Download Understand - Montezuma-Cortez School District

Curriculum Development Course at a Glance
Planning for High School Mathematics
Grade Level
Content Area
Mathematics
Course Name/Course Code
Geometry
Standard
Grade Level Expectations (GLE)
GLE Code
1.
Number Sense, Properties,
and Operations
1.
The complex number system includes real numbers and imaginary numbers
MA10-GR.HS-S.1-GLE.1
2.
Quantitative reasoning is used to make sense of quantities and their relationships in problem situations
MA10-GR.HS-S.1-GLE.2
Patterns, Functions, and
Algebraic Structures
1.
Functions model situations where one quantity determines another and can be represented algebraically,
graphically, and using tables
MA10-GR.HS-S.2-GLE.1
2.
Quantitative relationships in the real world can be modeled and solved using functions
MA10-GR.HS-S.2-GLE.2
3.
Expressions can be represented in multiple, equivalent forms
MA10-GR.HS-S.2-GLE.3
4.
Solutions to equations, inequalities and systems of equations are found using a variety of tools
MA10-GR.HS-S.2-GLE.4
1.
Visual displays and summary statistics condense the information in data sets into usable knowledge
MA10-GR.HS-S.3-GLE.1
2.
Statistical methods take variability into account supporting informed decisions making through
quantitative studies designed to answer specific questions
MA10-GR.HS-S.3-GLE.2
3.
Probability models outcomes for situations in which there is inherent randomness
MA10-GR.HS-S.3-GLE.3
1.
Objects in the plane can be transformed, and those transformations can be described and analyzed
mathematically
MA10-GR.HS-S.4-GLE.1
2.
Concepts of similarity are foundational to geometry and its applications
MA10-GR.HS-S.4-GLE.2
3.
Objects in the plane can be described and analyzed algebraically
MA10-GR.HS-S.4-GLE.3
4.
Attributes of two- and three-dimensional objects are measurable and can be quantified
MA10-GR.HS-S.4-GLE.4
5.
Objects in the real world can be modeled using geometric concepts
MA10-GR.HS-S.4-GLE.5
2.
3.
4.
Data Analysis, Statistics, and
Probability
Shape, Dimension, and
Geometric Relationships
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
High School
Page 1 of 22
Curriculum Development Course at a Glance
Planning for High School Mathematics
Colorado 21st Century Skills
Mathematical Practices:
Critical Thinking and Reasoning: Thinking
Deeply, Thinking Differently
Invention
Information Literacy: Untangling the Web
Collaboration: Working Together, Learning
Together
Self-Direction: Own Your Learning
Invention: Creating Solutions
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Unit Titles
Length of Unit/Contact Hours
Unit Number/Sequence
Introduction
3 weeks
1
Constructions
3 weeks
2
Triangle Properties
3 weeks
3
Polygon Properties
2 weeks
4
Circle Properties
2 weeks
5
Transformations
1 week
6
Area and Volume
2 weeks
7
Similarity
3 weeks
8
Pythagorean Theorem
3 weeks
9
Trigonometry
3 weeks
10
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 2 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Introduction
Focusing Lens(es)
Precision
Structure
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Expressing geometric properties with equations
Concepts
undefined terms (point, line, plane), definitions, distance, angle, coordinate plane, geometric relationships
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
3 weeks
MA10-GR.HS-S.4-GLE.1
MA10-GR.HS-S.4-GLE.5
Why should precise terms and definitions be used in mathematical discussions? Does this facilitate easier communication?
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
The notions of point, line, and plane create precision
definitions for geometric terms upon which concepts and
proofs are built. (MA10-GR.HS-S.4-GLE.1-EO.a.i)
How do we define geometric objects such as angle,
circle, line segment, parallel and perpendicular lines?
What makes a good definition of a shape? (MA10GR.HS-S.4-GLE.1-IQ.4)
Why is the use of correct symbol notation important
when labeling geometric figures such as angles,
lines, rays and line segments?
Why is it important to know and be able to give precise
definitions?
Why are point, line and plane considered undefined
terms?
Why are point, line and plane considered the building
blocks of geometry?
Geometric shapes, their measures, and their properties
can be used to describe objects. (MA10-GR.HS-S.4-GLE.5EO.a.i)
How are mathematical objects different from the
physical objects they model? (MA10-GR.HS-S.4GLE.5-IQ.1)
What makes a good geometric model of a physical
object or situation? (MA10-GR.HS-S.4-GLE.5-IQ.2)
Why is it important to be able to relate real-world
objects to mathematical descriptions?
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 3 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
around a circular arc. (MA10-GR.HS-S.4-GLE.1-EO.a.i); (CCSS: G-CO.1, supporting)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 1, sections 1-9 (pages 28-35, 38-87)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I can accurately used the undefined terms of point, line, and plane to define geometric shapes.
Academic Vocabulary:
angle, bisect, circle, definition, diagonal, diameter, distance, line, point, rectangle, square, triangle
Technical Vocabulary:
acute angle, arc, chord, collinear, complementary angles, congruent, consecutive parts, coplanar, equiangular, equilateral, isosceles triangle, kite, line
segment, linear pair of angles, major arc, midpoint, minor arc, obtuse angle, parallel lines, parallelogram, perimeter, perpendicular lines, point of
tangency, polygon, ray, rhombus, right angle, supplementary angles, tangent, trapezoid, undefined terms, vertical angles
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 4 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Constructions
Focusing Lens(es)
Precision
Relationships
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Congruence
Concepts
conjecture, geometric construction, rigid transformation
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
3 weeks
MA10-GR.HS-S.4-GLE.1
MA10-GR.HS-S.4-GLE.3
Why is it important to accurately construct a geometric shape rather than simply sketching or drawing it?
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
Formal geometric constructions are different than
sketches or drawings. (MA10-GR.HS-S.4-GLE.1-EO. d.i)
What is formal geometric construction?
How does a geometric construction differ from a
geometric drawing or sketch?
How does the construction of a perpendicular bisector
of a line segment help prove that all the points on
the bisector are equidistant from the endpoints of
the segment?
How do geometric constructions connect to terms and
definitions?
Why should geometric constructions be made with a
straight-edge rather than a ruler?
How can an altitude be constructed outside of a triangle?
There is a difference between an inscribed and
circumscribed circle of a triangle, and be able to describe
the difference. (MA10-GR.HS-S.4-GLE.2-EO. e.ii)
What does it mean for points to be concurrent?
What is the difference between inscribed and
circumscribed?
Why do only two lines need to be constructed to
determine the point of concurrency? What is the
benefit to constructing the third line?
How would a point of concurrency be used in a
construction project?
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 5 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…



Make formal geometric constructions with a variety of tools and methods. (MA10-GR.HS-S.4-GLE.1-EO.d.i); (CCSS: G-CO.12, supporting)
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle (MA10-GR.HS-S.4-GLE.1-EO.d.ii); (CCSS: G-CO.13, supporting)
Construct the inscribed and circumscribed circles of a triangle. . (MA10-GR.HS-S.4-GLE.2-EO.e.ii); (CCSS: G-C.3, additional)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 3, sections 1-7 (pages 149-166, 170-184)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I can construct congruent figures given the necessary line segment lengths and angle measurements.
Academic Vocabulary:
angle, arc, bisect, distance, intersect, line, parallel, perpendicular, point, side, square, triangle
Technical Vocabulary:
altitude, bisector, circumcenter, circumscribe, compass, concurrent, congruent, conjecture, geometric constructions, incenter, inscribe, line segment,
median, midpoint, midsegment, ray, straight edge
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 6 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Triangle Properties
Focusing Lens(es)
Precision
Relationships
Inquiry Questions
(EngagingDebatable):


Unit Strands
Geometry: Congruence
Concepts
congruence, corresponding angles, corresponding sides, proof
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
3 weeks
MA10-GR.HS-S.4-GLE.1
MA10-GR.HS-S.4-GLE.2
How would the idea of congruency be used outside of mathematics? (MA10-GR.HS-S.4-GLE.1-IQ.2)
What does it mean for two things to the same? Are there different degrees of sameness? (MA10-GR.HS-S.4-GLE.1-IQ.3)
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
Congruent triangles create six pairs of congruent
corresponding sides and angles. (MA10-GR.HS-S.4-GLE.1EO.b.iii)
What combinations of sides and angles are sufficient to
prove congruency of triangles?
Which combinations of congruent side and/or angle
pairs do not prove congruent triangles?
Why are triangles not congruent if all the angles are
congruent?
Why is SSA (AAS) not a congruency shortcut?
What is meant by corresponding angles and sides?
Why is three the fewest number of congruent sides
and/or angle pairs necessary to prove two triangles
congruent?
How are mathematical triangles different from built
triangles in the physical world? How are they the
same? (MA10-GR.HS-S.4-GLE.2-IQ.3)
Stated assumptions, definitions, and previously
established results help in the construction of proofs.
(MA10-GR.HS-S.4-GLE.1-EO.c)
How are assumptions and definitions used in proofs?
How can you prove relationships between angles
formed when transversal intersects parallel lines?
What is the relationship between vertical angles? A
linear pair of angles?
What are the different types of geometric proof?
Why are proofs an integral part of geometry?
How does writing a proof deepen your understanding of
geometric concepts?
Why is correct symbol notation for congruence necessary
for writing a proof?
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 7 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…







Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures. (MA10-GR.HS-S.4-GLE.1EO.b.i); (CCSS: G-CO.6, major)
Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (MA10-GR.HS-S.4-GLE.1-EO.b.ii); (CCSS: G-CO.6, major)
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. (MA10-GR.HS-S.4-GLE.1-EO.b.iii); (CCSS: G-CO.7, major)
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. ((MA10-GR.HS-S.4-GLE.1-EO.b.iv);
(CCSS: G-CO.8, major)
Prove theorems about lines and angles. (MA10-GR.HS-S.4-GLE.1-EO.c.i); (CCSS: G-CO.9, major)
Prove theorems about triangles. (MA10-GR.HS-S.4-GLE.1-EO.c.ii); (CCSS: G-CO.10, major)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (MA10-GR.HS-S.4-GLE2-EO.b.iii); (CCSS: G-SRT.5, major)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 4: sections 1-8 (pages 199-211, 215-248)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I can prove two triangles are congruent.
Academic Vocabulary:
alternate, angle, area, corresponding, exterior, height, interior, prove, side, triangle
Technical Vocabulary:
alternate angles, altitude, base angles, congruent, conjecture, consecutive parts, corresponding angles, corresponding parts, exterior angles,
hypotenuse, included, angle, included side, inequality, interior angles, isosceles triangle, linear pair of angles, midsegment, median, midpoint, proof,
right triangle, transversal, vertex, vertical angles
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 8 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Polygon Properties
Focusing Lens(es)
Interdependence
Inquiry Questions
(EngagingDebatable):


Unit Strands
Geometry: Congruence
Geometry: Expressing geometric properties with equations
Concepts
congruence, corresponding angles, corresponding sides
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
2 weeks
MA10-GR.HS-S.4-GLE.1
MA10-GR.HS-S.4-GLE.2
MA10-GR.HS-S.4-GLE.3
Why is it important to be able to classify polygons?
How are properties the same and different within different classifications? Why is there overlap?
Generalizations
My students will Understand that…
The coordinate plane models algebraically twodimensional geometric relationships. (MA10-GR.HS-S.4GLE.3-EO.a.ii)
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Guiding Questions
Factual
How can you determine the slope of line parallel or
perpendicular to a given line?
What does it mean for two lines to be parallel? (MA10GR.HS-S.4-GLE.3-IQ.1)
What information is needed to calculate the perimeters
of polygons and area of triangles and rectangles in
the coordinate plane?
What is the relationship of the slopes of parallel lines?
Of perpendicular lines?
How can any angle of a regular polygon be determined?
How are polygons classified?
Date Completed: _____________________________
Conceptual
Why is it helpful to model geometric relationships on the
coordinate plane?
How can two lines have the same slope yet not be
parallel?
Why is the product of the slopes of perpendicular lines
equal to -1?
Page 9 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…






Use coordinates to prove simple geometric theorems algebraically. (MA10-GR.HS-S.4-GLE.3-EO.a.ii.1); (CCSS: G-GPE.4, major)
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (MA10-GR.HS-S.4-GLE.3-EO.a.ii.2); (CCSS: G-GPE.5, major)
Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (MA10-GR.HS-S.4-GLE.3-EO.a.ii.3); (CSS: G-GPE.6, major)
Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles. (MA10-GR.HS-S.4-GLE.3-EO.a.ii.4); (CCSS: G-GPE.7,
major)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (MA10-GR.HS-S.4-GLE.2-EO.b.iii); (CCSS: G-SRT.5, major)
Prove theorems about parallelograms. (MA10-GR.HS-S.4-GLE.1-EO.c.iii); (CCSS: G-CO.11, major)
Resources



What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Algebra Skills 1 (pages 36-37)
Algebra Skills 2 (pages 135-136)
Chapter 5: sections 1-7, Algebra Skills 5 (pages 257-303)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I can use coordinates of the vertices of a quadrilateral to show it is a rectangle by calculating the slopes and lengths of
each side.
Academic Vocabulary:
angle, consecutive, diagonal, exterior, interior, height, parallel, rectangle, side, square, triangle
Technical Vocabulary:
altitude, base angles, bisector, concave, conjecture, convex, corresponding angle, corresponding side, equiangular, isosceles, exterior angle, isosceles,
kite, midpoint, midsegment, oblique polygon, parallel, parallelogram, perpendicular, polygon, polyhedron, regular polygon, rhombus, slope,
supplementary angles, trapezoid, vertex
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 10 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Circle Properties
Focusing Lens(es)
interdependence
Inquiry Questions
(EngagingDebatable):


Unit Strands
Geometry: Circles
Geometry: Expressing geometric properties with equations
Concepts
arc length, arc measure, center, central angle, chord, congruent, inscribed angle
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
2 weeks
MA10-GR.HS-S.4-GLE.2
Do perfect circles naturally occur in the physical world? If so, how do we model them? (MA10-GR.HS-S.4-GLE.2-IQ.4)
Why does modeling Earth as a sphere allow us to calculate measures such as diameter, circumference and surface area? (MA10-GR.HS-S.4-GLE.2-RA.1)
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
Arc length determines the interdependent relationship of
inscribed, circumscribed and central angles of a circle.
(MA10-GR.HS-S.4-GLE.2-EO.e)
What is the relationship between inscribed, central, and
circumscribed angles of a circle that subtend to the
same arc?
How does the measure of the central angle help you
find the area of the corresponding sector?
Why are inscribed, central, and circumscribed angles of a
circle independent with each other when they
subtend the same arc?
Why does the measure of a central angle equal the
measure of its intercepted arc?
The center and radius of the circle constrain the equation
by providing location and size. (MA10-GR.HS-S.4-GLE.3EO.a.i.1, 2)
What is equation of a circle?
Within the equation of the circle, where is the center
and the radius?
How does the Pythagorean Theorem define all points on
a circle with a given center and radius?
Why is the radius of a circle perpendicular to the tangent
where the radius intersects the circle?
The length of chords and their corresponding arcs vary
proportionally. (MA10-GR.HS-S.4-GLE.2-EO.f)
What is the longest chord in a circle and how do you
know?
How can you prove two tangent lines from a point
outside a circle are congruent?
Why does a radius that bisects an arc also bisect the
corresponding chord?
Why is the radius always perpendicular to the tangent
line at the point of tangency?
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 11 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…




Identify and describe relationships among inscribed angles, radii, and chords. (MA10-GR.HS-S.4-GLE.2-EO.e.i); (CCSS: G-C.2, additional)
Prove properties of angles for a quadrilateral inscribed in a circle. (MA10-GR.HS-S.4-GLE.2-EO.e.ii); (CCSS: G-C.3, additional)
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector. (MA10-GR.HS-S.4-GLE.2-EO.f.i); (CCSS: G-C.5, additional)
Derive the formula for the area of a sector. (MA10-GR.HS-S.4-GLE.2-EO.f.ii); (CCSS: G-C.5, additional)
Resources


What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 6: sections 1-7 (pages 309-344, 349-358)
Chapter 8: sections 5 & 6 (pages 449-457)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I know how the measure of a central angle of a circle relates to the measure of its corresponding inscribed angle.
I can determine the measure of an arc.
Academic Vocabulary:
angle, arc, center, circle, distance, hemisphere, parallel, perpendicular, segment, semicircle
Technical Vocabulary:
arc length, arc measure, bisector, central angle, chord, circumference, conjecture, cyclic quadrilateral, diameter, inscribed angle, intercepted arc,
major arc, minor arc, parallel lines, point of tangency, radius, sector, tangent
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 12 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Transformations
Focusing Lens(es)
Relationships
Transformations
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Congruence
Concepts
coordinate plane, direction, patterns, rigid transformation
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
1 week
MA10-GR.HS-S.4-GLE.1
MA10-GR.HS-S.4-GLE.3
What happens to the coordinates of the vertices of shapes when different transformations are applied in the plane? (MA10-GR.HS-S.4-GLE.3IQ.2)
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
A sequence of rigid transformation creates congruent
figures. (MA10-GR.HS-S.4-GLE.1-EO.b.i, ii)
How can you describe the sequence of transformation
that carry a geometric figure onto itself?
How can transformations be used to show to two
figures congruent without directly measure each
part of the figure?
How does the definition of congruence in terms of rigid
motion explain the criteria for triangle congruence?
Geometric transformations create functions that take
points in the plane as inputs and give unique
corresponding points as outputs. (MA10-GR.HS-S.4-GLE.1EO.a.iii)
What function operations work with transformations?
What happens to the coordinates of the vertices of
shapes when different transformations are applied
in the plane? (MA10-GR.HS-S.4-GLE.1-IQ.1)
Why are transformations functions?
How can rigid transformations be compared?
Rigid transformations preserve distance and angle.
(MA10-GR.HS-S.4-GLE.1-EO.a)
What do non-rigid transformations preserve?
How can I use transformations to prove to figures are
congruent?
What information is needed to correctly transform a
figure?
Why is it important that rigid transformations preserve
distance and angle?
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 13 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…







Represent transformations in the plane using appropriate tools. (MA10-GR.HS-S.4-GLE.1-EO.a.ii); (CCSS: G-CO.2, supporting)
Describe transformations as functions that take points in the plane as inputs and give other points as outputs. (MA10-GR.HS-S.4-GLE.1-EO.a.iii); (CCSS: G-CO.2, supporting)
Compare transformations that preserve distance and angle to those that do not. (MA10-GR.HS-S.4-GLE.1-EO.a.iv); (CCSS: G-CO.2, supporting)
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (MA10-GR.HS-S.4-GLE.1-EO.a.v); (CCSS: G-CO.3,
supporting)
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (MA10-GR.HS-S.4-GLE.1EO.a.vi); (CCSS: G-CO.4, supporting)
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools. (MA10-GR.HS-S.4-GLE.1-EO.a.vii); (CCSS: G-CO.5,
supporting)
Specify a sequence of transformations that will carry a given figure onto another. (MA10-GR.HS-S.4-GLE.1-EO.a.viii); (CCSS: G-CO.5, supporting)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 7: sections 1-4 (pages 368-393)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
Given a geometric figure and a rotation, reflection, or translation I can draw the transformed figure.
Academic Vocabulary:
angle, intersecting lines, line, mirror-image, parallel, slide, turn
Technical Vocabulary:
angle of rotation, center of rotation, conjecture, coordinate pair, line of reflections, parallel lines, point of rotation, pre-image, post-image, reflection,
rigid transformation, rotation, transformation, translation
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 14 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Area and Volume
Focusing Lens(es)
Modeling
Relationships
Structure
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Geometric measurement and dimension
Geometry: Modeling with geometry
Concepts
area, measurements, models, perimeter, volume
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
2 weeks
MA10-GR.HS-S.4-GLE.4
MA10-GR.HS-S.4-GLE.5
How might surface area and volume be used to explain biological differences in animals? (MA10-GR.HS-S.4-GLE.3-IQ.1)
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
Underlying and related structures of perimeter, area and
volume can reveal patterns within complex objects.
(MA10-GR.HS-S.4-GLE.4-EO.a, b)
How are 1-dimensional (perimeter), 2-dimensional
(area), or 3-dimensional (volume) measurements
different? How is that difference reflected in the unit
notation?
How does the relationship between the volumes of a
cone and its corresponding cylinder help us find the
volume of a pyramid?
How is the area of an irregular shape measured? (MA10GR.HS-S.4-GLE.4-IQ.2)
How can surface area be minimized while maximizing
volume? (MA10-GR.HS-S.4-GLE.4-IQ.3)
How can the relationship between area and volume be
explained through cross-sections and rotations?
Why is it important to be able to identify the base of a
pyramid or prism?
What is the relationship between a prism and a pyramid?
A cylinder and a cone?
Geometric models chosen and created with the use of
appropriate measurements deepen understandings of
empirical situations and improve decision-making. (MA10GR.HS-S.4-GLE.5-EO.a)
How can the geometric concepts of area and volume
model density?
Why does understanding areas and volumes enable
design and building? (MA10-GR.HS-S.4-GLE.5-RA.1)
Why are ratios an important component of geometric
modeling?
How does a container that maximizes volume and
minimizes surface area reduce costs and increase
efficiency? (MA10-GR.HS-S.4-GLE.5-RA.1)
How does knowing area help create a blueprint for a
building? (MA10-GR.HS-S.4-GLE.5-RA.1)
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 15 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…







Use geometric shapes, their measures, and their properties to describe objects. (MA10-GR.HS-S.4-GLE.5-EO.a.i); (CCSS: G-MG.1, major)
Apply concepts of density based on area and volume in modeling situations. (MA10-GR.HS-S.4-GLE.5-EO.a.ii); (CCSS-G-MG.2, major)
Apply geometric methods to solve design problems. (MA10-GR.HS-S.4-GLE.5-EO.a.iii); (CCSS: G-MG.3, major)
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. (MA10-GR.HS-S.4-GLE.4-EO.a.i);
(CCSS:G-GMD.1, additional)
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. (MA10-GR.HS-S.4-GLE.4-EO.a.ii); (CCSS:G-GMD.3, additional)
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional
objects. (MA10-GR.HS-S.4-GLE.4-EO.b.i); (CCSS:G-GMD.4, additional)
Use the distance formula on coordinates to compute perimeters of polygons and areas of triangles and rectangles. (MA10-GR.HS-S.4-GLE.3-EO.a.ii.4); (CCSS: G-GPE.7, major)
Resources


What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 8: sections 1-3 (pages 420-437)
Chapter 10: sections 1-6 (pages 519-561)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I can see how a human torso can be modeled as a cylinder for purposes of volume and surface area.
I understand that geometry is used to create simplified models of complex physical systems and that analyzing the
model helps to understand the system. (MA10-GR.HS-S.4-GLE.5-RA.1)
Academic Vocabulary:
area, circle, height, hemisphere, square, three-dimensional, triangle, two-dimensional, volume
Technical Vocabulary:
altitude, base, circumference, cone, conjecture, cylinder, density, diameter, edge, face, great circle, height, lateral, oblique, perimeter, prism,
pyramid, radius, regular polygon, polyhedron, right, side, sphere, trapezoid, vertex
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 16 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Similarity
Focusing Lens(es)
Similarity
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Similarity, right triangles, and trigonometry
Concepts
dilation, magnitude, proportionality, scale factor, transformation
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
3 weeks
MA10-GR.HS-S.4-GLE.2
What does it mean for two things to be the same? Are there different degrees of “sameness?” (MA10-GR.HS-S.4-GLE1-IQ.3)
Generalizations
My students will Understand that…
Guiding Questions
Factual
Conceptual
Dilations require a center from which the transformation
originates and a scale factor which describes magnitude
and direction. (MA10-GR.HS-S.4-GLE2-EO.a.i)
How does dilation prove all circles are similar?
Which properties are conserved during a rigid
transformation? Which are not?
Why do dilations create similar figures?
Why are angle measures preserved in dilation?
Mathematicians use similar triangles to prove
generalizable relationships. (MA10-GR.HS-S.4-GLE.2EO.b.i)
How can similar triangle be used to prove that a line
parallel to one side of a triangle divides the other
two proportionally?
How can you determine the measure of something that
you cannot measure physically? (MA10-GR.HS-S.4GLE.2-IQ.1)
Why are similar triangles the foundation for
mathematical proofs about side lengths of triangles?
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 17 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…









Verify experimentally the properties of dilations given by a center and a scale factor. (MA10-GR.HS-S.4-GLE2-EO.a.i); (CCSS: G-SRT.1, major)
Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (MA10-GR.HS-S.4GLE2-EO.a.i.1); (CCSS: G-SRT.1a, major)
Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. (MA10-GR.HS-S.4-GLE2-EO.a.i.2); (CCSS: G-SRT.1b, major)
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. (MA10-GR.HS-S.4-GLE2-EO.a.ii); (CCSS: G-SRT.2, major)
Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding
pairs of sides. (MA10-GR.HS-S.4-GLE2-EO.a. iii); (CCSS: G-SRT.2, major)
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (MA10-GR.HS-S.4-GLE2-EO.a. iv); (CCSS: G-SRT.3, major)
Prove theorems about similar triangles. (MA10-GR.HS-S.4-GLE2-EO.b.i); (CCSS:G-SRT.4, major)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (MA10-GR.HS-S.4-GLE2-EO.b.iii); (CCSS: G-SRT.5, major)
Prove that all circles are similar. (MA10-GR.HS-S.4-GLE2-EO.b.ii); (CCSS:G-C.1, additional)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 11: sections 1-7, Algebra Skills 11 (pages 577-630)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
The dilation is the only transformation that produces similar polygons because it stretches or shrinks line segments.
I can identify the lines of symmetry of an object and determine whether the figure has reflection symmetry,
rotational symmetry, or both.
Academic Vocabulary:
angle, corresponding, distance, height, line, side, triangle,
Technical Vocabulary:
conjecture, contraction, corresponding parts, dilation, expansion, indirect measurement, magnitude, non-rigid transformation, polygon, proportional,
ratio, rigid transformation, scale factor, similar
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 18 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Right Triangles
Focusing Lens(es)
Interdependence
Relationships
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Similarity, right triangles, and trigonometry
Geometry: Modeling with geometry
Concepts
right triangle, similar triangles, trigonometric functions
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
3 weeks
MA10-GR.HS-S.4-GLE.2
MA10-GR.HS-S.4-GLE.3
MA10-GR.HS-S.4-GLE.5
How does knowing right triangle relationships help in building construction? In what other fields would it also be useful?
Generalizations
My students will Understand that…
Mathematicians use similar triangles to prove
generalizable relationships. (MA10-GR.HS-S.4-GLE.2EO.b.i)
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Guiding Questions
Factual
What are the names for the sides of a right triangle?
Date Completed: _____________________________
Conceptual
How can you use right triangle similarity to prove the
Pythagorean Theorem?
Page 19 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…






Prove theorems about similar triangles. (MA10-GR.HS-S.4-GLE.1-EO.a.i); (CCSS: G-SRT.4, major)
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (MA10-GR.HS-S.4-GLE.1-EO.a.i); (CCSS: G-SRT.8, major)
Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles. (MA10-GR.HS-S.4-GLE.1-EO.a.i); (CCSS: G-GPE.7, major)
Use geometric shapes, their measures, and their properties to describe objects. (MA10-GR.HS-S.4-GLE.1-EO.a.i); (CCSS: G-MG.1, major)
Derive the equation of a circle of a given center and radius using the Pythagorean Theorem. (MA10-GR.HS-S.4-GLE.3-EO.a.i.1); (CCSS: G-GPE.1, additional)
Complete the square to find the center and radius of a circle given by an equation. (MA10-GR.HS-S.4-GLE.3-EO.a.i.2); (CCSS: G-GPE.1, additional)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 9: sections 1- 6 (pages 477-488, 491-511)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I can use the Pythagorean Theorem to calculate the length of the unknown side in a right triangle.
I can use the Pythagorean Theorem to determine if a triangle is a right triangle.
Academic Vocabulary:
angle, area, perpendicular, side, triangle
Technical Vocabulary:
hypotenuse, leg, Pythagorean Theorem, right angle/triangle
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 20 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Unit Title
Trigonometry
Focusing Lens(es)
Relationships
Inquiry Questions
(EngagingDebatable):

Unit Strands
Geometry: Similarity, right triangles, and trigonometry
Geometry: Modeling with geometry
Length of Unit
Standards and Grade
Level Expectations
Addressed in this Unit
2 weeks
MA10-GR.HS-S.4-GLE.2
How can you determine the measure of something that you cannot measure physically? (MA10-GR.HS-S.4-GLE.2-IQ.1)
Concepts
Generalizations
My students will Understand that…
The relationship between the side ratios and angles of a
right triangle define the trigonometric functions. (MA10GR.HS-S.4-GLE.2-EO.c)
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Guiding Questions
Factual
What are trigonometric ratios?
What is the relationship of the sine and cosine of
complementary angles?
How can you determine the measure of something that
you cannot measure physically? (MA10-GR.HS-S.4GLE.2-IQ.1)
Why is the does the adjacent side and opposite side
depend on which angle is being used in the
trigonometric function?
Date Completed: _____________________________
Conceptual
How does similarity explain that the side ratios in right
triangles are a function of the angles of the triangle?
How do we know that the sine of all 30 degree angles is
the same?
How does the knowledge of right triangle trigonometry
allow modeling and application of angle and distance
relationships such as surveying land boundaries,
shadow problems, angles in a truss, and the design of
structures? (MA10-GR.HS-S.4-GLE.3-RA.1)
Page 21 of 22
Curriculum Development Overview
Unit Planning for High School Mathematics
Key Knowledge and Skills:
My students will…



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. (MA10-GR.HSS.4-GLE.2-EO.c.i); (CCSS: G-SRT.6, major)
Explain and use the relationship between the sine and cosine of complementary angles. (MA10-GR.HS-S.4-GLE.2-EO.c.ii); (CCSS: G-SRT.7, major)
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (MA10-GR.HS-S.4-GLE.2-EO.c.iii); (CCSS: G-SRT.8, major)
Resources

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the
mathematics samples what students should know and do are combined.
Key Curriculum Press: Discovering Geometry – An Investigative Approach, 4th edition
Chapter 12: sections 1-3 (pages 639-660)
Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.
A student in Geometry can demonstrate the ability to
apply and comprehend critical language through the
following statement(s):
I know the sine and cosine of the acute angles in an isosceles right triangle are the same.
Academic Vocabulary:
angle, explain, prove, right triangle, side
Technical Vocabulary:
adjacent side, angle of depression, angle of elevation, complementary angles, cosine, hypotenuse, indirect measurement, leg, opposite side, ratio,
similar triangles, sine, tangent, trigonometric ratios
Authors of the Sample: Shelley Curtis (Montezuma-Cortez RE-1)
High School, Mathematics, Geometry
Date Completed: _____________________________
Page 22 of 22