W. Menard
Geometry: A Discovery Approach
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We will be using GeoGebra, patty paper, and the coordinate plane as tools to define, classify, and justify geometric ideas, postulates and theorems.
We will look at 2 column proof, but focus on explicitly stating all assumptions and working towards a logical narrative to describe our solution to any problem.
There will be a task or project for each unit, as well as regular problem-solving activities.
Assessments attached in separate document. Keep in mind Dana Center Assessments
TOOLS
• Interactive Notebooks
• Weekly Math Maintenance Sheets
• Patty Paper
• Geogebra
• Compass/Straight Edge constructions
• Mirrors
Unit/Driving
Questions
Unit 0
Unit 1:
Definition &
Classification
Because definitions
are so essential to
proof in geometry,
we will begin the
year by setting up
our interactive
notebooks with
glossaries.
8/27/14
Topics Covered
Tools for this course
How do we define and classify
geometric objects?
What are the basic terms we
need to talk about objects in
geometry?
Definitions of:
Point line plane line segment ray
angle midpoint bisect polygon
collinear coplanar non-collinear
non-coplanar skew adjacent circle
radius diameter concurrent arc
CCSSM/
SMP
MP-1
MP-5 Tools
MP-6
Precision
G-CO.1
G-CO.12,
13
G-MG.1
Learning Activities
Assessments
Ø Pick a Point (Sam Shah/Dan Meyer)
Ø Folding Paper Activity
Ø Exercise in description
o Work with a partner, describe a sketch
o Have partner draw sketch
o Swap
Ø Geogebra tutorial
Ø Marshmallow tower?
Ø What are the Standards of Mathematical Practice?
MATHOGRAPHY
mathography– Intro
to Google Docs and
Kaizena
Ø Exercise in classification: Arguing About Shapes
o Sort shapes, objects
o Describe how you sorted
o How do you fit new shapes into these?
o I NOTICE, I WONDER
Ø Exercise in definitions
o Pictionary?
o Modified Frayer Models Definitions and Frayer Models:
Examples/Counter Examples
o How accurate is your definition?
Ø Collinear/Coplanar exercise in Jacobs p. 63
Ø Intro to Geometric Construction: The Equilateral
Triangle (copying segments)
o Creative Constructions
Cost of a Great Looking
Floor? Cost of a Great
Looking Floor.pdf
Other
IDEA
http://www.inquirymaths.com/ho
me
4 COLUMN GLOSSARY TEMPLATE:
DEFINTION/SKETCH/EXAMPLE/NONEXAMPLE
INTRODUCTION TO
GEOGEBRA
Algebra Review
Ø Graphing on the
coordinate plane
Ø Integer operations
W. Menard
Unit 2: The Logic
of Geometry
Claim, Evidence,
Explanation
Conditional statements
(hypothesis/conclusion)
8-G.6,7
Converse of conditionals
MP-3
Construct
Arguments
Syllogisms: Chain of logic
Deductive versus Inductive logic
Postulates versus Theorems
Truth Value/Logical Equivalence
Ø
o
o
o
o
A-REI.1,3
MP-8
Repeated
Reasoning
Inverse and Contrapositive
Unit 3: Planes,
Lines and
Angles
Plane theorems
8-G.5
Measurement of line segments
and angles
A-SSE.1a
Angle/Segment addition
G-CO.1, 0
Bisecting
MP-2,3,5,7
Ø
Ø
A-CED.1
Ø
Angle Pairs
Parallel/Perpendicular (separate
unit?)
Ø
Ø
Ø
8/27/14
Follow Jacob’s Geometry:
The conditional and its converse
Syllogisms and proof
Indirect proof (3 Hats p. 56)
Deductive System
§ Postulates
§ Famous Theorems:
v Pythagorean Theorem How do we
correct the scarecrow? (Kaplinsky)
v Triangle Sum
Planes: Intersecting, Parallel, Coplanar ★
Lines
o Length/measurement/adding segments
o Intersecting, parallel, perpendicular, skew
What is an angle?
o Geogebra demo
o Measuring angles with protractors, patty paper
o Angle addition
BISECTION
o What does it mean to bisect?
§ A line?
• Perpendicular bisector vs. any bisector
§ An angle?
Angle Pairs
o Adjacent/non-adjacent, linear pairs
o Supplementary/Complementary
o Vertical
o Identifying all angle pairs – first on pictures,
then naming
o Linear versus Adjacent activity
o Finding angle measures
§ With numbers
§ With expressions
§ Writing equations
Parallel Lines intersected by a Transversal
o fawnnguyen.com/parallel-perpendicularlines/
o Explore angle pairs created with Patty
Paper/Geogebra
o Add definitions to glossary
o Alternate Interior, Alternate Exterior,
Logic Posters (revise)
Logic Poster
Proj_Global Cit
2013.doc
Algebra Review
Ø Solving equations
? inverse and contrapositive
? truth value
? logical equivalence
Use Geogebra
constructions as
assessment
Algebra Review
Ø Writing expressions
Ø Writing equations to
solve problems
★ Foldable:
Everybody is a
Genius Points Lines
and Planes
Picciotto: Geometry Labs 1.10
Soccer Angles
Equal Divisions in Carpentry
(http://wncpactivemath.wordpress
.com/)
ç
Idea for Activity/INBs:
angle pair flashcards
W. Menard
Corresponding, Consecutive (Same-Side)
Interior Angle pairs
Unit 4:
Properties of
Triangles
Defining and classifying triangles
8-G.5
Solving Angle Sum problems
G-CO.1, 10
Isosceles Triangles
A-CED/1
Exterior Angle Theorem
G-SRT.7,8
(if trig is
included)
Triangle Inequality Theorems
Triangle Segments and points of
concurrency
Ø
MP-2,3,6
Triangle midsegments
Ø
Ø
What’s so great about right
triangles?
Unit 5: Triangle
Congruence
How do we prove
that figures are
congruent? Is
there more than
one way?
Ø
Meaning of congruence
8-G.1-3
Congruence through
transformations
G-CO.2,6,7
Ø
Ø
MP-2,3,5
Congruence shortcuts
Congruent parts of congruent
figures
Ø
Introduction: Triangle quilt
http://drawingonmath.blogspot.com/2011/12/triangl
e-quilt.html (use as pre-assess on vocabulary and
basic concepts)
Vocabulary: acute, obtuse, right, scalene,
isosceles, equilateral, equiangular, median,
centroid, incenter, circumscribe, inscribe, exterior
angle, midsegment, leg, hypotenuse, base angle,
vertex angle
OPPOSITE VERSUS ADJACENT SIDES
Explore midsegments in Geogebra; construct,
make conjectures
Triangle personal ad
Project Personal Ad.pdf
What does congruent mean? Why is it important?
Exploration of congruence through transformations:
coordinate plane, patty paper, Geogebra
o Review types of transformations:
reflections, translations, rotations
Exploration of SSS through constructions, handson activity with straws?
Illustrative Mathematics
Partitioning Hexagons
(GGB)
https://www.illustrative
mathematics.org/illustr
ations/1200
Ø
(Claim, Evidence,
Explanation)
Unit 6: Polygons/
Quadrilaterals
What features do
all polygons
share? What
types of special
features create
different
polygons?
Interior angles of polygons
A-REI.6
Exterior angles of polygons
G-CO.11
Types of polygons
MP-2,5
Special quadrilaterals:
Ø
Ø
Ø
Ø
Exploration of polygon sum formula
Discovery of exterior angle sum
Use Geogebra to explore parallelogram family
(look at Bannard Exeter materials)
Ideas:
o “Can you make?” CanYouMake
Geoboards.doc
o Pattern Block Explorations Pattern Block
Riddles.doc
o Eyeballing game:
http://woodgears.ca/eyeball/
o Always Sometimes Never - Quads
(crazymathteacherlady)
Constructing a rhombus/square in a circle
Ø
Fixed distance from a point intro: cell phone tower:
Ø
Parallelograms
Rectangles
Rhombi
Squares
Trapezoids
Isosceles trapezoids
Unit 7: Locus
8/27/14
5 Locus Theorems
G-CO.1
Algebra Review
Ø Slope, writing
equations of lines
Silver Rectangles Silver Rectantgle
Activity:
Congruent Triangle
Quiz (see M. Pershan
post: ignoring-meaningof-feedback.html)
Proof Blocks
Algebra Review
Ø Slope of parallel and
perpendicular lines
Algebra Review
Ø Systems of linear
equations
Enrichment: Taxi Cab
Tie into constructions
W. Menard
Compound Locus
MP-4,5,6
Locus on the Coordinate Plane
Ø
how-i-introduced-locus-this-year
Locus Coloring Activity
Geometry; compare
locus definitions
Derivation of distance formula
Exploring Quads TAKS activity Parallelograms
TAKS activities.pdf
Create your own proof story
o Have students identify coordinates which
create a specific geometric object and justify
their findings.
Whitebeard’s Treasure
Solving problems with locus
Applications of locus
Unit 8:
Geometric
Properties on
the Coordinate
Plane
Equations of Lines
Pythagorean Theorem and
Distance
Midpoint
Slopes of Parallel and
Perpendicular Lines
G-GPE.1,47
Ø
Ø
A-CED.2
A-REI.7
MP-2,6-8
Ø
Linear Equation
Golf/Equations of Lines
on a Grid
Nested Boxes Group
Activity
Equations of Perpendicular
Bisectors
Justifying properties
algebraically
Equations of Circles
Literal Coordinates/Formulas
Equations of Parabolas
Quadratic Linear Systems
Unit 9:
Transformations
in Space and the
Coordinate
Plane
Isometries
G-CO.2,4,5
Translations
MP-5,7
Combine transformations worksheet combinetransformations_worksheet.pdf
NRICH Transformation challenges
Transformations_Challenges.pdf
Reflections
Over y and x axes, y=x
Rotational Designs rotationalDesigns.pdf rotationBy45.pdf
Over y=k or x=k
Patty Paper Explorations
Point Reflections
Rotations
Dilations
Glide Reflections
Coat of Arms project or
Art Reflections Lines of
Symmetry Reflection
Project.docx
Formative: Illustrative
Mathematics Rectangle
Reflections
https://www.illustrative
mathematics.org/illustr
ations/1229
Composition of Transformations
Unit 10:
Similarity
8/27/14
Ratio and Proportion Review
G-SRT.2-5
Worksheet: Tips4RM Unit Similar Triangles Unit.pdf
Similar Figures and Dilations
MP-1,2,7,8
Similar Triangle properties
Identifying Similar Triangles (Mathbits) Identifying Similar
Triangles.pdf
Similarity shortcuts
Geogebra
Triangle proportionality theorem
Floodlights Floodlights similarity.pdf
Midsegment Theorem
Exploring Right Triangle Similarity Curriculum Geometry
Grade 10 Applied p 20
Dana Center Tasks
Tree activity from
@marybourassa
W. Menard
Right Angle Altitude
Other Similar Figures
Unit 11: Circles
Parts of Circles: Segments &
Angles
G-C.2,3
Transformulas.org: Circles and Similarity Intuition (desmos)
MP-2,5,8
Patty paper explorations
Arcs and Chords
Geogebra explorations
Central Angles
Inscribed Angles and the Eye Project.chord properties.doc
Art, Arcs and Sectors
Use a photo for exploration:
Inscribed Angles
Angles created by: intersecting
chords, tangents and secants
HOW TO MAKE THIS UNIT LESS PROCEDURAL AND
MORE ACCESSIBLE TO ALL?
Segments: Diameters
perpendicular to chords lead to
exploration of:
Extra Credit: Wind Rose Construction
Intersecting chords [ta-da!]
Properties of tangent segments
Intersecting tangents and
secants
Unit 12: Volume
and Surface
Area
Properties of:
Ø
Ø
Ø
Ø
Ø
Prisms
Cylinders
Pyramids
Cones
Spheres
8-G.9
Explore 3D models; make conjectures
GGMD.1,3,4
G-MG.1
MP-4,6,7
How are volume and surface
area formulas derived? Problemsolving with 3D solids
Optimization with 3D solids
Design your dream home project - @algebranic1 or everybodyisagenius project
Resources for INBs:
CHORDS_SECANTS_AND_TANGENTS.doc
Analyzing Solids
8/27/14
The Wedding Cake
Problem
Sketch-Up
Exploring Nets and Folding
Making stellated icosahedra!