Download Course Guide - Washoe County School District

2015-2016
Course Guide
Geometry
Critical Areas in Geometry
The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from
the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric
relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry
course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in
this course. Close attention should be paid to the introductory content for the Geometry conceptual category found
in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the
content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that
makes use of their ability to make sense of problem situations. The critical areas, organized into six units are as
follows.
(1) 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 criteria,
based on analyses of rigid motions and formal constructions. They use 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.
(2) 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.
(3) 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 twodimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object
about a line.
(4) 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 by connecting the geometric and algebraic definitions of the parabola.
(5) In this unit students prove basic theorems about circles, such as a tangent line is 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 an 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.
(6) 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.
Possible book sections – Code Key
Red Book Text Sections referenced are from 2010 Glencoe Geometry - #.#
Blue Book Text Sections referenced are from 2008 Geometry concepts and Applications - #.#
On Core sections – OC
Explorations in Math sections referenced – check dropbox
Excellence in Education, Every Student, Every Day, to Graduation
Vision
As a courageous innovative leader in education, Washoe County School District will be one of the
nation’s top performing school districts, graduating all students college and/or highly skilled career
ready.
Mission
To create an education system where all students achieve academic success, develop personal
and civic responsibility, and achieve career and college readiness for the 21st century.
Core Beliefs
We believe:
•
•
•
•
•
•
All students will learn and be successful.
The achievement gap will be eliminated by ensuring every student is challenged to
learn at, or above grade level.
Effective teachers and principals, dedicated support staff, rigorous curriculum,
measurable outcomes, ongoing monitoring and assessment, collaboration,
professional development and a culture of continuous improvement will ensure
classroom success for all students.
Superior performance will be achieved through clear goals that set high expectations
and standards for all students and employees.
Family, school and community engagement will be required for student academic
success.
Leadership and passion, together with accountability and transparency, will be keys
to reform and success.
Philosophy of Mathematics Education
•
•
•
•
Mathematical understandings of new concepts are taught through a planned learning progression
through a coherent set of mathematics standards.
Student learning of new content is facilitated through the mathematical shifts; focus coherence and
rigor (conceptual understanding, application and procedural fluency).
Students learn through the use of good tasks/questions, mathematical discussions and regular checks
for understanding.
All students should have the opportunity to learn and engage in the mathematics that prepares them for
career and college.
WCSD Mathematics Message
Teachers in grades K-8, Algebra 1, Geometry and Algebra 2 are to teach to the Nevada Academic Content
Standards for their grade level. We realize that students may have gaps in understanding and may need
scaffolding to support conceptual development; yet we do not support a philosophy of only teaching
procedurally to the gaps. Teachers may want to consider focusing intervention time on addressing concerns
around the following domains; number sense, ratios and proportions and equations and expressions (this
includes fractions in the intermediate grades and rational numbers in the grades 7-12).
* Grade 6-8: Course Guides are designed to address topics within a unit of study and include reference to all of
the NVACS-M for each grade. Students should have the opportunity to engage in all of the content standards for
their grade and the amount of time spent on the topics and standards is recommended in the course guides.
Teachers are expected to use both their adopted textbook and approved supplemental resources to meet the
needs of their students.
* Grade 9-12: Course Guides are designed to address topics within a unit of study. The NVACS-M provide the
standards for Algebra 1, Geometry and Algebra 2. Students should have the opportunity to engage in all of the
content standards for their course and the amount of time spent on the topics and standards is recommended
in the course guides. Teachers are expected to use both their adopted textbook and approved supplemental
resources to meet the needs of their students Fourth year courses have been revised to engage students in the
fourth year standards from the NVACS-M and topics that best address the intent of the course.
* Grading recommendations are made for mathematics courses in grades 6-12. See course guide.
* The Curriculum and Instruction department have many professional development opportunities for teachers
currently posted on Professional Learning Café.
Please feel to contact Vicki Collaro [email protected] with questions regarding interpretations of the
standards or course guides and [email protected] for assessment particulars.
Assessment Philosophy
• Assessment is the continuous process of collecting information to make decisions about teaching
and learning
• A balanced assessment program is essential for determining and reporting the learning needs,
progress, and achievement of students at the state, district, and classroom levels
• The most impactful assessment comes from a teacher with well-planned units, lessons, and multiple
strategies designed to elicit evidence of student learning
• Assessments must begin with clear purpose, targets and design to provide information from which
valid inferences can be made
• Assessment use must always adhere to the intended purpose of the assessment
• Practicing for an assessment by focusing on specific items and tested skills does not promote lasting
student achievement, and the assessment itself should never be the learning target.
• Formative Assessment Processes including close, purposeful observation provide valuable data
integral to student learning at every grade level
• Self-assessment and self-monitoring activities enhance student self-efficacy
• Clear, specific, and timely feedback must accompany assessment
On the following pages are guidelines and information for planning instruction around the Common
Core State Standards.
Teachers should look for opportunities to incorporate the Mathematical Practices from the Common
Core State Standards (CCSS) into planning and instruction where appropriate. "What students can learn
at any particular grade level depends upon what they have learned before" (CCSS Introduction, p. 5).
Teachers are encouraged to communicate with other grade levels and align practices and goals to
support vertical transitions. Review the CCSS critical areas of focus for grades 6-12 to assist in planning
short and long-range goals for student learning and strengthening teaching of the Mathematics
Standards.
Students should engage in 60 minutes of daily math instruction (minimum).
Recommendations for 6-12 Grading in Mathematics
Note: Grading recommendations were established to provide those that need more specific
direction somewhere to start. The importance of the recommendations is that consistency is
established at a school site or between a feeder middle school and the high school. The PLC
group should decide on more specific grading policies for their school but should be in line
with the recommendations here.
1. Grading at any level should be consistent within the building for like grades and courses.
2. Assignments are assigned and are completed by the students on their own time or in class with
assistance. Individual performance on projects may be included in this category.
3. Quizzes/In Class Checks are evaluations of what the students know but could be used to inform
instruction or offer additional assistance to a student. Students may have multiple attempts to get
these points but each attempt needs to be completed by the individual without assistance.
4. Assessments are to determine what the students have learned and are summative in nature. These
are individual performance measures and should be monitored assessments. Students may have the
opportunity to take a retake. Grades recorded in assessment should reflect what the students know.
Caution should be given to practices that would inflate test grades.
6-12 Math Courses Grading Recommendations
Math 6, Math 7, Math 8, Math 7-8
Grading Recommendations:
• Assignments (independent work, projects, group work) – 10-15%
• Final – 10% (2015-16)
• Quizzes (monitored in class checks, individual performance) – 20-30%
• Assessments/Exams (individual performance) – 40-55%
Algebra 1, Geometry, Formal, Algebra 2, and non-honors 4th year courses
Grading Recommendations:
• Assignments (independent work, projects, group work) – 10-15%
• Final – 15% (2015-16)
• Quizzes (monitored in class checks, monitored individual performance) – 10-20%
• Assessments/Exams (monitored individual performance) – 45-60%
Honors Mathematics Courses
Grading Recommendations:
• Assignments (independent work, projects, group work) – 0-10%
• Final – 15-20% (2015-16)
• Quizzes (monitored in class checks, monitored individual performance) – 10-20%
• Assessments/Exams (monitored individual performance) – 60-70%
WCSD Recommended Pathways to Advanced Mathematics
Grade 7
Options
Grade 8
Options
Grade 9
Options
Grade 10
Options
Grade 11
Options
Grade 12
Options
Trigonometry
Pre-Calculus
Geometry
**
Math
7
Math
8
Algebra
2**
Prob/Stat
Discrete
Algebra 1
Geometry
and
Algebra 2
**
Adv.
Algebra 3
Pre-College
Math
EC Math 095
EC Math 096
Geometry
**
Math
7/8
Algebra
2**
Trigonometry
Pre-Calculus
Algebra
1*
EC Math 096
Alg Pre-Calc
AP Calculus
AB or BC
Geometry
and
Algebra 2
**
Trigonometry
Pre-Calculus
AP Calculus
AB or BC
AP Stats
College Level
Dual Credit
Course
* High School credit is not awarded for high school level courses taken prior to 9th grade. Students must earn a C
or better to progress on to the next course in sequence
** Students choose from two class options to fulfill this requirement – Geometry or Formal Geometry (H) and
Algebra 2 or Algebra 2 (H)
Primary Mathematics Resources
Algebra 1
McDougal Littell Algebra 1,
copyright 2007 by McDougal Littell, a division of Houghton Mifflin Company
TE-ISBN 978-0-618-59556-3; SE-ISBN 978-0-678-59402-3
On Core Mathematics Algebra 1,
copyright by Houghton Mifflin Harcourt Publishing Company
TE-ISBN 978-0-547-61723-7; SE-ISBN 978-0-547-57527-8
Geometry & Formal Geometry
Glencoe Geometry, (red book)
copyright 2010 by The McGraw Hill Companies, Inc.
TE-ISBN 978-0-078-88485-6; SE-ISBN 978-0-078-88484-9
Glencoe Geometry Concepts/Applications, (blue book)
copyright 2008 by The McGraw Hill Companies, Inc.
TE-ISBN 978-0-078-69932-0; SE-ISBN978-0-078-79914-3
On Core Mathematics Geometry,
copyright by Houghton Mifflin Harcourt Publishing Company
TE-ISBN 978-0-547-61724-4; SE-ISBN 978-0-547-57530-8
Algebra 2 & Honors Algebra 2
McDougal Littell Algebra 2,
copyright 2007 by McDougal Littell, a division of Houghton Mifflin Company
TE-ISBN 978-0-618-78280-2; SE-ISBN 978-0-618-85941-9
On Core Mathematics Algebra 2,
copyright by Houghton Mifflin Harcourt Publishing Company
TE-ISBN 978-0-547-61726-8; SE-ISBN 978-0-547-57529-2
Trigonometry/Pre-Calculus
Pearson Blitzer Pre-Calculus,
copyright 2010 by Pearson Education, Inc.
TE-ISBN 978-0-321-57538-8; SE-ISBN 978-0-131-36221-5
Probability/Statistics/Discrete
Pearson Elementary Statistics: Picturing the World,
copyright 2009 by Pearson Education, Inc.
TE-ISBN 978-0-132-06290-9; SE-ISBN 978-0-136-00720-3
Pearson Thinking Mathematically,
copyright 2008 by Pearson Education, Inc.
TE-ISBN 978-0-131-75206-1; SE-ISBN 978-0-131-34678-9
Advanced Algebra 3
Pearson Custom Publishing Advanced Algebra Applications,
copyright 2009 by Pearson Custom Publishing
TE-ISBN 978-0-133-65992-4; SE-ISBN 978-0-558-20908-7
Mathematical Practices
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at
all levels should seek to develop in their students. These practices rest on important “processes and
proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM
process standards of problem solving, reasoning and proof, communication, representation, and
connections. The second are the strands of mathematical proficiency specified in the National Research
Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding
(comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying
out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual
inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and
one’s own efficacy).
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry
points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the
form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students
might, depending on the context of the problem, transform algebraic expressions or change the viewing window on
their graphing calculator to get the information they need. Mathematically proficient students can explain
correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features
and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete
objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers
to problems using a different method, and they continually ask themselves, “Does this make sense?” They can
understand the approaches of others to solving complex problems and identify correspondences between different
approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They
bring two complementary abilities to bear on problems involving quantitative relationships: the ability to
decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols
as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to
pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the
units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using
different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established
results in constructing arguments. They make conjectures and build a logical progression of statements to explore the
truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use
counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that take into account the context from
which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible
arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—
explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made
formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all
grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to
clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life,
society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a
situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem
in the community. By high school, a student might use geometry to solve a design problem or use a function to
describe how one quantity of interest depends on another. Mathematically proficient students who can apply what
they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing
that these may need revision later. They are able to identify important quantities in a practical situation and map
their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze
those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the
context of the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
Mathematical Practices
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools
might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer
algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar
with tools appropriate for their grade or course to make sound decisions about when each of these tools might be
helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high
school students analyze graphs of functions and solutions generated using a graphing calculator. They detect
possible errors by strategically using estimation and other mathematical knowledge. When making mathematical
models, they know that technology can enable them to visualize the results of varying assumptions, explore
consequences, and compare predictions with data. Mathematically proficient students at various grade levels are
able to identify relevant external mathematical resources, such as digital content located on a website, and use them
to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of
concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in
discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including
using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling
axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express
numerical answers with a degree of precision appropriate for the problem context. In the elementary grades,
students give carefully formulated explanations to each other. By the time they reach high school they have learned
to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example,
might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of
shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression + 9 + 14, older students can see the
14as2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use
the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift
perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being
composed of several objects. For example, they can see 5– 3(– ) as 5 minus a positive number times a square
and use that to realize that its value cannot be more than 5 for any real numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for
shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same
calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation
of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students
might abstract the equation (– 2)/(– 1) = 3. Noticing the regularity in the way terms cancel when expanding
(– 1)( + 1), (– 1)( + + 1), and (– 1)( + + + 1) might lead them to the general formula for the
sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of
the process, while attending to the details. They continually evaluate the reasonableness of their intermediate
results.
Connecting the Standards for Mathematical Practice to the Standards for Mathematical
Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of
mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise
throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional
development should all attend to the need to connect the mathematical practices to mathematical content in mathematics
instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding.
Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the
content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which
to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply
the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics
accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a
lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those
content standards which set an expectation of understanding are potential “points of intersection” between the Standards
for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be
weighted toward central and generative concepts in the school mathematics curriculum that most merit the time,
resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment,
professional development, and student achievement in mathematics.
How to use the Course Guides
Middle School and High School
1. Examine the
Unit by
examining the
topics in that
unit.
2. Read for understanding the
standards
associated with
the topics in that
unit.
3. Set an explicit goal for the unit.
Create a learning progression for
the unit.
4. Examine the resources (secons in the book are referenced
because there are items in that secon that may be useful,
secons referenced should not be taught without making
connecons to the learning progression and standards). It is
important to realize that not all of the items in the secon listed
will need to be taught.
⇒
Addionally in planning, examine the course and standards for the grade level below and above to
ensure that teaching is Focused and coherent on the course standards.
⇒
Provide opportunies for student engagement through good quesoning and discussion techniques.
Discussion Tools: Ancipang, Monitoring, Selecng, Sequencing, and Connecng
⇒
Formavely assess regularly within the unit, adjust instrucon as needed to stay on track with your goals.
Formave Tools: Learning Targets/Criteria for Success, Feedback, Student Goal Se6ng,
Student Self-Assessment, Teacher Quesoning and Student Quesoning
⇒
Design summave assessments to reflect the Nevada Academic Content Standards in your course.
Assessment items should include conceptual understanding, procedural fluency and applicaon.
Plan-Do-Study-Act
6-12 Math
PLAN
DO
STUDY
ACT
Read through and review the enre Unit in the Course Guide:
•
Read the Nevada Academic Content Standards referenced in the Unit. Ensure the work of the lesson
reflects the shi)s required by the CCSS for Mathemacs.
•
Ensure that your learning goals reflect the Focus and Coherence of the Unit.
•
Present a balance of conceptual understanding, procedural fluency and applicaon presented in the
Nevada Academic Content Standards.
•
Prepare tasks/quesons that will promote discussions.
•
What formave pracce are you using throughout the unit to check for student understanding?
What misconcepons do you ancipate students will have? How will you know that the student
is successful in meeng the goals of the Unit?
Teach the Unit:
•
What quesoning strategies are being used to engage the learner in discussions?
•
What are you nocing about how students are reasoning about the mathemacs? How are the
Mathemacal Pracces supporng the learners engagement in mathemacs?
•
What evidence are you collecng to support how the learner is understanding and making
connecons with the mathemacs? How are misconcepons being addressed during instrucon?
•
Employ instruconal pracces that all students to master the content of the lesson.
Review formave pracce, quesoning prompts, feedback, classroom discussions and wri7en response items:
•
Based on the instruconal opportunies provided, which learners may need more support with
concept development and applicaon during this unit? Which learners may need deeper
opportunies to explore connecons?
•
What are you nocing about how the learners are engaging in the mathemacs? Are the mathemacal models accurate? Are classroom discussions producve? Does the environment support all
learners?
Now What?
•
How will you use intervenon and small group instrucon me to support these learners in building
or deepening understanding during this unit?
•
What quesons do you have about the Nevada Academic Content Standards that are arising as you
facilitate and build instruconal opportunies for your learners? What instruconal pracces are
you looking at focusing on and cra)ing to best support ALL learners?
CCSS
Standards
10 Days
Introduce
basic dila(ons
G.CO.2,
G.CO.4,
C.GO.5,
C.CO.6
Transla(ons G.CO.2,
G.CO.4,
C.GO.5,
C.CO.6
Rota(ons
G.CO.2,
G.CO.4,
C.GO.5,
C.CO.6
Composi(ons G.CO.6
of transforma(ons
Reflec(ons
Basic Terms G.CO.1,
and
G.CO.12
Construc(ons
TransforG.CO.2
ma(ons and
Rigid Mo(ons
Topics
1. Construc'ons &
Transforma'ons
5/5/15
Geometry : Semester 1
G.CO.2,
G.CO.9
Angle
Bisectors
Proofs about G.CO.9
Line Segments
and Angles
G.CO.1,
G.CO.12
CCSS
Standards
12 Days
Angles
Topics
2. Angles and
Angle Bisectors
G.CO.2,
G.CO.9
G.CO.1,
G.CO.12
CCSS
Standards
Proofs about G.CO.9
Parallel and
Perpendicular
Bisectors
Parallel and
Perpendicular
Lines
Perpendicular
Bisectors
Topics
15 Days
3. Parallel Lines and
Perpendicular Bisectors
CCSS
Standards
Slope and
G.GPE.5
Perpendicular
Lines
Slope and
G.GPE.5
Parallel Lines
Par((oning a G.GPE.6
Segment
Topics
10 Days
4. Parallel & Perpendicular Lines
G.CO.5,
G.CO.6
CCSS
Standards
15 Days
G.CO.7,
G.CO.8
G.SRT.5
G.CO.10
Using
Congruence
Criteria in
Proofs
The isosceles
Triangle
Theorem
G.CO.10
Developing
Congruence
Criteria
Angels in
Triangles
Congruence & G.CO.7
Triangles
Congruence
Topics
5. Congruence
G.CO.3
CCSS
Standards
12 Days
G.GPE.7
G.GPE.7
CCSS
Standards
Perimeter and G.GPE.7,
Area
G.MG.1,
G.MG.2
Perpendicular G.CO.1
Lines and
Distance
Coordinate
G.GPE.5,
Proof using
G.GPE.4
slope and
distance
Midpoint
Formula
Distance
Formula
Topics
12 Days
7. Coordinate Geometry
The “days” are RECOMMENDATIONS ONLY
and refer to a typical, 50 minute class
mee(ng 5 days a week. You will have to
adjust to your own schedule.
Special
G.CO.11,
Parallelograms
Criteria for
G.CO.11,
Parallelograms
Diagonals of G.CO.11,
Parallelograms
Sides & Angles G.CO.11,
of
G.SRT.5
Parallelograms
Symmetry
Topics
6. Quadrilaterals
2015-16 Geometry Units - Semester One
8. Dila'ons
Topics
20 Days
CCSS
Standards
G.SRT.1,
G.CO.2
Ra(os and
Propor(ons
Proper(es of
Dila(ons
G.CO.2
G.SRT.2, G.C.1
Drawing
Dila(ons
Similarity
G.SRT.5,
G.MG.3
G.SRT.2,
G.SRT.3
Solving
Problems
Using
Similarity
G.SRT.4,
G.SRT.5
Similarity &
Triangles
The Triangle
Propor(onal
Theorem
G.SRT.4,
G.SRT.5
Midsegment of G.CO.10,
a Triangle
G.GPE.4
Proving the
Pythagorean
Theorem
5/5/15
Geometry : Semester 2
G.SRT.6,
G.SRT.7
10 Days
10. Perimeter & Circum-
Topics
CCSS
Standards
Circumference G.GMD.1,
G.MG.1
7.G.6
Area of regular G.RST.8
Polygons
Review of
Surface Area
11. Volumes
CCSS
Standards
7.G.6
G.GMD.1,
G.GMD.3
Volume of
Prisms
Volumes of
Pyramids
8.G.9
G.GMD.1,
G.GMD.3,
Volume of
Cones
Solving
Design
Problems
Volume of
Spheres
8.G.9
G.GMD.1,
G.GMD.3
8.G.9
G.GMD.3,
G.MG.2
G.GMD.3,
G.MG.1,
G.MG.2,
G.MG.3
Volume of
Cylinders
Topics
15 Days
12. Circles
CCSS
Standards
G.CO.1, G.C.5
Area of Circles G.C.5,
and Sectors
G.GMD.1
Arc Length
Tangent Lines G.C.2, G.C.4+
Central Angles G.C.2
and Inscribed
Angles
Topics
10 Days
Condi(onal
Probability
Two-Way
Tables
Dependent
Events
10 Days
CCSS
Standards
Independent S.CP.2,
Events
S.CP.3,
S.CP.4, S.CP.5
S.CP.8+
S.CP.3,
S.CP.4,
S.CP.5, S.CP.6
S.ID.5
Probability
S.CP.1
and Set
Theory
Mutually
S.CP.7
Exclusive and
Overlapping
Events
Topics
13. Probability
2015-16 Geometry Units - Semester Two
G.SRT.6,
G.SRT.7
CCSS
Standards
10 Days
9. Trigonometric Ra'os &
Right Triangles
Topics
Special Right
Triangles
Inverse Trig
Func(ons
G.SRT.8
The Sine,
G.SRT.6,
Cosine &
G.SRT.7
Tangent Ra(os
Solving Right
Triangles
G.GMD.3,
G.MG.1,
G.MG.2,
G.MG.3
Surface Area G.RST.8
of threedimensional
figures with
regular
polygon bases
Visualizing
G.GMD.4
ThreeDimensional
Figures
Solving
Design
Problems
10 Days
CCSS
Standards
14. Construc'ons
Topics
Construc(ng G.C.3
Inscribed
Quadrilaterals
Construc(ng G.CO.13
Inscribed
Polygons
Construc(ng G.C.3
Circumscribed
Circles
Construc(ng G.C.3
Inscribed
Circles
Geometry Unit 1 - Semester 1
Topics
Basic Terms and Constructions (G.CO.1, G.CO.12)
Transformations and Rigid Motions (G.CO.2)
Reflections (G.CO.2, G.CO.4, C.GO.5, C.CO.6)
Translations (G.CO.2, G.CO.4, C.GO.5, C.CO.6)
Rotations (G.CO.2, G.CO.4, C.GO.5, C.CO.6)
Compositions of transformations (G.CO.6)
On Core Book
Red Formal
Blue Book
1.1
2.1
2.2
2.5
2.6
1.1, ext 1.1
4.7, 9.4
9.1
9.2
9.3
9.4
1.2,1.5
5.3
5.3, 16.4
5.3, 16.3
5.3, 16.5
Introduce basic dilations
•
See dropbox for additional resources for this unit
10 days
Standards to be taught
Experiment with transformations in the plane.
G.CO.1
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.
G.CO.2 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 angle to those that do not (e.g., translation versus
horizontal stretch).
G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g.,
graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a
given figure on to another.
Understand congruence in terms of rigid transformations.
G.CO.6 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.
Make geometric constructions.
G.CO.12 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; constructing perpendicular lines, including the perpendicular bisector
of a line segment; and constructing a line parallel to a given line through a point not on the line.
5.5.15
Geometry Unit 2 – Semester 1
Topics
On Core Book
Red Formal
Blue Book
Angles (classifying and naming, etc.) (G.CO.1, G.CO.12)
Angles (vertical, supplementary, complementary, etc.)
1.4
1.4-1.5
3.1, 3.2
Proofs About Line Segments and Angles (G.CO.9)
1.6
2.7, 2.8
Angle Bisectors (G.CO.2, G.CO.9)
2.3
1.4, 5.1
•
2.2, 3.3,
3.4
3.3
See dropbox for additional resources for this unit
12 days
Standards to be taught
Experiment with transformations in the plane.
G.CO.1 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.
G.CO.2 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 angle to those that do not (e.g., translation versus
horizontal stretch).
Prove geometric theorems.
G.CO.9 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 line segment are exactly those equidistant from the segment’s endpoints.
Make geometric constructions.
G.CO.12 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; constructing perpendicular lines, including the perpendicular bisector
of a line segment; and constructing a line parallel to a given line through a point not on the line.
5.5.15
Geometry Unit 3 – Semester 1
Topics
On Core Book
Parallel and Perpendicular Lines (G.CO.1, G.CO.12)
Perpendicular Bisectors (G.CO.2, G.CO.9)
Proofs About Parallel and Perpendicular Lines (G.CO.9)
•
1.5
2.4
1.7
Red Formal
Blue Book
3.6, 5.1
3.5
4.1
2.3
4.2-4.4
See dropbox for additional resources for this unit
15 days
Standards to be taught
Experiment with transformations in the plane.
G.CO.1 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.
G.CO.2 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 angle to those that do not (e.g., translation versus horizontal
stretch).
Prove geometric theorems.
G.CO.9 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 line segment are exactly those equidistant from the segment’s endpoints.
Make geometric constructions.
G.CO.12
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; constructing perpendicular lines, including the perpendicular bisector of a line segment; and
constructing a line parallel to a given line through a point not on the line.
Geometry Unit 4 – Semester 1
Topics
On Core Book
Partitioning a Segment (G.GPE.6)
8.3
Slope and Parallel Lines (G.GPE.5)
8.4
Slope and Perpendicular Lines (G.GPE.5)
8.5
•
Red Formal
3.3-3.4,
ext 3.4
3.6
Blue Book
4.5, 4.6
4.6
See dropbox for additional resources for this unit
10 days
Standards to be taught
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g.,
find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given
ratio.
5.5.15
Geometry Unit 5 – Semester 1
Topics
On Core Book
Red Formal
Blue Book
Congruence (G.CO.5, G.CO.6)
Congruence and Triangles (G.CO.7)
Angles in Triangles, Triangle Inequality & Angles in Two Triangles (G.CO.10)
Developing Congruence Criteria (G.CO.7, G.CO.8)
Proofs involving triangle congruence, corresponding parts of triangles.
Using Congruence Criteria in Proofs (G.SRT.5)
The Isosceles Triangle Theorem (G.CO.10)
3.1
3.2
3.5
4.7
4.1
4.2, 5.5, 5.6
5.3
5.4
5.2, 7.2
3.3
4.3
5.5, 5.6
3.4
3.6
4.4-4.5
4.6
5.5, 5.6
15 days
Standards to be taught
Experiment with transformations in the plane.
G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g.,
graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a
given figure on to another.
Understand congruence in terms of rigid motions.
G.CO.6 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.
G.CO.7 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.
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence
in terms of rigid motions.
Prove geometric theorems.
G.CO.10 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 the length; the medians of a triangle meet at a point.
Prove theorems involving similarity.
G.SRT.5
5.5.15
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
Geometry Unit 6 – Semester 1
Topics
On Core Book
Red Formal
Blue Book
Symmetry (G.CO.3)
4.1
10.6
Sides and Angles of Parallelograms (G.CO.11, G.SRT.5)
Prove theorems about parallelograms throughout Unit 6.
4.2
6.2
8.2
Diagonals of Parallelograms (G.CO.11, G.SRT.5)
4.3
6.2
8.2
Criteria for Parallelograms (G.CO.11, G.SRT.5)
4.4
6.3
8.3
Special Parallelograms (G.CO.11, G.SRT.5)
4.5
6.4-6.5
8.4
12 days
Standards to be taught
Experiment with transformations in the plane.
G.CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that
carry it onto itself.
Prove geometric theorems.
G.CO.11 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.
Prove theorems involving similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
5.5.15
Geometry Unit 7 – Semester 1
Topics
The Distance Formula (G.GPE.7)
(use the coordinate plane to derive distance through the Pythagorean
Theorem, use the formula with rational numbers)
The Midpoint Formula (G.GPE.7)
(use the midpoint formula to find the center of a circle in the
coordinate
plane, or find the endpoint of a diameter)
Perpendicular Lines and Distance (G.CO.1)
Coordinate Proofs using Slope and Distance (G.GPE.4, G.GPE.5)
Perimeter & Area the Coordinate Plane (G.GPE.7, G.MG.1, G.MG.2)
(previous knowledge: perimeter and area - the new standards to
perform those operation in the coordinate plane)
On Core Book
Red Formal
Blue Book
1.2
1.2
6.7
1.3
1.3
2.5
3.6
8.6
15.6
9.2
10.3
12 days
Standards to be taught
Experiment with transformations in the plane.
G.CO.1 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.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that 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 the origin and containing the point (0, 2).
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g.,
find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.*
Apply geometric concepts in modeling situations.
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or
a human torso as a cylinder).*
G.MG.2 Apply concepts of density based on area and volume in modeling situations
5.5.15
Geometry Unit 8 – Semester 2
Topics
On Core Book
Red Formal
Blue Book
Ratios and proportions
Properties of Dilations (G.SRT.1, G.CO.2)
Drawing Dilations (G.CO.2)
Similarity (G.SRT.2, G.C.1)
Similarity and Triangles (G.SRT.2, G.SRT.3)
Solving Problems Using Similarity (G.SRT.5, G.MG.3)
5.1
5.2
5.3
5.4
5.5
11.5, 7.6
9.6
ext 9.6
7.2
7.3
7.5
9.7
16.6
The Triangle Proportional Theorem (G.SRT.4, G.SRT.5)
Midsegment of a Triangle (G.CO.10, G.GPE.4)
Proving the Pythagorean Theorem (G.SRT.4, G.SRT.5)
5.6
3.8
5.7
7.4
7.4
ext 8.2, 8.2
9.2
9.3
9.2-9.4
9.3, 9.4
9.5
20 days
Standards to be taught
Understand and apply theorems about circles.
G.C.1
Prove that all circles are similar.
Experiment with transformations in the plane.
G.CO.2 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 angle to those that do not (e.g., translation versus horizontal stretch).
Prove geometric theorems.
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 the length; the medians of a triangle meet at a point.
Understand similarity in terms of similarity transformations.
G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
a. 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.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.2 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 and proportionality of all corresponding pairs of sides.
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity.
G.SRT.4 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.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
Apply geometric concepts in modeling situations.
G.MG.3 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).*
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that 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 the origin and containing the point (0, 2).
G.CO.10
5.5.15
Geometry Unit 9 – Semester 2
Topics
Special Right Triangles (G.SRT.6, G.SRT.7)
The Sine , Cosine and Tangent Ratios (G.SRT.6, G.SRT.7)
The Inverse Sine , Cosine and Tangent Ratios (G.SRT.6)
Solving Right Triangles (G.SRT.8)
On Core Book
Red Formal
Blue Book
6.3
6.1, 6.2
8.3
8.4
8.4
8.5
13.2, 13.3
13.4, 13.5
13.4
6.4
10 days
Standards to be taught
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.6 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.
G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
Geometry Unit 10 – Semester 2
Topics
On Core Book
Red Formal
Blue Book
9.3
10.1
11.5, 11.6
11.2, 11.3,
11.4
10.4, 10.5
12.2, 12.4
12.4
12.1
12.7
Circumference (G.GMD.1, G.MG.1)
Review Area and Area of regular Polygons (G.RST.8)
(Use trig ratios and the Pythagorean Theorem to solve right triangles in
applied problems)
Review Surface Area (use nets) (7.G.6)
Surface Area of three-dimensional figures with regular polygonal bases
Visualizing Three-Dimensional Figures (G.GMD.4)
10.1
12.2,12.3
12.3
12.1
Solving Design and Density Problems (G.GMD.3, G.MG.1, G.MG.2, G.MG.3)
10.6
12.8
10 days
Standards to be taught
Apply geometric concepts in modeling situations.
G.MG.1 Use geometric concepts of density based on area and volume in modeling situations (e.g., modeling a tree
trunk or a human torso as a cylinder).*
G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,
BTUs per cubic foot).*
G.MG.3 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).*
Explain volume formulas and use them to solve problems.
G.GMD.1 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 dissections arguments, Cavalieri’s Principle, and informal limit arguments.
Visualize relations between two-dimensional and three-dimensional objects.
G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects.
5.5.15
Geometry Unit 11 – Semester 2
Topics
On Core Book
Red Formal
Blue Book
Volume of Prisms (7.G.6)
Volume of Cylinders (8.G.9, G.GMD.1, G.GMD.3)
Volumes of Pyramids (G.GMD.1, G.GMD.3)
Volume of Cones (8.G.9, G.GMD.1, G.GMD.3)
Volume of Spheres (8.G.9, G.GMD.3)
Solving Design and Density Problems (G.GMD.3, G.MG.1, G.MG.2, G.MG.3)
10.2
10.2
10.3
10.4
10.5
10.6
12.4
12.4
12.5
12.5
12.6
12.8
12.3
12.3
12.5
12.5
12.6
12.7
15 days
Standards to be taught
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.6
Solve real-world and mathematical problems involving area, volume and surface are of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.
Solve real-life and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.9
Know the formulas for the volumes of cones, cylinders and spheres and use them to solve real-world and
mathematical problems.
Explain volume formulas and use them to solve problems.
G.GMD.1 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 dissections arguments, Cavalieri’s Principle, and informal limit arguments.
G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. *
Apply geometric concepts in modeling situations.
G.MG.1 Use geometric concepts of density based on area and volume in modeling situations (e.g., modeling a tree
trunk or a human torso as a cylinder).*
G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,
BTUs per cubic foot).*
G.MG.3 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).*
5.5.15
Geometry Unit 12 – Semester 2
Topics
Central Angles and Inscribed Angles (G.C.2)
(compare central angles, inscribed angles, and circumscribed angles, use
inscribed angles of common arcs to discuss similar and congruent
triangles within the circle)
Tangent Lines (G.C.2, G.C.4+)
Arc Length (G.CO.1, G.C.5)
Area of Circles and Sectors (G.C.5, G.GMD.1)
On Core Book
Red Formal
Blue Book
7.1
10.2, 10.4
14.1, 14.4
tangent
lines only
7.5
9.4
9.5
10.6
10.2
10.4
14.2
11.6
10 days
Standards to be taught
Understand and apply theorems about circles.
G.C.2
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G.C.4+
Construct a tangent line from a point outside a given circle to the circle.
G.C.5
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.
Experiment with transformations in the plane.
G.CO.1
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.
Explain volume formulas and use them to solve problems.
G.GMD.1 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 dissections arguments, Cavalieri’s Principle, and informal limit arguments.
5.5.15
Geometry Unit 13 – Semester 2
Topics
Probability and Set Theory (S.CP.1)
Mutually Exclusive and Overlapping Events (S.CP.7)
Conditional Probability (S.CP.3, S.CP.4, S.CP.5, S.CP.6)
Two-Way Frequency Tables (S.ID.5)
Independent Events (S.CP.2, S.CP.3, S.CP.4, S.CP.5)
Dependent Events (S.CP.8+)
•
On Core Book
Red Formal
11.1
11.5
13.1
13.6
Blue Book
11.6
11.7
11.8
13.5
Geometric probability involving shapes are appropriate
10 days
Standards to be taught
Understand independence and conditional probability and use them to interpret data.
S.CP.1 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”).
S.CP.2 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.
S.CP.3 Understand the conditional probability of A given B as ()⁄
(), 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 same as the probability of B.
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object
being classified. Use the two-way table as a sample space to decide if events are independent and to approximate
conditional probabilities.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and
everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance
of being a smoker if you have lung cancer.
S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret
the answer in terms of the model.
S.CP.7 Apply the Addition Rule, () = () + () − (), and interpret the answer in terms of the
model.
S.CP.8+ Apply the general Multiplication Rule in a uniform probability model
() = ()
(|) = ()
(|), and interpret the answer in terms of the model.
Summarize, represent and interpret data on two categorical and quantitative variables.
S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the
context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible
associations and trends in the data.
5.5.15
Geometry Unit 14 – Semester 2
Topics
Constructing Inscribed Quadrilaterals (G.C.3)
Constructing Inscribed Polygons (G.C0.13)
Constructing Circumscribed Circles (G.C.3)
Constructing Inscribed Circles (G.C.3)
On Core Book
Red Formal
Blue Book
7.2
7.3
7.4
7.6
10.5
10.5
ext10.5
ext10.5
11.3
11.4
11.4
11.4
10 days
Standards to be taught
Understand and apply theorems about circles.
G.C.3
Construct inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
Make geometric constructions.
G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
5.5.15