Transformation of Euclid? Revisiting
Geometry in Light of the CCSSM
Standards
NHTM Spring 2014 Conference
Teresa D. Magnus, Rivier University
March 17, 2014
Motivation for this Presentation
Investigate some of the rumors out there
regarding how CCSSM has/will change the
geometry we teach.
Did CCSSM take proof-writing out of
geometry?
What does a transformational approach mean?
Transformation of Euclid: Teresa
Magnus
Early Publication of CCSSM
Mathematical practices
K-8 grade level standards
Topical standards for high school
Natural tendency: Look at the standards
for the course(s) I teach. Interpret the list
as the topics to be covered.
Transformation of Euclid: Teresa
Magnus
From the Updated CCSSM Website
“The Common Core concentrates on a
clear set of math skills and concepts.
Students will learn concepts in a more
organized way both during the school year
and across grades. The standards encourage
students to solve real-world problems.”
http://www.corestandards.org/Math/
Transformation of Euclid: Teresa
Magnus
Major Shifts
1. Greater focus on fewer topics.
2. Linking topics and thinking across grades
3. Pursue conceptual understanding,
procedural skills and fluency, and
application with equal intensity
http://www.corestandards.org/other-resources/key-shifts-in-mathematics/
Transformation of Euclid: Teresa
Magnus
Previous Website?
I remember finding it difficult to find the
word “prove” when I first looked at the
CCSSM, but the words “rigor” and
“reason” were always there.
Now the website has been updated and it
does appear more prominently in places.
Transformation of Euclid: Teresa
Magnus
From the CCSSM Website
Understanding Mathematics
“These standards define what students should understand and be able to do in their study of mathematics. But asking a student to understand something also means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.”
http://www.corestandards.org/Math/
Transformation of Euclid: Teresa
Magnus
Themes in CCSSM Geometry
Content for the Middle Grades Geometrical Properties of polygons
Connections between coordinate and plane geometry
Connections between 2-dimensional and 3-dimensional figures
Concepts of congruence and similarity evolve from transformations (rotations, reflections, translations, dilations)
Problem solving, reasoning, constructing
Topics of congruence/similarity criteria, area and volume formulas, Pythagorean theorem remain
Transformation of Euclid: Teresa
Magnus
High School Geometry Overview
Congruence & Similarity
Experiment with transformations in the
plane.
Understand congruence in terms of rigid
motions and similarity in terms of dilations
Prove geometric theorems.
Make geometric constructions.
Understand trigonometric ratios and apply
trigonometry to general triangles.
Transformation of Euclid: Teresa
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High School Geometry Overview
Circles:
Understand and apply theorems about circles.
Find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations:
Translate between the geometric description and the equation for a conic section.
Use coordinates to prove simple geometric theorems algebraically
Transformation of Euclid: Teresa
Magnus
High School Geometry Overview
Geometric Measurement and Dimension
Explain volume formulas and use them to
solve problems
Visualize relationships between two-
dimensional and three-dimensional objects
Modeling with Geometry:
Apply geometric concepts in modeling
situations
Transformation of Euclid: Teresa
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Activity Given a shape drawn on a piece of paper, how
can you fold it so that you can cut out the figure
using a single snip with scissors?
www.artofmathematics.org
Transformation of Euclid: Teresa
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How did you do this?
Where did you fold the figure?
What other line(s) were needed to fold
the irregular figures?
How is this related to symmetry?
Transformation of Euclid: Teresa
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Geometrical Observations
If folding along a line through a vertex of a
angle maps each side of the angle onto the
other, ...
If folding along a mirror intersecting a line
maps each ray of the line onto the other, ...
Transformation of Euclid: Teresa
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And the Converses
Folding along an angle bisector takes each
side of the angle to the other.
Folding along a line m perpendicular to line l takes each ray of l to the other.
Transformation of Euclid: Teresa
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The Equilateral Triangle Case
Folding across the bisector of angle ABC, leaves ABF which is congruent to CBF. The two sides
and are now aligned as are their endpoints. In addition, and are now aligned since the angle bisector of ABC is also the
perpendicular bisector of . Then folding along the bisector of the vertex angle at A maps point F onto point F’ and segment onto segment . All three sides of ABF are now along one cutting
edge with vertices A and C together at one end and vertex B at the other.
AB
CB
AF CF
AC
AF
'AF
Transformation of Euclid: Teresa
Magnus
Scalene Case Less Clear
Reflecting across the angle bisector at vertex C, we get the rays and
to line up, but the vertices B and A are not together. In addition is not
perpendicular to so does not lie on .
CB CA
CFAB FAFB
Transformation of Euclid: Teresa
Magnus
Properties of reflected objects:
• What happens to a point on the
mirror?
• What happens to a point off the
axis of reflection? Where does
the segment connecting B and B’
intersect the mirror?
• What happens to a line segment?
• What happens to a triangle or
other polygon?
• What happens to the orientation
of the polygon?
Transformation of Euclid: Teresa
Magnus
Other Transformations: Rotation
150 rotation clockwise about D.
Use the figure on the left to see angle
and orientation properties.
The figure on the lower right
suggests a method of
determining the location of the
rotational center.
Transformation of Euclid: Teresa
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Other Transformation: Translation
Observations?
Transformation of Euclid: Teresa
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Glide reflection
Triangle ABC is
reflected across line
and then translated using
vector .
Observations?
DE
DE
Transformation of Euclid: Teresa
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Finding the vector:
Transformation of Euclid: Teresa
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Dilations
Transformation of Euclid: Teresa
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GeoGebra Activity
Open up Geogebra (check what’s under
the arrows on the lower right corner of
each icon).
Mystery Transformation 1
Mystery Transformation 2
Mystery Transformation 3
May also do on paper.
Transformation of Euclid: Teresa
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Sources for Tasks
www.illustrativemathematics.org
https://www.khanacademy.org/math/geometry/similarity/simila
rity-and-transformations/e/defining-similarity-through-angle-
preserving-transformations
http://www.artofmathematics.org/
Transformation of Euclid: Teresa
Magnus