Download NEKSDC CCSSM HS Geometry

PRESENTATION WILL INCLUDE…
• Overview of K – 8 Geometry
• Overarching Structure of HS Geometry Content
Standards
• Closer Look at Several Key Content Standards
• Discussion and Activities around Instructional Shifts
and Tasks to engage students in Geometry Content
Standards and reinforce Practice Standards
K – 6 GEOMETRY
STUDENTS BECOME FAMILIAR WITH GEOMETRIC
SHAPES
• THEIR COMPONENTS (Sides, Angles, Faces)
• THEIR PROPERTIES (e.g. number of sides, shapes of
faces)
• THEIR CATEGORIZATION BASED ON PROPERTIES (e.g.
A square has four equal sides and four right angles.)
K – 6 GEOMETRY
COMPOSING AND DECOMPOSING GEOMETRIC
SHAPES
The ability to describe, use, and visualize the effects of
composing and decomposing geometric regions is
significant in that the concepts and actions of
creating and then iterating units and higher-order
units in the context of construction patterns,
measuring, and computing are established bases
for mathematical understanding and analysis.
K-6 GEOMETRY PROGRESSIONS
SPATIAL STRUCTURING AND
SPATIAL RELATIONS IN GRADE 3
• Students are using abstraction when they conceptually structure
an array understand two dimensional objects and sets of objects
in two dimensional space as truly two dimensional.
• For two-dimensional arrays, students must see a composition
of squares (iterated units) and also as a composition of rows
or columns (units of units)
SPATIAL STRUCTURING AND
SPATIAL RELATIONS IN GRADE 5
• Students must visualize three-dimensional solids as being
composed of cubic units (iterated units) and also as a
composition of layers of the cubic units (units of units).
CLASSIFY TRIANGLES IN GRADE 4
By Side Length
Equilateral
Isosceles
Scalene
CLASSIFY TRIANGLES IN GRADE 4
By Angle Size
Acute
Obtuse
Right
ANGLES,
IN GRADE 4, STUDENTS
Understand that angles
are composed of two rays
with a common endpoint
Understand that
an angle is a rotation from a reference line
and that the rotation is measured in degrees
PERPENDICULARITY, PARALLELISM
IN GRADE 4, STUDENTS
Distinguish between lines and line segments
Recognize and draw
Parallel and perpendicular lines
COORDINATE PLANE
Plotting points in Quadrant I is introduced in Grade 5
By Grade 6, students understand the continuous nature of the 2-dimensional
coordinate plane and are able to plot points in
all four quadrants, given an ordered pair
composed of rational numbers.
ALTITUDES OF TRIANGLES
In Grade 6, students recognize that there are three altitudes in every triangle and that
choice of the base determines the altitude.
Also, they understand that an altitude can lie…
Outside the triangle
On the triangle
Inside the triangle
POLYHEDRAL SOLIDS
In Grade 6, students analyze, compose, and decompose polyhedral solids
They describe the shapes of the faces and the number of faces, edges, and vertices
VISUALIZING CROSS SECTIONS
In Grade 7, students describe cross sections parallel to the base of a polyhedron.
SCALE DRAWINGS
In Grade 7, students use their understanding of proportionality to solve problems
involving scale drawings of geometric
figures, including computing actual
lengths and areas from a scale drawing
and reproducing a scale drawing at a
different scale.
Scale: ¼ inch = 3 feet
UNIQUE TRIANGLES
They partake in discovery activities, and
form conjectures, but do not formally
prove until HS.
In Grade 7 students recognize when given conditions will result in a UNIQUE TRIANGLE
IMPOSSIBLE TRIANGLES
In Grade 7 students recognize when given side lengths will or will not result in a triangle
The triangle inequality theorem states
that any side of a triangle is always
shorter than the sum of the other two sides.
If the sum of the lengths of A and B is less than the length of C, then the 3 lengths will not form a
triangle.
If the sum of the lengths of A and B are equal to the length of C, then the 3 lengths will not form a
triangle, since segments A and B will lie flat on side C when they are connected.
GRADE 7 FORMULAS FOR CIRCLES
Know the formulas for the area and circumference of a
circle and use them to solve problems; give an
informal derivation of the relationship between the
circumference and area of a circle.
C = 2πr
A = πr 2
GRADE 7 ANGLE RELATIONSHIPS
Use facts about supplementary, complementary, vertical,
and adjacent angles in a multi-step problem to write
and solve simple equations for an unknown angle in a
figure.
GRADE 7 PROBLEMS INVOLVING 2-D AND 3-D SHAPES
Solve real-world and mathematical problems involving
area, volume and surface area of two- and threedimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
Find the volume
and surface area
GRADE 8 TRANSFORMATIONS
Understand congruence and similarity using physical models,
transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and
translations:
 Lines are taken to lines, and line segments to line segments of
the same length.
 Angles are taken to angles of the same measure.
 Parallel lines are taken to parallel lines.
GRADE 8 TRANSFORMATIONS
Describe the effect of dilations, translations, rotations,
and reflections on two-dimensional figures using coordinates.
GRADE 8 CONGRUENCE VIA RIGID TRANSFORMATIONS
Understand that a two-dimensional figure is congruent to another if
the second can be obtained from the first by a sequence of
rotations, reflections, and translations; given two congruent
figures, describe a sequence that exhibits the congruence
between them.
GR. 8 SIMILARITY VIA NON-RIGID AND RIGID TRANSFORMATIONS
Understand that a two-dimensional figure is similar to another if the second can
be obtained from the first by a sequence of rotations, reflections,
translations, and dilations; given two similar two-dimensional figures,
describe a sequence that exhibits the similarity between them.
Enlarge PQR by a factor of 2.
GRADE 8 ANGLES
c
b
a
Use informal arguments* to establish facts about:
• the angle sum and exterior angle of triangles,
• the angles created when parallel lines are cut by a transversal
• the angle-angle criterion for similarity of triangles.
*For example, arrange three copies of the same triangle so that the
sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
GRADE 8 PYTHAGOREAN THEOREM
 Understand and apply the Pythagorean Theorem.
 Explain a proof of the Pythagorean Theorem and its converse.
Here is one of
many proofs of
the Pythagorean
Theorem.
How does this
prove the
Pythagorean
Theorem?
GRADE 8 PYTHAGOREAN THEOREM
 Apply the Pythagorean Theorem to determine unknown side
lengths in right triangles in real-world and mathematical problems
in two and three dimensions.
 Apply the Pythagorean Theorem to find the distance between two
points in a coordinate system.
From
Kahn
Academy
GRADE 8 VOLUME
Solve real-world and mathematical problems involving volume of
cylinders, cones, and spheres.
 Know the formulas for the volumes of cones, cylinders, and
spheres and use them to solve real-world and mathematical
problems.
http://www.math.com
TURN AND TALK TO YOUR NEIGHBOR
What concepts and skills that HS Geometry have
traditionally spent a lot of time on are now being
introduced in middle school?
How does that change your ideas for focus in HS
Geometry?
What concepts and skills do you predict will be areas of
major focus in HS Geometry?
STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS
Congruence (G-CO)
Similarity, Right Triangles, and Trigonometry (G-SRT)
Circles (G-C)
Expressing Geometric Properties with Equations (G-GPE)
Geometric Measurement and Dimension (G-GMD)
Modeling with Geometry (G-MG)
STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS
Congruence (G-CO)
• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions
• Prove geometric theorems (required theorems listed)
• Theorems about Lines and Angles
• Theorems about Triangles
• Theorems about Parallelograms
 Make geometric constructions (variety of tools and
methods…by hand and using technology) (required
constructions listed)
STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS
Similarity, Right Triangles, and Trigonometry (G-SRT)
• Understand Similarity in terms of similarity
transformations
• Prove theorems involving similarity
• Define trigonometric ratios and solve problems
involving right triangles
• (+) Apply trigonometry to general triangles
• Law of Sines
• Law of Cosines
STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS
Circles (G-C)
Understand and apply theorems about circles
• All circle are similar
• Identify and describe relationships among inscribed
angles, radii, and chords.
• Relationship between central, inscribed, and
circumscribed angles
• Inscribe angles on a diameter are right angles
• The radius of a circle is perpendicular to the tangent where
the radius intersects the circle
Find arc lengths and sectors of circles
STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS
Expressing Geometric Properties with Equations (G-GPE)
• Translate between the geometric description and the
equation for a conic section
• Use coordinates to prove simple geometric theorems
algebraically
STRUCTURE OF THE HS GEOMETRY CONTENT STANDARDS
Geometric Measurement and Dimension (G-GMD)
• Explain volume formulas and use them to solve
problems
• Visualize relationships between two-dimensional and
three-dimensional objects
Modeling with Geometry (G-MG)
• Apply geometric concepts in modeling situations
HS GEOMETRY CONTENT STANDARDS
Primarily Focused on Plane Euclidean Geometry
Shapes are studied Synthetically & Analytically
• Synthetic Geometry is the branch of geometry which makes
use of axioms, theorems, and logical arguments to draw
conclusions about shapes and solve problems
• Analytical Geometry places shapes on the coordinate
plane, allowing shapes to defined by algebraic equations,
which can be manipulated to draw conclusions about
shapes and solve problems.
FINDING ANGLES
Work through this “synthetic” geometry
problem. What definitions, axioms, and
theorems do students need to know?
What algebraic skills?
FINDING ANGLES
The next three shapes and the
previous one were taken from a
site filled with rich Geometry
problems.
http://donsteward.blogspot.com/
In addition to being used to find
angles, students can be asked to
create a copy of each shape
using GeoGebra, which reinforces
many of the Practice Standards
as well as knowledge of
transformations.
FINDING ANGLES
FINDING ANGLES
FORMAL DEFINITIONS AND PROOF
HS Students begin to formalize the experiences with
geometric shapes introduced in K – 8 by
• Using more precise definitions
• Developing careful proofs
When you hear the word “proof”, what do
you envision?
FORMAL DEFINITIONS AND PROOF
In a triangle, the segment connecting the midpoints of two
sides is parallel to the third side and has a length that is
half the length of the third side.
Given the verbal statement of a
theorem, what are the steps that
students need to take in order to
prove the theorem?
SCAFFOLDING
PROOFS
How has the proof of the theorem
already been scaffolded at this
step?
Geometry, Proofs, and the Common Core Standards, Sue Olson, Ed.D,
UCLA Curtis Center Mathematics Conference March 3, 2012
WAYS TO SCAFFOLD THIS SYNTHETIC* PROOF
Easiest to Most Challenging:
• Provide a list of statements and a list of reasons to
choose from and work together as a class
• The above, but no reasons provided
• The above, but done individually
• No list of statements or reasons and done individually
*As opposed to Analytic (using coordinates)
CHANGE IT TO AN ANALYTIC APPROACH
Easiest to Hardest
Use the methods of coordinate geometry to prove that the segment
connecting the midpoints of a triangle with vertices
A (8, 10), B (14, 0), and C (0, 0)
is parallel to the third side and has a length that is one-half the length of
the third side.
Start by drawing a diagram.
Would this method result in a proof? Why or why not?
CHANGE IT TO AN ANALYTIC APPROACH
Harder:
Use the methods of coordinate geometry to prove that the segment
connecting the midpoints of a triangle with vertices
A (2b, 2c), B (2a, 0), and C (0, 0)
is parallel to the third side and has a length that is one-half the length of
the third side.
Would this method result in a proof? Why or why not?
CHANGE IT TO AN ANALYTIC APPROACH
Most Challenging
Use the methods of coordinate geometry to prove that the segment
connecting the midpoints of any triangle is parallel to the third
side and has a length that is one-half the length of the third side.
What could help make this less challenging?
INSTRUCTIONAL SHIFT:
MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE
Congruence, Similarity, and Symmetry are understood
from the perspective of
Geometric Transformation
extending the work that was started in Grade 8
INSTRUCTIONAL SHIFT:
MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE
Rigid Transformations (translations, rotations,
reflections) preserve distance and angle and therefore
result in images that are congruent to the original
shape.
G-C0 Cluster Headings Revisited
• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions
• Prove geometric theorems
• Make geometric constructions
TRANSFORMATIONS AS FUNCTIONS
Using an Analytical Geometry lens, transformations can be
described as functions that take points on the plane as
inputs and give other points on the plane as outputs.
What transformations do these functions imply?
Will they result in congruent shapes?
(x,y)  (x + 3, y)
(x, y)  (y, x)
(x,y)  (x,-y)
(x, y)  (-y, x)
(x,y)  (2x, 2y)
(x, y)  (3x + 2, 3y + 2)
(x, y)  (.5x, y)
(x, y)  (x – 1, y – 1)
TRANSFORMATIONS AS FUNCTIONS
(x,y)  (x + 3, y)*
Turn and talk to your neighbor:
*Compare and contrast the notation above that
communicates a right shift of 3 and the
function notation f(x – 3) used to indicate the function f(x)
is shifted 3 to the right.
INSTRUCTIONAL SHIFT:
MORE FOCUS ON TRANSFORMATIONAL PERSPECTIVE
Two shapes are defined to be congruent to each other if there
is a sequence of rigid motions that
carries one onto the other.
Prove these triangles are congruent
by writing the sequence of rigid
transformations
CONGRUENCY BY TRANSFORMATION
Prove these shapes are congruent
by describing the sequence of
rigid transformations
PROVING SIMILARITY VIA TRANSFORMATIONS
Dilation is a Non-Rigid Transformation that preserves
angle, but involves a scaling factor that affects the
distance, which results in images that are similar to
the original shape.
G-SRT Cluster Headings dealing with Similarity:
• Understand Similarity in terms of similarity
transformations
• Prove theorems involving similarity
PROVING SIMILARITY VIA TRANSFORMATIONS
From a transformational perspective…
Two shapes are defined to be similar to each other if
there is a sequence of rigid motions followed by a nonrigid dilation that carries one onto the other.
A dilation formalizes the idea of scale factor studied in
Middle School.
ANIMATION SHOWING DILATIONS OF LINES AND CIRCLES
Link to Charles A. Dana Center Mathematics Common Core Toolbox
Click on the link
Go to Standards for Mathematical Content
Go to Key Visualizations
Go to Geometry
Discuss how this visualization could be used in the classroom.
What would be a good follow-up activity?
PROVE SIMILARITY BY TRANSFORMATIONS
What non-rigid transformation
proves that these triangles
are similar?
What is the center of dilation?
What is the scale factor of the
Dilation?
FIND SCALE FACTORS GIVEN A TRANSFORMATION
www.ck12.org Similarity Transformations Created by: Jacelyn O'Roark
TOOLS FOR CREATING TRANSFORMATIONS
Using
• Compass
• Ruler
• Protractor
• Transparencies
Task: Leaping Lizards
TOOLS FOR CREATING TRANSFORMATIONS
Using manipulatives such as a set of Tangrams
What shapes do you see?
How are they related?
Can you compose the
shapes to form other
congruent or similar shapes?
Rachel McAnallen's Tangram Activities
TANGRAM PARTNER ACTIVITY
Switch partner roles between “creator” and “maker”
Place a file folder between the partners so they
can’t see each other’s shape.
Each partner has a white sheet of paper marked
N, S, E, W on the appropriate edges.
1st couple of rounds:
The creator creates a shape using all 7 pieces.
Then stands up and gives directions while
watching the “maker” create the shape.
2nd couple of rounds: Creator doesn’t watch the maker.
What Practice
Standards are being
used?
TANGRAM PARTNER ACTIVITY
Using two sets of tangrams, show an illustration
of the Pythagorean Theorem.
What Practice
Standards are being
used?
M.C. ESCHER HTTP://WWW.MCESCHER.COM/
GROUP ACTIVITY
Go to the M.C Escher website and choose Picture Gallery and Symmetry.
Choose a picture. Describe the transformations as clearly as you can.
What transformations do you see. Are there more than one?
What Practice Standards
did you use?
TOOLS FOR CREATING TRANSFORMATIONS
• GeoGebra
• Geometer’s Sketchpad
• Other Dynamic Geometric Software
• Roman Mosaic
Work with a partner or a group to create
this mosaic using GeoGebra.
Discuss the Practice Standards and Content
Standards that were used.
C-C 5. ARC LENGTHS AND SIMILARITY
Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the
constant of proportionality.
http://www.themathpage.com/atrig/arc-length.htm
C-C 5. ARC LENGTHS AND SIMILARITY
http://www.themathpage.com/atrig/arc-length.htm
C-C 5. ARC LENGTHS AND SIMILARITY
The arc length s is proportional to
the radius r. The radian measure θ is
the constant of proportionality
http://www.themathpage.com/atrig/arc-length.htm
C-C 5. ARC LENGTHS AND SIMILARITY
The arc length s is proportional to
the radius r. The radian measure θ is
the constant of proportionality
http://www.themathpage.com/atrig/arc-length.htm
RIGHT TRIANGLE TRIGONOMETRY
Understand that by similarity, side ratios in right triangles
are properties of angles in the triangle, leading to
definitions of trigonometric ratios for acute angles.
Explain and use the relationships between the sine and
cosine of complementary angles.
Relationship between sine and cosine in complementary
angles
CIRCLES IN ANALYTIC GEOMETRY
G-GPE (Expressing Geometric Properties with Equations)
 Derive the equation of a circle given center (3,-2) and radius 6 using the
Pythagorean Theorem
 Complete the square to find the center and radius of a circle with
equation x2 + y2 – 6x – 2y = 26
Think of the time spent in Algebra I on factoring
Versus completing the square to solve quadratic
Equations. What % of quadratics can be solved
by factoring? What % of quadratics can be
Solved by completing the square?
Is completing the square using the area model
more intuitive for students?
CONIC SECTIONS – CIRCLES AND PARABOLAS
• Translate between the geometric description and the equation for a conic section
• Derive the equation of a parabola given a focus and directrix
• Parabola – Note: completing the square to find the vertex of a parabola is in
the Functions Standards
(+) Ellipses and Hyperbolas in Honors or Year 4
Sketch and derive the equation for the parabola with
Focus at (0,2) and directrix at y = -2
Find the vertex of the parabola with equation
Y = x2 + 5x + 7
VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS
• Identify the shapes of 2-dimensional cross sections of 3dimensional objects
VISUALIZE RELATIONSHIPS BETWEEN 2-D AND 3-D OBJECTS
• Identify 3-dimensional shapes generated by rotations
of 2-dimensional objects
http://www.math.wpi.edu/Course_Materials/MA1022C11/volrev/node1.html
LINKS FOR TASKS ON CONSTRUCTION
CCSSI Math Tools Part 1
CCSSI Math Tools Part 2
http://ccssimath.blogspot.fr/2012/12/mathematical-tools-part-3.html
RICH HS GEOMETRY TASK
http://www.illustrativemathematics.org/illustrations/607
This modeling task involves several different types of geometric knowledge and
problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric
ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure
involving multiple circular arcs into parts whose areas can be found (MP.7).
Teachers who wish to use this problem as a classroom task may wish to have
students work on the task in cooperative learning groups due to the high
technical demand of the task. If time is an issue, teachers may wish to use the
Jigsaw cooperative learning strategy to divide the computational demands of the
task among students while requiring all students to process the mathematics in
each part of the problem.