geometry grade 9-12 curriculum guide final draft...trapezoids and kites areas of trapezoids,...

Hackettstown Public Schools

Geometry Grade 9-12

CURRICULUM GUIDE

FINAL DRAFT

May 2017

Mr. David C. Mango, Superintendent Ms. Nadia Inskeep, Director of Curriculum & Instruction

Developed by: Stevie Klie

This curriculum may be modified through varying techniques,

strategies and materials, as per an individual student’s Individualized Education Plan (IEP).

Approved by the Hackettstown Board of Education At the regular meeting held on

And Aligned with the New Jersey Student Learning Standards


Table of Contents

Component Page

Philosophy and Rationale:

3

Scope and Sequence:

4-5

Mission Statement:

6

Units:

7-38

NJ Content Standards Link:

39

21st Century Skills Link:

39


Philosophy and Rationale The mission of the Hackettstown Mathematics Department is to design and

implement a mathematics curriculum that stresses the key ideas identified in the New Jersey Student Learning Standards. This goal is to be achieved by continually returning to the organizing principles of mathematics and the laws of algebraic and geometric structure to reinforce those concepts.

Through a variety of real world scenarios and applications, our students will be challenged to think critically, apply, evaluate, and communicate the ideas and concepts presented in each course as it relates to the field of mathematics. Students will also be challenged to analyze their performance in an effort to become independent problem solvers.

Our students’ performance will be assessed using formative techniques such as homework, quizzes, and tests. Additionally, our students will be assessed using state and national assessment tools based upon the branch of mathematics being studied.


Scope and Sequence Unit 1

Points, Lines, Planes

Measuring Segments

Measuring Angles

Exploring Angle Pairs

Basic Constructions

Midpoint and Distance in the Coordinate Unit 2

Reasoning in Algebra and Geometry

Proving Angles Congruent

Lines and Angles

Properties of Parallel Lines

Proving Lines Parallel

Parallel and Perpendicular Lines

Parallel Lines and Triangles

Equations of a Line in a Coordinate Plane

Slopes of Parallel and Perpendicular Lines Unit 3

Congruent Figures

Triangle Congruence by SSS and SAS

Triangle Congruence by ASA and AAS

Using Corresponding Parts of Congruent Traingles

Isosceles and Equilateral Triangles

Congruence in Right Triangles

Congruence in Overlapping Triangles Unit 4

Midsegments of Triangles

Perpendicular and Angle Bisectors

Inequalities in One Triangle

Inequalities in Two Triangles

Bisectors in Triangles

Medians and Altitudes Unit 5

Ratios and Proportions

Similar Polygons

Proving Triangles Similar

Similarity in Right Triangles

Proportions in Triangles Unit 6

The Pythagorean Theorem and Its Converse

Special Right Triangles

Trigonometry

Angles of Elevation and Depression


Law of Sines

Law of Cosines Unit 7

Properties of Parallelograms

Proving That a Quadrilateral Is a Parallelogram

Properties of Rhombuses, Rectangles, and Squares

Conditions of Rhombuses, Rectangles, and Squares

Trapezoids and Kites

Areas of Trapezoids, Rhombuses, and Kites Unit 8

The Polygon Angle-Sum Theorem

Areas of Regular Polygons

Trigonometry and Area

Polygons in the Coordinate Plane

Applying Coordinate Geometry

Proofs Using Coordinate Geometry Unit 9

Circles and Arcs

Areas of Circles and Sectors

Tangent Lines

Chords and Arcs

Inscribed Angles

Angle Measures and Segment Lengths

Circles in the Coordinate Plan

Equations of Circles Unit 10

Translations

Reflections

Rotations

Composition of Isometries

Congruence Transformations

Dilations

Similarity Transformations Unit 11

Surface Areas of Prisms and Cylinders

Surface Areas of Pyramids and Cones

Volumes of Prisms and Cylinders

Volumes of Pyramids and Cones

Surface Areas of Volumes of Spheres

Perimeters and Areas of Similar Figures

Areas and Volumes of Similar Solids


Mission Statement

Building on tradition and success, the mission of the Hackettstown School District is to educate and inspire students through school, family and community partnerships so that all become positive, contributing members of a global society, with a life-long commitment to learning.


Unit 1 – Tools of Geometry

Stage 1: Desired Results Content Standards 12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 12.G.GPE.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). 12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles

and rectangles, e.g., using the distance formula.★

Essential Questions

What are the building blocks of geometry?

How can you describe the attributes of a segment or angle? Enduring Understandings

Geometry is a mathematical system built on accepted facts, basic terms, and definitions.

Definitions establish meanings and remove possible misunderstandings.

Number operations are used to find and compare the lengths of segments and measures of angles.

Attributes such as length, area, volume, and angle measure are measurable.

Special angle pairs can help identify geometric relationships and find angle measures.

Special geometric tools are used to make a figure that is congruent to an original figure without measuring.

Formulas are used to find the midpoint and length of any segment in the coordinate plane.

Knowledge and Skills (SWBAT embedded course proficiencies) By the end of this unit students will be able to:

Understand basic terms and postulates of geometry.

Find and compare lengths of segments and measures of angles.

Identify special angle pairs and use their relationships to find angle measures.


Make basic constructions using a straightedge and a compass.

Find the midpoint of a segment and the distance between two points in the coordinate plane.

Stage 2: Evidence of Understanding, Learning Objectives and Expectations

Benchmarks (embedded student proficiencies) Students will:

Understand basic terms and postulates of geometry.

Find and compare lengths of segments and measures of angles.

Identify special angle pairs and use their relationships to find angle measures.

Make basic constructions using a straightedge and a compass.

Find the midpoint of a segment and the distance between two points in the coordinate plane.

Assessment Methods

Formative:

Warm up

Sample problems in class

PARCC practice problems

Homework

Quizzes

Unit test Summative:

Performance based tasks

Essays Other Evidence and Student Self-Assessment:

Homework groups

Evaluation of answers and determine feasibility of them in groups and pairs

Stage 3: Learning Plan In this unit, the 21st century skills of managing goals and time will be realized through the summer assignment and the beginning of geometry skills. Students will be required to complete a summer assignment prior to the start of school and be prepared to ask questions when they return. This assignment will be differentiated by class level and students who need help are recommended to seek help prior to the start of the geometry topics. Differentiation: The summer assignment will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course.


Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment:

Honors – approximately 16 days

CPA – approximately 10 days

CP – approximately 15 days Resources

Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 2 – Linear Relationships in Geometry

Stage 1: Desired Results Content Standards 12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid

systems based on ratios).★

Essential Questions

How to prove that two lines are parallel?

What is the sum of the measures of the angles of a triangle?

How do you write an equation of a line in the coordinate plane? Enduring Understandings

Algebraic properties of equality are used in geometry, they help solve problems and justify each step.

Using given information, definitions, properties, postulates and previously proven theorems help as reasons in a proof.

Not all lines and not all planes intersect.

The special angle pairs formed by parallel lines and a transversal are congruent, supplementary, or both.

Using certain angles pairs to decide whether two lines are parallel.

Using the relationship of two lines to a third line decide whether the two lines are parallel or perpendicular to each other.

The sum of the angle measures of a triangle is always the same.

Graphing a line and writing its equation when you know certain facts about the line, such as its slope and a point on the line.


Determining whether two lines are parallel or perpendicular by comparing their slopes.

Knowledge and Skills (SWBAT embedded course proficiencies) By the end of the unit, students will be able to:

Connect reasoning in algebra and geometry

Prove and apply theorems about angles

Identify relationships between figures in space and angles formed by two lines and a transversal

Prove theorems about parallel lines and use properties of parallel lines to find angle measures

Determine whether two lines are parallel

Relate parallel and perpendicular lines

Use parallel lines to prove a theorem about triangles and find measures of angles of triangles

Graph and write linear equations

Relate slope to parallel and perpendicular lines

Stage 2: Evidence of Understanding, Learning Objectives and Expectations Benchmarks (embedded student proficiencies) Students will:

Connect reasoning in algebra and geometry

Prove and apply theorems about angles

Identify relationships between figures in space and angles formed by two lines and a transversal

Prove theorems about parallel lines and use properties of parallel lines to find angle measures

Determine whether two lines are parallel

Relate parallel and perpendicular lines

Use parallel lines to prove a theorem about triangles and find measures of angles of triangles

Graph and write linear equations

Relate slope to parallel and perpendicular lines Assessment Methods

Formative:

Warm up



Homework

Quizzes



Essays


Other Evidence and Student Self-Assessment:

Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of reason effectively will be used. Students must use skills they have learned in Algebra I and apply them to this unit. Students will be asked to use and modify their algebra reasoning skills for geometry concepts. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment


CPA and CP – approximately 12 days Resources

Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 3 – Congruence

Stage 1: Desired Results Content Standards 12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Essential Questions

How do you identify corresponding parts of congruent triangles?

How do you show that two triangles are congruent?

How can you tell whether a triangle is isosceles or equilateral? Enduring Understandings

Determine whether two figures are congruent by comparing their corresponding parts.

Prove that two triangles are congruent without having to show that all corresponding parts are congruent.

If two triangles are congruent, then every pair of their corresponding parts is also congruent.

The angles and sides of isosceles and equilateral triangles have special relationships.

Use the congruent corresponding parts of one pair of congruent triangles to prove another pair of triangles congruent.

Knowledge and Skills (SWBAT embedded course proficiencies) By the end of the unit, students will be able to:

Recognize congruent figures and their corresponding parts.

Prove two triangles congruent using the SSS, SAS, ASA Postulates and the AAS Theorem.

Use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent.

Use and apply properties of isosceles and equilateral triangles.

Prove right triangles congruent using the Hypotenuse-Leg Theorem.

Identify congruent overlapping triangles and prove two triangles congruent using congruent triangles.


Stage 2: Evidence of Understanding, Learning Objectives and Expectations

Benchmarks (embedded student proficiencies) Students will:

Recognize congruent figures and their corresponding parts.

Prove two triangles congruent using the SSS, SAS, ASA Postulates and the AAS Theorem.

Use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent.

Use and apply properties of isosceles and equilateral triangles.

Prove right triangles congruent using the Hypotenuse-Leg Theorem.

Identify congruent overlapping triangles and prove two triangles congruent using congruent triangles.

Assessment Methods

Formative:

Warm up



Homework

Quizzes




Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of use systems thinking will be used. Students will use their understanding from the first lesson, that all parts of congruent figures are congruent, to assist them in understanding each lesson after that. The students must be able to understand how the parts of the entire figures allow for the whole outcome of congruent figures. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student


feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment



Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 4 – Triangle Relationships

Stage 1: Desired Results Content Standards 12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Essential Questions

How do you use coordinate geometry to find relationships within triangles?

How do you solve problems that involve measurements of triangles?

How do you write indirect proofs? Enduring Understandings

There are two special relationships between a midsegment of a triangle and the third side of the triangle.

There is a special relationship between the points on the perpendicular bisector of a segment and the endpoints of the segment.

For any triangle, certain sets of lines are always concurrent. Two of these sets of lines are the perpendicular bisectors of the triangle’s three sides and the bisectors of the triangle’s three angles.

A triangle’s three medians are always concurrent.

The angles and sides of a triangle have special relationships that involve inequalities.

In triangles that have two pairs of congruent sides, there is a relationship between the included angles and the third pair of sides.

Knowledge and Skills (SWBAT embedded course proficiencies) By the end of this unit, students will be able to:

Use properties of midsegments to solve problems.

Use properties of perpendicular bisectors and angle bisectors.

Identify properties of perpendicular bisectors and angle bisectors.


Identify properties of medians and altitudes of a triangle.

Use inequalities involving angles and sides of triangles.

Apply inequalities in two triangles.


Use properties of midsegments to solve problems.

Use properties of perpendicular bisectors and angle bisectors.

Identify properties of perpendicular bisectors and angle bisectors.

Identify properties of medians and altitudes of a triangle.

Use inequalities involving angles and sides of triangles.

Apply inequalities in two triangles. Assessment Methods

Formative:

Warm up



Homework

Quizzes



Essays Other Evidence and Self-Assessment:

Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of making judgments and decisions will be used. Students must gather information and decide, based on their best judgments, which point of concurrency would be best used for each situation at hand. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for


themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment



Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 5 – Similarity and Right Triangles

Stage 1: Desired Results Content Standards 12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Essential Questions

How do you use proportions to find side lengths in similar polygons?

How do you show two triangles are similar?

How do you identify corresponding parts of similar triangles? Enduring Understandings

Write a ratio to compare two quantities.

Use ratios and proportions to decide whether two polygons are similar and to find unknown side lengths of similar figures.

Show that two triangles are similar when you know the relationships between only two or three pairs of corresponding parts.

Drawing the altitude to the hypotenuse of a right triangle, forms three pairs of similar right triangles.

When two or more parallel lines intersect other lines, proportional segments are formed.


Write ratios and solve proportions.

Identify and apply similar polygons.

Use the AA ~ Postulate and the SAS~ and SSS~ Theorems.

Use similarity to find indirect measurements.

Find and use relationships in similar right triangles.

Use the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem.


Write ratios and solve proportions.

Identify and apply similar polygons.

Use the AA ~ Postulate and the SAS~ and SSS~ Theorems.

Use similarity to find indirect measurements.


Find and use relationships in similar right triangles.

Use the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem. Assessment Methods

Formative:

Warm ups



Homework

Quizzes




Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of communicating clearly will be used. Students must clearly be able to communicate why triangles are similar. Not only do they have to solve problems numerically, they must then explain their reasoning clearly. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment



Geometry textbook

njctl.org

graphing calculator


SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 6 – Right Triangle and Trigonometry

Stage 1: Desired Results Content Standards 12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. 12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 12.G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. 12.G.SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). 12.G.MG.1 Use geometric shapes, their measures, and their properties to describe

objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

Essential Questions

How do you find a side length or angle measure in a right triangle?

How do trigonometric ratios relate to similar right triangles? Enduring Understandings

If the lengths of any twos sides of a right triangle are known, then the length of the third side is found by using the Pythagorean Theorem.

Certain right triangles have properties that allow you to use shortcuts to determine side lengths without using the Pythagorean Theorem.

Certain combinations of side lengths and angle measures of a right triangle helps find other side lengths and angle measures.

Use the angles of elevation and depression as the acute angles of right triangles formed by a horizontal distance and a vertical height.

Knowing the measures of two angles and the length of a side, or two side lengths and the measure of a nonincluded obtuse angle help find all the other measures of the triangle.

Knowing the measures of two side lengths and measure of the included angle, or all three side lengths help find all the other measures of the triangle.


Use the Pythagorean Theorem and its converse.

Use the properties of 45-45-90 and 30-60-90 triangles.

Use the sine, cosine, and tangent ratios to determine side lengths and angle measures in right triangles.

Use angles of elevation and depression to solve problems.

Apply the Law of Sines and Law of Cosines



Use the Pythagorean Theorem and its converse.

Use the properties of 45-45-90 and 30-60-90 triangles.

Use the sine, cosine, and tangent ratios to determine side lengths and angle measures in right triangles.

Use angles of elevation and depression to solve problems.

Apply the Law of Sines and Law of Cosines Assessment Methods

Formative:

Warm ups



Homework

Quiz




Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of collaborating with others. Students will be asked to conduct a lab and apply the concepts of this unit. For the lab, students will be working with partners and groups where they must collaborate. They will receive the same grade, therefore they are responsible for their collaboration. The teacher will identify issues of students not using their collaborative skills. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having.


Time Allotment



Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 7 – Quadrilaterals

Stage 1: Desired Results Content Standards 12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 12.G.GPE.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). 12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles

and rectangles, e.g., using the distance formula.★

12.G.MG.1 Use geometric shapes, their measures, and their properties to describe


Essential Questions

How can you classify quadrilaterals?

How can you use coordinate geometry to prove general relationships? Enduring Understandings

Parallelograms have special properties regarding their sides, angles, and diagonals.

Decide whether a quadrilateral is a parallelogram if its sides, angles, and diagonals have certain properties.

The special parallelograms have basic properties about their sides and angles that identify them. The diagonals of these parallelograms also have certain properties.

Determine whether a parallelogram is a rhombus or a rectangle based on the properties of its diagonals.

The angles, sides and diagonals of a trapezoid and kite have certain properties.

Classify figures in the coordinate plane using the formulas for slope, distance and midpoint.

Use variables to name the coordinates of a figure, this shows that relationships are true for a general case.

Find the area of a trapezoid when you know its height and lengths of its bases.

Find the area of a rhombus or a kite when you know the lengths of its diagonals. Knowledge and Skills (SWBAT embedded course proficiencies) By the end of this unit, students will be able to:

Use relationships among sides, angles and diagonals of parallelograms.

Determine whether a quadrilateral is a parallelogram.

Determine and classify special types of parallelograms.

Use properties of diagonals of rhombuses and rectangles.


Determine whether a parallelogram is a rhombus or rectangle.

Verify and use properties of trapezoids and kites.

Classify polygons in the coordinate plane.

Name coordinates of special figures by using their properties.

Find the area of a trapezoid, rhombus or kite.


Use relationships among sides, angles and diagonals of parallelograms.

Determine whether a quadrilateral is a parallelogram.

Determine and classify special types of parallelograms.

Use properties of diagonals of rhombuses and rectangles.

Determine whether a parallelogram is a rhombus or rectangle.

Verify and use properties of trapezoids and kites.

Classify polygons in the coordinate plane.

Name coordinates of special figures by using their properties.

Find the area of a trapezoid, rhombus or kite. Assessment Methods

Formative:

Warm ups



Homework

Quiz




Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of reason effectively will be used. Students will need to reason whether or not a parallelogram is a rectangle, rhombus or square with the appropriate reasons, conditions and/or properties. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and


holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment



Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 8 – Polygons

Stage 1: Desired Results Content Standards 12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 12.G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 12.G.MG.1 Use geometric shapes, their measures, and their properties to describe


Essential Questions

How can you find the sum of the measures of polygon angles?

How do you find the area of a polygon? Enduring Understandings

The sum of the interior angle measures of a polygon depends on the number of sides the polygon has.

The area of a regular polygon is related to the distance from the center to a side.

Use trigonometry to find the area of a regular polygon when you know the length of a side, radius or apothem.


Find the sum of the measures of the interior and exterior angles of a polygon.

Find the area of a regular polygon.

Find the areas of regular polygons and triangles using trigonometry.


Find the sum of the measures of the interior and exterior angles of a polygon.

Find the area of a regular polygon.

Find the areas of regular polygons and triangles using trigonometry. Assessment Methods

Formative:

Warm ups



Homework

Quizzes

Unit test


Summative:



Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of solving problems will be used. Students must solve problems in order to understand polygons and their properties. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment



Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 9 – Circles

Stage 1: Desired Results Content Standards 12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 12.G.C.1 Prove that all circles are similar. 12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 12.G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle. 12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. 12.G.GPE.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. Essential Questions

How do you find the circumference and area of a circle?

How can you prove relationships between angles and arcs in a circle?

When lines intersect a circle or within a circle, how do you find the measures of resulting angles, arcs, and segments?

How do you find the equation of a circle in the coordinate plane? Enduring Understandings

Find the length of part of a circle’s circumference by relating it to an angle in the circle.

Find the area of a circle when its radius is known. Use the area of a circle to find part of a circle formed by two radii and the arc the radii form when they intersect with the circle.

A radius of a circle and the tangent that intersects the endpoint of the radius on the circle have a special relationship.

Use information about congruent parts of a circle to find information about other parts of the circle.

Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept.

Angles formed by intersecting lines have a special relationship to the related arcs formed when the lines intersect a circle.

The information in the equation of a circle allows the circle to be graphed. The equation of a circle can be written if its center and radius are known.



Find the measures of central angles and arcs.

Find the circumference and arc length

Find the areas of circles, sectors and segments of circles.

Use properties of a tangent to a circle.

Use congruent chords, arcs and central angles.

Use perpendicular bisectors to chords.

Find the measure of an inscribed angle, and an angle formed by and tangent and a chord.

Find measures of angles formed by chords, secants, and tangents.

Find the lengths of segments associated with circles.

Write the equation of a circle and find the center and radius of a circle.


Find the measures of central angles and arcs.

Find the circumference and arc length

Find the areas of circles, sectors and segments of circles.

Use properties of a tangent to a circle.

Use congruent chords, arcs and central angles.

Use perpendicular bisectors to chords.

Find the measure of an inscribed angle, and an angle formed by and tangent and a chord.

Find measures of angles formed by chords, secants, and tangents.

Find the lengths of segments associated with circles.

Write the equation of a circle and find the center and radius of a circle. Assessment Methods

Formative:

Warm ups



Homework

Quiz




Homework groups



Stage 3: Learning Plan In this unit, the 21st century skills of using systems thinking will be used. Students will recognize that once they understand a whole circle, they will then be able to figure out every part of a circle. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment



Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 10 – Transformations (CPA and Honors Independent Unit)

Stage 1: Desired Results

Content Standards 12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 12.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Essential Questions

How can you change a figure’s position without changing its size and shape? How can you change a figure’s size without changing its shape?

How can you represent a transformation in the coordinate plane?

How do you recognize congruence and similarity in figures? Enduring Understandings

Change the position of a geometric figure so that the angle measures and the distance between any two points of a figure stay the same.

When a figure is reflected across a line, each point of the figure maps to another point the same distance from the line but on the other side.

Rotations preserves distance, angle measures and orientation of figures.

All isometries can be expressed as compositions of reflections.


Use compositions of rigid motions to understand congruence.

Use a scale factor to make a larger or smaller copy of a figure that is also similar to the original figure.

Use compositions of rigid motions and dilations to help understand the properties of similarity.


Identify rigid motions.

Find translation images of figures.

Find reflection images of figures.

Draw and Identify rotation images of figures.

Find compositions of isometries, including glide reflections.

Classify isometries.

Identify congruence transformations.

Prove triangle congruence using isometries.

Understand dilation images of figures.

Identify similarity transformations and verify properties of similarity.


Identify rigid motions.

Find translation images of figures.

Find reflection images of figures.

Draw and Identify rotation images of figures.

Find compositions of isometries, including glide reflections.

Classify isometries.

Identify congruence transformations.

Prove triangle congruence using isometries.

Understand dilation images of figures.

Identify similarity transformations and verify properties of similarity. Assessment Methods

Formative:

Warm ups



Homework

Quizzes



Essays


Other Evidence and Student Self-Assessment

Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of working independently will be used. Students will be directed to find time on their own to learn and understand this unit with the assistance of resources. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having. Time Allotment

Honors – no in-class days

CPA – approximately 3 in-class days

CP – approximately 11 days Resources

Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


Unit 11 – Surface Area and Volume (Honors Independent Unit)

Stage 1: Desired Results Content Standards 12.G.MG.1 Use geometric shapes, their measures, and their properties to describe


12.G.MG.2 Apply concepts of density based on area and volume in modeling situations

(e.g., persons per square mile, BTUs per cubic foot).★

12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 12.G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve

problems.★

Essential Questions

How do you find the surface area and volume of a solid?

How do perimeters and areas of similar polygons compare?

How do the surface areas and volumes of similar solids compare? Enduring Understandings

To find the surface area of a three-dimensional figure, find the sum of the areas of all the surfaces of the figure.

Find the volume of a prism or a cylinder when its height and the area of its base are known.

The volume of a pyramid is related to the volume of a prism with the same base and height.

The volume of a cone is related to the volume of a cylinder with the same base and height.

Find the surface area and the volume of a sphere when its radius is known.

Use ratios to compare the perimeters and areas of similar figures.

Use ratios to compare the areas and volumes of similar solids. Knowledge and Skills (SWBAT embedded course proficiencies) By the end of this unit, students will be able to:

Find the surface area of a prism and a cylinder.

Find the surface area of a pyramid and a cone.

Find the volume of a prism and the volume of a cylinder.

Find the volume of a pyramid and of a cone.

Find the surface area and volume of a sphere.

Find the perimeters and areas of similar polygons.

Compare and find the areas and volumes of similar solids.



Find the surface area of a prism and a cylinder.

Find the surface area of a pyramid and a cone.

Find the volume of a prism and the volume of a cylinder.

Find the volume of a pyramid and of a cone.

Find the surface area and volume of a sphere.

Find the perimeters and areas of similar polygons.

Compare and find the areas and volumes of similar solids. Assessment Methods

Formative:

Warm ups



Homework

Quizzes




Homework groups


Stage 3: Learning Plan In this unit, the 21st century skills of manage projects. Students will be assigned a project that they must complete. The students will receive a rubric and must set goals and manage their time, in order to complete the assignment by the due date. Differentiation: Assignments will vary in content and depth according to the class level. Students at higher levels are required to put in more time outside of class. Technology: Students are required to have a graphing calculator to use when necessary in class, on assessments and for homework. The online textbook is free and holds various resources, as well as Khan Academy and YouTube. Google classroom is used as a hub for all documents and materials necessary for the course. Self-assessment: From the beginning of the course, the students are aware that they are responsible for understanding when they are in need of assistance. If a student feels in need of help in class during notes or practice problems they must advocate for themselves in class or during the teacher’s office hours. If a student feels they are in need of help on their homework, they must reach out to their teacher and identify the difficulties they are having.


Time Allotment

Honors – no in-class days


Geometry textbook

njctl.org

graphing calculator

SmartBoard

Khan Academy

YouTube

Google Forms

compass

protractor


New Jersey Student Learning Standards http://www.state.nj.us/education/cccs/

Integration of 21st Century Theme(s) The following websites are sources for the following 21st Century Themes and Skills: http://www.nj.gov/education/code/current/title6a/chap8.pdf http://www.p21.org/about-us/p21-framework . http://www.state.nj.us/education/cccs/standards/9/index.html 21st Century Interdisciplinary Themes (into core subjects) • Global Awareness • Financial, Economic, Business and Entrepreneurial Literacy • Civic Literacy • Health Literacy • Environmental Literacy Learning and Innovation Skills • Creativity and Innovation • Critical Thinking and Problem Solving • Communication and Collaboration Information, Media and Technology Skills • Information Literacy • Media Literacy • ICT (Information, Communications and Technology) Literacy Life and Career Skills • Flexibility and Adaptability • Initiative and Self-Direction • Social and Cross-Cultural Skills • Productivity and Accountability • Leadership and Responsibility

Integration of Digital Tools ● Classroom computers/laptops ● Technology Lab ● FM system ● Other software programs

http://www.state.nj.us/education/cccs/

http://www.nj.gov/education/code/current/title6a/chap8.pdf

http://www.p21.org/about-us/p21-framework

http://www.state.nj.us/education/cccs/standards/9/index.html

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