumference
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Tennis Balls
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Circumference If I double the RADIUS of a circle what happens
to the circumference?
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The circumference is three and a bit times as large as the
diameter. http://tedcoe.com/math/circumference
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What is an angle? Angles
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Using objects at your table measure the angle Angles
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CCSS, Grade 4, p.31
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Measure the length of s. Choose your unit of measure carefully.
Measure the angle. Choose your unit carefully. s
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Define: Area
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Area: Grade 3 CCSS
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What about the kite?
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Area of whole square is 4r^2 Area of red square is 2r^2 Area of
circle is
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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
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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.
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Lets 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
http://www.cut-the-knot.org/pythagoras/index.shtml
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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
http://www.cut-the-knot.org/pythagoras/index.shtml
Slide 41
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
http://www.cut-the-knot.org/pythagoras/index.shtml
Slide 42
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
http://www.cut-the-knot.org/pythagoras/index.shtml
Slide 43
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
http://www.cut-the-knot.org/pythagoras/index.shtml
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1 23 45 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
http://www.cut-the-knot.org/pythagoras/index.shtml
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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.
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Is this a proof?
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Area of blue square: c 2 a b Area of whole (red) square: (a +
b)(a + b) b a OR c This means that: (a + b)(a + b) = 2ab + c 2 a 2
+ ab + ab + b 2 = 2ab + c 2 a 2 + 2ab + b 2 = 2ab + c 2 a 2 + b 2 =
c 2