Download GEOMETRY Semester 1 Unit 1- Congruence, Proof, and

GEOMETRY
Semester 1
Unit 1- Congruence, Proof, and Constructions
Critical Area: In previous grades, students were asked to draw triangles based on given
measurements. They also have prior experience with rigid motions: translations, reflections,
and rotations and have used these to develop notions about what it means for two objects to
be congruent. In this unit, students establish triangle congruence as a familiar foundation for
the development of formal proof. Students prove theorems-using a variety of formats-and
solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to
complete geometric constructions and explain why they work.
Main Idea
Point, line, line segment, ray, angle, parallel lines, perpendicular lines,
circle
● Precise definitions/descriptions
● constructions
● apply theorems and properties
Distance formula
Circumference
Standard
G.CO.1, G.CO.12
Textbook
Geo 1.1-1.4, pg.33-34, 1.7, 2.2, 3.1, 10.1
Outside
Resources
TI-Nspire activity: Points, Lines and Planes
Main Idea
Perform and explain constructions using a variety of tools and methods:
● Copy a segment and an angle
● Bisect a segment and an angle
● Construct perpendicular lines, perpendicular bisector of a line
segment,
● Construct a line parallel to a line through a point not on the line
Apply properties, theorems and definitions to support constructions
Standard
G.CO.12
Textbook
Geo pg. 33-34, Activity 3.2, Activity 3.6,
Outside
MathIsFun constructions (Copyright issue?)
Resources
Mimio Gallery
Main Idea
Transformations including translations, reflections, rotations:
● Demonstrate and describe the different transformations using a variety
of tools (software, transparancies, etc.)
● Demonstrate and describe transformations that map a polygon onto
itself.
● Describe with coordinate notation (input/output)
● Compare rigid motion (isometry) with dilations.
● Develop definitions of transformations in terms of angles, circles,
perpendicular lines, parallel lines and line segments
● Draw the transformation using a variety of tools and specify the
sequence of transformations that carry a given figure onto another.
Standard
G.CO.2, G.CO.3, G.CO.4, G.CO.5
Textbook
4.8, 6.7, 9.1, 9.3-9.5, 9.7, pg.616-618 Geometry SCCSS Supplement 4.2
Outside
Resources
IEFA Native American Designs
IEFA: Beading Patterns Using Reflections
Illuminations: Reflections activity
Illuminations Translations Activity
Illuminations: Rotations activity
Illuminations Intro to Translations Activity
Illuminations Mirror Tool for Isometry
TI-84 : Transformational puppet
TI-Inspire activity: Tessellations
TI-84 activity: Tessellations
TI-INspire: Reflections and Rotations
SIMMS Crazy Cartoon Module
MARS Activity Transformations
Main Idea
Rigid Motion and Congruence (Use transformations to define congruence)
● Determine if two figures are congruent using the definition of
congruence in terms of rigid motions
● Use geometric descriptions of rigid motions to transform figures and
predict the effect of a given motion on a given figure.
Standard
G.CO.6
Textbook
Geo 9.1, Geometry CCSS Supplement Activity 4.2
Outside
Resources
Company Logo Unit
SIMMS Crazy Cartoon Module
Main Idea
Triangle Congruence:
● Use the definition of congruence in terms of rigid motions to show two
triangles are congruent if and only if corresponding parts are congruent
● Explain informally how the criteria for triangle congruence (ASA, SAS,
and SSS) follow from the definition of congruence in terms of rigid
motions (they omit HL and AAS??)
Standard
G.CO.7, G.CO.8, Geometry CCSS Supplement 4.5
Textbook
Geo 4.2-4.6, 6.3
Outside
Resources
Illumination Triangle Congruence Activity
Texas Instruments Activity
Why Does ASA Work?
Why Does SAS Work?
Why Does SSS Work?
When Does SSA Work to Determine Triangle Congruence?
Main Idea
Prove theorems (using a variety of methods) about lines and angles when a
transversal crosses a pair of parallel lines
● Vertical angles are congruent
● Alternate interior angles are congruent
● corresponding angles are congruent
● Use the principle that corresponding parts of congruent triangles are
congruent to solve problems
Prove (using a variety of methods)points on a perpendicular bisector of a line
segment are equidistant to the line’s endpoints.
Identify Hypothesis and Conclusion of a theorem.
Standard
G.CO.9, G.CO.10, G.CO.11
Textbook
Geo 1.3,2.2, 3.1-3.3,5.2
Outside
Resources
Mathwarehouse Exploration
Ti-Nspire activity
Midpoints of the Sides of a Parallelogram
Main Idea
Prove Theorems about Triangles
● interior angles add to 180
● base angles of isosceles triangles are congruent
● midsegment theorem
● concurrency of medians,
● concurrency of perpendicular bisectors, and angle bisectors(as
preparation for G.C.3)
Standard
G.CO.10
Textbook
Geo 1.5, 2.6, 4.7,5.1-5.4
Outside
Resources
Illuminations Triangle Incenter Activity
Illuminations Perpendicular Bisector Activity
Main Idea
Prove Theorems about Parallelograms
● opposite sides are congruent
● opposite angles are congruent
● diagonals bisect each other
Classify Quadrilaterals
Standard
G.CO.11
Textbook
Geo 8.2-8.6,4.6
Outside Resources
Activity: Properties of quadrilaterals
Illuminations Quadrilateral Diagonals Activity
Main Idea
Construct polygons inscribed in a circle
● equilateral triangle
● square
● regular polygon
● hexagon
Make formal geometric constructions, including those representing Montana
American Indians, with a variety of tools and methods.
Standard
G.CO.12, G.CO.13
Textbook
Geo 10.4
Outside
Resources
Making a Star Quilt
Critical Area/Unit 2 – Similarity, Proof, and Trigonometry
Critical Area: Students apply their earlier experience with dilations and proportional reasoning
to build a formal understanding of similarity. They identify criteria for similarity of triangles,
use similarity to solve problems, and apply similarity in right triangles to understand right
triangle trigonometry, with particular attention to special right triangles and the Pythagorean
theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of
general (not necessarily right) triangles, building on students’ work with quadratic equations
done in the first course. They are able to distinguish whether three given measures (angles or
sides) define 0,1,2, or infinitely many triangles.
Main Idea
• Define image, pre-image, scale factor, center,
dilation(enlargement/reduction), congruence
•Verify experimentally, through construction or graphing, the properties of
dilations
Standard
G. SRT. 1
Textbook
Activity 6.7 and Geo. 6.7 Geometry CCSS Supplement 6.3 A/B
Outside
Resources
CC Supp?
SIMMS: “Crazy Cartoons” Activity 1
Constructing a Dilation
Dilating a Line
Main Idea
• Given two figures, decide if they are similar by using the definition of
similarity in terms of similarity transformations
Standard
G. SRT. 2
Textbook
Geo. 6.1-6.3
Outside
Resources
Are They Similar?
Main Idea
• Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar
Standard
G. SRT. 3
Textbook
Geo. 6.4 and Geometry CCSS Supplement 6.4
Outside
Resources
SIMMS: “A New Angle on an Old Pyramid” Activity 1
Main Idea
Recall postulates, theorems, and definitions to prove theorems about
triangles
• A line parallel to one side of a triangle divides the two proportionally
• Pythagorean theorem proved using triangle similarity.
Prove theorems involving similarity about triangles
Standard
G. SRT. 4
Textbook
Geo. 5.1,6.6,7.3, pg.493, and Activity 6.6,
Outside
Resources
Youtube: prove pythagorean theorem using similar triangles
Main Idea
•Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures
•Apply and implement theorems in real-world situations.
Standard
G. SRT. 5
Textbook
6.3-6.7, 9.1, mixed review pg. 416
Outside
Resources
Main Idea
•Use similarity of right triangles to develop definitions of trig ratios.
•Define parts of a right triangle as related to an acute angle
•Explain and use the relationship between the sine and cosine of
complementary angles.
Standard
G. SRT. 6, G.SRT.7
Textbook
Geo. 7.1,7.2,7.4
Outside
Resources
SIMMS: “A New Angle on an Old Pyramid” Activity 2
Example Worksheet Trig Ratios
Example Worksheet Complementary Angles
Main Idea
•Define and apply the sine, cosine, and tangent ratios
•Use Pythagorean Theorem, sine, cosine, and tangent to solve right triangles.
(Including real world applications)
Standard
G.SRT.8, and G. MG. 1
Textbook
Geo. 7.5-7.7 and Mixed Review p. 492
Outside
Resources
SIMMS: “A New Angle on an Old Pyramid” Activities 3&4
Modeling a Montana American Indian tipi as a geometric shape
Main Idea
•Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side
Standard
G. SRT. 9-(+)
Textbook
Alg. 2 13.5
Outside
Resources
Example Worksheet
Main Idea
•Prove the Laws of Sines and Cosines and use them to solve problems
•Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g. surverying
problems, resultant forces)
Standard
G. SRT. 10-(+), G.SRT.11-(+)
Textbook
Alg. 2 13.6
Outside
Resources
Illuminations Law of Sines Activity
Illuminations Law of Cosines Activity
Illuminations: Vectors activity
Main Idea
• Define Density
• Apply concepts of density based on area and volume to model real-life
situations
• Describe a typographical grid system
• Apply geometric methods to solve design problems(e.g., designing an object
or structure to satisfy physical constraints or minimize cost; working with
typographic grid systems based on ratios)
Standard
G. MG. 2, G. MG. 3
Textbook
Outside
Resources
Density Activity
Density Activity Map
Density Activity Table
Critical Area/Unit 3 – Extending to Three Dimensions
Students’ experience with two-dimensional and three-dimensional objects is extended to
include informal explanations of circumference, area and volume formulas. Additionally,
students apply their knowledge of two-dimensional shapes to consider the shapes of crosssections and the result of rotating a two-dimensional object about a line.
Main Idea
Give an informal argument for the formulas for the circumference and area
of a circle.
Standard
G.GMD.1
Textbook
Geo1-7
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
SIMMS: Level1 “A New Look at Boxing” & “What Will We Do When the Well Runs
Dry”
IEFA Tipi Geometry & Trigonometry
Main Idea
Constructing Nets to help visualize relationships between 2-D and 3-D
objects.
Standard
G.GMD.4
Textbook
Geo12.2, 12.2 Activity: Investigating Surface Area (Pg. 802)
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
Nets Activity
Nets Activity 2
Main Idea
Given an informal argument for the formulas for the volume of a cylinder,
pyramid, and cone.
Standard
G.GMD.1
Textbook
Geo 12.4, 12.5
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
IEFA Tipi Geometry & Trigonometry
Main Idea
•Use volume formulas for cylinders, pyramids, cones, and spheres to solve
contextual problems
•Utilize the appropriate formula for volume, depending on the figure
Standard
G.GMD.3
Textbook
Geo12.4, 12.5, 12.6
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
IEFA Tipi Geometry & Trigonometry
Doctor's Appointment
Centerpiece
Main Idea
Relate the shapes of two-dimensional cross-sections to their threedimensional objects
Standard
G.GMD.4
Textbook
Geo12.1, 12.6
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
Tennis Balls in a Can
Popcorn Anyone?
Main Idea
Use dissection arguments, Cavalieri’s principle, and informal limit
arguments
Standard
G.GMD.1
Textbook
Geo12.1, 12.4, 12.5, Activity: Investigate the Volume of a Pyramid (Pg.
828)
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
IEFA Tipi Geometry & Trigonometry
Popcorn Anyone?
Main Idea
Discover three-dimensional objects generated by rotations of twodimensional objects
Standard
G.GMD.4
Textbook
none
Outside
Resources
Common Core State Standards Curriculum Companion: Solids of Revolution
Activity Pg CC32-CC33
Applet on Solids of Revolution "The Lathe" :)
Volumes of Solids of Revolution
Main Idea
•Given a real world object, classify the object as a known geometric shape –
use this to solve problems in context this to solve problems in context
• Focus on situations well modeled by trigonometric ratios for acute angles
• Use measure and properties of geometric shapes to describe real world
objects
Standard
G.MG.1
Textbook
Geo12.4, 12.5, 12.6, Mixed Review (pg. 855), Standard Test Practice (pg.
864),
Problem Solving Wk.Shop (Pg. 826-827)
Outside
Resources
SIMMS Level 2 Volume 2 “There’s No Place Like Home”
IEFA Tipi Geometry & Trigonometry
Building 3-D objects with Appropriate Volume
Toilet Roll
Semester 2
Unit 4- Connecting Algebra and Geometry Through
Coordinates
Approximately __30__ days
Critical Idea: Building on their work with the Pythagorean theorem in 8th grade to find
distances, students use a rectangular coordinate system to verify geometric relationships,
including properties of special triangles and quadrilaterals and slopes of parallel and
perpendicular lines, which relates back to work done in the first course. Students continue
their study of quadratics and algebraic definitions of the parabola.
Main Idea
(5 days)
Distance and Midpoint Formulas
● Find base, height, radius on coordinate plane
● Find area, perimeter, and circumference on coordinate plane
● Example: Prove or disprove that a figure defined by four given points in
the coordinate plane is a rectangle
● Derive the equation of a line through 2 points using similar right
triangles.
Standard
G.GPE.4
Textbook
Geo 1.3, Alg2 9.1
Outside
Resources
http://map.mathshell.org/materials/download.php?fileid=1202
Classzone.com (animations, chapter 1-archaeology)
Smart Exchange (midpoint and distance) and (perimeter and area with
coordinate geometry)
Derive Equation of a Line from Similar Right Triangles
G.GPE.7
Main Idea
(10 days)
Slope and Equations of Lines
● Recognize and prove slopes of Parallel and Perpendicular Lines
Standard
G.GPE.4
Textbook
Geo L3.4, Geo L3.5, Alg1 5.5, Alg2 2.4
Outside
http://map.mathshell.org/materials/download.php?fileid=703
G.GPE.5
Resources http://www.clackamasmiddlecollege.org/documents/Parallel+and+Perpendicular+lines.pdf
Main Idea
Ratios in Coordinate Geometry
● given a line segment and a ratio, find the scaled point
Standard
G.GPE.6
Textbook
Geo L6.1, Geo L6.2, Geo L6.7, Geometry CCSS Supplement lesson 6.7
Outside Resources
Investigate Geometry Activity 6.7 Dilations (pg408)
Main Idea
Parabola
● Write Equation of parabola given Focus and Directrix
Standard
G.GPE.2
Textbook
Alg1 L10.2, Alg2 L9.2, Geo pg499, Geo pg882-883
Outside Resources
Smart Exchange (focus and directrix)
Unit 5- Circles with and without Coordinates
Aproximately _30__ days
Critical Area: In this unit, students prove basic theorems about circles, such as a tangent line in
perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and
tangents dealing with segment lengths and angle measures. They study relationships among
segments on chords, secants, and tangents as an application of similarity. In the Cartesian
coordinate system, students use the distance formula to write the equation of a circle when
given the radius and the coordinates of its center. Given and equation of a circle, they draw the
graph in the coordinate plane, and apply techniques for solving quadratic equations, which
relates back to work done in the first course, to determine intersections between lines and
circles or parabolas and between two circles.
Main Idea
Prove that all circles are similar
● suggestion: develop definition of pi
Standard
G. C. 1
Textbook
none
Outside Resources
Illuminations Pi Line
Main Idea
Identify and describe the properties of different angles and segments in a
circle
Standard
G. C. 2
Textbook
Geo. 5.2,5.3,10.1,10.2,10.4
Outside
Resources
http://brightstorm.com/math/geometry/circles/
http://brightstorm.com/math/geometry/constructions/
http://map.mathshell.org/materials/download.php?fileid=1194
Two Wheels and a Belt
Main Idea
Construct the inscribed and circumscribed circles of a triangle
Proves properties of angles for a quadrilateral inscribed in a circle
Standard
G. C. 3
Textbook
Geo. 5.2,5.3,10.4
Outside Resources
http://map.mathshell.org/materials/download.php?fileid=1194
Locating a Warehouse
Placing a Fire Hydrant
Main Idea
Construct a tangent line from a point outside a given circle to the circle
Standard
G. C. 4 (+)
Textbook
Geometry CCSS Supplement lesson 10.4
Outside Resources
http://brightstorm.com/math/geometry/circles/
Main Idea
Derive the formulas for arc length and area of a sector using proportionality
Define and use the radian measure of an angle.
Verify that the constant of a proportion is the same as the radian measure, Θ,
of the given central angle. Conclude s = r Θ
Standard
G. C. 5
Textbook
Geo. 11.4,11.5; Alg2 13.2
Outside
Resources
Geometry CCSS Supplement Extension 11.4
Main Idea
Derive the equation of a circle given center and radius using Pythagorean
Theorem
Complete the square of a quadratic equation to find the center and radius of
a circle
Standard
G. GPE. 1
Textbook
Geo. 10.7
Outside
Resources
http://brightstorm.com/math/geometry/pythagorean-theorem/equation-of-acircle/
Slopes and Circles
Main Idea
Use coordinates to prove simple geometric theorems algebraically
● Derive simple proofs involving circles.
Standard
G. GPE. 4
Textbook
Geo. 5.1,8.3
Outside Resources
A Midpoint Miracle
Main Idea
Use and apply the properties of circles to real world situations (do this
throughout the unit)
Standard
G. MG. 1
Textbook
Outside
MARS activity: Security camera
Resources
Setting up Sprinklers
Critical Area/Unit 6-Applications of Probability
Approximately __20___ Days
Critical Area: Building on probability concepts that began in the middle grades, students use
the languages of set theory to expand their ability to compute and interpret theoretical and
experimental probabilities for compound events, attending to mutually exclusive events,
independent events, and conditional probability. Students should make use of geometric
probability models wherever possible. They use probability to make informed decisions.
Main Idea
• Define unions, intersections and complements of events
• Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events
Standard
S.CP.1
Textbook
Alg1: Extension after 2-1 (Pg. 71-72) & Section 13-1
Alg2: 10-4
Outside
Resources
SIMMS: Level 2 Vol. 1 “What Are My Child’s Chances?”
Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
IEFA Ko'ko'hasenestôtse
Main Idea
•Compute Theoretical and Experimental Probabilities
• Recall previous understandings of probability
• Use probabilities to make fair decisions (ie drawing by lots, random
number generator)
Standard
S.MD.6(+)
Textbook
Alg1 13-1; Alg2 10-3
Outside
Resources
SIMMS: Level 2 Vol. 1 “What Are My Child’s Chances?”
Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
IEFA Ko'ko'hasenestôtse
Main Idea
•Identify situations that are permutations and those that are combinations
• Use permutations and combinations to compute probabilities of
compound events and solve problems
Standard
S.CP.9 (+)
Textbook
Alg1 13-2 & 13-3; Alg2 10-1 & 10-2; Geo Pg. 891-892
Outside
Resources
The Random Walk III
The Random Walk IV
SIMMS Level 4: “Everyone Counts” Pg. 275-293
Main Idea
• Categorize events as independent or not using the characterization that two events A
and B are independent when the probability of A and B occurring together is the
product of their probabilities
Standard
S.CP.2
Textbook
Alg2 13-4, 10-5; Geo Pg. 893
Outside
Resources
SIMMS Level 2 Vol. 1 “What Are My Child’s Chances?”
Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
IEFA Ko'ko'hasenestôtse
Main Idea
•Use the multiplication rule with correct notation and interpret the answer
• Apply the general Multiplication Rule in a uniform probability model P(A
and B) = P(A)P(B|A) = P(B)P(A|B)
Standard
S.CP.8 (+)
Textbook
Alg1 13-4; Alg2 10-5
Outside
Resources
SIMMS: Level 2 Vol. 1 “What Are My Child’s Chances?”
Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
IEFA Ko'ko'hasenestôtse
Main Idea
•Use the Additional Rule, P(A or B) = P(A) + P(B) – P(A and B)
• Interpret the answer in terms of the model
Standard
S.CP.7
Textbook
Alg1: 13-4; Alg2: 10-4
Outside Resources
SIMMS: Level 2 Vol. 1 “What Are My Child’s Chances?”
Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
IEFA Ko'ko'hasenestôtse
Main Idea
• Categorize events as independent or not using the characterization that two
events A and B are independent when the probability of A and B occurring
together is the product of their probabilities
Standard
S.CP.2
Textbook
Alg1: 13-4, Alg2: 10-5, Geometry pg. 893
Outside
Resources
SIMMS: Level 2 Vol. 1 “What Are My Child’s Chances?”
Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
IEFA Ko'ko'hasenestôtse
Main Idea
•Know the conditional probability of A given B as P(A and B)/P(B)
• Interpret independence of A and B as saying that the conditional probability of A
given B is the same as the probability of A, and the conditional probability of B given A
is the same as the probability of B
Standard
S.CP.3
Textbook
Alg2: 10-5
Outside
Resources
IEFA Ko'ko'hasenestôtse
SIMMS: Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
Main Idea
• Use the two-way table as a sample space to decide if events are independent and
to approximate conditional probabilities
• Build on work with two way tables from Algebra 1 Unit 3 S-ID.5 to develop
understanding of conditional probability and independence
• Interpret two-way frequency tables of data when two categories are associated
with each object being classified
Standard
S.CP.4
Textbook
Alg1 13-1;
Alg2 10-5
Outside
Resources
IEFA Ko'ko'hasenestôtse
Main Idea
• Recognize the concepts of conditional probability and independence in
everyday language and everyday situations
• Explain the concepts of conditional probability and independence in
everyday language and everyday situations
Standard
S.CP.5
Textbook
Alg2 10-5
Outside
Resources
IEFA Ko'ko'hasenestôtse
SIMMS: Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!”
Main Idea
Recall previous understandings of probability
• Analyze decisions and strategies using probability concepts
• Extend to more complex probability models. Include situations such as those
involving quality control, or diagnostic tests that yield both false positive and
false negative results
Standard
S.MD.7(+)
Textbook
Alg2 10-5
Outside
Resources
IEFA Ko'ko'hasenestôtse
SIMMS: Level 2 Vol.2 “Hurry! Hurry! Hurry! Step Right Up!” Activity 3