Download I34 Pre IB Geometry Mathematics Curriculum Essentials Document

I34
Pre IB Geometry
Mathematics
Curriculum Essentials
Document
Boulder Valley School District
Department of Curriculum and Instruction
January 2012
Boulder Valley School District Mathematics – An Introduction to The Curriculum Essentials
Document
Background
The 2009 Common Core State Standards (CCSS) have brought about a much needed move towards consistency in mathematics
throughout the state and nation. In December 2010, the Colorado Academic Standards revisions for Mathematics were adopted
by the State Board of Education. These standards aligned the previous state standards to the Common Core State Standards to
form the Colorado Academic Standards (CAS). The CAS include additions or changes to the CCSS needed to meet state
legislative requirements around Personal Financial Literacy.
The Colorado Academic Standards Grade Level Expectations (GLE) for math are being adopted in their entirety and without
change in the PK-8 curriculum. This decision was made based on the thorough adherence by the state to the CCSS. These new
standards are specific, robust and comprehensive. Additionally, the essential linkage between the standards and the proposed
2014 state assessment system, which may include interim, formative and summative assessments, is based specifically on these
standards. The overwhelming opinion amongst the mathematics teachers, school and district level administration and district
level mathematics coaches clearly indicated a desire to move to the CAS without creating a BVSD version through additions or
changes.
The High School standards provided to us by the state did not delineate how courses should be created. Based on information
regarding the upcoming assessment system, the expertise of our teachers and the writers of the CCSS, the decision was made
to follow the recommendations in the Common Core State Standards for Mathematics- Appendix A: Designing High
School Math Courses Based on the Common Core State Standards. The writing teams took the High School CAS and
carefully and thoughtfully divided them into courses for the creation of the 2012 BVSD Curriculum Essentials Documents (CED).
The Critical Foundations of the 2011 Standards
The expectations in these documents are based on mastery of the topics at specific grade levels with the understanding that the
standards, themes and big ideas reoccur throughout PK-12 at varying degrees of difficulty, requiring different levels of mastery.
The Standards are: 1) Number Sense, Properties, and Operations; 2) Patterns, Functions, and Algebraic Structures; 3) Data
Analysis, Statistics, and Probability; 4) Shape, Dimension, and Geometric Relationships. The information in the standards
progresses from large to fine grain, detailing specific skills and outcomes students must master: Standards to Prepared
Graduate Competencies to Grade Level/Course Expectation to Concepts and Skills Students Master to Evidence Outcomes. The
specific indicators of these different levels of mastery are defined in the Evidence Outcomes. It is important not to think of these
standards in terms of ―introduction, mastery, reinforcement.‖ All of the evidence outcomes in a certain grade level must be
mastered in order for the next higher level of mastery to occur. Again, to maintain consistency and coherence throughout the
district, across all levels, adherence to this idea of mastery is vital.
In creating the documents for the 2012 Boulder Valley Curriculum Essentials Documents in mathematics, the writing teams
focused on clarity, focus and understanding essential changes from the BVSD 2009 standards to the new 2011 CAS. To maintain
the integrity of these documents, it is important that teachers throughout the district follow the standards precisely so that each
child in every classroom can be guaranteed a viable education, regardless of the school they attend or if they move from
another school, another district or another state. Consistency, clarity and coherence are essential to excellence in mathematics
instruction district wide.
Components of the Curriculum Essentials Document
The CED for each grade level and course include the following:
-A-Glance page containing:
o approximately ten key skills or topics that students will master during the year
o the general big ideas of the grade/course
o the Standards of Mathematical Practices
o assessment tools allow teachers to continuously monitor student progress for planning and pacing needs
o description of mathematics at that level
with additional topics or more in-depth coverage of topics included in bold text.
teachers should be familiar and comfortable
using during instruction. It is not a comprehensive list of vocabulary for student use.
-12 Prepared Graduate Competencies
-12 At-A-Glance Guide from the CAS with notes from the CCSS
-12
Explanation of Coding
In these documents you will find various abbreviations and coding used by the Colorado Department of Education.
MP – Mathematical Practices Standard
PFL – Personal Financial Literacy
CCSS – Common Core State Standards
Example: (CCSS: 1.NBT.1) – taken directly from the Common Core State Standards with an reference to the specific CCSS
domain, standard and cluster of evidence outcomes.
NBT – Number Operations in Base Ten
OA – Operations and Algebraic Thinking
MD – Measurement and Data
G – Geometry
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Standards for Mathematical Practice from
The Common Core State Standards for Mathematics
The Standards for Mathematical Practice have been included in the Nature of Mathematics section in
each Grade Level Expectation of the Colorado Academic Standards. The following definitions and
explanation of the Standards for Mathematical Practice from the Common Core State Standards can be
found on pages 6, 7, and 8 in the Common Core State Standards for Mathematics. Each Mathematical
Practices statement has been notated with (MP) at the end of the statement.
Mathematics | Standards for Mathematical Practice
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
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BVSD Curriculum Essentials
3
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.
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 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive
property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 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(x – y)2 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.
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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 (y – 2)/(x – 1) = 3.
Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1),
and (x – 1)(x3 + x2 + x + 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.
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21st Century Skills and Readiness Competencies in Mathematics
Mathematics in Colorado’s description of 21 st century skills is a synthesis of the essential abilities
students must apply in our rapidly changing world. Today’s mathematics students need a repertoire of
knowledge and skills that are more diverse, complex, and integrated than any previous generation.
Mathematics is inherently demonstrated in each of Colorado 21st century skills, as follows:
Critical Thinking and Reasoning
Mathematics is a discipline grounded in critical thinking and reasoning. Doing mathematics involves
recognizing problematic aspects of situations, devising and carrying out strategies, evaluating the
reasonableness of solutions, and justifying methods, strategies, and solutions. Mathematics provides
the grammar and structure that make it possible to describe patterns that exist in nature and society.
Information Literacy
The discipline of mathematics equips students with tools and habits of mind to organize and interpret
quantitative data. Informationally literate mathematics students effectively use learning tools,
including technology, and clearly communicate using mathematical language.
Collaboration
Mathematics is a social discipline involving the exchange of ideas. In the course of doing mathematics,
students offer ideas, strategies, solutions, justifications, and proofs for others to evaluate. In turn, the
mathematics student interprets and evaluates the ideas, strategies, solutions, justifications and proofs
of others.
Self-Direction
Doing mathematics requires a productive disposition and self-direction. It involves monitoring and
assessing one’s mathematical thinking and persistence in searching for patterns, relationships, and
sensible solutions.
Invention
Mathematics is a dynamic discipline, ever expanding as new ideas are contributed. Invention is the key
element as students make and test conjectures, create mathematical models of real-world
phenomena, generalize results, and make connections among ideas, strategies and solutions.
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Colorado Academic Standards
Mathematics
The Colorado academic standards in mathematics are the topical organization of the concepts and
skills every Colorado student should know and be able to do throughout their preschool through
twelfth-grade experience.
1. Number Sense, Properties, and Operations
Number sense provides students with a firm foundation in mathematics. Students build a deep
understanding of quantity, ways of representing numbers, relationships among numbers, and
number systems. Students learn that numbers are governed by properties and understanding
these properties leads to fluency with operations.
2. Patterns, Functions, and Algebraic Structures
Pattern sense gives students a lens with which to understand trends and commonalities.
Students recognize and represent mathematical relationships and analyze change. Students
learn that the structures of algebra allow complex ideas to be expressed succinctly.
3. Data Analysis, Statistics, and Probability
Data and probability sense provides students with tools to understand information and
uncertainty. Students ask questions and gather and use data to answer them. Students use a
variety of data analysis and statistics strategies to analyze, develop and evaluate inferences
based on data. Probability provides the foundation for collecting, describing, and interpreting
data.
4. Shape, Dimension, and Geometric Relationships
Geometric sense allows students to comprehend space and shape. Students analyze the
characteristics and relationships of shapes and structures, engage in logical reasoning, and use
tools and techniques to determine measurement. Students learn that geometry and
measurement are useful in representing and solving problems in the real world as well as in
mathematics.
Modeling Across the Standards
Modeling links classroom mathematics and statistics to everyday life, work, and decision-making.
Modeling is the process of choosing and using appropriate mathematics and statistics to analyze
empirical situations, to understand them better, and to improve decisions. When making mathematical
models, technology is valuable for varying assumptions, exploring consequences, and comparing
predictions with data. Modeling is best interpreted not as a collection of isolated topics but rather in
relation to other standards, specific modeling standards appear throughout the high school standards
indicated by a star symbol (*).
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I34 Pre-IB Geometry Course Overview
Course Description
PIB Geometry will cover the concepts of Geometry with more
emphasis on problem solving strategies. PIB Geometry
presents a thorough study of the structure of the postulate
system and development of formal two-column proof. It
considers the topics of congruence, parallelism,
perpendicularity, properties of polygons, similarity, and the
relationships of circles, spheres, lines, and planes with
respect to space as well as the plane. Basic principles of
probability will be introduced. The use of algebraic skills is
expected. As an advanced course, this course goes beyond
the curriculum expectations of a standard course and
addresses specific prerequisite skills needed for the study of
further International Baccalaureate mathematics by
increasing the depth and complexity. Students are engaged in
dynamic, high‐level learning.


Assessments
Teacher created assessments
Common Assessments and Finals
Grade Level Expectations
Standard
1.
2.
3.
Big Ideas for Pre IB Geometry
1. Number sets and their properties
form a basis for algebraic number
sense.
2. Patterns,
Functions, &
Algebraic
Structures
1. Properties of the real number
system can be applied
algebraically.
2. The coordinate plane allows us to
apply algebraic understandings to
Geometric concepts.
1. Probability models outcomes for
situations in which there is inherent
randomness.
1. The use of Geometric definitions
and deductive logic form the
foundation for expanding
understanding to applied
Geometry.
2. Objects in the plane can be
transformed, and those
transformations can be described and
analyzed mathematically.
3. Concepts of similarity are foundational
to geometry and its applications.
4. Logic and the study of reason provide
the processes for students to
formulate a strategic plan for
problem-solving.
5. Attributes of two- and threedimensional objects are measurable
and can be quantified.
6. Objects in the real world can be
modeled using geometric concepts.
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Topics at a Glance
Congruence
Similarity
Angles and Triangles
Coordinate Geometry
Areas and Volumes
Introduction to Trigonometry
Parallel and Perpendicular Lines
Geometric Inequalities
Circles & Spheres
Lines, Planes, Separation and Space
Logic & The study of reasoning
Probability
Standards for Mathematical Practice
1. Number Sense,
properties, and
operations
3. Data Analysis,
Statistics, &
Probability
4. Shape,
Dimension, &
Geometric
Relationships












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. 7.
Look for and make use of structure.
Look for and express regularity in
repeated
reasoning.
Notes on this document:
 Plain text indicates standards which are
part of the Colorado Academic Standards
as defined by the Common Core State
Standards for the general (M41) Geometry
course.
 Evidence Outcomes in italics are items
that have been added or changed from the
Colorado Academic Standards as needed
to fit the specific needs of BVSD students.
 Evidence outcomes in bold indicate
items that were added to (M41)
Geometry standards for additional
depth and understanding in Advanced
Geometry.
 Items in bold blue italic font indicate
specific topics added to meet the
prerequisites for the International
Baccalaureate program in
mathematics.
BVSD Curriculum Essentials
8
Course Information:
Students enrolling in PIB Geometry should have successfully completed an advanced Algebra I course.
Students wishing to take this course should be highly motivated and hard working. PIB Geometry is
designed to prepare students for success in both IB and AP classes later on in high school.
The Grade Level Expectations and Evidence Outcomes listed in this course for Standard 3: Data
Analysis, Statistics and Probability are a part of the Geometry standards for the Colorado Academic
Standards and should be addressed as a part of this course because students will not receive exposure
to these concepts elsewhere.
Pre-International Baccalaureate Geometry Curriculum Map: Semester 1 Semester 2
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1.
Number Sense, Properties, and Operations
Number sense provides students with a firm foundation in mathematics. Students build a deep understanding of quantity, ways of
representing numbers, relationships among numbers, and number systems. Students learn that numbers are governed by properties, and
understanding these properties leads to fluency with operations.
Prepared Graduates
The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado
education system must master to ensure their success in a postsecondary and workforce setting.
Prepared Graduate Competencies in the Number Sense, Properties, and Operations
Standard are:
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
Understand the structure and properties of our number system. At their most basic level
numbers are abstract symbols that represent real-world quantities

Understand quantity through estimation, precision, order of magnitude, and comparison.
The reasonableness of answers relies on the ability to judge appropriateness, compare,
estimate, and analyze error

Are fluent with basic numerical and symbolic facts and algorithms, and are able to select
and use appropriate (mental math, paper and pencil, and technology) methods based on
an understanding of their efficiency, precision, and transparency

Make both relative (multiplicative) and absolute (arithmetic) comparisons between
quantities. Multiplicative thinking underlies proportional reasoning

Understand that equivalence is a foundation of mathematics represented in numbers,
shapes, measures, expressions, and equations

Apply transformation to numbers, shapes, functional representations, and data
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Content Area: Mathematics - Pre-IB Geometry
Standard: 1. Number Sense, Properties, and Operations
Prepared Graduates:
Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent
real-world quantities.
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
1. Number sets and their properties form a basis for algebraic number sense.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
a. Learn the meaning and correct notations for universal
set, null set, subset, member or element of a set.
b. Be able to explain the union, intersection, and
complement of sets using proper notation.
c. Use Venn Diagrams to solve set problems, specifically
classification in 2 and 3 set problems and finding the
number of members in a set.
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Inquiry Questions:
1. When you extend to a new number systems (e.g., from
integers to rational numbers and from rational numbers
to real numbers), what properties apply to the
extended number system?
Relevance and Application:
1. The understanding of the closed system of real, rational
and irrational numbers is the foundation for the truth of
properties and operations throughout mathematical
development.
2. Venn Diagrams are a useful means of modeling set
theory and problem solving complex situations with
overlapping characteristics.
Nature of the Discipline:
1. Mathematicians build a deep understanding of quantity, ways
of representing numbers, and relationships among numbers
and number systems.
2. Mathematics involves making and testing conjectures,
generalizing results, and making connections among ideas,
strategies, and solutions.
3. Mathematicians look for and make use of structure. (MP)
4. Mathematicians look for and express regularity in repeated
reasoning. (MP)
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2. Patterns, Functions, and Algebraic Structures
Pattern sense gives students a lens with which to understand trends and commonalities. Being a student of mathematics
involves recognizing and representing mathematical relationships and analyzing change. Students learn that the structures
of algebra allow complex ideas to be expressed succinctly.
Prepared Graduates
The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who
complete the Colorado education system must have to ensure success in a postsecondary and workforce setting.
Prepared Graduate Competencies in the 2. Patterns, Functions, and Algebraic Structures Standard are:
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
Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate
(mental math, paper and pencil, and technology) methods based on an understanding of their efficiency,
precision, and transparency

Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures,
expressions, and equations

Make sound predictions and generalizations based on patterns and relationships that arise from numbers,
shapes, symbols, and data

Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by
relying on the properties that are the structure of mathematics

Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present
and defend solutions
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Content Area: Mathematics - Pre-IB Geometry
Standard: 2: Patterns, Functions, and Algebraic Structures
Prepared Graduates:
Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and
pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
1. Properties of the real number system can be applied algebraically.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students Can:
a. Review Algebra I Concepts to be Used in Geometry
i.
Real number system
ii.
Number line
iii.
Commutative, associative, distributive properties
iv.
Properties of equality
v.
Properties of inequality
vi.
Trichotomy property
vii.
Absolute value
Inquiry Questions:
1. When you extend to a new number systems (e.g., from
integers to rational numbers and from rational numbers to real
numbers), what properties apply to the extended number
system?
2. How is the number system extended and applied to geometric
figures and properties?
3. Why don’t we have a different number system for shapes and
angles?
Relevance and Application:
1. The understanding of the closed system of real, rational and
irrational numbers is the foundation for the truth of properties
and operations throughout mathematical development.
Nature of the Discipline:
Standards for Mathematical Practice.
1. Reason abstractly and quantitatively.
2. Model with mathematics.
3. Use appropriate tools strategically.
4. Attend to precision.
5. Look for and make use of structure.
6. Look for and express regularity in repeated reasoning.
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Content Area: Mathematics - Pre-IB Geometry
Standard: 2: Patterns, Functions, and Algebraic Structures
Prepared Graduates:
Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data
GRADE LEVEL EXPECTATION
Concepts and skills students master:
2. The coordinate plane allows us to apply algebraic understandings to Geometric concepts.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students Can:
a. Explore algebra and geometry relationships concerning
coordinate geometry
b. Understand the history behind coordinate geometry and
its usefulness today
c. Cartesian coordinate system (parts and notation)
i.
Label points on a graph
ii.
Find projection of a point to the x and y axis
d. Understand a three dimensional coordinate system
i.
Ordered triple, octants, graphing, projections, to
XY, YZ, XZ planes
e. Be able to discuss fallacies in graphing and improper
graphing techniques
f. Be able to graph inequalities and absolute value
relationships on a coordinate plane to parallel and no
parallel lines
g. Express and understand of slope; verbally, graphically
and computationally
h. Know the difference between positive, negative, zero,
and undefined slope
i. Prove the theorems for the slope of two parallel and
two perpendicular lines and apply in problems with
quadrilaterals and triangles.
j. Prove the distance and midpoint formula and apply
them in two and three dimensional problems.
k. Prove previously proven theorems in geometry by
coordinate geometry using slope, distance and midpoint
formulas
l. Describe a line by an equation
i.
Be familiar with slope intercept form, point slope
form, and standard form for a linear equation
ii.
Write the equation of a line given:
1. Slope, Y-intercept
Inquiry Questions:
1. How can the 2 and 3-dimensional coordinate planes be used to
systematize applications of Geometric properties?
Relevance and Application:
1. Visualization and the use of coordinate Geometry is used in
professions such as architecture, robotics, animation, film and
computer graphics, navigation, manufacturing, engineering,
urban planning, interior design, construction management and
military design.
Nature of the Discipline:
Standards for Mathematical Practice.
1. Reason abstractly and quantitatively.
2. Construct viable arguments and critique the reasoning of
others.
3. Model with mathematics.
4. Use appropriate tools strategically.
5. Attend to precision.
6. Look for and make use of structure.
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* Indicates a part of the standard connected to the mathematical
practice of Modeling.
BVSD Curriculum Essentials
14
m.
n.
o.
p.
q.
r.
s.
2. Slope, A point
3. 2 points
4. ║or ┴ another line through a given point
Know how to graph using slope and intercept the
intercept method or a table of values
Demonstrate graphing 3 variable equations
Equation of circles in the coordinate plane
Be able to write the equation of a circle at the origin
given the radius and center (and vice versa)
Be able to write the equation of a circle not at the origin
given the center and radius (vice versa)
Be able to change an equation of a circle in vertex form
to standard form (vice versa)
Identify whether the graph of the equation
X2+Y2+AX+BY+C=0 is a circle a point or the empty set.
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BVSD Curriculum Essentials
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3. Data Analysis, Statistics, and Probability
Data and probability sense provides students with tools to understand information and uncertainty. Students ask
questions and gather and use data to answer them. Students use a variety of data analysis and statistics
strategies to analyze, develop and evaluate inferences based on data. Probability provides the foundation for
collecting, describing, and interpreting data.
Prepared Graduates
The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students
who complete the Colorado education system must master to ensure their success in a postsecondary and
workforce setting.
Prepared Graduate Competencies in the 3. Data Analysis, Statistics, and Probability Standard are:
6/21/2012

Recognize and make sense of the many ways that variability, chance, and randomness appear in a
variety of contexts

Solve problems and make decisions that depend on understanding, explaining, and quantifying the
variability in data

Communicate effective logical arguments using mathematical justification and proof. Mathematical
argumentation involves making and testing conjectures, drawing valid conclusions, and justifying
thinking

Use critical thinking to recognize problematic aspects of situations, create mathematical models, and
present and defend solutions
BVSD Curriculum Essentials
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Content Area: Mathematics - Pre-IB Geometry
Standard: 3. Data Analysis, Statistics, and Probability
Prepared Graduates:
Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts.
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
1. Probability models outcomes for situations in which there is inherent randomness.
21st Century Skills and Readiness Competencies
Evidence Outcomes
Students can:
Inquiry Questions:
a. Understand independence and conditional probability and use them
1. Can probability be used to model all types of uncertain
to interpret data. (CCSS: S-CP)
situations? For example, can the probability that the 50th
i.
Describe events as subsets of a sample space5 using
president of the United States will be female be
characteristics (or categories) of the outcomes, or as
determined?
unions, intersections, or complements of other events.6
2. How and why are simulations used to determine
(CCSS: S-CP.1)
probability when the theoretical probability is unknown?
ii.
Explain that two events A and B are independent if the
3. How does probability relate to obtaining insurance? (PFL)
probability of A and B occurring together is the product of
Relevance and Application:
their probabilities, and use this characterization to
1. Comprehension of probability allows informed decisiondetermine if they are independent. (CCSS: S-CP.2)
making, such as whether the cost of insurance is less
iii.
Using the conditional probability of A given B as P(A and
than the expected cost of illness, when the deductible on
B)/P(B), interpret the independence of A and B as saying
car insurance is optimal, whether gambling pays in the
that the conditional probability of A given B is the same as
long run, or whether an extended warranty justifies the
the probability of A, and the conditional probability of B
cost. (PFL)
given A is the same as the probability of B. (CCSS: S-CP.3)
2. Probability is used in a wide variety of disciplines including
iv.
Construct and interpret two-way frequency tables of data
physics, biology, engineering, finance, and law. For
when two categories are associated with each object being
example, employment discrimination cases often present
classified. Use the two-way table as a sample space to
probability calculations to support a claim.
decide if events are independent and to approximate
Nature of the Discipline:
conditional probabilities.7 (CCSS: S-CP.4)
1. Some work in mathematics is much like a game.
v.
Recognize and explain the concepts of conditional
Mathematicians choose an interesting set of rules and
probability and independence in everyday language and
then play according to those rules to see what can
everyday situations.8 (CCSS: S-CP.5)
happen.
b. Use the rules of probability to compute probabilities of compound
2. Mathematicians explore randomness and chance through
events in a uniform probability model. (CCSS: S-CP)
probability.
i.
Find the conditional probability of A given B as the fraction
3. Mathematicians construct viable arguments and critique
of B’s outcomes that also belong to A, and interpret the
the reasoning of others. (MP)
answer in terms of the model. (CCSS: S-CP.6)
4. Mathematicians model with mathematics. (MP)
ii.
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and
5
B), and interpret the answer in terms of the model. (CCSS:
The set of outcomes. (CCSS: S-CP.1)
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6
S-CP.7)
"Or," "and," "Not". (CCSS: S-CP.1)
7
For example, collect data from a random sample of students in
8
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your school on their favorite subject among math, science, and
English. Estimate the probability that a randomly selected
student from your school will favor science given that the
student is in tenth grade. Do the same for other subjects and
compare the results. (CCSS: S-CP.4)
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. (CCSS: S-CP.5)
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4. Shape, Dimension, and Geometric Relationships
Geometric sense allows students to comprehend space and shape. Students analyze the characteristics and
relationships of shapes and structures, engage in logical reasoning, and use tools and techniques to determine
measurement. Students learn that geometry and measurement are useful in representing and solving problems
in the real world as well as in mathematics.
Prepared Graduates
The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all
students who complete the Colorado education system must master to ensure their success in a postsecondary
and workforce setting.
Prepared Graduate Competencies in the 4. Shape, Dimension, and Geometric
Relationships standard are:
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
Understand quantity through estimation, precision, order of magnitude, and comparison.
The reasonableness of answers relies on the ability to judge appropriateness, compare,
estimate, and analyze error

Make sound predictions and generalizations based on patterns and relationships that arise
from numbers, shapes, symbols, and data

Apply transformation to numbers, shapes, functional representations, and data

Make claims about relationships among numbers, shapes, symbols, and data and defend
those claims by relying on the properties that are the structure of mathematics

Use critical thinking to recognize problematic aspects of situations, create mathematical
models, and present and defend solutions
BVSD Curriculum Essentials
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Content Area: Mathematics - Pre-IB Geometry
Standard: 4. Shape, Dimension, and Geometric Relationships
Prepared Graduates:
Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are
the structure of mathematics
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
1. The use of Geometric definitions and deductive logic to form the foundation for expanding understanding to applied
Geometry.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
Inquiry Questions:
Introduction to Geometry
1. Does the postulate system of Geometry lead to more or less
a. State precise definitions of angle, circle, perpendicular
uniformity of thought?
line, parallel line, and line segment, based on the
2. Why do we need to know the formal definitions, properties,
undefined notions of point, line, distance along a line,
postulates and theorems to be able to apply logic to
and distance around a circular arc. (CCSS: G-CO.1)
Geometry?
i.
A Deductive System of Reasoning
3. How can mathematical concepts be ―undefined‖? What does
ii.
Learn the 3 undefined terms of geometry- point,
this mean for our understanding of other concepts that
line, plane
depend on the undefined?
iii.
Definitions- be able to write accurate definitions
4. When is deductive reasoning a more appropriate tool than
and recognize faulty definitions
inductive reasoning?
iv.
Distinguish between postulates and theorems
Relevance and Application:
v.
Beginning Postulates in Geometry
1. The understanding of foundational definitions and properties
vi.
Distance postulate
allows for the development of strategic problem-solving skills.
vii.
Ruler postulate
Nature of the Discipline:
viii.
Line postulate
b. Betweeness Property (segment addition)
c. Formal Definitions
i.
Segment, length, endpoint
ii.
Ray, opposite ray, midpoint, bisector
d. Point Plotting Theorem
e. Midpoint Theorem
i.
Lines, Planes, Separation and Space
f. Describe how Points, Lines and Planes are Related (2
Dimensional and 3 Dimensional)
i.
Know the definition of space
ii.
Be able to draw 3 dimensional figures (one point
and two point perspective)
iii.
Line postulate
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Plane-space postulate
v.
Discuss the intersection of 2 lines and line with a
plane
vi.
vii.
viii.
g.
h.
i.
j.
k.
l.
Plane postulate
Intersection of 2 planes postulate
Determination of a plane
1. 3 noncolinear points
2. line and point not on the line
3. two intersecting lines
Describe How Planes and Space are Separated by Lines
and Planes
i.
Definition convex set
ii.
Plane separation postulate
iii.
Space separation postulate
iv.
Angles and Triangles
Define Basic Terms using Correct Notation
i.
Angle
ii.
Triangle
iii.
Interior
iv.
Exterior
v.
Perimeter
Use Postulates for Measuring Angles
i.
Angle measurement postulate
ii.
Angle addition postulate
iii.
Angle construction postulate
iv.
Supplement postulate
Discuss the Relationship Between Angles and How They
are Related
i.
Right angles
ii.
Obtuse angles
iii.
Acute angles
iv.
Perpendicular angles
v.
Vertical angles
vi.
Adjacent angles
vii.
Linear pair
viii.
Supplementary angles
ix.
Complimentary angles
Know the Equivalence Relation for Angles
i.
Symmetric property
ii.
Reflexive property
iii.
Transitive property
Use Theorems Concerning Angles in problems
i.
Supplement theorem
ii.
Complement theorem
iii.
Vertical angles theorem
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7
8
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.
(CCSS: G-CO.12)
Compass and straightedge, string, reflective devices, paper
folding, dynamic geometric software, etc. (CCSS: G-CO.12)
BVSD Curriculum Essentials
21
m. Proof- Using Angles
i.
Hypothesis- conclusion
ii.
Acceptable forms
n. Acceptable reasons used in proofs
o. Make geometric constructions. (CCSS: G-CO)
p. Make formal geometric constructions7 with a variety of tools
and methods.8 (CCSS: G-CO.12)
q. Construct an equilateral triangle, a square, and a regular
hexagon inscribed in a circle. (CCSS: G-CO.13)
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Content Area: Mathematics - Pre-IB Geometry
Standard: 4. Shape, Dimension, and Geometric Relationships
Prepared Graduates:
Apply transformation to numbers, shapes, functional representations, and data.
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
2. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
Transformations
a. Experiment with transformations in the plane. (CCSS: G-CO)
b. State 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. (CCSS: G-CO.1)
c. Represent transformations in the plane using1 appropriate
tools. (CCSS: G-CO.2)
d. Describe transformations as functions that take points in the
plane as inputs and give other points as outputs. (CCSS: GCO.2)
e. Compare transformations that preserve distance and angle to
those that do not.2 (CCSS: G-CO.2)
f. Given a rectangle, parallelogram, trapezoid, or regular
polygon, describe the rotations and reflections that carry it
onto itself. (CCSS: G-CO.3)
g. Develop definitions of rotations, reflections, and translations in
terms of angles, circles, perpendicular lines, parallel lines, and
line segments. (CCSS: G-CO.4)
h. Given a geometric figure and a rotation, reflection, or
translation, draw the transformed figure using appropriate
tools.3 (CCSS: G-CO.5)
i. Specify a sequence of transformations that will carry a given
figure onto another. (CCSS: G-CO.5)
Congruence
j. Understand congruence in terms of rigid motions. (CCSS: GCO)
k. Use geometric descriptions of rigid motions to transform
figures and to predict the effect of a given rigid motion on a
given figure. (CCSS: G-CO.6)
l. Given two figures, use the definition of congruence in terms of
rigid motions to decide if they are congruent. (CCSS: G-CO.6)
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Inquiry Questions:
1. What happens to the coordinates of the vertices of shapes
when different transformations are applied in the plane?
2. How would the idea of congruency be used outside of
mathematics?
3. What does it mean for two things to be the same? Are there
different degrees of ―sameness?‖
4. What makes a good definition of a shape?
5. What conditions create unique polygons?
6. What does it mean for two lines to be parallel?
7. How can slope and distance be used to create and
investigate the relationship of the triangles created by
joining the mid-segments of a triangle?
Relevance and Application:
1. Comprehension of transformations aids with innovation and
creation in the areas of computer graphics and animation.
2. Knowledge of right triangle trigonometry allows modeling and
application of angle and distance relationships such as
surveying land boundaries, shadow problems, angles in a
truss, and the design of structures.
3. Points of concurrency are used in fields such as
architecture, engineering and physics.
Nature of the Discipline:
1. Geometry involves the investigation of invariants. Geometers
examine how some things stay the same while other parts
change to analyze situations and solve problems.
2. Mathematicians construct viable arguments and critique the
reasoning of others. (MP)
3. Mathematicians attend to precision. (MP)
4. Mathematicians look for and make use of structure. (MP)
5. Geometry involves the investigation of invariants. Geometers
examine how some things stay the same while other parts
BVSD Curriculum Essentials
23
m.
Use Congruence Postulates for Triangles in Proofs
i.Understand one to one correspondence
ii.Know the definition of congruent triangles, segments
and angles
iii.Definition- included side, included angle
iv.Equivalence relation- triangles
v.Work with the SAS, ASA, and SSS postulates in
triangle proofs
vi.
Use the Angle Bisector Theorem in Proofs
vii.
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. (CCSS: G-CO.7)
viii.
Explain how the criteria for triangle congruence (ASA,
SAS, and SSS) follow from the definition of congruence
in terms of rigid motions. (CCSS: G-CO.8)
n. Name the parts of Isosceles and Equilateral Triangles
o. Use Quadrilaterals in Proofs
ix.Rectangle, squares
p. Use Medians in Proofs
x.
Know definition- median
q. Prove geometric theorems. (CCSS: G-CO)
r. Prove theorems about lines and angles.4 (CCSS: G-CO.9)
s. Prove theorems about triangles.5 (CCSS: G-CO.10)
i. Use the Isosceles Triangle Theorem and its
Converse in Proofs
ii. Use the Properties of Equilateral and Equiangular
Triangle in Proofs
1. Classify triangles by sides and angles
2. Understand the nature of corollaries
3. Work with overlapping triangles in proofs
t. Prove theorems about parallelograms (rectangles,
rhombuses, squares).6 (CCSS: G-CO.11)
u. Use circle properties (involving chords and angles) to
prove theorems about triangles.
v. Points of concurrency
w. Segments of irrational length (e.g. using equilateral
triangles and squares)
x. Use specific properties of quadrilaterals and triangles,
to construct each. Examples: Construct a parallelogram
given lengths of diagonals. Given lengths of two sides
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change to analyze situations and solve problems.
6. Mathematicians make sense of problems and persevere in
solving them. (MP)
7. Mathematicians construct viable arguments and critique the
reasoning of others. (MP)
*Indicates a part of the standard connected to the mathematical
practice of Modeling
1 e.g., Transparencies and geometry software. (CCSS: G-CO.2)
2 e.g., Translation versus horizontal stretch. (CCSS: G-CO.2)
3 e.g., Graph paper, tracing paper, or geometry software. (CCSS: GCO.5)
4 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. (CCSS: G-CO.9)
5 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. (CCSS:G-CO.10)
6 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. (CCSS: G-CO.11)
11 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 the origin and
containing the point (0, 2). (CCSS: G-GPE.4)
12 e.g., Find the equation of a line parallel or perpendicular to a
given line that passes through a given point. (CCSS: G-GPE.5)
BVSD Curriculum Essentials
24
and a non-included angle, construct two triangles.
i.
Determine uniqueness.
Coordinate Geometry
a. Express Geometric Properties with Equations. (CCSS: G-GPE)
b. Translate between the geometric description and the equation
for a conic section. (CCSS: G-GPE)
i.
Derive the equation of a circle of given center and
radius using the Pythagorean Theorem. (CCSS: GGPE.1)
ii.
Complete the square to find the center and radius of a
circle given by an equation. (CCSS: G-GPE.1)
iii.
Derive the equation of a parabola given a focus and
directrix. (CCSS: G-GPE.2)
c. Use coordinates to prove simple geometric theorems
algebraically. (CCSS: G-GPE)
d. Use coordinates to prove simple geometric theorems11
algebraically. (CCSS: G-GPE.4)
e. Prove the slope criteria for parallel and perpendicular lines and
use them to solve geometric problems.12 (CCSS: G-GPE.5)
f. Find the point on a directed line segment between two given
points that partitions the segment in a given ratio. (CCSS: GGPE.6)
g. Use coordinates and the distance formula to compute
perimeters of polygons and areas of triangles and rectangles.★
(CCSS: G-GPE.7) Extend to solve problems with similar
figures (e.g. triangles created by connecting midpoints
of sides of a triangle)
h. Use distance and slope to further investigate and
informally prove properties of points of concurrency in
triangles (orthocenter, incenter, circumcenter, centroid)
Geometric Inequalities
a. Parts Theorem
i.Know the definition of less than for angles and
segments
b. Use the Exterior Angle Theorem in Inequality Proofs
i.Know definition of remote interior and exterior
angles
c. Be able to use AAS and HL Congruence Postulates in
Proof
d. Work With Inequalities in a Single Triangle
i.Larger angle opposite longest side and converse
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BVSD Curriculum Essentials
25
e. Use the First Minimum Theorem in Proofs
i.Know the definition of distance between a line and
an external point
f. Understand the Triangle Inequality Theorem in
Problems and Planes
g. Work With the Hinge Theorem and its Converse in
Proofs
h. Know the definition of an altitude
i.Use of altitudes in proofs
Lines and Planes in Space
a. Perpendicular lines and planes in space
i.
Learn definition – line and plane perpendicular in
space
ii.
Understand the basic theorem on perpendiculars
and its corollary
iii.
Using theorems in Unit 6, be able to discuss the
relationships between intersecting and parallel
lines to a plane in space
iv.
Learn the second minimum theorem and the
definition of distance from a point to a plane
b. Parallel lines and planes in space
i.
Know the definition of Parallel planes
ii.
Understand the relationship between parallel
planes in space, intersecting other lines or planes
iii.
Know the definition , parts and notation for a
dihedral angle
iv.
Discuss the projection of a point or line onto a
plane
c. Parallel Lines in a Plane
i. Understand the facts about parallel lines that are
not dependent on the parallel postulate
ii. Know the definition of parallel and skew line
iii. Be able to name the four ways to determine a plane
(emphasize 2 lines perpendicular to a third line)
iv. Discuss the existence and uniqueness of line
through a point parallel to another line
v. Identify transversals and angles created by 2 lines
and a transversal
vi. If 3 parallel lines intercept congruent segments on
one transversal; they intersect congruent segments on
any other transversal.
vii. Know conditions which guarantee two parallel lines
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BVSD Curriculum Essentials
26
and the proofs of those conditions.
1. AIP Theorem
2. CAP Theorem
3. Same side Interior angles
supplementary
4. 2 lines ┴ to 3rd line (in a plane only)
viii. Be able to solve problems finding angle measures
using the above theorems
ix. Understand the parallel postulate and the facts about
parallel lines that are dependent upon the parallel
postulate
x. Given the parallel postulate know the conditions and
the proofs of those conditions that allow you to find
angle relationships and measures
1. PAI Theorem
2. PCA Theorem
3. ║ Lines → same side interior angles
supplementary
4. Lines ┴ to one of two parallel lines is ┴ to
the other
xi. Prove triangle relationships that use the parallel
postulate and be able to use these relationships to solve
angle measurements
1. Sum of interior angles of a triangle equals 180◦
2. Two angles of a triangle congruent to
corresponding parts of another →triangle 3rd
angles congruent
3. Acute angles of a right triangle are
complementary
4. Exterior angle theorem (equality)
xii. Using the knowledge of Parallel lines learn some
fact about quadrilaterals and triangles
xiii. Know the definition of a quadrilateral, diagonal,
opposite, and consecutive sides and angles.
1. Know the definition of a;
a. Parallelogram
b. Trapezoid
c. Rectangle
d. Rhombus
e. Square
f. Kite
xiv. Name the six properties of a parallelogram and be
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BVSD Curriculum Essentials
27
able to prove the ones which are theorems
(5!)
Proofs with Quadrilaterals
a. Given a quadrilateral, name the four ways to prove the
quadrilateral is parallelogram.
b. Know the midline theorem, its proof and be able to
solve segment length and angle measure problems
using the theorem.
c. Know all the properties of rectangles rhombus squares
trapezoids and kites using the family “tree” of
quadrilaterals and be able to justify them by proof.
d. Use the knowledge of quadrilaterals to prove facts and
right triangles and medians of triangles and trapezoids
e. Median to the hypotenuse is half as long as the
hypotenuse.
f. If the acute of a right triangle has a measure of 3o
(degrees), the opposite side is half as long as the
hypotenuse.
g. Medians of a triangle are concurrent. Point of
intersection is 2/3 the way from the vertex along the
median.
h. Median of a trapezoid is parallel to the bases and the
length is one half the sums of the bases and the length
is one half the sums of the bases
i. Line bisects one side of a triangle and is parallel to a
second side, and then it bisects the third side
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BVSD Curriculum Essentials
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Content Area: Mathematics - Pre-IB Geometry
Standard: 4. Shape, Dimension, and Geometric Relationships
Prepared Graduates:
Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions.
GRADE LEVEL EXPECTATION
Concepts and skills students master:
3. Concepts of similarity are foundational to geometry and its applications.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
Inquiry Questions:
a. Understand similarity in terms of similarity transformations.
1. What happens to the coordinates of the vertices of shapes
(CCSS: G-SRT)
when different transformations are applied in the plane?
i. Verify experimentally the properties of dilations given by a
2. How would the idea of congruency be used outside of
center and a scale factor. (CCSS: G-SRT.1)
mathematics?
1. Show that a dilation takes a line not passing through the
3. What does it mean for two things to be the same? Are there
center of the dilation to a parallel line, and leaves a line
different degrees of ―sameness?‖
passing through the center unchanged. (CCSS: G-SRT.1a)
Relevance and Application:
2. Show that the dilation of a line segment is longer or
1. Comprehension of transformations aids with innovation and
shorter in the ratio given by the scale factor. (CCSS: Gcreation in the areas of computer graphics and animation.
SRT.1b)
2. Special triangles are commonly used in engineering,
ii. Given two figures, use the definition of similarity in terms of
surveying and physics.
similarity transformations to decide if they are similar. (CCSS: Nature of the Discipline:
G-SRT.2)
1. Geometry involves the investigation of invariants. Geometers
iii. Explain using similarity transformations the meaning of
examine how some things stay the same while other parts
similarity for triangles as the equality of all corresponding
change to analyze situations and solve problems.
pairs of angles and the proportionality of all corresponding
2. Mathematicians construct viable arguments and critique the
pairs of sides. (CCSS: G-SRT.2)
reasoning of others. (MP)
a. Definition of ratio and proportion
3. Mathematicians attend to precision. (MP)
b. Properties for ratio
4. Mathematicians look for and make use of structure. (MP)
c. Similar units (exception-rates)
d. Notation (fraction, semicolon)
*Indicates a part of the standard connected to the mathematical
b. Properties for proportions
practice of Modeling
1. Cross product
2. Inverting
3. Numerators over denominator
9 Theorems include: a line parallel to one side of a triangle divides the
4. Adding, subtracting numerators to
other two proportionally, and conversely; the Pythagorean Theorem
denominators (vice versa)
proved using triangle similarity. (CCSS: G-SRT.4)
iv. Use the properties of similarity transformations to establish
10Include the relationship between central, inscribed, and
the AA criterion for two triangles to be similar. (CCSS: Gcircumscribed angles; inscribed angles on a diameter are right
SRT.3)
angles; the radius of
b. Prove theorems involving similarity. (CCSS: G-SRT)
a circle is perpendicular to the tangent where the radius intersects
c. Prove theorems about triangles.9 (CCSS: G-SRT.4)
the circle. (CCSS: G-C.2)
d. Prove that all circles are similar. (CCSS: G-C.1)
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e. Use congruence and similarity criteria for triangles to solve
problems and to prove relationships in geometric figures.
(CCSS: G-SRT.5)
f. Use similarity criteria to discover Pythagorean triplet
patterns.
g. Review the algebraic properties for the geometric mean
h. Definition of geometric mean, arithmetic mean
i. Similarity for triangles
a. Definition similarity for triangles
b. Find lengths of sides and angle measurements in
triangles using the definition of similar triangles
Proofs with Similarity
a. Proportionality theorem (in a triangle and parallel lines)
b. Prove and apply the “basic proportionality” theorem
and its converse triangle
c. Prove and apply the “angle bisector” proportional
theorem to a triangle
d. Prove and apply the “parallels proportional segment”
theorem for 3 or more parallel lines
e. Methods to prove triangles similar
f. AAA, similarity theorem and AA corollary
g. Prove; apply proportion properties to solve various
problems involved segment length, angle measure,
ECT…
h. Line parallel to one size of a triangle intersects the
other two sides creating a triangle similar to the
original triangle
i. Similarity of triangles is and equivalence relation →
transit for triangle similarity.
j. SAS similarity and SSS similarity (prove and apply in
problems)
k. Similarity in Right Triangles
l. Altitude to the hypotenuse separates the triangle into
two triangles similar to each other and to the original
triangle
i.
Altitude to the hypotenuse is the geometric mean
between the segments it separates on the
hypotenuse.
ii.
leg of a right triangle is the geometric mean
between the hypotenuse and the segment of the
hypotenuse adjacent to the legs (projection!!)
m. Prove the Pythagorean Theorem using geometric mean
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n. Know the relationship between are, perimeter and
lengths of side, altitudes, and medians for similar
triangles and be able to solve problems
6/21/2012
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Content Area: Mathematics - Pre-IB Geometry
Standard: 4. Shape, Dimension, and Geometric Relationships
Prepared Graduates:
Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are
the structure of mathematics.
GRADE LEVEL EXPECTATION
Concepts and skills students master:
4. Logic and the study of reason provide the processes for students to formulate a strategic plan for problem-solving.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
Logic- Study of Reasoning
a. Identify Mathematical Sentences
i.Open
ii.Closed
iii.Truth value
b. Understand the Connectives used in Logic Statements
i.
And
ii.
Or
iii.
If-then
iv.
If and only if
c. Truth Value and Truth Tables
i.
Negations
ii.
Conjunctions
iii.
Disjunctions
iv.
Conditional
v.
BI- conditional
vi.
Compound statements
d. Identify Tautologies
i.
Conjunctive statements- disjunctive statements
e. Use the Law of Detachment (Modus Poens) in Logical
arguments
i.Hidden conditional
f. Use the Law of the Contrapositive in Logical arguments
i.Inverse, converse
ii.Logically equivalent statements
g. Use the Law of Modus Tollens in logical arguments
h. Recognize Invalid Arguments
i.
Conditional and converse
ii.
Conditional and inverse
i. Compare the Chain Rule used in Logic with the transitive
property in algebra
Inquiry Questions:
1. Why is the study of logic important for building a personal
life-long problem-solving schematic?
Relevance and Application:
1. The study of reason forms the foundation for problem-solving
in law, science, computer programming and research.
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Nature of the Discipline:
Standards for Mathematical Practice.
1. Reason abstractly and quantitatively.
2. Construct viable arguments and critique the reasoning of
others.
3. Look for and make use of structure.
4. Look for and express regularity in repeated reasoning.
BVSD Curriculum Essentials
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j.
k.
l.
m.
n.
o.
p.
q.
r.
s.
t.
u.
v.
Negations and DE Morgan’s Laws
i.
Double negation
ii.
Negation of a conjunction
iii.
Negation of a disjunction
Law of Conjunction
Law of Simplification
Law of Disjunctive Addition
Quantifiers Used in Logic Statements
i.Universal quantifier
ii.Existential quantifier
iii.Negation of universal and existential quantifier
Writing Logic Proofs
More Techniques in Proofs
Performance Standards and Objectives
Learn to Write Indirect Proofs
i.
Review the deductive system of reasoning
ii.
Understand the parts of an indirect proof
Statement to be proved
Assumption or supposition
Conclusion from assumption
Know contradictory fact to conclusion
i.
Work with existence and uniqueness proofs
ii.
Use the perpendicular bisector theorem and its
corollary in proofs
iii.
Understand auxiliary sets and their contribution in
proofs
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Content Area: Mathematics - Pre-IB Geometry
Standard: 4. Shape, Dimension, and Geometric Relationships
Prepared Graduates:
Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are
the structure of mathematics.
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
5. Attributes of two- and three-dimensional objects are measurable and can be quantified.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
a. Visualize relationships between two-dimensional and threedimensional objects. (CCSS: G-GMD)
b. Identify the shapes of two-dimensional cross-sections of
three-dimensional objects, and identify three-dimensional
objects generated by rotations of two-dimensional objects.
(CCSS: G-GMD.4)
Polygonal regions and their areas
a. Learn the postulates for areas polygonal regions
b. Learn the definition of triangular region, polygonal
region and identify polygonal regions
i.
Area postulates
ii.
Congruent postulate
iii.
Area addition postulate
iv.
Unit postulate (area of a square region)
c. Learn the formulas and be able to prove them for
rectangles, triangles, parallelograms, rhombuses, and
kites
d. Be able to solve area problems using the appropriate
the appropriate formulas
e. Be able to solve area ratio problems involving changes
in bases and height
f. Use ratios to solve problems within different dimensions
(e.g. given the ratio of the surface area of two similar
solids, find the ratio of their volumes and corresponding
lengths).
Circles and Spheres
a. Basic definitions
b. Explain volume formulas and use them to solve problems.
(CCSS: G-GMD)
a. Give an informal argument13 for the formulas for the
circumference of a circle, area of a circle, volume and
Inquiry Questions:
1. How might surface area and volume be used to explain
biological differences in animals?
2. How is the area of an irregular shape measured?
3. How can surface area be minimized while maximizing volume?
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Relevance and Application:
1. Understanding areas and volume enables design and building.
For example, a container that maximizes volume and
minimizes surface area will reduce costs and increase
efficiency. Understanding area helps to decorate a room, or
create a blueprint for a new building.
Nature of the Discipline:
1. Mathematicians use geometry to model the physical world.
Studying properties and relationships of geometric objects
provides insights in to the physical world that would otherwise
be hidden.
2. Mathematicians make sense of problems and persevere in
solving them. (MP)
3. Mathematicians construct viable arguments and critique the
reasoning of others. (MP)
4. Mathematicians model with mathematics. (MP)
*Indicates a part of the standard connected to the mathematical
practice of Modeling
13Use dissection arguments, Cavalieri’s principle, and informal limit
arguments. (CCSS: G-GMD.1
10
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. (CCSS: G-C.2)
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c.
d.
e.
f.
g.
h.
surface area of a cylinder, pyramid, sphere and cone.
(CCSS: G-GMD.1)
b. Use volume formulas for cylinders, pyramids, cones,
and spheres to solve problems.★ (CCSS: G-GMD.3)
c. Extend prior knowledge of surface area to
cylinder, cones, and spheres.
Understand and apply theorems about circles. (CCSS: G-C)
Identify and describe relationships among inscribed angles,
radii, and chords.10 (CCSS: G-C.2)
Construct the inscribed and circumscribed circles of a triangle.
(CCSS: G-C.3)
Prove properties of angles for a quadrilateral inscribed in a
circle. (CCSS: G-C.3)
Find arc lengths and areas of sectors of circles. (CCSS: G-C)
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. (CCSS: G-C.5)
Derive the formula for the area of a sector. (CCSS: G-C.5)
Derive the formula for the surface area of a cone.
i.
j.
k.
l. Circle, sphere, diameter, radius, chord, secant, tangent,
interior and exterior of a circle, great circle, externally
and internally tangent circles, the following theorems in
circles problems.
a. Line ┴ to the radius at its outer endpoint is
tangent to the circle
b. Every tangent to the circle is ┴ to the radius at
the point of contact
c. The perpendicular from the center of a circle to a
chord bisects the chord
i. The segment from the center of a circle to
the midpoint of a chord (not a diameter) is
┴ to the chord.
d. In the plane of a circle, the ┴ bisector of a chord
pasts through the center of the circle
e. In the same circle or congruent circles, congruent
chords are equal is that from the center
(converse is true)
f. Apply the theorems above to spheres and tangent
planes
m. Measurement of arcs and angles
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n. Learn definitions:
a. Minor arc
b. Major arc
c. Semicircle
d. Central angle
e. Inscribed angle
f. Tangent -secant angles
g. 2 Chord angles
h. 2 secant, secant -tangent, 2 tangent angles
o. Prove the theorems for the above angles and arcs and
apply in problems
p. Know the difference between an angle intercepting an
arc and being inscribed in an arc
q. Understand what it means to inscribe or circumscribe
polygons and circles
a. define secant segment, tangent segment
b. learn and prove the “power” theorems and apply
in problems to find the segment length
i. 2 Tangent theorem
ii. 2 Secant theorem
iii. Tangent- secant theorem
iv. 2 chord theorem
r. Learn common tangents (external and internal) and
apply in problems in circles.
Areas of circles and sectors (using polygons)
a. Learn the definition of a polygon and the angle sum
formulas
b. Know the names for polygons with 3 to 10 sides (most
commonly used!)
c. Learn the definition of convex polygon
d. Develop the formulas for the following;
i. Number of diagonals in a polygon with n
sides
ii. Sum of the interior angles of a polygon
with n sides
iii. Sum of exterior angles (one angle at each
vertex) of a polygon with n sides
e. Work with regular polygons finding areas, lengths of
sides, and measurement of angles
f. Definition of a regular polygon and the apothem
g. Learn the formula for the area of a regular polygon
h. Develop the formulas for circumference and area of a
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BVSD Curriculum Essentials
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circle
Prove these formulas using inscribed regular polygons
and the concept of a limit
j. Apply in problems;
i.
Find the circumference and area of inscribed and
circumscribed circles given regular polygons
(especially 3, 4, 6 sides)
ii.
Find the area of an annulus
iii.
Understand the relationship between area,
circumference, and radii in ratio problems
k. Develop the formulas for length of an area of a sector
and area of a segment
l. Define length of an arc and compare this definition with
the measure of an arc
m. Define sector and segment
n. Apply these formulas in problems (especially the
“continuous belt” program
Solids and their volumes
a. Study the properties of solid figures, base areas,
volumes and surface area’s
b. Learn the definition of a prism
i.
Know the terms right prism, base, altitude, and
cross section
i.
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Content Area: Mathematics - Pre-IB Geometry
Standard: 4. Shape, Dimension, and Geometric Relationships
Prepared Graduates:
Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions.
GRADE LEVEL EXPECTATION:
Concepts and skills students master:
6. Objects in the real world can be modeled using geometric concepts.
Evidence Outcomes
21st Century Skills and Readiness Competencies
Students can:
a. Apply geometric concepts in modeling situations. (CCSS: GMG)
b. Use geometric shapes, their measures, and their properties to
describe objects.14 ★ (CCSS: G-MG.1)
c. Apply concepts of density based on area and volume in
modeling situations.15 ★ (CCSS: G-MG.2)
d. Apply geometric methods to solve design problems.16 ★ (CCSS:
G-MG.3)
Right Triangle Trigonometry
a. Define trigonometric ratios and solve problems involving right
triangles. (CCSS: G-SRT)
b. Explain 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. (CCSS: G-SRT.6)
c. Explain and use the relationship between the sine and cosine
of complementary angles. (CCSS: G-SRT.7)
d. Use trigonometric ratios to discover the ratio properties
of special right triangles.
e. Use trigonometric ratios and the Pythagorean Theorem to
solve right triangles in applied problems.★ (CCSS: G-SRT.8)
f. Prove and apply trigonometric identities. (CCSS: F-TF)
g. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. (CCSS:
F-TF.8)
h. Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ)
given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
(CCSS: F-TF.8)
i. Prove and use the Pythagorean Theorem for right
triangles
j. Be able to give several different proofs of the
Pythagorean Theorem
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Inquiry Questions:
1. How are mathematical objects different from the physical
objects they model?
2. What makes a good geometric model of a physical object or
situation?
3. How are mathematical triangles different from built triangles
in the physical world? How are they the same?
Relevance and Application:
1. Geometry is used to create simplified models of complex
physical systems. Analyzing the model helps to understand the
system and is used for such applications as creating a floor
plan for a house, or creating a schematic diagram for an
electrical system.
Nature of the Discipline:
1. Mathematicians use geometry to model the physical world.
Studying properties and relationships of geometric objects
provides insights in to the physical world that would otherwise
be hidden.
2. Mathematicians make sense of problems and persevere in
solving them. (MP)
3. Mathematicians construct viable arguments and critique the
reasoning of others. (MP)
4. Mathematicians model with mathematics. (MP)
e.g., Modeling a tree trunk or a human torso as a cylinder. (CCSS:
G-MG.1)
15 e.g., Persons per square mile, BTUs per cubic foot. (CCSS: GMG.2)
16 e.g., Designing an object or structure to satisfy physical constraints
or minimize cost; working with typographic grid systems based on
ratios. (CCSS: G-MG.3)
14
BVSD Curriculum Essentials
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k. Solve problems using the Pythagorean Theorem
involving the length of sides, altitude, and area with
quadrilaterals and triangles.
l. Prove the relationship between the sides of a 30-60-90
right triangle and a 45-45-90 right triangle using the
Pythagorean Theorem.
m. Solve area problems using special right triangles
(including Isosceles and equilateral triangle problems)
n. Develop the basic trigonometric ratios
a. Sine, cosine, tangent ratios
b. Develop the above ratios for special right
triangles
o. Develop the ability to use a trigonometry table and a
calculator to evaluate trig ratios
p. Find sides and angles missing in triangle problems
q. Be able to solve angle elevation and depression
problems
r. (optional) understand the process of interpolation in
estimating trig values
s. Develop the ratios to the winding function (the unit
circle)
t. Distinguish between degree measure and radian
measure on the unit circle
u. Understand directed angles, terminal and initial sides
v. Use the definition of the sine, cosine, and tangent of
directed angles to find the value of various trigametric
functions of numbers.
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*Indicates a part of the standard connected to the mathematical
practice of Modeling
BVSD Curriculum Essentials
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Pre- IB Geometry Academic Vocabulary for Students
Standard 1: complement of a set, element of a set, intersection, null set, union, universal set, Venn
diagram
Standard 2: associative property, commutative property, distance formula, distributive property,
midpoint formula, parallel lines, perpendicular lines, Real numbers, Trichotomy property, undefined slope
Standard 3: bivariate data, box plot, compound events, dot plot, frequency table, first quartile,
Independently combined probability models, independent events, inter-quartile range, line plot, mean,
mean absolute variation, median, probability distribution, probability, probability model, sample space,
scatter plot, third quartile, uniform probability model
Standard 4: associative property, inverses, commutative property, Centroid, Circumcenter, congruent,
dilation, identity property of 0, Incenter, Median, Midsegment, multiplicative inverses, Kite, Orthocenter,
Points of Concurrency, properties of equality, properties of inequality, properties of operations, rectilinear
figure, Rhombus, rigid motion, similarity transformations, transitivity principle for indirect measurement,
theorem, transformations, trigonometric ratios, Tessellation, Unique
Reference Glossary for Teachers
* These are words available for your reference. Not all words below are listed above
because these are words that you will see above are for students to know and use while the
list below includes words that you, as the teacher, may see in the standards and materials.
Word
Additive inverses
Associative property
of addition
Associative property
of multiplication
Bivariate data
Box plot
Centroid
Circumcenter
Commutative property
Complement of a set
Compound events
Congruent
Dilation
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Definition
Two numbers whose sum is 0 are additive inverses of one another. Example:
3/4 and – 3/4 are additive inverses of one another because
3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.
The SUM is the same no matter what way you group the addends.
In general, the associative property of addition can be written as:
(a + b) + c = a + (b + c).
Notice that the PRODUCT is the same no matter what way you group
the factors. In general, the associative property of multiplication can
be written as: (a × b) × c = a × (b × c).
Pairs of linked numerical observations. Example: a list of heights
and weights for each player on a football team.
A method of visually displaying a distribution of data values by using
the median, quartiles, and extremes of the data set. A box shows the middle
50% of the data.1
That point where the medians of a triangle intersect.
That point where any two perpendicular bisectors of the sides of a polygon
inscribed in the circle intersect.
The Commutative Property of Addition states that changing the order of
addends does not change the sum, i.e. if a and b are two real numbers,
then a + b = b + a.
In set theory, a complement of a set A refers to things not in (that is, things outside
of), A.
A combination of multiple simple events, can be independent or dependent.
Two plane or solid figures are congruent if one can be obtained from
the other by rigid motion (a sequence of rotations, reflections, and
translations).
A transformation that moves each point along the ray through the
point emanating from a fixed center, and multiplies distances from the
center by a common scale factor.
BVSD Curriculum Essentials
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Distance Formula
The distance between (x1) and (x2) is the length of the line segment between them:
Distributive property
Multiplication of real numbers distributes over addition of real numbers. Ex: 2 × (1 + 3) =
(2 × 1) + (2 × 3).
A data display method which records frequency using a ―•‖ notation as
shown below.
Dot plot
First quartile
For a data set with median M, the first quartile is the median of the data
values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15,
22, 120}, the first quartile is 6.2 See also: median, third quartile, interquartile range.
Frequency table
A table that lists items and uses tally marks to show the number of times
they occur.
Identity property of addition states that the sum of zero and any number or
variable is the number or variable itself. For example, 4 + 0 = 4, - 11 + 0 =
- 11, y + 0 = y are few examples illustrating the identity property of
addition.
That point where the bisectors of the angles of a triangle or of a regular
polygon intersect. The point where the three angle bisectors of a
Identity property of 0
Incenter
triangle meet.
Independent events
Independently
combined probability
models
Inter-quartile Range
Intersection
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Two events, A and B, are independent if the fact that A occurs does not
affect the probability of B occurring.
Two probability models are said to be combined independently if the
probability of each ordered pair in the combined model equals the product of
the original probabilities of the two individual outcomes in the ordered pair.
A measure of variation in a set of numerical data, the inter-quartile range is
the distance between the first and third quartiles of the data set. Example:
For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the inter-quartile
range is 15 – 6 = 9. See also: first quartile, third quartile.
A set that contains elements shared by two or more given sets.
BVSD Curriculum Essentials
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Kite
A quadrilateral that has two distinct pairs of consecutive equilateral sides.
Line plot
A method of visually displaying a distribution of data values where
each data value is shown as a dot or mark above a number line. Also known
as a dot plot
A measure of center in a set of numerical data, computed by adding the
values in a list and then dividing by the number of values in the list.4
Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is
21.
A measure of variation in a set of numerical data, computed by adding the
distances between each data value and the mean, then dividing by the
number of data values. Example: For the data set {2, 3, 6, 7, 10,12, 14, 15,
22, 120}, the mean absolute deviation is 20.
A measure of center in a set of numerical data. The median of a list of
values is the value appearing at the center of a sorted version of the list—or
the mean of the two central values, if the list contains an even number of
values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the
median is 11.
The segment connecting the vertex of an angle in a triangle to the midpoint
of the side opposite it.
Mean
Mean absolute
deviation
Median (Statistical)
Median of a Triangle
Midpoint formula
Midpoint of a line segment is the point that is halfway between the endpoints of the
In two-dimensional coordinate plane, the midpoint
of a line with coordinates of its endpoints as (x1, y1) and
line segment.
(x2, y2) is given by
Midsegment
Multiplicative inverses
Null set
Orthocenter
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A line segment joining the midpoints of two sides of a triangle.
Two numbers whose product is 1 are multiplicative inverses of one another.
Example: 3/4 and 4/3 are multiplicative inverses of one
another because 3/4 × 4/3 = 4/3 × 3/4 = 1.
a set that is empty; a set with no members
The point where the three altitudes of a triangle intersect.
BVSD Curriculum Essentials
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Parallel lines
Parallel Lines are distinct lines lying in the same plane and they never
intersect each other. Parallel lines have the same slope.
In the figure below, lines PQ and RS are parallel and the lines l and m are
parallel.
Perpendicular lines
Perpendicular lines are lines that intersect at right angles. If two lines are
perpendicular to each other, then the product of their slopes is equal to – 1.
In the figure shown below, the lines AB and EF are perpendicular to each other.
Points of Concurrency
Probability
Probability
distribution
Probability model
Real number
Rectilinear figure
Rigid motion
Sample space
The place where three or more lines, rays, or segments intersect at the
same point. See point H in diagram above.
A number between 0 and 1 used to quantify likelihood for processes that
have uncertain outcomes (such as tossing a coin, selecting a person at
random from a group of people, tossing a ball at a target, or testing for a
medical condition).
The set of possible values of a random variable with a probability assigned
to each.
A probability model is used to assign probabilities to outcomes of a chance
process by examining the nature of the process. The set
of all outcomes is called the sample space, and their probabilities sum to 1.
See also: uniform probability model.
The real numbers are sometimes thought of as points on an infinitely long line
named number line or real line.
Rectilinear figures are figures bounded by straight lines.
A transformation of points in space consisting of a sequence of one or more
translations, reflections, and/or rotations. Rigid motions are here
assumed to preserve distances and angle measures.
In a probability model for a random process, a list of the individual outcomes
that are to be considered
Scatter plot
A graph of plotted points that show the relationship between two sets
of data (bivariate).
Similarity
transformation
A rigid motion followed by a dilation.
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BVSD Curriculum Essentials
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Tessellation
A covering of a plane with congruent copies of the same region with no holes
or overlaps.
Theorem
An important mathematical statement which can be proven by postulates,
definitions, and previously proved theorems.
For a data set with median M, the third quartile is the median of the data
values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15,
22, 120}, the third quartile is 15. See also: median, first quartile,
interquartile range.
Informally, moving a shape so that it is in a different position, but still has
the same size, area, angles and line lengths. Formally, a correspondence
between two sets of points such that each point in the pre-image has a
unique image and that each point in the image has exactly one pre-image.
Third quartile
Transformations
Transitivity principle
for indirect
measurement
Trichotomy property
Trigonometric ratios
Undefined slope
Uniform probability
model
Union
Unique
Universal Set
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If the length of object A is greater than the length of object B, and the
length of object B is greater than the length of object C, then the length of
object A is greater than the length of object C. This principle applies to
measurement of other quantities as well.
The property that for natural numbers a and b , either a is less
than b , a equals b , or a is greater than b .
A ratio that describes the relationship between a side and an angle of a
triangle. Sine, Cosine, Tangent
The "slope" of a vertical line. A vertical line has undefined slope because
all points on the line have the same x-coordinate. As a result
the formula
used for slope has a denominator of 0, which makes the
slope undefined.
A probability model which assigns equal probability to all outcomes. See
also: probability model
A set, every member of which is an element of one or another of two or more given
sets.
Limited to a single outcome or result; without alternative possibilities.
A set containing all elements of a problem under consideration.
BVSD Curriculum Essentials
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Venn Diagram
Diagrams that show all hypothetically possible logical relations between a
finite collection of sets (aggregation of things).
Definitions adapted from:
Boulder Valley School District Curriculum Essentials Document, 2009.
―Math Dictionary‖ www.icoachmath.com/math_dictionary/mathdictionarymain.html. Copyright © 1999
- 2011 HighPoints Learning Inc. December 30, 2011.
―The Mathematics Glossary.‖ Common Core Standards for Mathematics.
http://www.corestandards.org/the-standards/mathematics/glossary/glossary/ Copyright
2011. June 23, 2011.
―Thesaurus.‖ http://www.thefreedictionary.com/statistical+distribution . Copyright © 2012 Farlex, Inc.
January 5, 2012.
(n.d.). Retrieved from sas.com.
Statistics Glossary (n.d.). Retrieved from University of Glasgow:
http://www.stats.gla.ac.uk/steps/glossary/index.html.
―Illustrated Mathematics Dictionary.‖ http://www.mathsisfun.com/definitions/Copyright © 2011
MathsIsFun.com. January 14, 2012.
―Math Open Reference.‖ http://www.mathopenref.com/triangleincenter.html 2009 Copyright Math
Open Reference. January 15, 2012.
6/21/2012
BVSD Curriculum Essentials
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