Transformation of Euclid? Revisiting Geometry in Light of

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

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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.


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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/


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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/


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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.


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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/


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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


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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.


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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


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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


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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


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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?


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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, ...


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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.


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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


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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


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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?


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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.


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Other Transformation: Translation

Observations?


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Glide reflection

Triangle ABC is

reflected across line

and then translated using

vector .

Observations?

DE

DE


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Finding the vector:


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Dilations


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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



May also do on paper.


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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/


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http://www.illustrativemathematics.org/

https://www.khanacademy.org/math/geometry/similarity/similarity-and-transformations/e/defining-similarity-through-angle-preserving-transformations




















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Transformation of Euclid? Revisiting Geometry in Light of