Download MA30 Geometry Sequenced Units for Arizona’s College and Career Ready Standards

Sequenced Units for
Arizona’s College and Career Ready Standards
MA30 Geometry
Year at a Glance
Semester 1
Unit 1: Basics of Geometry (12 days)
Unit 2: Reasoning and Proofs (13 days)
Unit 3: Parallel and Perpendicular Lines (12 days)
Unit 4: Transformations (13 days)
Unit 5: Congruent Triangles (17 days)
Unit 6: Relationships Within Triangles (15 days)
Semester 2
Unit 7: Quadrilaterals and Other Polygons (13 days)
Unit 8: Similarity (11 days)
Unit 9: Right Triangles and Trigonometry (17 days)
Unit 10: Circles (18 days)
Unit 11: Circumference, Area, and Volume (19 days)
2015-2016
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Students began their study of geometric concepts in middle school mathematics. They studied area, surface area, and volume and informally
investigated lines, angles, and triangles. Students in middle school also explored transformations, including translations, reflections, rotations, and
dilations.
The Geometry course outlined in this document begins with developing the tools of geometry, including transformations, proof, and constructions.
These tools are used throughout the course as students formalize geometric concepts studied in earlier courses and extend those ideas to new
concepts presented in the high school standards. There is a focus on modeling, problem solving, and proof throughout the course.
This document reflects our current thinking related to the intent of Arizona’s College and Career Ready Standards for Mathematics and assumes 160
days for instruction, divided among 11 units. The number of days suggested for each unit assumes 45-minute class periods and is included to convey
how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the
mathematical content described in the standards; however, the order of the standards included in any unit does not imply a sequence of content
within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units,
depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and
comments are included throughout the document to clarify and provide additional background for each unit.
Throughout Geometry, students should continue to develop proficiency with Arizona’s College and Career Ready Standards’ eight Standards for
Mathematical Practice:
1.
2.
3.
4.
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.
5.
6.
7.
8.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be
understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. In a high
school geometry course, reasoning and developing viable arguments are particularly important, as are use of strategic tools and precision of
language. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be
interpreted to mean that other practices should be neglected in those units.
When using this document to help in planning your district's instructional program, you will also need to refer to the standards document, relevant
progressions documents for the standards, and the appropriate assessment consortium framework.
Mesa Public Schools
1
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 1: Basics of Geometry
Suggested number of days: 12
Students will be familiar with the basic geometric terms of point, line, segment, ray, and Plane from previous courses. They should also be familiar with how each is represented, meaning
the notation used to identify these geometric objects. Sketching geometric relationships is challenging for many students, therefore specific steps are given throughout the unit to help
students. Ensure you have lots of common objects around the room for students to reference when, for example, they are asked to describe the intersection of two lines, or the intersection
of a line and a plane.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
A. Experiment with transformations in the plane
1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
D. Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector
of a line segment; and constructing a line parallel to a given line through a point not on the line.
Expressing Geometric Properties with Equations — G-GPE
B. Use coordinates to prove simple geometric theorems algebraically
7.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or
a human torso as a cylinder).
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
2
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 2: Reasoning and Proofs
Suggested number of days: 13
In this unit, students are introduced to inductive and deductive reasoning as well as conditional statements written in if-then form. Students examine when conditional statements are true
and false and connections are made to the reasoning students did in an algebra setting when solving equations, as there is always justification for the steps taken in solving an equation,
although we do not always write them. Students are introduced to two-column, flowchart, and paragraph proofs. These different formats are practiced while proving statements about
segments and angles.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
C. Prove geometric theorems
9.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal
crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points
on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals.
Similarity, Right Triangles, and Trigonometry — G-SRT
B. Prove theorems involving similarity
4.
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
3
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 3: Parallel and Perpendicular Lines
Suggested number of days: 12
Students have been introduced to the idea of an axiomatic system and have a growing list of theorems, postulates, definitions, and properties. In this unit, the list will grow some more,
and students will continue to develop their deductive skills. This unit begins with line relationships (parallel, skew, coplanar, intersecting) and angle-pair relationships (corresponding,
alternate interior, alternate exterior, and consecutive interior) and then statements about parallel lines cut by a transversal and the relationships between the pairs of angles formed are
presented. The converse of statements is explored and there is a greater focus on proofs as the unit progresses. It is in this unit that students often use circular reasoning, using a theorem
in the process of trying to prove the theorem. Skills from algebra are connected, allowing for a coordinate approach to justifying that two lines are parallel or two lines are perpendicular.
Throughout the unit, students should not lose sight of the bigger picture with regard to writing proofs. The goal is to write a logical argument based on known information or information
that can be deduced from given information.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
A. Experiment with transformations in the plane
1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
C. Prove geometric theorems
9.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal
crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points
on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
D. Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector
of a line segment; and constructing a line parallel to a given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Expressing Geometric Properties with Equations — G-GPE
B. Use coordinates to prove simple geometric theorems algebraically
5.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g.,
find the equation of a line parallel or perpendicular to a given line that passes through a given point).
6.
Find the point on a directed line segment between two given points that partitions the segment in a given
ratio.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
4
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 4: Transformations
Suggested number of days: 13
Students should have a conceptual understanding of transformations from middle school, where they studied translations, reflections, rotations and have been introduced to glide
reflections. The focus on plane transformations in the Arizona Common Core State Standards makes sense in terms of the continuity from middle school to high school. In middle school,
the conditions for triangle congruence were informally explored. In high school, “once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can
be used to prove theorems about triangles, quadrilaterals, and other geometric figures.” The criteria for triangle similarity are established through similarity transformations. In this unit,
key postulates and theorems relating to rigid motions are presented. Translations, reflections, glide reflections, and rotations are all postulated to be rigid motions.
While all could be proven and established as theorems, the authors of the resource have chosen to treat them as postulates. Dilations are introduced as non-rigid transformations where
the scale factor k results in an enlargement (k > 1) or a reduction (0 < k < 1). The composition of a dilation with rigid motions results in a similarity transformation. Similar figures are
defined in terms of similarity transformations and the use of dynamic geometry software for the explorations and formal lessons is highly encouraged. This tool provides students the
opportunity to explore and make conjectures, mathematical practices we want to develop in all students.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
A. Experiment with transformations in the plane
2.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs. Compare
transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal
stretch).
3.
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that
carry it onto itself.
4.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
5.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g.,
graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a
given figure onto another.
B. Understand congruence in terms of rigid motions
6.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid
motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to
decide if they are congruent.
Similarity, Right Triangles, and Trigonometry — G-SRT
A. Understand similarity in terms of similarity transformations
1.
Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line
passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2.
Given two figures, use the definition of similarity in terms of similarity transformations to decide i f they are
similar; 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.
continued on next page
Mesa Public Schools
5
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 4: Transformations
Suggested number of days: 13
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios).
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
6
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 5: Congruent Triangles
Suggested number of days: 17
In this unit, students will work with a variety of proof formats as they investigate triangle congruence. Methods for establishing triangle congruence (SAS, SSS, ASA, and AAS) are
established using rigid motions. The proof of each congruence criteria for triangles is done by composing transformations. This means a sequence of rigid motions maps one triangle onto
another triangle. Other proof styles presented in this unit include the two-column proof, the paragraph or narrative proof, and finally, the coordinate proof. In addition to working with
proofs, there are properties of equilateral and isosceles triangles that are proven. The use of dynamic geometry software for the explorations is highly encouraged. This tool provides
students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
B. Understand congruence in terms of rigid motions
7.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
8.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in
terms of rigid motions.
C. Prove geometric theorems
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles
of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third
side and half the length; the medians of a triangle meet at a point.
D. Make geometric constructions
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, and Trigonometry — G-SRT
B. Prove theorems involving similarity
5.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Expressing Geometric Properties with Equations — G-GPE
B. Use coordinates to prove simple geometric theorems algebraically
4.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by
four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at
the origin and containing the point (0, 2).
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios). 
Mesa Public Schools
7
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 6: Relationships Within Triangles
Suggested number of days: 15
This unit uses the deductive skills developed in the last unit, Unit 5, to explore special segments in a triangle. These segments include perpendicular bisectors, angle bisectors, medians,
altitudes, and mid-segments. Students are able to discover special properties of these segments by using dynamic geometry software. To prove these relationships, a variety of proof formats
and approaches are used: transformational, synthetic, analytic, and paragraph. This unit includes inequalities within one triangle and in two triangles. The indirect proof is introduced and
used to prove several of the theorems and triangles have been the geometric structure used to help students develop their deductive reasoning skills. In the next unit, students will study
quadrilaterals and other polygons.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
C. Prove geometric theorems
9.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal
crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points
on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel
to the third side and half the length; the medians of a triangle meet at a point.
D. Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector
of a line segment; and constructing a line parallel to a given line through a point not on the line.
Circles — G-C
A. Understand and apply theorems about circles
3.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or
a human torso as a cylinder). 
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios). 
Mesa Public Schools
8
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 7: Quadrilaterals and Other Polygons
Suggested number of days: 13
Students have studied the special quadrilaterals in middle school and should be familiar with their definitions and some of their properties. Special quadrilaterals include parallelograms,
rectangles, rhombuses, squares, trapezoids, isosceles trapezoids, and kites. Students have investigated the properties of these special quadrilaterals using inductive reasoning. In this unit,
students are able to investigate the properties of special quadrilaterals by using dynamic geometry software. The properties are then proven using a variety of proof formats:
transformational, synthetic, analytic, and paragraph.
The unit begins with the angle measures in a polygon where students first derive a formula for the sum of the interior angles and then a formula for the sum of the exterior angles. The unit
continues by studying special quadrilaterals. In Unit 5, students proved triangles congruent given different hypotheses. The reasoning used in that unit is applied to proofs involving
quadrilaterals in this unit.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
C. Prove geometric theorems
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals.
Similarity, Right Triangles, and Trigonometry — G-SRT
B. Prove theorems involving similarity
5.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or
a human torso as a cylinder).
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios). 
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
9
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 8: Similarity
Suggested number of days: 11
This is a short unit that revisits similarity, a concept first introduced in Unit 4. This unit begins with what it means for two polygons to be similar: corresponding sides are proportional and
corresponding angles are congruent. Then methods for proving two triangles are similar are presented. One of the methods involves only angles (AA), with the other two methods involving
only sides (SSS) or sides and the included angle (SAS). Unlike congruency methods, the corresponding sides must be proportional versus congruent. Several proportionality theorems, mostly
connected to triangles, are presented and properties of similar triangles will be needed in Unit 9 when trigonometric ratios are defined.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Similarity, Right Triangles, and Trigonometry — G-SRT
A. Understand similarity in terms of similarity transformations
2.
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are
similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
3.
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
B. Prove theorems involving similarity
4.
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
5.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
Expressing Geometric Properties with Equations — G-GPE
B. Use coordinates to prove simple geometric theorems algebraically
5.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g.,
find the equation of a line parallel or perpendicular to a given line that passes through a given point).
6.
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or
a human torso as a cylinder).
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios). 
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
10
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 9: Right Triangles and Trigonometry
Suggested number of days: 17
This unit introduces students to right triangle trigonometry. Students will encounter a more in-depth study of trigonometry in Algebra 2. The Pythagorean Theorem will not be completely
new to students who will have familiarity with this theorem from middle school. Students will use knowledge of similar triangles to investigate relationships in special right triangles (30°-60°90° and 45°-45°-90°) as well as similar triangles that are formed when the altitude to the hypotenuse is drawn in a right triangle. Being familiar with these relationships and solving for
segment lengths in triangles will be helpful in subsequent lessons. Presented are the tangent, sine, and cosine ratios with the focus being to solve for parts of a right triangle. Many real-life
applications are presented. The Law of Sines and the Law of Cosines are learned so that non-right triangles can be solved.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Similarity, Right Triangles, and Trigonometry — G-SRT
B. Prove theorems involving similarity
4.
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
5.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
C. Define trigonometric ratios and solve problems involving right triangles
6.
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading
to definitions of trigonometric ratios for acute angles.
7.
Explain and use the relationship between the sine and cosine of complementary angles.
8.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied pr oblems.
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios).
Mesa Public Schools
11
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 10: Circles
Suggested number of days: 18
This unit is all about circles. Many of the theorems will be investigated using dynamic geometry software. Vocabulary and symbols related to circles are introduced first, followed by
looking at circular arcs that are intercepted by chords. All of the angle relationships that occur when two chords, secants, or tangents intersect a circle are investigated, as well as the
investigation of segment relationships that occur when two chords, secants, or tangents intersect a circle. Circles are then presented in the coordinate plane where the standard
form of the equation is derived. Much of the earlier units has been devoted to lines, triangles, and polygons.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
A. Experiment with transformations in the plane
1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
D. Make geometric constructions
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Circles — G-C
A. Understand and apply theorems about circles
1.
Prove that all circles are similar.
2.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between
central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
3.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Expressing Geometric Properties with Equations — G-GPE
A. Translate between the geometric description and the equation for a conic section
1.
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to
find the center and radius of a circle given by an equation.
B. Use coordinates to prove simple geometric theorems algebraically
4.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure
defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the
circle centered at the origin and containing the point (0, 2).
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios).
Mesa Public Schools
12
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School Geometry
Unit 11: Circumference, Area, and Volume
Suggested number of days: 19
This unit on circumference, area, and volume finishes the study of measurement of solids that began in middle school. Students will come to this unit with knowledge of many formulas for
surface area and volume. These will be reviewed and a few new formulas added to the list. Different from junior high school is that students now have a greater ability to solve equations.
Now that students also know the Pythagorean Theorem and trigonometry, so they are able to solve for measures that previously had to be told to them. In this unit, students will do
additional work with circles involving arc length and area of Sectors, as well as find the area of regular polygons.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Congruence — G-CO
A. Experiment with transformations in the plane
1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
Circles — G-C
B. Find arc lengths and areas of sectors of circles
5.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and
define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a
sector.
Geometric measurement and dimension — G-GMD
A. Explain volume formulas and use them to solve problems
1.
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder,
pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
3.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
B. Visualize relationships between two-dimensional and three- dimensional objects
4.
Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional
objects generated by rotations of two-dimensional objects.
Modeling with Geometry — G-MG
A. Apply geometric concepts in modeling situations
1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).
2.
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per
cubic foot).
3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical
constraints or minimize cost; working with typographic grid systems based on ratios).
Mesa Public Schools
13
May 2015