Download Geometry 1 - Skyline Prep High School

Integrated Geometry 1, Curricular Guide
1
Geometry 1
This course explores beginning Geometry concepts, Math Modeling, Basic Formulae, Composing
Proofs and Methods, Sequences, Area and Volume of Shapes, Angles, Lines, Circles, and
Transformations. Students will use problem-solving strategies to prepare solutions to authentic
situations involving algebra and geometry through algorithmic thinking, logic, and problemsolving skills. Competency (70% or above) in Integrated Beginning Algebra 1 and 2 is a prerequisite for this course. This course meets one of the four math requirements for university
admission and Arizona State Graduations requirements.
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Integrated Geometry 1, Curricular Guide
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Integrated Geometry 1, Curricular Guide
3
An Introduction to Curriculum Mapping and Standards Log
Objectives are mapped according to when they should be introduced and when they should be assessed
throughout the month (K-4), block (5-8), or course (7-12). A record of when all objectives are
introduced and assessed is to be kept through the course map and log, using the month, day, and year
introduced. Objectives only have to be reviewed if assessment is not 80% students at 80% mastery.
**In some cases, it is not necessary to teach the standards if 80% students are at 80% mastery when
pretested. However, if less than 80% students achieve 80% mastery, it is necessary to give instruction
and a posttest.**
The curriculum is standards-based, and it is the Skyline philosophy to use “Backwards Design” when
lesson planning. Backwards Design starts with standards, and from there, an assessment is created in
alignment with the standards; next, the instruction for that assessment and those standards is created.
Also, all standards addressed for instruction and assessment should be visibly posted in the classroom,
along with student-friendly wording of the objectives.
Assessments for mastery are to be summative, or cumulative in nature. Formative assessments are
generally quick-assessments where the teacher can gauge whether or not student-learning is acquired.
Curriculum binders are set up to have a master of each grade or content level, as well as a teacher’s
copy, which is to serve as a working document. Teachers may write in the teacher’s binder to log
standards, suggest remapping, adjust timing, and so on.
The curriculum mapping may be modified or adjusted as necessary for individual students and classes,
as well as available resources, within reason. Major changes are to be submitted to the school’s
Professional Learning Community, Administration, and the Board.
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Integrated Geometry 1, Curricular Guide
Suggested Methods of Activity and Instruction
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Teacher Modeling
Learning Centers
Learning Stations
Anchor Activities
Group Work
Small Group Discussion
Independent Study
Mentor Study
Think/Pair/Share
Total Physical Response
Graphic Organizers
Tiered Assignments
Literature Circles
Experiment
Rigor/Relevance: Quadrant “D” Learning
Drama/Skits/Plays
Arts Integration Projects
Simulations
Data Collection
Lecture
Whole Group Debate
Learning Games
Learning Contracts
Curriculum Compacting
Flexible Pacing
Self-Directed Learning
Problem-Based Learning
Conferencing
Seminars
Real-World Scenarios
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Integrated Geometry 1, Curricular Guide
5
Suggested Methods of Assessment
FORMATIVE
(Grades are not necessarily assigned for all
formative assessments)
 Quick-write
 Quick-draw
 Verbal response
 Asking questions
 Interaction during activities
 Pretests
 Learning games
 Web/Computer-based assessments
 Homework/Class Work
 Notes
 Pop quizzes
 Criteria and goal setting
 Teacher observations
 Self and peer assessment
 Student record keeping
 Graphic Organizers
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SUMMATIVE
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Standardized Tests
State Assessments
Student Portfolio
Interdisciplinary projects
Student-Teacher conference narratives
Posttests
District/School/Course/Content tests
Chapter/Unit Tests
Integrated Geometry 1, Curricular Guide
6
Curriculum Mapping and Standards Log
Understanding Look-up Codes.
All the standards follow the state codes for tracking general concepts and topics.
Number and Quantity
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The Real Number System (N-RN)
Quantities (N-Q)
The Complex Number System (N-CN)
Vector and Matrix Quantities (N-VM)
Geometry
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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)
Algebra
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Seeing Structure in Expressions (A-SSE)
Arithmetic with Polynomials and Rational Expressions (A-APR)
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Functions
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Interpreting Functions (F-IF)
Building Functions (F-BF)
Linear, Quadratic, and Exponential Models (F-LE)
Trigonometric Functions (F-TF)
Modeling
Statistics and Probability
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Interpreting Categorical and Quantitative Data (S-ID)
Making Inferences and Justifying Conclusions (S-IC)
Conditional Probability and the Rules of Probability (S-CP)
Using Probability to Make Decisions (S-MD)
Contemporary Mathematics
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Discrete Mathematics (CM-DM)
Honors Extensions listed after the standards are suggested projects or explorations a student wishing to earn honors credit may elect to do with permission from
the administration in addition to their normal class work.
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Integrated Geometry 1, Curricular Guide
7
Block 1
Lookup
Code
HS.F-BF.1b
HS.M.BS
HS.A-CED.4
HS.A-REI.1
HS.F-BF.2
Descriptor
Combine standard function types using
arithmetic operations. For example, build a
function that models the temperature of a cooling
body by adding a constant function to a decaying
exponential, and relate these functions to the
model.
Modeling Standards are spread through the rest
of the standards. Anything that requires
composing a organized form of data, graphic, or
formula is modeling.
Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V
= IR to highlight resistance R.
Explain each step in solving a simple equation as
following from the equality of numbers asserted
at the previous step, starting from the
assumption that the original equation has a
solution. Construct a viable argument to justify a
solution method.
Write arithmetic and geometric sequences both
recursively and with an explicit formula, use them
to model situations, and translate between the
two forms.
Determine an explicit expression, a recursive
process, or steps for calculation from a context.
HS.F-BF.1a
HS.N-Q.1
HS.N-Q.2
Use units as a way to understand problems and
to guide the solution of multi-step problems;
choose and interpret units consistently in
formulas; choose and interpret the scale and the
origin in graphs and data displays.
Define appropriate quantities for the purpose of
descriptive modeling
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Connections
ETHS-S6C103;ETHSS6C2-03
ETHS-S6C103;ETHSS6C2-03;910.RST.7;
11-12.RST.7
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Block 2
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Block 3
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Block 4
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Integrated Geometry 1, Curricular Guide
8
Block 1
Lookup
Code
HS.S-ID.9
HS.G-CO.1
HS.G-CO.2
HS.G-CO.6
HS.G-CO.7
HS.G-CO.8
HS.G-CO.9
HS.G-CO.10
Descriptor
Distinguish between correlation and causation.
Know precise definitions of angle, circle,
perpendicular line, parallel line, and line
segment, based on the undefined notions of
point, line, distance along a line, and distance
around a circular arc.
Represent transformations in the plane using,
e.g., transparencies and geometry software;
describe transformations as functions that take
points in the plane as inputs and give other
points as outputs. Compare transformations that
preserve distance and a
Use geometric descriptions of rigid motions to
transform figures and to predict the effect of a
given rigid motion on a given figure; given two
figures, use the definition of congruence in terms
of rigid motions to decide if they are congruent.
Connections
9-10.RST.9
Use the definition of congruence in terms of rigid
motions to show that two triangles are congruent
if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
910.WHST.1e
Explain how the criteria for triangle congruence
(ASA, SAS, and SSS) follow from the definition
of congruence in terms of rigid motions.
910.WHST.1e
Prove theorems about lines and angles.
Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a lin
. Prove theorems about triangles. Theorems
include: measures of interior angles of a triangle
sum to 180°; base angles of isosceles triangles
are congruent; the segment joining midpoints of
two sides of a triangle is parallel to the third side
and half th
ETHS-S1C201;910.WHST.1a1e
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Cannot be reproduced without permission
9-10.RST.4
ETHS-S6C103
ETHS-S1C201;910.WHST.1e
ETHS-S1C201;910.WHST.1a1e
Presented
Assessed
Block 2
Presented
Assessed
Block 3
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Block 4
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Integrated Geometry 1, Curricular Guide
9
Block 1
Lookup
Code
HS.G-CO.11
HS.G-CO.12
HS.G-CO.13
HS.G-SRT.2
HS.G-SRT.3
HS.G-SRT.4
Descriptor
. Prove theorems about parallelograms.
Theorems include: opposite sides are congruent,
opposite angles are congruent, the diagonals of
a parallelogram bisect each other, and
conversely, rectangles are parallelograms with
congruent diagonals.
. Make formal geometric constructions with a
variety of tools and methods (compass and
straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting
a segment; bisecting an angle
. Construct an equilateral triangle, a square, and
a regular hexagon inscribed in a circle.
Given two figures, use the definition of similarity
in terms of similarity transformations to decide if
they are similar; explain using similarity
transformations the meaning of similarity for
triangles as the equality of all corresponding
pairs of angles
Use the properties of similarity transformations to
establish the AA criterion for two triangles to be
similar.
Prove theorems about triangles. Theorems
include: a line parallel to one side of a triangle
divides the other two proportionally, and
conversely; the Pythagorean Theorem proved
using triangle similarity.
Use congruence and similarity criteria for
triangles to solve problems and to prove
relationships in geometric figures.
Connections
910.WHST.1a1e
Understand that by similarity, side ratios in right
triangles are properties of the angles in the
triangle, leading to definitions of trigonometric
ratios for acute angles.
ETHSS6C1-03
HS.G-SRT.5
HS.G-SRT.6
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Cannot be reproduced without permission
ETHS-S6C103
ETHSS6C1-03
ETHS-S1C201;910.RST.4;910.WHST.1c
ETHS-S1C201;910.RST.7
ETHS-S1C201;910.WHST.1a1e
ETHS-S1C201;910.WHST.1a1e
Presented
Assessed
Block 2
Presented
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Block 3
Presented
Assessed
Block 4
Presented
Assessed
Integrated Geometry 1, Curricular Guide
10
Block 1
Lookup
Code
HS.G-SRT.8
Descriptor
Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied
problems.
Prove that all circles are similar.
Connections
ETHS-S6C2-03;910.RST.7
Construct the inscribed and circumscribed
circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
ETHS-S6C1-03
ETHS-S1C2-01;910.WHST.1a-1e
HS.G-C.1
HS.G-C.3
HS.G-C.5
HS.G-GPE.4
HS.G-GPE.6
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; derive
the formula for the area of a sector.
Use coordinates to prove simple geometric
theorems algebraically. For example, prove or
disprove that a figure defined by four given
points in the coordinate plane is a rectangle;
prove or disprove that the point (1, √3) lies on the
circle centered at th
Find the point on a directed line segment
between two given points that partitions the
segment in a given ratio.
Use coordinates to compute perimeters of
polygons and areas of triangles and rectangles,
e.g., using the distance formula.
ETHS-S1C2-01;1112.RST.4
ETHS-S1C2-01;910.WHST.1a-1e;1112.WHST.1a-1e
ETHS-S1C2-01;910.RST.3
ETHS-S1C2-01;910.RST.3;11-12.RST.3
HS.G-GPE.7
Give an informal argument for the formulas for
the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use
dissection arguments, Cavalieri’s principle, and
informal limit arguments.
HS.GGMD.1
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Cannot be reproduced without permission
9-10.RST.4;910.WHST.1c; 910.WHST.1e;1112.RST.4;1112.WHST.1c;1112.WHST.1e
Presented
Assessed
Block 2
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Block 3
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Block 4
Presented
Assess
Integrated Geometry 1, Curricular Guide
11
Block 1
Lookup
Code
HS.GGMD.2
HS.GGMD.3
HS.GGMD.4
HS.G-MG.1
HS.G-MG.3
HS.S-IC.2
HS.S-IC.3
Descriptor
Give an informal argument using Cavalieri’s
principle for the formulas for the volume of a
sphere and other solid figures.
Use volume formulas for cylinders, pyramids,
cones, and spheres to solve problems.
Identify the shapes of two-dimensional crosssections of three-dimensional objects, and
identify three-dimensional objects generated by
rotations of two-dimensional objects.
Use geometric shapes, their measures, and their
properties to describe objects (e.g., modeling a
tree trunk or a human torso as a cylinder).
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).
Decide if a specified model is consistent with
results from a given data-generating process,
e.g., using simulation. For example, a model
says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row
cause you to question th
Recognize the purposes of and differences
among sample surveys, experiments, and
observational studies; explain how randomization
relates to each.
Evaluate reports based on data.
HS.S-IC.6
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Connections
9-10.RST.4;910.WHST.1c;910.WHST.1e;1112.RST.4;1112.WHST.1c;1112.WHST.1e
9-10.RST.4
ETHS-S1C201
ETHS-S1C201;910.WHST.2c
ETHS-S1C201
ETHS-S6C203;910.WHST.2d;910.WHST.2f
11-12.RST.9;1112.WHST.2b
1112.RST.4;1112.RST.5;1112.WHST.1b;1112.WHST.1e
Presented
Assessed
Block 2
Presented
Assessed
Block 3
Presented
Assessed
Block 4
Presented
Assessed
Integrated Geometry 1, Curricular Guide
12
Block 1
Lookup
Code
HS.S-CP.1
HS.S-CP.2
HS.S-CP.3
HS.S-MD.7
Descriptor
. 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
(“or,” “and,” “not”).
. Understand that two events A and B are
independent if the probability of A and B
occurring together is the product of their
probabilities, and use this characterization to
determine if they are independent.
Understand the conditional probability of A given
B as P(A and B)/P(B), and 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
Analyze decisions and strategies using
probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the
end of a game).
Study the following topics related to vertex-edge
graphs: Euler circuits, Hamilton circuits, the
Travelling Salesperson Problem (TSP), minimum
weight spanning trees, shortest paths, vertex
coloring, and adjacency matrices.
Connections
1112.WHST.2e
Understand, analyze, and apply vertex-edge
graphs to model and solve problems related to
paths, circuits, networks, and relationships
among a finite number of elements, in real-world
and abstract settings.
ETHS-S6C203;1112.RST.9;1112.WHST.1b;1112.WHST.1e;
1112.WHST.1e
11-12.RST.5;1112.WHST.1e
ETHS-S1C201;ETHS-S6C203
ETHS-S6C203;11-12.RST.4;
11-12.RST.5;1112.RST.9;1112.WHST.1b;1112.WHST.1e
HS.CMDM.1
HS.CMDM.2
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Cannot be reproduced without permission
Presented
Assessed
Block 2
Presented
Assessed
Block 3
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Block 4
Presented
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Integrated Geometry 1, Curricular Guide
13
Block 1
Lookup
Code
Descriptor
Devise, analyze, and apply algorithms for
solving vertex-edge graph problems.
Connections
ETHS-S6C2-03;1112.RST.3; 1112.RST.4;1112.RST.9;1112.WHST.1a;1112.WHST.1b;1112.WHST.1e
Extend work with adjacency matrices for graphs,
such as interpreting row sums and using the nth
power of the adjacency matrix to count paths of
length n in a graph.
ETHS-S6C2-03;1112.RST.4; 1112.RST.5;1112.RST.9;1112.WHST.1a;1112.WHST.1b;1112.WHST.1e
HS.CMDM.3
HS.CMDM.4
HS.GSRT.1a
HS.GSRT.1b
A dilation takes a line not passing through the
center of the dilation to a parallel line, and leaves
a line passing through the center unchanged.
The dilation of a line segment is longer or
shorter in the ratio given by the scale factor.
Practices Applied in all Math Classes
Mathematical Practices (MP)
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
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Cannot be reproduced without permission
Presented
Assessed
Block 2
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Block 4
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Integrated Geometry 1, Curricular Guide
14
Suggested Coursework and Pacing
The course is planned as an 8 week course with padded time for review and testing for midterms and finals.
Week
Course Material
Week
Course Material
Week 1
Sequences and Patterns
Basic Logic Of Conditional Statements
Designating Assumptions in a Proof
Week 5
Angle Chasing In complex Designs
Week 2
Bi-Conditional Statements and Converses
Laws of Logic (Syllogism and Detachment)
Steps of Proof
Week 6
Transformations On a graph
Translation and Reflection
Week 3
Using Proof on Given Forms
Defining Triangles and Polygons
Rules of Angles (Complementary and Supplementary)
Week 7
Transformations on a Graph
Rotation and Dilation
Week 4
Angles with Lines
Parallel Line Proofs
Triangle and Polygon Sum Theorem
Week 8
Basic Rules of Circles
Area, Circumference
Radius and Arcs
Chords
Regular Shapes
Suggested Honors Extensions
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Simple design and construction concepts. Using proofs to support ideas. Problems with written answers from the Math Modeling Competition.
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Programming Basics (Python or Basic Stamps) with conditional statement and logic design. Robotics with the BOEbot (Parallax.com) with navigation and
decision making programming. Possibly design a Mini RoboSumo program (Parallax.com)
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Integrated Geometry 1, Curricular Guide
15
Online Resources for Content
AZ/ADE Comprehensive Links for AIMS, Standards, Vision, Vocabulary, Rubrics, etc.:
http://www.ade.az.gov/K12Literacy/langarts.asp
Arizona ELP Information and Standards:
http://www.ade.state.az.us/oelas/
Online Resources for Instructional Methods
Rigor and Relevance Framework:
http://www.leadered.com/rrr.html
http://rigor-relevance.com/
http://www.edteck.com/wpa/index.htm
www.leadered.com/pdf/Academic_Excellence.pdf
21st Century Leaner:
http://www.ala.org/
http://www.p21.org/
http://dpi.wi.gov/cal/iecouncil.html
Character Education:
http://goodcharacter.com/
http://charactercounts.org/
http://www.ade.state.az.us/charactered/
Bloom’s Taxonomies:
http://www.nwlink.com/~Donclark/hrd/bloom.html
Multiple Intelligences:
http://www.thomasarmstrong.com/multiple_intelligences.htm
http://www.infed.org/thinkers/gardner.htm
http://literacyworks.org/mi/assessment/findyourstrengths.html
Project-based Learning:
http://www.edutopia.org/project-based-learning-research
http://pblchecklist.4teachers.org/
http://en.wikipedia.org/wiki/Project-based_learning
http://www.pbl-online.org/
http://www.bie.org/index.php/site/PBL/overview_pbl/
http://www.edutopia.org/project-based-learning-research
Power Point Games:
http://jc-schools.net/tutorials/PPT-games/
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