· Web viewGeometry 1-2. General Information. Course Goal: Geometry 1-2 provides students the opportunity to formulate conjectures using inductive reasoning and to construct proofs

Geometry 1-2General Information

Course Goal:

Geometry 1-2 provides students the opportunity to formulate conjectures using inductive reasoning and to construct proofs using deductive reasoning. Students in this course will study the properties of points, lines, planes, angles, triangles, quadrilaterals, polygons, and circles with a strong emphasis on logical arguments using paragraph proofs, two-column proofs, and indirect proofs. This course provides extensive work using the Pythagorean Theorem in both two-dimensional and three-dimensional space. This course also covers area of two-dimensional figures and surface area and volume of three-dimensional figures. The course concludes with a strong review of crucial concepts from Algebra 1-2 using coordinate geometry. Throughout the course, students will be able to manipulate figures and discover geometric properties using Geometer’s Sketchpad software.

Organization of Course:

This year-long course is structured into fifteen content-based units.

Units include:1. Introduction to Geometry, Proofs, and Logic 2. Basic Concepts and Proofs3. Congruent Triangles4. Indirect Proof, Parallel Lines, and Angle Relationships5. Quadrilaterals6. Polygons7. Lines and Planes in Space8. Similar Polygons9. Triangle Inequalities10. The Pythagorean Theorem11. Special Right Triangles and Trigonometry12. Circles13. Area14. Surface Area and Volume15. Coordinate Geometry

Units 1 – 3 should be covered first quarter, Units 4 – 9 in second quarter, Units 10 – 12 in third quarter, and Units 13 – 15 in fourth quarter. As time permits, the students will explore geometric constructions using a straightedge and compass in the last several days of second semester.

Instructional Methods:

Although interactive lectures will be the primary means of instruction, the course will also include collaborative group work, discovery activities (often using Geometer’s Sketchpad software), and projects.

Primary Resources:

Textbook, Geometry for Enjoyment and Challenge by Richard Rhoad, George Milauskas, and Robert Whipple ©1997 McDougall, Littell and Company.

CD containing teacher-generated daily lesson packets (to facilitate note-taking), supplemental assignments, Geometer’s Sketchpad labs, projects, quizzes, and tests. (Mrs. Best, Mrs. Kiefer, Mrs. Linsley, and Mrs. Woodnorth have binders with hard copies of this material in their classrooms.)

Geometer’s Sketchpad software Exploring Geometry with Geometer’s Sketchpad ; ©2002 Key Curriculum Press

Evaluation of Student Performance:

Students in Geometry will be assessed in a variety of ways. To foster each student’s personal responsibility for learning, homework will be checked for completion and accuracy on a weekly basis. Typically, the students will have a quiz over the material covered in the middle of each unit, and they will have a test at the end of each unit. In addition, the students will be evaluated on their completion of Geometer’s Sketchpad labs as well as their performance on three projects.

Standards for Mathematics

School District of Whitefish Bay - 2006

We believe that proficiency in mathematics is based on content and process standards including:

a) Number and Operation. Students will understand numbers, develop meanings of operations, and compute with fluency. Students should be able to perform computations in different ways, explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general.

b) Algebraic Relationships. Students will discover, describe, and generalize simple and complex patterns and functional relationships. In the context of real-world problem situations, the student will use algebraic notation and techniques to define and describe the problems to determine and justify appropriate solutions.

c) Geometry. Students will be able to analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems.

d) Measurement Students will understand the attributes, units, systems, and processes of measurement as well as apply the techniques, tools, and formulas to determine measurements to a specific degree of accuracy.

e) Statistics and Probability. Students will reason statistically both descriptively and inferentially. They will formulate questions and collect, organize, and display relevant data to address these questions. Additionally, they will use statistical methods to analyze data, make inferences and predictions based on data, and understand and use the basic concepts of probability.

f) Problem Solving. Students will formulate and solve a wide variety of problems. They will reflect on their thinking during the problem-solving process so that they can apply and adapt the strategies they develop to other problems and in other contexts. By solving mathematical problems, students will acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that serve them well outside the mathematics classroom.

g) Reasoning and Proof. Students will reason, think analytically, and recognize patterns, structure, or regularities in both real-world and mathematical conjectures. They will develop and evaluate formal and informal mathematical arguments and proofs. Students will see and expect that mathematics make sense.

h) Communication. Students will communicate the results of their thinking to others orally and in writing using clear, convincing, and precise mathematical language. Students will listen to others’ explanations and engage in conversations to develop their own understandings and explore multiple perspectives.

i) Connections. Students will connect mathematical ideas and view mathematics as a coherent whole. They will relate mathematics to other subjects, real-world situations, and their own interests and experience.

j) Representation. Students will represent mathematical ideas in a variety of ways such as pictures, concrete materials, tables, graphs, number and letter symbols, and spreadsheets. They will acquire a set of tools to model and interpret physical, social, and mathematical phenomena.

Belief Statements

School District of Whitefish Bay Mathematics Review Committee - 2006

We believe all students can demonstrate proficiency in mathematics given effective curriculum, instruction, and assessment.

1. We believe that effective mathematics curricula:a) Must engage and build upon students’ existing understanding and experience.

b) Must balance the roles of factual knowledge and conceptual understanding and develop the relationship between them.

c) Must foster meta-cognition whereby students are able to think about their own thinking, take control of their own learning, consciously define learning goals, and monitor progress toward those goals.

d) Must transition effectively from grade to grade and level to level.

2. We believe that effective mathematics instruction:a) Must develop understanding of mathematical concepts, operations and relations.

b) Must develop fluency in carrying out skills and procedures flexibly, accurately, efficiently, and appropriately.

c) Must develop the ability to reason logically, reflect on one’s thinking, explain and justify processes and products.

d) Must develop the ability to formulate, represent, and solve a variety of mathematical problems and apply problem solving skills in a variety of contexts.

e) Must develop a positive attitude toward mathematics whereby mathematics is seen as sensible, useful and worthwhile.

f) Must develop student’s sense that through effort and the use of effective strategies, they are able to learn mathematics.

3. We believe that effective mathematics assessment:a) Is a continuous process of helping students understand standards, their performance as

it relates to those standards, and providing opportunities to close the gap between the two.

b) Includes self-assessment as a means to helping students reflect on their current skills, understanding and overall progress toward standards.

Whitefish Bay Schools Unit Design Template

Course: Geometry 1-2

Unit Title: Introduction to Geometry, Proofs, and Logic

Author(s): Best

Approx Instructional Time: 17 days

What enduring understandings are desired? What are the overarching essential questions? A proof is a systematic method of documenting

mathematical ideas with a chain of reasoning. What is proof and why is learning to write a

proof important?

Geometric concepts can be modeled by algebraic equations; conversely, geometry gives visual meaning to algebra.

Why is modeling a geometric concept with an algebraic equation helpful (as opposed to using other methods like guess and check)?

Being logical can help us understand our world. What does it mean to be logical?

Smaller pieces like points, lines, and planes are put together to create complex shapes.

In what context will the understanding be applied? What “essential” and “unit” questions will focus this unit?

First-day-of-school index card activity where the students have to work collaboratively and present basic geometric terms and notation to their classmates

How can the concepts of congruence, bisection, substitution, and addition be used in an effective, coherent proof?

As always, throughout this course, when new theorems are introduced, they will be proven (discussed and explained) so that students deeply understand the theorems (and do not just memorize them).

What type of geometric situation calls for the “piece + piece = whole” algebraic equation and what type calls for an algebraic equation where the two quantities are set equal to one another?

How are the symbols of logic used to represent statements and how are those statements combined to create logical arguments?

Standards: What grade level outcomes/benchmarks are taught and assessed?

Geometry. Students will be able to analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems.

Uses properties of and relationships among figures to solve mathematical and real-world problems.

Present convincing arguments by means of demonstration, informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent) and generalizations (e.g., the Pythagorean Theorem holds for all right triangles).

Communicate logical arguments and clearly show

- why the reasoning is or is not valid- an understanding of the difference between

examples that support a conjecture and a proof of the conjecture

Develop effective oral and written presentations employing correct mathematical terminology, notation, symbols, and conventions for mathematical arguments and display of data.

Understands the difference between a statement that is verified by mathematical proof (i.e., a theorem) and one that is assumed because it can be verified empirically using examples or data (i.e., postulate).

Reasoning and Proof. Students will reason, think analytically, and recognize patterns, structure, or regularities in both real-world and mathematical conjectures. They will develop and evaluate formal and informal mathematical arguments and proofs. Students will see and expect that mathematics make sense.

Communication. Students will communicate the results of their thinking to others orally and in writing using clear, convincing, and precise mathematical language. Students will listen to others’ explanations and engage in conversations to develop their own understandings and explore multiple perspectives.

Rubrics to articulate quality aspects of desired understanding targets. (if developed)

Student Self-Assessment (if developed)

none Homework checks and daily corrections are an on-going self-assessment tool.

Review packets are provided before every quiz and test so students can practice the necessary skills and familiarize themselves with the format and directions they will encounter on the assessment. An answer key is provided so students can assess how well they have mastered the material before the actual quiz or test.

Given the targeted understandings, other unit goals, and the assessment evidence identified, what knowledge and skills are central to this unit?Students will need to know… (Information) Students will need to be able to do… (Skills)

Important new geometric terms: point, endpoint, midpoint, vertex, segment, ray, line, angle, triangle, parallel lines, perpendicular, right angles, union, intersection, acute angle, obtuse angle, straight angle, adjacent, congruence, collinear, postulate, theorem, proof (two-column versus paragraph), bisect, trisect, angle bisector, angle trisector,

Important logic terms: inductive reasoning, deductive reasoning, negation, conjunction, disjunction, truth table, conditional statement, converse, inverse, contrapositive, valid arguments, invalid arguments, converse error, inverse error

How to correctly name a segment, line, ray, angle, etc. using proper notation.

A right angle measures 90 degrees, and straight angles measures 180 degrees.

Given a diagram, identify the union or intersection of various combinations of rays, segments, angles, etc.

All straight angles are congruent. All right angles are congruent

How to correctly mark segments and angles congruent and how to correctly identify which segments and angles are congruent based on those markings.

What can be assumed (straight lines, etc.) and what cannot be assumed from a diagram (right angles, etc.)

Apply the definition of right angle, straight angle, or congruent angles to write and solve an equation to find a missing angle

Definition of Betweenness: On a segment, piece plus piece equals whole (e.g. if A, B, and C are collinear and B is between A and C, then AB + BC = AC)

Convert an angle written as a decimal to degrees/minutes/seconds (and vice versa)

Angle Addition Property Be able to add or subtract angles in degree/minute/second form.

The difference between substitution and transitive

Be able to find the angle that the hands of a clock make at a certain time.

The steps allowed in an algebraic equation (e.g., can add to both sides, multiply both sides, etc.)

Apply the definition of betweenness to write and solve an equation to find the length of a segment.

When is a conjunction true When is a disjunction true When is a conditional statement true

Apply the Angle Addition Property to write and solve an equation to find the measure of an angle.

An argument is valid when it is in the form:If A, then B.ATherefore: B. (modus ponens)

or If A, then B.If B, then C.Therefore: if A, then C. (chain of reasoning)

Apply the definition of midpoint or angle bisector to write and solve an equation to find the length of a missing segment or the measure of a missing angle.

Given a key geometric term, be able to correctly deduce what is known (e.g., given that M is the midpoint of segment ST, you could deduce that segment SM is congruent to segment MT by definition of midpoint).

Apply the key definitions from Unit 2 (congruence, midpoint, angle bisector, etc.) and the key properties from Unit 2 (substitution, transitive, betweenness, angle addition, etc.) to write a two-column proof to make step-by-step deductions to a final conclusion.

Write a paragraph proof in order to justify a conclusion.

Classify a real-life situation as either deductive or inductive reasoning.

How to write the negation of statement involving “all” (universal statements) or “some” (existential statements)

Complete a truth table involving negations, conjunctions, disjunctions, and conditional statements

Rewrite a statement in if-then (conditional) form.

Given a conditional statement, write the converse, inverse, and contrapositive.

Use logic symbols to rewrite an argument and determine if the argument is valid (possibly using the contrapositive or by reordering the statements) or if the argument is invalid (converse error or inverse error)

Be able to logically deduce the correct conclusion based on given statements.

Someone who has developed key skills and understandings in this unit will be able to….List performance assessments, key assignments and quizzes/tests:

Section 1.1 and 1.2 “Fun with Geometry” worksheet

Section 1.4 Basic Proofs Packet Section 1.5 Proof Packet Quiz on Sections 1.1 – 1.5 Logic Worksheets #1 – 4 End-of-unit test.

Effective homework problems from the text that emphasize problem-solving:

1.1 #1 – 4, 6, 7, 9 – 11, 141.2 #5, 9, 10, 16, 18, 20, 23 1.3 #1, 3, 4, 6, 7 – 13 1.6 #1, 2, 4, 6 – 8

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Chapter 1: Geometry for Enjoyment and Challenge ©1997 McDougall,

Littell and Company

Teacher-generated CD containing handouts, labs, quizzes, tests, etc.

Note cards for the first-day-of-school intro to geometry activity

Please attach any supporting documents (rubrics, assignment sheets etc.) that you’ve developed to this template. (See Geometry CD)



Unit Title: Basic Concepts and Proofs

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? Geometry is all around us. Why are supplementary and complementary

angles so prevalent in the real world? Where are they found?

A proof is a systematic method of documenting mathematical ideas with a chain of reasoning.

What is proof and why is learning to write a proof important?



Drawing a geometric diagram and marking the known information effectively is an important problem-solving technique.

How do you mark known information on a geometric diagram?


Geometer’s Sketchpad Lab How can the concepts involving complementary angles, supplementary angles, addition, subtraction, and the transitive property be used to write an effective, coherent proof?

As always, throughout this course, when new theorems are introduced, they will be proven (discussed and explained) so that students deeply understand the theorems (and do not just memorize them)

What type of geometric situation calls for an algebraic equation that is equal to 90 versus one that is set equal to 180 (versus one in which two quantities are set equal to each other)?

In-class discovery (using a ruler and protractor) of the relationship between vertical angles

How do you choose an appropriate method to solve a system of algebraic equations and execute that method effectively?

What can and cannot be assumed from a geometric diagram?



Identify, describe, and analyze properties of figures, relationships among figures, and relationships among their parts by drawing precisely with paper and pencil or computer software.

Uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among figures.

Uses properties of and relationships among figures to solve

mathematical and real-world problems. Present convincing arguments by means of demonstration,

informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent) and generalizations (e.g., the Pythagorean Theorem holds for all right triangles).

Identify, describe, and analyze properties of 2 and 3 dimensional figures, relationships among their parts (e.g., parallel, perpendicular and congruent sides, diagonals, various types of angles and triangles, complementary and supplementary angles, sum of angles in a triangle).

Understands and applies the concept of a mathematical proof. Constructs logical verifications or counter-examples to test

conjectures and to justify algorithms and solutions to problems (i.e., uses deductive reasoning).


Problem Solving. Students will formulate and solve a wide variety of problems. They will reflect on their thinking during the problem-solving process so that they can apply and adapt the strategies they develop to other problems and in other contexts. By solving mathematical problems, students will acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that serve them well outside the mathematics classroom.






Important new geometric vocabulary: perpendicular, complementary angles, supplementary angles, complement, supplement, linear pairs, perpendicular pairs, vertical angles, opposite rays

Find the supplement or complement of an angle that is written in degree/minute/second form (i.e., be able to add and subtract angles in this form).

Right angles are congruent. Be able to use horizontal and vertical distances

in order to find the missing coordinate of a rectangle plotted on the coordinate plane.

What can be assumed (straight lines, etc.) and what cannot be assumed from a diagram (right angles, etc.)

Apply the definitions of complementary, supplementary, perpendicular, linear pairs, etc. to write and solve a system of equations to find a missing angle.

Congruent Complements and Supplements Theorems (Section 2.4):

If angles are supplementary to the same angle (or congruent angles), then they are congruent to each other.

If angles are complementary to the same angle (or congruent angles), then they are congruent to each other.

Directly translate a word problem involving supplements and complements to set up an algebraic equation to find a missing angle.

Addition and Subtraction Theorems (Section 2.5): If a segment (or angle) is added to the

congruent segments (or angles), the sums are congruent.

If congruent segments (or angles) are added to congruent segments (or angles), the sums are congruent.

If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent.

If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent.

Given a key geometric term, be able to correctly deduce what is known (e.g., given that M is the midpoint of segment ST, you could deduce that segment SM is congruent to segment MT by definition of midpoint).

Transitive and Substitution Properties Apply the Congruent Complements and Supplements Theorems in two-column proofs or to set up an algebraic equation in order to find a missing angle.

Vertical angles are congruent. Write a paragraph proof in order to justify a conclusion.

Apply the Addition and Subtraction Theorems in two-column proofs or to set up an algebraic equation in order to find a missing side or angle.

Apply the Transitive Property in a two-column proof.

Apply “vertical angles are congruent” in a two-column proof or to set up an algebraic equation in order to find a missing angle.

Be able to correctly explain all the theorems from this unit in thorough, coherent sentences.


Section 2.4 Packet on Congruent Complements and Supplements

Quiz: Sections 2.1 – 2.4 Geometer’s Sketchpad Lab: Intro to

Geometer’s Sketchpad Section 2.5 Packet on the Addition and

Subtraction Properties Section 2.7 Packet on the Transitive and

Substitution Properties Section 2.8 Packet on Vertical Angles Chapter 2 Proof Practice Packet End-of-unit test Enrichment worksheet: geometry

crossword puzzles


2.1 #2 – 8, 10 – 14 2.2 #1 – 5, 10 – 13, 17 – 19, 21 – 23 2.3 #1 – 6, 8 – 10, 12, 13


Littell and Company


Exploring Geometry with Geometer’s Sketchpad ; ©2002 Key Curriculum Press A classroom set of computers installed with Geometer’s Sketchpad software




Unit Title: Congruent Triangles

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? Congruent figures have the same size and

shape. What do you know if two triangles are

congruent?


How are the measures of angles and the lengths of sides in triangles connected?

Everything you can do mathematically (i.e., every step in a proof, every step of an algebraic equation) can be justified.


Drawing a geometric diagram and marking the known information effectively is an important problem-solving technique.

When you read given information, how can it be represented on a geometric diagram and used logically in a proof?


Section 3.2 cooperative group activity using rulers and protractors to discover the “shortcuts” for proving triangles congruent (SSS, ASA, SAS)1

What information is required to be able to prove that triangles are congruent? (What are the shortcuts for proving triangles congruent?)

Section 3.2 cooperative group activity where each step of a proof is on a slip of paper and the students need to put the steps in order (CRISS strategy) 2

How do you determine which angle (or which side) in a triangle is the largest (or smallest)?

How do you classify a triangle by its sides or angles?

How can the concepts of CPCTC, altitude, and median be used to write an effective, coherent proof?

When you read information about triangles, how do you mark the picture effectively, prove the triangles congruent, and describe the process in your own words effectively? (See the P3 Project.)



Identify, describe, and analyze properties of figures, relationships among figures, and relationships among their parts using appropriate transformations (translations, reflections, etc.) and by drawing precisely with paper and pencil or computer software.

Uses geometric constructions (e.g., the parallel to a line through a given point, line segment congruent to a given line segment) to complete simple proofs, to model, and to solve mathematical and real-world problems.




Use transformations and symmetry to solve problems. Create and use representations to organize, record, and

communicate mathematical ideas. Communicate mathematical ideas and reasoning using the

vocabulary of mathematics in a variety of ways (e.g., using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).

Understands and applies the concept of a mathematical proof.






Rubric for the P 3 (Picture, Process, Proof) Project (attached)

Homework checks and daily corrections are an on-going self-assessment tool.


GEOMETRY: Picture, Process, Proof – Project Grading Rubric

Possible Points

Student-Assessed

Score

Teacher-Assessed

ScorePictureDo you have a series of at least 4 pictures?Are the corresponding parts of the congruent triangles clearly marked at each stage of the proof?

4

ProcessDid you clearly explain how and why you did each step of the proof?What were you given and what could you say next?How did this help you to reach the conclusion?

4

Proof

Is your proof done accurately and with sound mathematical language, logic, and reasoning?Is your notation correct?Have you used theorems and definitions accurately and appropriately?

4

Degree of Difficulty 1, 2, or 3 See list on the back of this rubric 3

PROJECT TOTAL 15


Important geometric vocabulary: congruent figures, circle, equidistant, center, radius (radii), median, altitude, auxiliary line, scalene triangle, isosceles triangle, legs, base, vertex angle, base angles, equilateral triangle, equiangular, triangle, acute triangle, obtuse triangle, right triangle, hypotenuse

Slide a figure on the coordinate plane horizontally or vertically.

Reflexive Postulate: Any segment or angle is congruent to itself

Reflect a figure on the coordinate plane over the x- or y- axis

The postulates for proving triangles congruent: SSS, SAS, or ASA

Identify which method can be used to prove two triangles congruent (and when there is not enough information to prove two triangles congruent)

CPCTC: Corresponding parts of congruent triangles are congruent

Given congruent triangles, determine the corresponding parts that are congruent (focusing on overlapping triangles)

The formulas for area and circumference of a circle

Write a two-column proof to prove triangles congruent using SSS, SAS, or ASA.

All radii of a circle are congruent After proving triangles congruent, apply CPCTC to prove other facts about the figure

Postulate: Two points determine a line (or ray or segment)

Be able to clearly explain the process of proving triangles congruent (as on P3 Project)

Theorem (Section 3.6): If two sides of a triangle are congruent, then the angles opposite them are congruent. And conversely, if two angles are congruent, then the sides opposite them are congruent.

How to draw and appropriately label medians and altitudes of triangles. (Medians are always on the interior of triangles, but altitudes can be on the outside of a triangle.)

If the two sides of a triangle are not congruent, then the angles opposite those sides are not congruent. The largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

Apply the definitions of altitude and median in a proof or to set up an algebraic equation to find a missing side or angle

The sum of the three angles in a triangle is 180 degrees; therefore, any two angles in a triangle must sum to less than 180 degrees.

Describe a triangle using all terms that apply (scalene, isosceles, equilateral, equiangular, acute, obtuse, right)

HL Postulate Apply the definitions of special triangles (isosceles, right, equilateral, etc.) in a proof or to set up an algebraic equation to find a missing side or angle.

Apply the Section 3.6 Theorem (if angles, then sides) in a proof or to set up an algebraic equation to find a missing side or angle.

Determine which angle (or which side) in a triangle is the largest (or smallest).

Know when and how to apply the HL Postulate in a two-column proof


P 3 (Picture, Process, Proof) Project Take-home mini-quiz on Sections 3.1 – 3.5 End-of-unit test.


3.1 #1 – 5 3.2 #1, 2, 10, 203.2 #3, 4, 11, 14 – 16, 22 3.3 #3, 4, 6, 7, 10, 11, 13 – 15 3.4 #1, 2, 4 – 6, 8, 10, 113.6 #1 – 3, 5, 7- 10, 12 – 14 3.7 #2, 4 – 8, 11, 12, 14, 16 3.8 #2 – 4, 6, 10, 12


Littell and Company


Shared transparencies of the SSS, ASA, and SAS triangles so that the students can check their work on the discovery activity1

Rulers and protractors Cut-up proof puzzles that the students can put together in order 2




Unit Title: Indirect Proof, Parallel Lines, and Angle Relationships

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? A proof is a systematic method of documenting

mathematical ideas with a chain of reasoning. What is proof and why is learning to write a

proof important?

There are different forms of proof. When is best to use a two-column proof as opposed to a indirect proof or a paragraph proof?



Geometry is all around us. Why are parallel lines so prevalent in the real world? Where are they found?


Teacher-generated handouts to facilitate note taking, including the proof of why a triangle has 180 degrees (Section 7.1)

What is the method to write a logical indirect proof?

Investigative Geometer’s Sketchpad Lab How can the concepts involving parallel lines (and angles formed by a transversal intersecting parallel lines) be used to write an effective, coherent proof?

What information is required to prove lines parallel? And conversely, if lines are parallel, what do you know?

How do you apply the theorems involving parallel lines (and angles formed by a transversal intersecting parallel lines) to set up an algebraic equation to find an unknown angle measure?




Uses geometric constructions (e.g., the parallel to a line through a given point, line segment congruent to a given line segment) to complete simple proofs, to model, and to solve mathematical and real-world problems.

Uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among figures (e.g., the relationship among interior angles of parallel lines cut by a transversal).



Use reason and logic to evaluate information, perceive patterns, identify relationships, formulate questions, pose problems, and make and test conjectures.

Understands and applies the concept of a mathematical proof.








Important geometric vocabulary: plane, coplanar, noncoplanar, transversal, alternate interior angles, alternate exterior angles, corresponding angles, exterior angle of a triangle, remote interior angle of a triangle

Write an indirect proof for a conclusion that cannot be proved directly (e.g., lines not parallel, angles not congruent, segments not congruent etc.).

The difference between parallel lines and skew lines

Given congruent (or supplementary) angles, determine which theorem justifies which lines are parallel.

Given parallel lines, determine which theorem justifies which angles are congruent (or supplementary)

The midpoint formula Write a two column proof using the theorems involving parallel lines (the theorems from Sections 5.2 and 5.3).

If two angles are both supplementary and congruent, then they are right angles

Use the theorems involving parallel lines to set up an equation to find the measure of a missing angle.

The components (method) of an indirect proof Apply the Exterior Angle Inequality Theorem to determine the restrictions on an exterior or remote interior angle.

The Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle

Apply the theorems from Section 7.1 to find the missing angle in a figure or to set up an equation in order to solve for a missing angle (including solving a system of equations).

Important theorems (Sections 5.2 and 5.3): Alternate interior (or exterior) angles are

congruent if and only if the lines are parallel. Corresponding angles are congruent if and

only if the lines are parallel. Same side interior (or exterior) angles are

supplementary if and only if the lines are parallel.

If two coplanar lines are perpendicular to a third line, the lines are parallel.

The Parallel Postulate: Through a point not on a line, there is exactly one parallel line to the given line.

Transitive Property of Parallel Lines: If 2 lines are parallel to a third line, then they are parallel to each other.

The sum of the three angles of a triangle is 180 degrees. (Section 7.1)

The measure of an exterior angle of a triangle is equal to the sum of the measure of the two remote interior angles. (Section 7.1)


Geometer’s Sketchpad Lab: Midpoint, Perpendicular, Parallel

End-of-unit test


4.5 #1 – 55.1 #1, 3 – 5, 9, 10, 135.2 #1, 2, 4, 6, 8, 14, 16 – 18, 205.3 #2, 4, 5, 7 – 12, 16, 22 7.1 #2 – 6, 10 – 12, 14, 15, 18

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Sections 4.5, 5.1 – 5.3, and 7.1: Geometry for Enjoyment and Challenge

©1997 McDougall, Littell and Company






Unit Title: Quadrilaterals

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? A quadrilateral is a polygon with four sides. What are the different types of quadrilaterals?

The coordinate plane can be used to model geometric figures and concepts effectively.

When a figure is plotted on the coordinate plane, what type of slopes do parallel sides have? What type of slopes do perpendicular sides have?

Finding the slope of lines can indicate if lines are parallel or perpendicular.


A proof is a systematic method of documenting mathematical ideas with a chain of reasoning


Create-definition-activity: What is a polygon? 1 What are the properties of specific quadrilaterals (parallelograms, rectangles, etc.)?

Cooperative group activity where students use a ruler and protractor to discover the properties of 6 important quadrilaterals (parallelogram, rectangle, square, rhombus, isosceles trapezoid, and kite) and present those properties to the class 2

How are the various quadrilaterals specifically alike and how are they different?

Investigative Geometer’s Sketchpad Lab How can the coordinate plane and slopes be used to determine the most descriptive name of a quadrilateral?

Always/Sometimes/Never partner worksheet to help establish the hierarchy of quadrilaterals

What are the different methods to prove a quadrilateral is a parallelogram in a coherent two-column proof?





Uses properties of and relationships among figures to solve

mathematical and real-world problems (e.g., uses the property that the sum of the angles in a quadrilateral is equal to 360 degrees to square up the frame for a building).


Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results.

Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g., using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).








Important geometric definitions: polygon, convex polygons, diagonals of polygons, quadrilateral, parallelogram, rectangle, rhombus, kite, square, trapezoid, isosceles trapezoid

Use Geometer’s sketchpad and a ruler and protractor to discover the important properties of the quadrilaterals.

The properties of each quadrilateral. For example, in a parallelogram, opposite sides are parallel (this is the definition), opposite sides are congruent, opposite angles are congruent, the diagonals bisect each other, and consecutive angles are supplementary. The properties of all the quadrilaterals are nicely summarized in Section 5.5.

Apply the properties of the quadrilaterals to set up an equation to find a missing side or angle.

The hierarchy of quadrilaterals (e.g., a square is a rectangle, but a rectangle is not necessarily a square)

Write a two-column proof the uses the properties of the quadrilaterals.

There are 5 ways to prove a quadrilateral is a parallelogram. Show that:

- Both pairs of opposite sides are parallel.- Both pairs of opposite sides are congruent.- Both pairs of opposite angles are congruent.- One pair of opposite sides are both parallel and congruent.- The diagonals of the quadrilateral bisect each other.

Prove that a quadrilateral is a parallelogram (or rectangle, a parallelogram with one right angle).

The slope formula and midpoint formula. Find the most descriptive name of a quadrilateral plotted on the coordinate plane by analyzing the slope of opposite and consecutive sides.

Parallel lines have the same slope; perpendicular lines have opposite reciprocal slopes.

When the midpoints of the sides of a quadrilateral are connected consecutively, a parallelogram is formed.


Geometer’s Sketchpad Lab: Quadrilaterals

End-of-unit test


5.4 #1 – 3, 6 – 12, 16, 18 – 205.5 #1 – 3, 5 – 7, 11, 12 5.5 #14 – 17, 19, 20, 23, 27 5.6 #1 – 3, 5 – 6, 8, 10 – 12, 14, 175.7 #2, 3, 6, 8, 10, 11, 14, 23, 28

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Sections 5.4 – 5.7: Geometry for Enjoyment and Challenge ©1997

McDougall, Littell and Company



Shared transparencies of examples and non-examples for the What is a Polygon? activity 1

Worksheets with drawings of parallelograms, rectangles, etc. that were drawn using Geometer’s Sketchpad. (The students need accurate shapes in order to correctly discover the important properties.) 2

rulers and protractors




Unit Title: Polygons

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? In mathematics, using a formula is often a

useful and efficient way to solve a problem. How can techniques like drawing a picture and

marking the known information help you derive a formula? What other techniques are useful when deriving formulas?

Formulas do not need to be memorized; they can be derived.

What does it mean to be able to work with a formula “forwards and backwards”?

Problems in mathematics can frequently be solved in more than one way.

Why can two different formulas be applied to a situation and yield the same result?





In what context will the understanding be applied?

What “essential” and “unit” questions will focus this unit?

Teacher-led derivations of the formulas for the sum of the interior angles of a polygon, the sum of the exterior angles of a polygon, and the number of diagonals in a polygon (Section 7.3)

How can dividing a polygon into triangles help you derive the formula for the sum of the interior angles?

Geometer’s Sketchpad Lab How can the concepts of AAS and the No Choice Theorem be used to write a coherent proof of a specific geometric idea?

Good class discussion of the No Choice Theorem and why it leads to the fact that AAS is another shortcut for proving triangles congruent (Section 7.2 Lesson)

How do you solve a quadratic equation?

How do you analyze given information about a polygon’s angles or diagonals and choose the right formula and solve it correctly to find a missing quantity?








Measurement. Students will understand the attributes, units, systems, and processes of measurement as well as apply the techniques, tools, and formulas to determine measurements to a specific degree of accuracy.

Determine measurements indirectly using geometric relationships and properties of polygons (e.g., size of central angles).

Uses a variety of strategies (e.g., identify a pattern, use equivalent representations) to understand new mathematical content and to develop more efficient solution methods or problem extensions.





Given the targeted understandings, other unit goals, and the assessment evidence identified, what knowledge and skill are central to this unit?Students will need to know… (Information) Students will need to be able to do… (Skills)

The formulas for…- the sum of the interior angles of a polygon- the number of diagonals in a polygon- the central angle measure of a regular

Apply the appropriate formula to find the missing interior angle measure, exterior angle, measure, the sum of the interior angles, the number of diagonals, the number of sides, or

polygon- the interior angle measure of a regular polygon

(The students should memorize these formulas or be able to derive these formulas quickly.)

the measure of the central angle of a regular polygon.

Important geometric vocabulary: polygon, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, n-gon, central angle of a regular polygon, interior angle and exterior angle of a polygon

Apply the Midline Theorem to find the missing angles or missing side lengths in a figure.

The sum of the exterior angles of a polygon is always 360 degrees.

Write a two-column proof using the No Choice and AAS Theorems

The Midline Theorem (Section 7.1) Be able to explain why the formula for the sum of the interior angles of a polygon works.

The interior angle of a polygon is supplementary to its adjacent exterior angle.

No Choice Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

AAS is another shortcut for proving triangles congruent


Geometer’s Sketchpad Labs: Regular Polygons

End-of-unit test.


7.3 #1 – 4, 6 – 8, 10 – 13, 177.1 #7, 8, 17 and 7.3 #57.4 #1 – 5, 10 – 12, 14, 17Chapter 7 Review: #2, 3, 7 – 17, 20, 21, 24, 25

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Sections 7.1 – 7.4: Geometry for Enjoyment and Challenge ©1997







Unit Title: Lines and Planes in Space

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? Geometry is not just two-dimensional.

Geometry is all around us in our three-dimensional world.

How can a real-life three-dimensional object be represented by a smaller 3-D model and by a two-dimensional drawing?

Using concrete models can be helpful to interpret, understand, and visualize a geometric concept.




Teacher-generated handouts to facilitate note taking; the theorems involving points, lines, and planes can be modeled using real-life objects (e.g., point – tennis ball, line – ruler, plane – piece of poster board)

How do you draw a two-dimensional drawing of a three-dimensional object?

3-D Project How do you interpret a two-dimensional drawing of a three-dimensional object in order to construct a concrete model of that object that is mathematically accurate?

In-class partner Always/Sometimes/Never worksheet to work on visualization skills

How do you use the concepts of skew, foot, and plane (and the intersection of lines and planes in 3-D space) to write a coherent two-column proof?



Identify, describe, and analyze properties of figures, relationships among figures, and relationships among their parts by constructing physical models and by drawing precisely with paper and pencil or computer software.

Use geometric models to solve mathematical and real-world problems.

Visualize 3-dimensional figures in problem solving situations.

Present convincing arguments by means of demonstration, informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent) and generalizations (e.g., the Pythagorean

Theorem holds for all right triangles).

Representation. Students will represent mathematical ideas in a variety of ways such as pictures, concrete materials, tables, graphs, number and letter symbols, and spreadsheets. They will acquire a set of tools to model and interpret physical, social, and mathematical phenomena.


Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g., using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).

Read and understand mathematical texts and other instructional materials.





3-D Project Grading Rubric (attached) Homework checks and daily corrections are an on-going self-assessment tool.


Geometry 3-D Project Rubric:

Possible Points Student-Assessed Score

Teacher-Assessed Score

Proof Degree of Difficulty Basic - 6 steps or less 3 2 triangles congruent, CPCTC Intermediate - 7-10 steps 4 More than one set of triangles congruent Advanced - 11 or more steps 5 Multiple sets of triangles congruent Uses new theorems from Chapter 6 Proof Degree of Difficulty Total 5

Proof

Problem identified2

Two-dimensional drawing included and is well-drawn by hand

4

Overall concept of proof 5 Accuracy and completeness of proof 5 Proof is neat and easy to read 4

Proof Total20

Model Degree of Difficulty Basic – constructs easily 3 One plane Figure made of 2 triangles Intermediate – more difficult construction 4 One plane or two parallel planes 3 or more intersecting triangles 3 triangles that create a solid Advanced – very difficult construction 5 3 or more parallel planes 4 or more triangles that create a solid Unique construction features Model Degree of Difficulty Total 5

Model Dimensions of the base of the model no larger than 18” 2

Lines are straight 2 Planes are flat and solid 2 Planes, lines, etc. connected well; model is sturdy and well-

constructed3

Figure is accurately represented 4 Points are labeled & visible from all angles 2 Your name is written clearly on the model 1 Visual appearance (use of color, creativity, artistic qualities, use

of creative building materials – not just foam board)4

Model Total 20

PROJECT TOTAL 50


Important geometric vocabulary: plane, foot, skew

Use postulates and theorems from this unit to prove a relationship about a 3-D figure (with special emphasis on proving triangles congruent). (As an extension the students can type their proof using Equation Editor.)

There are 4 ways to determine a plane: three noncollinear points a line and a point not on the line two intersecting lines two parallel lines

Build a accurate, sturdy 3-D model of a diagram

If a line intersects a plane not containing it, then the intersection is exactly one point

Draw a two-dimensional diagram to represent a 3-D model. (As an extension the students can also draw the diagram using Geometer’s Sketchpad.)

If two planes intersect, their intersection is exactly one line.

If a line is perpendicular to a plane, then it is perpendicular to every line in the plane that passes through its foot.

If a line is perpendicular to two distinct lines that lie in the plane and that pass through its foot, then it is perpendicular to the plane.

If a plane intersects two parallel planes, the lines of intersection are parallel.

Transitive property of parallel lines and planes


3-D Project (Description Sheet, Rubric,, Approximate Point Values, and Equation Editor Direction Sheet)


6.1 #1 – 6, 11, 12


Littell and Company



Ruler and protractors




Unit Title: Similar Polygons

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? Similar figures have the same shape but not

necessarily the same size. What do you know if two figures are similar?

(The corresponding angles are congruent, and the corresponding sides are proportional.)

Ratios are used in the real-world to compare two quantities, and a proportion is when two ratios are set equal to one another.

How do similar figures compare to congruent figures?


How do you solve a proportion?




Ratio and proportion are easily applied to real-life (ratio of boys to girls in class, distances on a map, finding lengths of shadows

What information is needed to prove triangles similar?

In Section 8.3, good class discussion on why AA is the shortcut to prove triangles similar and not AAA.

Why is AA (and not AAA) sufficient to prove triangles similar? Why is it insufficient to prove triangles congruent?

How do you use the concepts involving similarity to write a coherent two-column proof?

How do you determine the correspondence between similar figures in order to set up a proportion that can be solved algebraically to find a missing side?



Use proportional reasoning to solve congruence and similarity problems (e.g., scale drawings and similar geometric figures).



Understands and applies the concept of a mathematical proof. Connect mathematics to the real world, as well as within

mathematics. Selects and uses an appropriate algorithm for multi-step and non-

routine problems. Reasoning and Proof.

Students will reason, think analytically, and recognize patterns, structure, or regularities in both real-world and mathematical conjectures. They will develop and evaluate formal and informal mathematical arguments and proofs. Students will see and expect that mathematics make sense.

Connections. Students will connect mathematical ideas and view mathematics as a coherent whole. They will relate mathematics to other subjects, real-world situations, and their own interests and experience.






Important geometric vocabulary: ratio, proportion, means, extremes, arithmetic mean, geometric mean, transversal

Compute a ratio about a real-life situation and reduce it to lowest terms.

Means-Extremes Product Theorem (cross multiplication)

Compute the arithmetic mean (average) and geometric mean between two numbers

In similar (symbol: ~) polygons, the corresponding angles are congruent and the corresponding sides are proportional.

Apply the Means-Extremes Product Theorem to algebraically solve for a ratio like x:y

AA (Angle-Angle) is a valuable shortcut for proving triangles similar

Set up a proportion to solve a real-life problem.

If two figures are similar, the ratio of their corresponding sides is the same as the ratio of

Set up a proportion to find the missing side of a

their perimeters. pair of similar figures

Three Theorems Involving Proportions (Section 8.5):

If a line is parallel to one side of a triangle and intersects the other two sides, it divides them proportionally. (Side Splitter Theorem)

If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides.

Determine which two triangles are similar in a given diagram (paying careful attention to the correct correspondence among the sides). Then, be able to justify why AA was the method used to prove the triangles similar. Finally, knowing that corresponding sides of similar triangles are proportional, set up a proportion to find the missing piece of information.

Apply the Section 8.5 Theorems to set up a proportion to algebraically find the missing side of a figure

Write a two-column proof using Chapter 8 Theorems and concepts.


Snowflake Projection/Dilation Activity End-of-unit test.


8.1 #3, 4, 6, 8, 11, 14, 17, 19, 20, 22, 268.2 #2, 3, 5, 7 – 10, 12 – 15, 188.3 #1 – 5, 8, 10, 12, 16, 198.4 #2 -4, 6 – 9, 11, 12, 16 – 20 8.5 #3, 4, 6 – 8, 11 – 15, 20, 22, 26


Littell and Company


ruler and protractors




Unit Title: Triangle Inequalities

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? In a triangle, there are relationships among the

length of the sides and the size of the angles. How are the lengths of sides and the measures

of angles opposite those sides connected?







Teacher-generated handouts to facilitate note taking; demonstrate the hinge theorem using two chalkboard compasses (they make great hinges!)

What is the Hinge Theorem?

When looking at a triangle, how do you determine which angle (or side) is the largest (or smallest)?

How do you know when the Hinge Theorem applies and when it is “longest side is opposite largest angle”?

How can you incorporate the Hinge Theorem and other concepts involving exterior angles of a triangle and the sum of the angles of a triangle in order to prove a geometric concept?

How can you apply the Exterior Angle Theorem and/or the fact that the sum of the angles of a triangle is 180 degrees in order to set up an algebraic equation to solve for a missing angle?

Standards: What grade level outcomes/benchmarks are taught and assessed? Geometry. Students

will be able to analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use


Present convincing arguments by means of demonstration, informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent) and generalizations (e.g., the Pythagorean Theorem holds for all right

visualization, spatial reasoning, and geometric modeling to solve problems.

triangles). Understands and applies the concept of a mathematical proof. Constructs logical verifications or counter-examples to test conjectures

and to justify algorithms and solutions to problems (i.e., uses deductive reasoning).







Triangle Inequality Postulate (Section 15.2) Apply the Triangle Inequality Postulate to determine if a triangle with the given sides exists

The Pythagorean Theorem Test:

If a2+b2=c2

, the triangle is right.

If a2+b2<c2 , the triangle is obtuse.

If a2+b2>c2 , the triangle is acute.

Apply the Pythagorean Theorem Test to determine if a triangle is right, obtuse, or acute

Exterior Angle Inequality Theorem (Section 15.2)

Apply the Triangle Inequality Postulate to determine a possible range of values for the

missing side of a triangle

In a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. (Section 15.2)

Apply the Exterior Angle Inequality Theorem to determine a possible range of values for a missing remote interior angle or exterior angle of a triangle

Important geometric vocabulary: exterior angle of a triangle, remote interior angle of a triangle

Write a two-column proof using the theorems and postulates from Sections 15.2 and 15.3.

The sum of the angles of a triangle is always 180 degrees.

Be able to list the sides and angles of a triangle in increasing or decreasing order*

The Hinge Theorem and the Converse Hinge Theorem (Section 15.3)

Apply the Hinge Theorem and its converse to determine which angle (or side) is larger*

*Overall, the students must be able to identify when they are looking at one triangle and when they are comparing two triangles. An important part of this unit is being able to justify when the Hinge Theorem is used and when the concept of “largest angle is opposite longest side” is being used.


15.3 Inequalities packet and 15.3 More Practice handout for the students to work on collaboratively in class

End-of-unit test.


15.2 #1 – 5, 7 – 12, 14, 18, 2015.3 #1 – 7, 9, 12, 15, 16

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Sections 15.2 and 15.3: Geometry for Enjoyment and Challenge ©1997






Unit Title: The Pythagorean Theorem

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? The Pythagorean Theorem is the most

important theorem in geometry. What is the Pythagorean Theorem and why is it

so useful in solving real-life problems?

Geometric concepts and figures can be represented on the coordinate plane.

How can the coordinate plane help to demonstrate that the distance between two points is really just the Pythagorean Theorem?

There are many ways to solve a quadratic equation.

When faced with a quadratic equation, how do you decide what method to use to solve it?


Teacher-generated handouts to facilitate note taking, including two teacher-led derivations of the Pythagorean Theorem (Section 9.4 of the text).

How do you apply the Pythagorean Theorem and its converse?

Cooperative group mini-exploration of the distance formula (Section 9.5 of the text).

What is a Pythagorean triple and how can it make you work more efficiently?

Investigative Geometer’s Sketchpad Labs When a triangle is plotted on the coordinate plane, how can you verify that the triangle is right by using slope and the distance formula?

How do you solve a quadratic equation by factoring, by taking square roots, and by using the Quadratic Formula?

Standards:What grade level outcomes/benchmarks are taught and assessed?




Uses the Pythagorean Theorem and its converse to solve mathematical problems.

Use the two-dimensional rectangular coordinate system and algebraic procedures to describe and characterize geometric properties and relationships (e.g., slope, parallelism, perpendicularity, Pythagorean Theorem, distance formula).


Solve and analyze routine and non-routine problems. Understands connections between equivalent representations and

corresponding procedures of the same problem situation or mathematical concept (e.g., connections between algebraic and graphing solutions).







Simplest radical form Quadratic Formula

Perform operations involving square roots and write the answer in simplest radical form.

New geometric terms: arc, circumference, area, arc measure, arc length, sector, central angle, inscribed angle, altitude, hypotenuse.

The measure of an inscribed angle is half the measure of the intercepted arc.

The Pythagorean Theorem The distance formula Perpendicular lines form right angles and the

slope of perpendicular lines are opposite reciprocals.

Pythagorean triples

Solve a quadratic equation (and simplify the answer completely)

- by finding square roots - by factoring - by using the quadratic formula Find the arc measure, arc length, and area of a

sector of a circle. Use similar triangle proportions (altitude-on-

hypotenuse theorems) to find the length of an altitude (leg) or hypotenuse of a right triangle.

Use the Pythagorean Theorem to find the length of the unknown leg or hypotenuse in a right triangle.

Apply the Pythagorean Theorem to more advanced problem-solving situations (e.g., find the altitude of an isosceles trapezoid, the perimeter of a rhombus, the missing side of a spiral, etc.)

Find the distance between two points on the coordinate plane.

Verify that a triangle is right by comparing slopes and by using the distance formula and the Pythagorean Theorem.

Recognize that Pythagorean triples (and multiples of those triples) can be useful to know in order to speed up mathematical processes.


Worksheets to review algebra skills involving square roots

Geometer’s Sketchpad Labs: The Pythagorean Theorem and The Square Root Spiral

End-of-unit test.


9.1 #1 – 4, 9, 12, 139.1 #5 – 8, 10, 119.2 #1 – 12, 14 – 16 9.3 #1, 3abc, 4ab, 8 – 14 9.3 #3def, 4cd, 6, 7, 16, 17, 219.4 #2 – 5, 7, 9, 12 – 14, 16 – 18, 229.5 #1 – 3, 5 – 7, 10, 14, 15, 17, 219.6 #1 – 8, 10 – 12, 15, 16, 21

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Chapter 9 (Sections 9.1 – 9.6): Geometry for Enjoyment and Challenge







Unit Title: Special Right Triangles and Trigonometry

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions?

The Pythagorean Theorem is the most useful formula in geometry, because right triangles are everywhere you look.

Why are drawing pictures, dropping altitudes, and labeling unknown sides as x helpful techniques in deriving relationships and formulas? What are other helpful techniques?

The Pythagorean Theorem can be used to derive other important theorems and relationships.

Why do the trig ratios stay the same regardless of the size of the right triangle?

When in doubt about how to approach a problem, break a complex geometric figure into smaller, simpler figures.

What is SOHCAHTOA and when can it be applied?

When in doubt about how to approach a problem, look for right triangles in order to apply the Pythagorean Theorem.

When do you use inverse sine, cosine, or tangent to solve a problem?

Right triangles are the basis of sine, cosine, and tangent (right triangle trigonometry); trigonometry is essential in solving real-life problems.


Teacher-generated handouts to facilitate note taking, including teacher-led derivations of the 30-60-90 Triangle Theorem and the 45-45-90 Triangle Theorem (Section 9.7 of the text) and the derivation of the formula for the diagonal of a box (Section 9.8 of the text).

How can you use the Pythagorean Theorem to derive the special right triangle relationships?

Investigative Geometer’s Sketchpad Lab How is the formula for finding the length of a diagonal of a box just an example of using the Pythagorean Theorem twice?

What are the relationships among the sides of the 30-60-90 and 45-45-90 special right triangles and how do you apply those relationships to find an unknown side?

How do you apply SOHCAHTOA to find the unknown side in a geometric or real-life situation?

How do you apply inverse trig functions in order to find the unknown angle in a geometric or real-life situation?





Uses the Pythagorean Theorem and its converse and properties of special right triangles (e.g., 30-60-90 triangle) to solve mathematical problems.

Understands the basic concepts of right triangle trigonometry (sine, cosine, and tangent) and uses right triangle trigonometric ratio methods to solve mathematical and real-world problems.

Uses geometric models to solve mathematical and real-world problems.



Selects and uses an appropriate algorithm for multi-step and non-routine problems.

Connect mathematics to the real-world, as well as within mathematics.





Review packets are provided before every quiz and test so students can practice the necessary skills and familiarize themselves with the format and directions they will encounter on the assessment. An answer key is provided so students can assess how well

they have mastered the material before the actual quiz or test.


New geometric terms (with regard to space figures): face, edge, diagonal, base, pyramid, lateral edge, slant height

Three primary trigonometry ratios: sine, cosine, and tangent

Angle of elevation and angle of depression Inverse sine, cosine, and tangent are used to

find the angle of a right triangle when two sides are known.

Use the Pythagorean Theorem to derive the special right triangle relationships and the formula for the diagonal of a box.

Use the special right triangle relationships to find the missing sides of 30-60-90 and 45-45-90 triangles effectively.

Apply the special right triangle relationships to more advanced problem-solving situations (e.g., finding the perimeter of an equilateral triangle, trapezoid, etc.)

Find the length of the diagonal of a box. Apply the formula for the diagonal of a box to

more advanced problem-solving situations. Use the Pythagorean Theorem to find the

length of the altitude, slant height, lateral edge, and base edge of a pyramid.

Find the sine, cosine, and/or tangent of an angle.

Use sine, cosine, and tangent to find the missing side of a right triangle (including real-life applications involving angle of elevation and depression).

Use inverse sine, cosine, and tangent to find the missing angle of a right triangle (including real-life applications involving angle of elevation and depression).


Introductory Geometer’s Sketchpad Lab to explore special right triangles.

Worksheets to supplement mastery of special right triangle relationships (for use after Sections 9.7 and 9.8).

Worksheets with more in-depth real-life applications of trigonometry (for use after Sections 9.9 and 9.10).

End-of-unit test.


9.7 #1 – 5, 7 – 9, 14 – 199.8 #1, 3 – 6, 13 – 17, 209.9 #1, 3 – 9, 12 9.9 #13 – 17, 22 9.10 #4 – 7, 9 – 11

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Chapter 9 (Sections 9.7 – 9.10): Geometry for Enjoyment and Challenge







Unit Title: Circles

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? When in doubt about problem situations

involving circles, look for (and or create) right angles in order to apply the Pythagorean Theorem.

What is meant by the distance from the center of a circle to a chord? What connection does this have to the Pythagorean Theorem?






Discovery activity of the Congruent Chords Theorem using a ruler and protractor (Section 10.2)

What are the relationships among radii, tangents, and chords of circles that allow us to apply the Pythagorean Theorem?

Discovery activity involving arcs of circles using a ruler and protractor (Section 10.3)

How can you use the theorems and relationships involving chords, arcs, and angles of circles to write an effective two-column proof?

Teacher-generated handout of two externally tangent circles to facilitate learning the common tangent procedure (Section 10.4)

How can you use the theorems and relationships involving chords, arcs, and angles of circles to set up an equation to algebraically solve for a missing piece of information?

In-class partner worksheet (Section 10.4) How can the walk-around method be applied to solve for a missing piece of information involving circles?

Cooperative group discovery worksheet involving inscribed and circumscribed polygons and coordinate geometry (Section 10.7)

How do you apply the common external tangent procedure to find the length of the common external tangent of two circles (when given the circles’ radii)?

Investigative Geometer’s Sketchpad Lab


Geometry. Students will be able to analyze characteristics of geometric shapes and make mathematical arguments


Uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among

about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems.

figures. Uses geometric models to solve mathematical and real-world

problems. Uses properties of and relationships among figures to solve

mathematical and real-world problems. Present convincing arguments by means of demonstration,

informal proof, counter-examples, or any other logical means to show the truth of statements (e.g., these two triangles are not congruent) and generalizations (e.g., the Pythagorean Theorem holds for all right triangles).


Understands and applies the concept of a mathematical proof. Selects and uses an appropriate algorithm for multi-step and

non-routine problems.







The geometric definition of a circle (the set of all points equidistant from a fixed point called the center)

Complete a diagram and apply the Sections 10.1 – 10.4 Theorems to find the length of a chord, radius, tangent, or distance from the chord to the center of a circle. (Overall, the students need to be able to apply any of the circle theorems in order to find a missing piece of information; this often involves creating a right triangle.)

All radii of a circle are congruent. Use the walk-around method to find missing information (e.g., the radii of three externally tangent circles or the perimeter of a quadrilateral circumscribed about a circle)

The distance from the center of a circle to a chord of the same circle is the perpendicular distance from the center to the chord.

Find the length of the common external tangent of two circles when given the circles’ radii

New geometric vocabulary: concentric circles, chord, common chord, circumscribed, inscribed, arc, major arc, minor arc, semicircle, secant, tangent, point of tangency, common external tangent, common internal tangent, externally tangent, internally tangent, central angle, inscribed angle, tangent-chord angle, chord-chord angle, secant-secant angle, secant-tangent angle, tangent-tangent angle

Complete a two-column proof using circle theorems and be able to describe the theorems in one’s own words

New theorems involving circles: If a radius is perpendicular to a chord,

then it bisects the chord. (10.1) Two chords are congruent if and only if

they are equidistant from the center. (10.2)

Two central angles are congruent if and only if their intercepted arcs (or corresponding chords) are congruent. (10.3)

Two-tangent theorem (two tangents drawn from the same external point to the same circle are congruent). (10.4)

The measure of an inscribed angle or tangent-chord angle is ½ the measure of the intercepted arc. (10.5)

The measure of a chord-chord angle is ½ the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle .(10.5)

The measure of a secant-secant angle, secant-tangent angle, or tangent-tangent angle is ½ the difference of the measures of the intercepted arcs. (10.5)

Use the inscribed angle theorem to find the measures of angles and arcs associated with a quadrilateral inscribed in a circle.

Two inscribed (or tangent-chord) angles that intercept the same (or congruent) arcs are congruent. (10.6)

An angle inscribed in a semi-circle is a right angle. (10.6)

The sum of the measures of a tangent-tangent angle and its minor arc is 180. (10.6)

A quadrilateral inscribed in a circle has opposite angles that are supplementary. (10.7)

A tangent line is perpendicular to the radius drawn to the point of tangency

Apply the Section 10.5, 10.6, and 10.7 Theorems to find the measure of angles and arcs.

Walk-around method Find the radius of a circle circumscribed about or inscribed in a regular polygon

The measure of a central angle of a circle is the same as its intercepted arc.

Find the perimeter of a figure comprised of circles, semicircles, rectangles, squares, triangles, etc.

The formula for the circumference of a circle Find the arc length of a circle

The arc length is a fraction of the circumference, dependent upon the measure of the central angle


Quiz: Sections 10.1 – 10.4 Geometer’s Sketchpad Lab: Tangents to a

Circle End-of-unit test.


10.1 #2, 4 – 6, 8 – 12, 14, 15, 20, 2310.2 #2 – 4, 7, 8, 10 – 13 10.3 #2 – 13, 16 – 19 10.4 #1 – 5, 7 – 12 10.4 #13 – 18, 2210.5 #1 – 1410.5 #15, 16, 19 – 27 10.6 #4, 5, 7, 9, 11 – 15, 17, 23, 24, 2710.7 #1 – 3, 5, 6, 9 – 12, 14 – 16, 2010.9 #2ab, 3ac, 4 – 8, 10 – 17

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Chapter 10 (excluding 10.8): Geometry for Enjoyment and Challenge




ruler and protractors




Unit Title: Area

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? When in doubt about how to approach a

problem, break a complex geometric figure into smaller, simpler figures.

What’s the most efficient method for finding the area of any given shape?

When in doubt about how to approach a problem, look for right triangles in order to apply the Pythagorean Theorem.

Why can the area of shapes be found in different ways?

There are multiple approaches in problem solving.

When do you add areas and when do you subtract areas?

Areas can be added (as in finding the area of two combined shapes) or subtracted (as in finding a “leftover” area, the area of a shaded region).

How is perimeter different from area?

Area is measured in square units and is the amount of two-dimensional space a figure occupies.

If you forget an area formula, how can dropping an altitude or breaking the figure into simpler shapes like triangles and rectangles help you?

Every area formula can be derived and follow from the area of a rectangle.

What is a tessellation?

Geometry has fabulous connections to art.



Teacher-generated handouts to facilitate note taking, including visual/algebraic derivations of the area formulas of parallelograms, triangles (Section 11.2), trapezoids (Section 11.3), kites (11.4), equilateral triangles and regular polygons (11.5)

How do you find the area of a parallelogram, triangle, trapezoid, kite, etc.?

Tessellation Project How can you use area to work backwards to find a missing height or other missing piece of information?

Investigative Geometer’s Sketchpad Lab How do you derive the formulas for the area of a parallelogram, triangle, trapezoid, kite, etc.?

What are the different methods for tessellating figures? How do you create a tessellation?



Identify, describe, and analyze properties of figures, relationships among figures, and relationships among their parts using appropriate transformations (translations, rotations, reflections) and by drawing precisely with paper and pencil or computer software.



Use transformations and symmetry to solve problems.


Understands procedures for basic indirect measurements (e.g., using grids to estimate area of irregular figures).

Understands formulas for finding measures (e.g., area, surface area, volume).

Determine the perimeter/area of two-dimensional figures. Determine measurements indirectly using geometric

relationships and properties of circles (e.g., area of a sector of a circle).

Determines measurements indirectly using geometric formulas to derive length, area, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids).

Connect mathematics to the real world, as well as within mathematics.

Solve an analyze routine and non-routine problems.




Rubric for the Tessellation Project (attached) Homework checks and daily corrections are an on-going self-assessment tool.


TESSELLATION PROJECT

Possible Points

Student Score

Teacher Score Description

5Rough draft is submitted on time and approved

5Tessellation cutout and template are turned in with final project

5Degree of difficulty – to help assess this, please complete the following:

Original shape ________________________

Types of tessellations (check all that apply) ______ Translation

______ Rotation about a midpoint

______ Rotation about a vertex

______ Other method discovered by student

Other factors that are considered include each side altered in some manner (no long flat sides), simple vs. complicated designs, and combining tessellation methods.

5Accuracy and completeness of tessellation – shape remains constant, no gaps, overlaps, or distortion, tessellated to the edges of the paper. (Your teacher cannot check the accuracy if you do not turn in your template.)

2Tessellation is creatively titled

Title: _____________________________________________

5Your shape should be something (i.e. not just a blob) or have an artistic theme (Be original and creative!)

5Coloring is neat and artistic, details are added

3Each tessellation shape is outlined so that it stands out(outlining in pencil is not enough – dark colored pencil is okay)

35Grade:


The difference between perimeter/circumference (linear units) and area (square units)

How to find the area (and perimeter) of a- square- rectangle- parallelogram- triangle (all types)- circle- trapezoid - kite- rhombus- regular polygon

(Note: The students should also be able to find the area of a shape that is a combination of any of the shapes above.)

The area formulas for a- square, rectangle, parallelogram - triangle - circle- trapezoid* - kite, rhombus* - regular polygon*- equilateral triangle*

* The area of these figures can also be found by dropping altitudes and/or breaking the figure up into more manageable rectangles and triangles, but using the formula is often the most efficient method.

Given the area of a shape (trapezoid, triangle, etc.), work “backwards” to find the missing dimension

New geometric vocabulary: inscribed, circumscribed, apothem, sector, segment of a circle, semiperimeter, cyclic quadrilateral

Use counting (subtraction) to find the horizontal distance (base) and vertical distance (height) in order to find the area of a triangle plotted on the coordinate plane

Hero’s Formula (Section 11.8) Use subtraction to find the area of a shaded region (e.g., the sector of a circle)

Brahmagupta’s Formula (Section 11.8) Calculate the ratio of the areas of triangles with the same height (and be able to explain why it is the same as the ratio of the bases)

Calculate the ratio of the areas of two similar triangles

Use Hero’s Formula to find the area of any triangle

Use Brahmagupta’s Formula to find the area of a cyclic quadrilateral


Quiz: Sections 11.1 – 11.5 Geometer’s Sketchpad Labs: Trapezoid

Area Exploration, Tessellations Tessellation Project (description of project,

description of methods, rubric, supplies) 1

End-of-unit test.


11.1 #1, 3, 6 – 13, 16 11.2 #3, 4, 7, 8, 10, 11b, 13 – 19, 22c, 23, 26, 3111.3 #2 – 8, 11 – 16 11.5 #2bd, 3bd, 5, 8, 10 – 13, 15, 1711.6 #2 – 4, 5cde, 6, 10, 11ab, 14abc, 15, 18a, 20b11.7 #1bd, 2ad, 3 – 5, 7 – 16 11.8 #1ce, 2, 3ab, 4 – 7, 9, 10, 12

Instructional Resources: Text, media, resources, manipulatives, etc. Geometry textbook Chapter 11: Geometry for Enjoyment and Challenge ©1997 McDougall, Littell

and Company



Note cards, tape, scissors, and starting shapes (square, rhombus, equilateral triangle, rectangle, hexagon) for Tessellation Project 1




Unit Title: Surface Area and Volume

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? Total surface area is the sum of the areas of

each face of a figure. What is the difference between surface area

and volume?

Volume is measured in cubic units and is the amount of space a three-dimensional shape occupies.

How many cones fill a cylinder of the same height and radius?

In general, volume is based on the concept of the area of the base times the height.

How many square pyramids fill a square prism with the same height and base edge?

Surface area and volume have a lot of important real-life applications.

How are the volumes of figures with one base connected? How are the volumes of figures with two bases connected?

Breaking a more complex geometric figure into smaller, more manageable pieces is extremely important.

Why are the area formula for triangles, rectangles, and circles so important in finding total surface area?

Drawing diagrams is an essential part of the visualization process in geometry.


In-class demonstration of surface area of a cylinder using a model; teacher-led derivation of the surface area of a cone1 (Section 12.3)

How do you find the volume of prisms, cylinders, pyramids, cones, and spheres?

In-class demonstration of the volume of a prism/cylinder using stacking models (Section 12.4); real-life demonstration of a problem like #5 on the Section 12.4 Lesson2

How do you find the surface area of prisms, cylinders, pyramids, cones, and spheres?

In-class demonstration using water to show why the volume of a pyramid/cone is one-third the volume of a prism/cylinder3

How do you use surface area or volume to work backwards to finding a missing piece of information (including in real-life problem situations)?

How can you derive the formula for the surface area of a cylinder by visualizing the cylinder cut apart and flattened out into a rectangle and two circles?

What is a frustum and how is it connected to a cone?

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

How do you draw an appropriate sphere, cylinder, prism, etc. that models a real-life

scenario that is comprised of those figures?



Uses geometric models to solve mathematical and real-world problems.

Visualize 3-dimensional figures in problem solving situations. Uses properties of and relationships among figures to solve

mathematical and real-world problems.


Understands formulas for finding measures (e.g., area, surface area, volume).

Determine the area/volume of three-dimensional figures. Solves problems involving there-dimensional measures (e.g.,

volume, surface area). Determines measurements indirectly using geometric formulas

to derive length, area, or volumes of shapes and objects (e.g., cones, parallelograms, cylinders, pyramids).


Analyze non-routine problems and arrive at solutions by various means, including models and simulations, often starting with provisional conjectures and progressing, directly or indirectly, to a solution, justification, or counter-example.

Selects and uses an appropriate algorithm for multi-step and non-routine problems.








The difference between lateral area and total surface area

Find the lateral area, total surface area, and volume of

- prisms- cylinders (including fractions of cylinders) - pyramids - cones- frustums- spheres (including hemispheres)

The students should also be able to find the surface area and volume of figures that are made up of a combination of any of the above figures.

The difference between surface area and volume

Work “backwards”: given the surface area or volume, find the requested dimension (Note: this may require solving a quadratic equation by factoring or solving a cubic equation by taking a cube root.)

The similarities and differences between pyramids and prisms and between cones and cylinders

Draw an appropriate diagram of a prism, cylinder, pyramid, cone, frustum, or sphere in order to label the given information and find the surface area and volume

Important geometric vocabulary: polyhedron, prism, slant height, base edge, lateral edge, regular pyramid, frustum, net

Explain the derivation of the formula for the total surface area of a cylinder

The two-dimensional area formulas from the Area Unit (Chapter 11); students should either memorize these formulas or be able to derive them.

Given the net of a rectangular prism, draw the box that results

The geometric definition of a sphere Solve a multi-step real-life problem (e.g., find the height of the water that is left over after a cylinder of water fills a rectangular pan)

The appropriate units for length (linear) versus surface area (square) versus volume (cubic)


In-class review sheet of Sections 12.1 – 12.3 to work on with a partner

Quiz: Sections 12.1 – 12.3 End-of-unit test.


12.1 #1b, 2b, 3, 4b, 5b, 6 – 9 12.2 #1 – 8, 1012.3 #1 – 4, 6 – 10, 12, 1312.4 #1 – 7, 9, 10 12.5 #1, 3 – 6, 8, 9, 1212.6 #2 – 5, 7, 9 – 11 12.5 #14, 18 – 20 and 12.6 #12


Littell and Company


Cylinder model (with removable paper to represent the lateral area); paper cones to cut up to demonstrate the lateral area of a cone1

Stacks of paper, CD’s etc. to demonstrate the volume of a prism/cylinder; water, cylinder, and small prism to demonstrate #5 from the 12.4 Lesson2

Cone, cylinder, water, bucket to demonstrate the volume of a pyramid/cone3




Unit Title: Coordinate Geometry

Author(s): Best


What enduring understandings are desired? What are the overarching essential questions? The coordinate plane can be used to model

geometric figures and concepts effectively. What is a coordinate proof?

There are many types of proof. What is the three-dimensional coordinate plane?

Algebra and geometry are strongly connected. What is the algebraic technique of completing the square and why is it useful?

Lines are one of the building blocks of geometric figures but they also have algebraic equations that can be written in different forms.

What are the different methods to graph a line and how can you determine which method is most effective?

What are the different forms of linear equations and which form is the most effective?



The Coordinate Geometry unit works perfectly as the last unit of this course. This unit reviews a lot of the important geometric concepts covered throughout second semester with an algebraic twist. It is the perfect unit to segue into a second-year algebra course.

How do you write the equation of a line that is an altitude, median, or perpendicular bisector of a triangle or of a line that is tangent to a circle at a given point?

Investigative Geometer’s Sketchpad Lab How do you plot a 3-D point?

Using rulers to draw 3-D points How do you find the distance between two points in 3-D space?

Fun connection of reflections and angles to mini-golf

How do you use completing the square to write the equation of a circle in a recognizable form so that you can graph it and identify its center and radius?

Derivation of the equation for standard form of a circle

How do you draw and label an appropriate diagram on the coordinate plane in order to write an organized, logical coordinate proof?

Coordinate Proof partner assignment




Use the two-dimensional rectangular coordinate system and algebraic procedures to describe and characterize geometric properties and relationships (e.g., slope, parallelism, perpendicularity, Pythagorean Theorem, distance formula).

Visualize 3-dimensional figures in problem solving situations. Understands that objects and relations in geometry correspond

directly to objects and relations in algebra. Uses synthetic (i.e., pictorial) representations and analytic (i.e.,

coordinate) methods to solve problems involving symmetry and transformations of figures (e.g., problems involving distance, midpoint, and slope; determination of symmetry with respect to a point or line).

Algebraic Relationships. Students will discover, describe, and generalize simple and complex patterns and functional relationships. In the context of real-world problem situations, the student will use algebraic notation and techniques to define and describe the problems to determine and justify appropriate solutions.

Solve linear and quadratic equations, linear inequalities, and systems of linear equations and inequalities graphically and symbolically (including use of the quadratic formula).

Uses the rectangular coordinate system to model and to solve problems.

Solves linear equations using concrete, informal, and formal methods (e.g., using properties, graphing ordered pairs, using slope-intercept form).

Describe, recognize, interpret, and translate graphical representations of mathematical and real-world phenomena on coordinate grids (e.g., slope, intercepts, etc.).



Understands connections between equivalent representations and corresponding procedures of the same problem situation or mathematical concept (e.g., connections between algebraic and graphing solutions).





none Homework checks and daily corrections are an

on-going self-assessment tool. Review packets are provided before every quiz

and test so students can practice the necessary skills and familiarize themselves with the format and directions they will encounter on the assessment. An answer key is provided so students can assess how well they have mastered the material before the actual quiz or test.


The definition of slope; the slope formula Put a linear equation in slope-intercept form

Parallel lines have the same slope and perpendicular lines have opposite reciprocal slopes

Graph a line using - a table of values- the slope and y-intercept- the x and y-intercepts- a point and the slope

Slope-intercept, point-slope, and standard form of a line

Write the equation of a line in slope-intercept, point-slope, and standard form in a wide variety of situations (e.g., given its graph, given two points on the line, etc.)

Important geometric vocabulary: altitude, median, perpendicular bisector, tangent line

Write the equation of a horizontal or vertical line

Standard form of the equation of a circle Write the equation of the line that is an altitude, median, or perpendicular bisector of a triangle

Solve a system of linear equations graphically or algebraically (using the substitution method or the elimination method)

Graph the solution set to a system of linear inequalities and given the graph, write the system of inequalities

Plot a 3-D point

Find the distance between two points in 3-D space

Use reflections to determine where to hit a ball in mini-golf to make it into the hole using 2 banks

Given the equation of a circle in standard form, graph it; label the center, radius, and 4 other key points

Use completing the square to put the equation of a circle in standard form

Given a circle’s center and radius, write the equation of the circle in standard form, verify if a point lies on that circle, and write the equation of the line tangent to the circle at a specified point

Find the area of a figure that has been plotted on a coordinate grid

Draw and label an appropriate diagram to write a coordinate proof involving distance, slope (parallel or perpendicular), and/or midpoint


Quiz: 13.1 – 13.4 Geometer’s Sketchpad Lab: Pool Table

Problem Coordinate Proofs partner assignment End-of-unit test.


13.1 #2 – 8, 13, 15 – 19, 2413.2 #2cd, 3 – 6, 7cd, 813.2 #9, 10, 13 – 19, 21 13.3 #1, 3b, 4a, 5, 9 – 12, 1413.4 #1, 2, 513.5 #1 – 7, 9 – 12, 14 13.6 #1, 2, 3ab, 4, 5, 7, 8bd, 9bd, 10, 11, 12ac, 1513.7 #1, 3, 5, 7, 12 – 17, 19


Littell and Company



Rulers for drawing three dimensional points and for mini-golf problems involving reflection (Section 13.5)


Documents

· Web viewGeometry 1-2. General Information. Course Goal: Geometry 1-2 provides students the opportunity to formulate conjectures using inductive reasoning and to construct proofs