Curriculum Management System

MONROE TOWNSHIP SCHOOLS

Course Name: Dynamics of Geometry Grade: High School Grades 10-12

For adoption by all regular education programs Board Approved: November 2014 as specified and for adoption or adaptation by all Special Education Programs in accordance with Board of Education Policy # 2220.

Table of Contents Monroe Township Schools Administration and Board of Education Members Page 3

Mission, Vision, Beliefs, and Goals Page 4

Core Curriculum Content Standards Page 5

Scope and Sequence Pages 6-9

Goals/Essential Questions/Objectives/Instructional Tools/Activities Pages 10-102

Quarterly Benchmark Assessment Pages 103-106

Monroe Township Schools Administration and Board of Education Members

ADMINISTRATION Mr. Dennis Ventrillo, Interim Superintendent Ms. Dori Alvich, Assistant Superintendent

BOARD OF EDUCATION Ms. Kathy Kolupanowich, Board President Mr. Doug Poye, Board Vice President Ms. Amy Antelis Ms. Michele Arminio Mr. Marvin I. Braverman Mr. Ken Chiarella Mr. Lew Kaufman Mr. Tom Nothstein Mr. Anthony Prezioso

Jamesburg Representative Mr. Robert Czarneski

WRITERS NAME Ms. Samantha Grimaldi

CURRICULUM SUPERVISOR Ms. Susan Gasko

Mission, Vision, Beliefs, and Goals

Mission Statement

The Monroe Public Schools in collaboration with the members of the community shall ensure that all children receive an exemplary education by well-trained committed staff in a safe and orderly environment. Vision Statement

The Monroe Township Board of Education commits itself to all children by preparing them to reach their full potential and to function in a global society through a preeminent education. Beliefs

1. All decisions are made on the premise that children must come first. 2. All district decisions are made to ensure that practices and policies are developed to be inclusive, sensitive and meaningful to our diverse population. 3. We believe there is a sense of urgency about improving rigor and student achievement. 4. All members of our community are responsible for building capacity to reach excellence. 5. We are committed to a process for continuous improvement based on collecting, analyzing, and reflecting on data to guide our decisions. 6. We believe that collaboration maximizes the potential for improved outcomes. 7. We act with integrity, respect, and honesty with recognition that the schools serve as the social core of the community. 8. We believe that resources must be committed to address the population expansion in the community. 9. We believe that there are no disposable students in our community and every child means every child.

Board of Education Goals

1. Raise achievement for all students paying particular attention to disparities between subgroups. 2. Systematically collect, analyze, and evaluate available data to inform all decisions. 3. Improve business efficiencies where possible to reduce overall operating costs. 4. Provide support programs for students across the continuum of academic achievement with an emphasis on those who are in the middle. 5. Provide early interventions for all students who are at risk of not reaching their full potential. 6. To Create a 21st Century Environment of Learning that Promotes Inspiration, Motivation, Exploration, and Innovation.

Common Core State Standards (CSSS)

The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

Links: 1. CCSS Home Page: http://www.corestandards.org 2. CCSS FAQ: http://www.corestandards.org/frequently-asked-questions 3. CCSS The Standards: http://www.corestandards.org/the-standards 4. NJDOE Link to CCSS: http://www.state.nj.us/education/sca 5. Partnership for Assessment of Readiness for College and Careers (PARCC): http://parcconline.org

Quarter 1

Unit Topics(s)

I. Points, Lines, Planes h. Distance Between Two Parallel Lines a. Collinear and Coplanar b. Lines, Segments, and Rays

c. Space d. Distance Formula e. Midpoint Formula f. Segment Addition g. Angles and Bisectors h. Types of Angles i. Angle and Segment Bisectors j. Angle Pair Relationships k. Polygons l. Convex vs. Concave m. Regular vs. Irregular

II. Proof and Reasoning a. Conjectures based on Patterns b. Hypothesis and Conclusion c. If-Then form, Converse, Inverse, Contrapositive d. Postulates vs. Theorems e. Algebraic Proofs f. Two Column Proofs

III. Parallel and Perpendicular Lines a. Transversals b. Identifying Angles c. Parallel Lines and Transversals d. Angle Relationships e. Algebraic Reasoning f. Proof of Parallel Lines g. Slope of a Line

Quarter 2

Unit Topic(s)

I. Triangle Congruence a. Classifying Triangles b. Isosceles vs. Equilateral c. Angle Sum Theorem d. Exterior Angle Theorem e. Congruency Theorems: SSS, SAS, ASA, AAS f. Right Triangle Congruency: LL, HA,LA, HL g. Proofs using above Theorems II. Relationships in Triangles a. Perpendicular Bisectors b. Angle Bisectors c. Medians d. Altitudes (inside and outside the triangle) e. Inequalities f. Angle Measure vs. Side Measure g. Greatest Side or Angle of a Triangle h. Acute, Right or Obtuse by Side Measures III. Similarity a. Proportions b. Application and Reasoning c. Scale Factor d. Similar Polygons and Similar Figures e. Similarity Postulates for Triangles: AA, SSS, SAS

Quarter 3

Unit Topic(s)

I. Right Triangles and Trigonometry b. Proving Quadrilaterals to be Parallelograms a. Perfect Squares c. Rectangles, Rhombi, and Squares b. Radical Simplification d. Trapezoids c. Geometric Mean e. Properties of Trapezoids d. Pythagorean Theorem and its Converse f. Medians of Trapezoids (Midsegments) e. Proof of Pythagorean Theorem g. Coordinate Geometry f. Special Right Triangles h. Other Quadrilaterals (Kites) g. Trigonometric Ratios h. Angles of Elevation and Depression i. Law of Sine’s II. Circles Parts and Properties a. Area and Circumference b. Arcs c. Properties d. Arc Measure vs. Arc Length e. Arcs and Chords f. Inscribed Angles g. Concentric Circles h. Tangents and Secants i. Equations III. Transformations and Symmetry a. Translations b. Reflections c. Rotations d. Congruence Transformations e. Dilations f. Similarity Transformations

IV. Quadrilaterals a. Properties of Parallelograms

Quarter 4

Unit Topic(s)

I. Area of Polygons a. Regular vs. Irregular b. Sum of the Interior Angles of a Polygon c. Finding each Interior Angle in a Regular Polygon d. Sum of the Exterior Angles of a Regular Polygon e. Triangles f. Quadrilaterals g. Irregular Figures h. Regular Polygons i. Coordinate Geometry j. Perimeter vs. Area on a Coordinate Plane II. Surface Area and Volume a. Nets b. Prisms c. Cylinders d. Pyramids e. Cones f. Spheres III. Probability and Measurement a. Permutations and Combinations b. Fundamental Counting Principal c. Experimental Probability d. Theoretical Probability e. Geometric Probability

Unit 1: Points, Lines, and Planes Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.1 Know precise definitions of -Understand Geometry is a mathematical system built on accepted facts, basic terms, and angle, circle, perpendicular lines, parallel definitions. line, and line segment, based on the undefined notions of point, line, distance -Show segments, rays, and lines are very similar but each have their own properties and along a line, and distance around a circular can be combined to form larger figures in the geometric world. arc. - Use formulas to find the midpoint and length of any segment on a coordinate plane. CC9-12.G.CO.4 Develop definitions of -Apply special angle pairs to identify geometric relationships and to find angle measures. rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

CC9-12.G.CO.12 Make formal geometric Meaning constructions with a variety of tools and UNDERSTANDINGS ESSENTIAL QUESTIONS methods (compass and straightedge, string, Students will understand that… -What are the building blocks of geometry? reflective devices, paper folding, dynamic -A line is made up of an infinite amount of geometric software, etc.). points. Two lines can intersect a point, two -How can you describe the attributes of a planes can intersect to form a line, and three segment or angle? CC9-12.G.GPE.7 Use coordinates to compute planes intersect at a point in space. perimeters of polygons and areas of -Why are units of measure important? triangles and rectangles, e.g., using the - Linear measure is the distance between distance formula. two points. The formal definition of between is used often in geometry and applies to segments directly. Segment addition is the idea of adding two connected segments together to get the length of the larger segment formed. Measurements should be as precise as possible to ensure the most accurate dimensions.

- The distance and midpoint formula can be applied to points on a coordinate plane.

- A ray is a part of a line. Two opposite rays form a line, which is 180 degrees. Two rays with the same endpoint form and angle. Angles can be measured using a protractor. An angle can be named three different ways (vertex, three points, or a number). An angle bisector cuts the angle into two perfectly congruent angles. Angles can be congruent or can be added together to find the measure of the larger angle formed.

- A perpendicular can be a line, segment, or ray that intersects another line, segment, or ray to form a 90-degree angle. Tick marks and arc marcs are a vital part of geometry and must be identified in order to solve problems effectively.

- The definition of a polygon is a closed figure whose sides are all segments whose endpoints only intersect two other segments at their endpoints. A polygon can be concave, convex, regular, and irregular. Certain polygons have special names where others are referred to as n-gons.

Acquisition Students will know… Students will be skilled at… -Definitions: _ -Modeling each concept using coordinate  Point geometry.  Line  Ray -Comparing and contrasting the similarities  Segment and differences between each undefined  Plane term.  Collinear -Generating a philosophical conversation on  Coplanar the idea of space and how it applies to that  Space around them.  Right Angles  Obtuse Angles -Writing and solving algebraic expressions  Acute Angles using segment addition and angle addition.  Adjacent Angles  Vertical Angles - Using the distance formula or Pythagorean  Linear Pair Theorem to determine the length of a  Complementary Angles segment on a coordinate plane.  Supplementary Angles  Perpendicular -Using the midpoint formula to find the  Polygon exact middle of a line segment with two -The Distance Formula endpoints given. Be able to find the -The Midpoint Formula endpoint of a segment when the midpoint -Segment Addition Postulate and one other endpoint are given. -Angle Addition Postulate -Naming an angle three different ways and determine if there is a best way when given various examples.

- Using protractors to determine the measure of each angle.

-Constructing angle and perpendicular bisectors using geometry tools.

-Finding the perimeter of a regular convex figure when given the length of one side.

Finding the perimeter of a irregular figure on a coordinate plane by finding the length of each segment it’s formed by.

Stage 2 – Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each problem on any summative assessment. Students will be assessed through a formative assessment that may contain but is not limited to the following tasks: 4 – Student demonstrates a complete • Identifying a point, line, plane, ray, or segment when given a diagram. They are understanding of the concept with a correct responsible for using the proper symbolic notation as outlined by the instructor. solution. • Understanding points that are coplanar and comparing them to points that are collinear. 3 – Solution contains one minor arithmetic • Distinguishing the difference between space and planes. error but demonstrates complete • Using angle pair relationships such as linear pairs, vertical angles, complementary understanding of the concept. angles, and supplementary angles, to solve real life application problems. 2 – Solution contains more than one minor • Using the Angle Addition Postulate or Segment Addition Postulate in algebraic arithmetic error but demonstrates partial problem solving. understand of the concept. • Applying the Pythagorean Theorem, Distance Formula and Midpoint Formula as needed on a coordinate plane. 1 – Solution contains multiple arithmetic • Comparing and contrasting polygons that are regular or irregular, convex or errors while demonstrating minimal concave. understanding of the concept.

0 – Incorrect answer demonstrating no understanding of the concept. The following is a student rubric to assess OTHER EVIDENCE: individual understanding during class activities. • Homework/Classwork • Written response to one of the essential questions using vocabulary, mathematical 4 – I understand completely and I can teach properties, and insights it to a classmate. • Formative assessments such as individual or pair quizzes

• Peer and self-assessment 3 – I understand the concept but I do not • Collaborative group work

think I can explain it to a classmate. • Geometer’s Sketchpad Explorations

2 – I can complete the task with assistance.

1 – I need help. Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Determine if a plane exists for points non-coplanar. • Represent the intersection of two planes intersecting. • Find examples of skew lines in real life • Use explorations to further the understanding of collinear and coplanar.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 1 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: The teacher scavenger hunt is now updated to include the iPad! Students will use QR codes to get a brief description of a polygon in which they have to identify and draw using the clue given. Once completed students will compare their drawings and findings to demonstrate how a polygon although has the same amount of sides can look various ways and still be considered that polygon. An example of the worksheet is seen below. Any scanning app can be used for this activity.

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on identifying and contrasting lines, segments, rays, and planes as well as naming them.

Constructions: Construct a perpendicular bisector using the given directions.

1. Begin with line segment XY.

2. Place the compass at point X. Adjust the compass radius so that it is more than (1/2)XY. Draw two arcs as shown here.

3. Without changing the compass radius, place the compass on point Y. Draw two arcs intersecting the previously drawn arcs. Label the intersection points A and B.

4. Using the straightedge, draw line AB. Label the intersection point M. Point M is the midpoint of line segment XY, and line AB is perpendicular to line segment XY.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP, NCTM

Illuminations www.illuminations.nctm.org, Khan Academy www.khanacademy.org, Learn 360 www.learn360.com, Regents Prep Center www.regentsprep.org, Jeopardy Games www.jeopardylabs.com, PARCC Online www.parcconline.org, Desmos Graphing Calculator www.desmos.com/calculator

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 2: Proof and Reasoning Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.9 Prove theorems about lines -Decipher patterns and determine the next logical term requires the analysis and synthesis and angles. of numbers, figures, and various objects. Finding the truths in these can lead to the proof of the hypothesis for the next term. CC9-12.G.CO.10 Prove theorems about triangles. -Understand conditional statements are the first form of proofs, when the hypothesis and conclusion are moved around the statement may or may not still be true.

-Assume postulates to be geometric truths we assume to be true, they can be used to prove or disprove theories of general approaches.

Meaning UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… -How can you make a conjecture and prove -If-then statements are often referred to as that it’s true? conditional statements and can be transformed into the inverse, converse, and -Is there a “best practice” to proving an contrapositive by moving the hypothesis answer is correct? and conclusion around and using negations. These various statements can be true or false.

- Postulates are geometric relationships assumed to be true that are used to solve various proofs including informal proofs called paragraph proofs.

- Properties of Equality for Real Numbers

are used to complete formal two- column algebraic and geometric proofs.

Two-column proofs involving segments can be solved using the Segment Addition Postulate as well as the Segment Congruence Properties.

-Two-column proofs involving angle relationships can be solved using the Angle Addition Postulate as well as the Angle Congruence Properties, Supplement Theorem and Complement Theorem.

Acquisition Students will know… Students will be skilled at… -Definitions: -Making conjectures about the next term in  Conjecture a pattern.  Counterexample  Hypothesis -Finding counterexamples to prove  Conclusion statements to be false.  Conditional Statement  Converse -Writing the converse, inverse, and  Inverse contrapositive of a sentence when given the  Contrapositive conditional statement and determine  Algebraic Proof whether each of these are true or false.  Flow Proof  Two-Column Proof -Using postulates involving segments and  Paragraph Proof lines as well as the Midpoint Theorem to complete several paragraph proofs.- -Properties of Equalities

-Segment Congruence Properties -Using Equality Properties to set up and -Angle Congruence Properties complete a two-column algebraic proof. -Supplement Congruence Theorem -Complement Congruence Theorem -Showing angle congruency on a clock by -Midpoint Theorem using equality properties and definitions of -Angle Bisector Theorem congruency in a two-column proof.

- Completing proofs involving segment addition by using the Segment Addition Postulate and Segment Congruence Postulates.

-Completing proofs about angle relationships by using the Angle Addition Postulate and Angle Congruence Postulates, Supplement Theorem and Complement Theorem

Stage 2 - Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each problem on any summative assessment. Students will be assessed through a formative assessment that may contain but is not limited to the following tasks: 4 – Student demonstrates a complete • Identifying conjectures and stating whether they are true or false. If false a counterexample understanding of the concept with a correct is required. solution. • Understanding the hypothesis and conclusion of a conditional statement can be transposed and negated to form other statements with their own truth-values. These statements are 3 – Solution contains one minor arithmetic call the converse, inverse, and contrapositive. error but demonstrates complete • Writing algebraic proofs by using the two-column format and the properties of equalities. understanding of the concept. • Writing proofs (two-column or flow proofs) using the Segment Congruence Properties as well as the Segment Addition Postulate and Midpoint Theorem. 2 – Solution contains more than one minor • Writing proofs (two-column or flow proofs) using the Angle Congruence Properties, Supplement Congruence Theorem, Complement Congruence Theorem, and Angle Bisector arithmetic error but demonstrates partial Theorem. understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

• Peer and self-assessment 3 – I understand the concept but I do not • Collaborative group work think I can explain it to a classmate. • Geometer’s Sketchpad Explorations 2 – I can complete the task with assistance.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Develop conditional statements that are tautologies. • Use truth tables to prove a statement logical. • Create proofs using various theorems, postulates, and definitions to solve. • Compare and contrast methods of proof.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 2 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Students will create an iMovie in which they use the various techniques to prove theorems. Each student is responsible to prove 3 different theorems learned in this unit. Their movie must include a flow proof, paragraph proof, and two-column proof. These proofs must also include content involving Algebraic Proof, Segment Proof, and Angle Proof.

White Board Activity: In groups of four or five, the student groups will be given a proof via PowerPoint and create a two-column proof . One student at a time is to write a line of the proof and the whiteboard and markers will rotate throughout the members of the group. A group member may edit the entry before his or hers but no one else’s. Once proof is complete the instructor will display the answer for the groups to check their solutions for accuracy.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

NCTM Illuminations www.illuminations.nctm.org, Khan Academy www.khanacademy.org, Learn 360 www.learn360.com, Regents Prep Center www.regentsprep.org, Jeopardy Games www.jeopardylabs.com, PARCC Online www.parcconline.org, Desmos Graphing Calculator www.desmos.com/calculator

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; PearsonEducation Inc., 2012

Unit 3: Parallel and Perpendicular Lines Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.1 Know precise definitions of -Understand not all lines and all planes intersect. When a line intersects two or more lines, angle, circle, perpendicular lines, parallel the angles formed at the intersection points create special angle pairs. line, and line segment, based on the undefined notions of point, line, distance -Use properties including special angle pairs formed by parallel lines and a transversal in along a line, and distance around a circular proof. arc. -Prove angle pairs can be used to decide whether two lines are parallel. CC9-12.G.CO.9 Prove theorems about lines -Show the relationships of two lines to a third line can be used to decide whether two lines and angles. are parallel or perpendicular to each other.

CC9-12.G.CO.12 Make formal geometric -Derive the theorem that states the sum of the angles of a triangle is always the same. constructions with a variety of tools and methods (compass and straightedge, string, - Create the equation of a line when certain facts about the line, such as its slope and a reflective devices, paper folding, dynamic point on the line are known. geometric software, etc.). -Compare the slopes of two lines can show whether the lines are parallel or perpendicular. CC9-12.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 Meaning perpendicular to a given line that passes UNDERSTANDINGS ESSENTIAL QUESTIONS through a given point) Students will understand that… -How do you write the equation of a line in a -Parallel lines are coplanar and skew lines coordinate plane? are non-coplanar. Planes can also be parallel. -What information is necessary prove that two lines are parallel or perpendicular? -When two lines are intersected at two different places by a line this line is called a transversal. A transversal creates four special angle pair relationships. If the two lines are parallel these special angle pairs

are either congruent or supplementary. If the transversal is also a perpendicular all 90 degree angles are formed.

- Slope is the ratio of its vertical rise to horizontal run. Slope can also be considered the rate of change describing how a quantity changes over time. Slopes with the same value are parallel, slopes that are opposite reciprocals of one another are perpendicular and slopes with no relation to one another are neither.

- The slope-intercept formula and point- slope formula can be used to write the equation of a line when given specific information.

- Formal two column proofs can be written to prove lines parallel when given a diagram and specific information.

- The shortest distance from a point to a line is a perpendicular drawn from the point to that line. Two lines are parallel if every point on one line is equidistant from the corresponding point on the second line.

Acquisition Students will know… Students will be skilled at… -Definitions: Naming segments that are skew and  Parallel Lines segments that are parallel when given a  Skew Lines prism. Identifying transversals if lines of  Parallel Planes

 Transversal prisms are extended.  Auxiliary Line  Alternate Interior Angles - Completing error analysis when  Alternate Exterior Angles identifying special angle pair relationships.  Same-Side (Consecutive) Interior Angles  Corresponding Angles -Proving lines parallel using various converse’s of special angle pair theorems -Corresponding Angles Postulate and it’s and postulates. Converse -Alternate Interior Angle Theorem and it’s - Writing and using algebraic expressions to Converse find the value of angle measures including -Alternate Exterior Angle Theorem and it’s those in which an auxiliary line must be Converse created. -Same Side Interior Angle Theorem and it’s Converse -Using the slope formula to determine if two -Slope Formula lines are parallel, perpendicular, or neither. -Slope-Intercept Formula -Point Slope Formula - Deciphering information and using linear -Perpendicular Transversal Theorem equations to solve application problems in -Parallel Postulate the everyday world. -Perpendicular Postulate -Finding the distance between two parallel lines by using the understanding that the shortest distance between two lines is the perpendicular between them.

3 – Solution contains one minor arithmetic parallel. error but demonstrates complete • Proving two or more lines parallel by using the Converse of the Alternate Interior Angle understanding of the concept. Theorem, Alternate Exterior Angle Theorem, Corresponding Angle Postulate, and Same-Side Interior Angle Theorem. 2 – Solution contains more than one minor • Using the slope formula to determine whether two lines are parallel, perpendicular, or have arithmetic error but demonstrates partial no relationship to each other. • Writing equations of lines that are parallel or perpendicular to each other. understand of the concept. • Using the Perpendicular Postulate to solve algebraic problem solving real world application questions. 1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Develop conditional statements that are tautologies.

• Use truth tables to prove a statement logical. • Create proofs using various theorems, postulates, and definitions to solve.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 3 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Students often find it difficult to rearrange equations in order to determine the slope of a line. This Khan Academy video is used to assist in determining the accuracy of the student ability to rearrange equations. The instructor can use this activity as a pre- assessment to determine grouping for a tiered lesson by ability. Students can attempt questions on their iPad and write down their comfort level based on the scales determined by the geometry instructors. https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/e/line_relationships

11 Graphing Calculator Activity: Parallel Lines l and m are cut by a transversal t. The equations of l, m, and t are yx=−=−4, yx 6, 22 and yx=−+21, respectively. Use a graphing calculator to determine the points of intersection of t with l and m.

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on using congruence and supplementary angles as it applies to parallel lines and/or determining the equation of a line parallel or perpendicular to a given line through a given point.

Construction: Given a point and a line, construct a line parallel to the line given through the point given.

1. Begin with point P and line k.

2. Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q. Now the task is to construct an angle with vertex P, congruent to the angle of intersection.

3. Center the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc.

4. Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R.

5. Line PR is parallel to line k.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 4: Triangle Congruence Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.6 Use geometric descriptions -Understand that classification is used in all facets of mathematics and can be used as a first of rigid motions to transform figures and to step to dissecting a problem. predict the effect of a given rigid motion on a given figure; given two figures, use the Prove the sums of the angles of a triangle are always 180 degrees. definition of congruence in terms of rigid -Use visualization, tick marks, and arc marks to show corresponding sides and angles in motions to decide if they are congruent. congruent triangles. CC9-12.G.CO.8 Explain how the criteria for -Prove two triangles congruent using the Congruence Postulates (SSS, SAS, ASA, AAS, HL). triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in -Distinguish the various differences between isosceles and equilateral triangles. terms of rigid motion. -Derive the relationships involving perpendicular bisectors and angle bisectors in triangles. CC9-12.G.CO.9 Prove theorems about lines and angles.

CC9-12.G.CO.10 Prove theorems about Meaning triangles UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… -Where do we see classification used in -Triangles are classified by their side concepts involving mathematics? lengths and angle measures. -How do you identify corresponding parts of - The Triangle Angle Sum Theorem states congruent triangles? that any triangle has three angles that will add up to 180 degrees. -How do you show that two triangles are congruent? -The Exterior Angle Theorem states the exterior angle of a triangle is congruent to -Can an equilateral triangle be classified the sum of the remote interior angles inside isosceles? Vice-versa? the triangle. -How do you use coordinate geometry to - Congruent triangles have corresponding find relationships within triangles?

congruent sides and angles. Congruence is preserved through transformations such as -How do you solve problems that involve slides, flips, and turns. measurements of triangles?

- The Side-Side-Side (SSS) postulate can be used to prove two triangles congruent when no angles are given and only side measures are shown.

-The Side-Angle-Side (SAS) postulate can be used to prove two triangles congruent when two corresponding sides are congruent and their included angle is the same measure. The included angle is the angle formed by the two congruent sides.

- The Angle-Side-Angle (ASA) postulate can be used to prove two triangles congruent when two corresponding congruent angles are given and the corresponding sides between them are congruent.

-The Angle-Angle-Side (AAS) theorem can be used to prove two triangles congruent when two corresponding angles are congruent and a corresponding side that follows is congruent.

-The Hypotenuse-Leg (HL) postulate is only used in right triangles. Two right triangles can be proven congruent if the corresponding hypotenuse and leg are congruent in both triangles.

- Isosceles triangles have two congruent sides called legs. The side not congruent is

called the base. The angles across from the congruent angles are called base angles and are also congruent. The remaining angle is called the vertex angle. These properties can be used to prove a triangle to be isosceles. .

Acquisition Students will know… Students will be skilled at… Definitions: -Classifying triangles by using the side  Legs classification and angle classifications.  Bases  Congruence -Using the properties of an equilateral or  Congruence Statement isosceles triangle to set up algebraic  Corresponding equations and solve for missing side  Included Angle measures.  Included Side  Vertex Angle - Recalling the distance formula is used to  Isosceles find the lengths of the sides of a triangle on  Equilateral a coordinate plane and can assist in  Equiangular classifying the triangle by its side lengths. -SAS Postulate -SSS Postulate -Writing and solving algebraic equations -ASA Postulate using the Angle Sum Theorem to find the -AAS Theorem measures of the angles in a triangle. -HL Theorem -Isosceles Triangle Theorem -Writing and solving algebraic equations -Properties and Corollaries of Equilateral using the Exterior Angle Theorem to find Triangles the measure of the exterior angle and remote interior angles.

-Writing congruency statements when given two congruent figures. - Completing two column proofs to show

three triangles congruent.

-Verifying triangle transformations on a coordinate plane by using the distance formula.

-Recognizing triangles in which can be proven congruent by using the SSS postulate, SAS Postulate, ASA Postulate, AAS Theorem, and HL Theorem.

-Using the congruence postulates and theorems to develop two column proofs regarding congruent triangles.

-Completing a two column proof, which proves the two angles across from the congruent sides are also congruent.

-Using properties of isosceles base angles to find missing angle measures in triangles.

-Writing and solve algebraic equations using the properties of an isosceles triangle to find missing side and angle measures.

solution. problem solving. • Identifying congruent triangles using SSS, SAS, ASA, AAS, and HL. 3 – Solution contains one minor arithmetic • Writing congruence statements based off the congruent triangles found. error but demonstrates complete • Using the Isosceles Triangle Theorem and the properties of equilateral triangles in real life understanding of the concept. application questions.

2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Solve for angle measures using the Triangle Sum Theorem and the Exterior Angle Theorem. • Prove the AAS Theorem and HL Theorem using the SSS, SAS, and ASA postulates. • Prove the Isosceles Triangle Theorem and its Converse. • Use coordinate geometry to determine if a triangle is isosceles or equilateral.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 4 Solve It! and Lesson Check from the Algebra 1 Common Core textbook.

iPad Activity: Students have now completed four units of study. They will be asked to use their notes and eBooks to develop a brochure highlighting the important concepts and big ideas from the first four units. The students may use the Pages app to develop this or Creative Book Builder. The brochure should include examples from the text, class notes, or tests and quizzes. This will be used at the end of every set of four units as a review tool.

In Class Study Guide: It’s your job to create a study guide for a friend who doesn’t quite understand triangles and congruence as much as you do! You are going to make a foldable for your friend to cover each topic in classifying and determining congruent triangles!

To make a foldable:

1. Take a stack of paper and hold it long-ways. 2. Fold right hand corner over to opposite side of the paper to create a right triangle. 3. Cut excess paper off bottom. 4. Staple across the hypotenuse.

You want to label each page of your foldable. Make sure you include notes, definitions, examples, and any helpful hints in your foldable

journal so your friend has the ultimate study guide! You can also add your personal thoughts on each topic whether you thought it was easy and grasped it right away or had trouble and it’s something your friend should study very hard to understand. Make sure your foldable is neat, precise, and easily understood.

Geometer’s Sketchpad Activity: Use the following instructions to construct the figure needed. Use Geometers SketchPad to construct and Construct BC to form Construct a line parallel to BC that intersects and at points D and E to form

1) Are the three angles in congruent to the three angles in Manipulate the figure to change the positions of DE and BC

2) Do the corresponding angles of the triangles remain congruent? 3) Are the two new triangles congruent? 4) Can the two triangles be congruent?

Constructions: Given a line segment as one side, construct an equilateral triangle.

1. Begin with line segment TU.

2. Center the compass at point T, and set the compass radius to TU. Draw an arc as shown.

3. Keeping the same radius, center the compass at point U and draw another arc intersecting the first one. Let point V be the point of intersection.

4. Draw line segments TV and UV. Triangle TUV is an equilateral triangle, and each of its interior angles has a measure of 60°.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 5: Relationships in Triangles Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.9 Prove theorems about lines Understand that triangles play a key role in relationships involving perpendicular bisectors and angles. and angle bisectors.

CC9-12.G.CO.10 Prove theorems about -Use special parts of a triangle that are always concurrent. A triangle’s three perpendicular triangles bisectors are always concurrent, as are a triangles three angle bisectors, its three medians, and its three altitudes. CC9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve -Find that the measures of angles of a triangle are related to the lengths of the opposite problems and to prove relationships in sides. geometric figures.

Meaning UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… -How do you use coordinate geometry to -When three or more lines intersect a find relationships within triangles? common point these lines are called concurrent line and they meet at a point of -How do you solve problems that involve concurrency. measurements of triangles?

-Three perpendicular bisectors meet at the -When can we apply points of concurrencies circumcenter . The circumcenter is in real life? equidistance from each of the vertices of the triangle.

-Three angle bisectors intersect to meet at the incenter. The incenter is equidistant from each of the sides of the triangle.

-A median is a segment whose endpoints are

a vertex of a triangle and the midpoint of the opposite side. Three medians meet at a centroid. The centroid is located 2/3 the distance from the vertex to the midpoint of the opposite side.

-An altitude of a triangle is a segment from a vertex to the opposite side forming a right angle. Three altitudes meet at the orthocenter.

-Inequalities can be written when comparing two quantities of different size.. The length of the side corresponds to the size of the angle. The longest side is across from the largest angle, the shortest side is across from the smallest. An inequality can be written to compare side lengths and angle measures.

-The Triangle Inequality Theorem states the sum of any two sides of a triangle is greater than the length of the third side. An inequality can be written to determine the length of the third side of a triangle when given the other two sides.

Acquisition Students will know… Students will be skilled at… Definitions: -Identifying different concurrent lines and  Midsegment their points of concurrency.  Perpendicular Bisector  Angle Bisector -Using the properties of the points of  Median concurrency to solve algebraic problems.  Altitude  Concurrent Lines -Using systems of equations, distance  Point of Concurrency formula, slope formula, and midpoint  Circumcenter formula to find points of concurrency on a  Incenter coordinate plane.  Centroid  Orthocenter -Forming tables in which they special  Inequality segments in a triangle are listed, the type of -Triangle Midsegment Theorem segment is identified, and their point of -Perpendicular Bisector Theorem and it’s concurrency is determined. Converse -Angle Bisector Theorem and it’s Converse Writing and using basic inequalities. -Circumcenter Theorem -Performing algebraic operations on -Incenter Theorem inequalities. -Centroid Theorem -Orthocenter Theorem -Using deductive reasoning to write -The Exterior Angle Inequality Theorem inequalities involving angle measures and -The Triangle Inequality Theorem the exterior angle.

-Using the properties of a triangle to list the sides and angles in ascending or descending order.

-Using the relationship between side length and angle measure to complete real life application problems.

-Identifying whether measures given can be

the side lengths of a triangle.

-Using standardized test questions to determine a possible side length of a triangle when only given two measurements.

-Writing and solving application problems involving the triangle inequality.

Stage 2 - Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each PERFORMANCE TASK(S): problem on any summative assessment. Students will be assessed through a formative assessment that may contain but is not 4 – Student demonstrates a complete limited to the following tasks: understanding of the concept with a correct • Identifying points of concurrency in diagrams and using their properties in algebraic solution. problem solving. • Completing PARCC like application problems involving the Circumcenter Theorem. 3 – Solution contains one minor arithmetic • Using the Triangle Midsegment Theorem to develop a two-column proof. error but demonstrates complete • Writing inequalities using the Triangle Inequality Theorem and the Exterior Angle understanding of the concept. Inequality Theorem.

2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

0 – Incorrect answer demonstrating no understanding of the concept.

The following is a student rubric to assess OTHER EVIDENCE: individual understanding during class activities. • Homework/Classwork • Written response to one of the essential questions using vocabulary, mathematical 4 – I understand completely and I can teach properties, and insights it to a classmate. • Formative assessments such as individual or pair quizzes

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Identify points of concurrency on a coordinate plane. • Construct perpendicular bisectors, angle bisectors, altitudes, and medians using a protractor and compass. • Create formal proofs of the Triangle Inequality. • Apply the range of possible side lengths of a triangle to perimeter questions.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 5 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Students will create a business in which they have three locations and are looking for the most efficient location for a headquarters. Students will use the points of concurrency and latitude and longitude points to find their perfect headquarters location. Students will put together an infomercial using iMovie to advertise their business, product, and the opening of their new headquarters!

Each infomercial must explain why they pick their headquarters to be where it was!

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on identifying points of concurrency, writing and solving algebraic equations based on the properties of points of concurrencies, determining if lengths can form a triangle, and possible range of side lengths for the third side of a triangle.

Geometer’s Sketchpad: Complete the following activity using Geometer’s SketchPad.

-Construct a triangle and the three perpendicular bisectors of its sides.

-Construct a triangle and its three angle bisectors.

-Construct a triangle. Through the vertex of the triangle construct a segment that is perpendicular to the line containing the side opposite that vertex. This is called the altitude. Construct altitudes for the other two vertices.

- Construct a triangle and the three medians of the triangle.

a) What property does the lines containing altitudes and the medians seem to have? b) State your conjectures about the lines containing the altitudes and about the medians. c) Think about acute, right, and obtuse triangles. Fill in the table below with inside, on, or outside to describe the location of each point of concurrency.

Perpendicular Angle Bisectors Lines Containing the Medians Bisectors Altitudes Acute Triangle

Right Triangle

Obtuse Triangle

d) What observations, if any, can you make about the special segments for isosceles triangles? For equilateral triangles?

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 6: Similarity Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.9 Prove theorems about lines -An equation can be written stating that two ratios are equal, and if the equation contains a and angles. variable, it can be solved to find the value of the variable.

CC9-12.G.CO.10 Prove theorems about - Ratios and proportions can be used to decide whether two polygons are similar and the triangles find unknown side lengths of similar figures.

CC9-12.G.SRT.2 Given two figures, use the -When two or more parallel lines intersect other lines, proportional segments are formed. definition of similarity in terms of similarity transformations to decide if they are -Triangles can be similar based on the relationship of two or three corresponding parts. similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the Meaning proportionality of all corresponding pairs of UNDERSTANDINGS ESSENTIAL QUESTIONS sides. Students will understand that… -How do you use proportions to find side CC9-12.G.SRT.5 Use congruence and -A ratio is a comparison of two quantities. A lengths in similar polygons? proportion is an equation stating that two similarity criteria for triangles to solve ratios are equal. -How do you show two triangles are problems and to prove relationships in similar? geometric figures. - Polygons are similar if their corresponding angles are congruent and corresponding -How do you identify corresponding parts of CC9-12.G.SRT.6 Understand that by sides proportional. similar triangles? similarity, side ratios in right triangles are properties of the angles in the triangle, -Similarity statements can be written to leading to definitions of trigonometric ratios show congruent angles and corresponding for acute angles. proportional sides. Scale factor is the ratio in which two similar figures are compared

by.

- Triangles can be similar when their angles are congruent and sides are proportional.

We can determine similarity of triangles by writing proportions, using the AA, SSS, and SAS Similarity.

-Triangles can be formed using parallel lines and transversals. The parallel lines split the triangle into proportional parts.

-When two triangles are similar their perimeters are proportional and the special segments inside of them are proportional. Angle bisectors can also create proportional parts.

Acquisition Students will know… Students will be skilled at… Definitions: -Using ratios to find angle measures in a  Ratio triangle and to find the missing side lengths  Extended Ratio when given the perimeter of a triangle.  Proportion  Means -Cross multiplying the means and the  Extremes extremes to determine if the cross product  Similar Figures of a proportion is equal.  Similar Polygons  Scale Factor -Setting up and solving proportions in  Indirect Measurement which a variable needs to be found. -Means and Extremes Property -The Hinge Theorem (SAS Inequality) and -Writing proportions of corresponding parts it’s Converse (SSS Inequality) to determine if two figures are similar. -AA Similarity Postulate - Triangle Proportionality Theorem -Using scale factor in application problems -Triangle Angle Bisector Theorem to determine the size of scaled models.

-Writing an informal proof using the Hinge

Theorem (SAS) to prove a comparison in

two triangles.

-Using the AA, SSS, and SAS Similarity Theorems to write proportions to find whether two or more triangles are similar..

-Completing application problems using indirect measurement.

-Completing a two-column proof in which the Triangle Proportionality Theorem is used.

-Determining if two lines in a triangle are parallel by completing a proportion.

-Using coordinate geometry to find coordinates of the midsegment.

-Using coordinate geometry to determine whether two lines are parallel in a triangle.

-Using previous knowledge of special segments in triangles to write proportions and solve for corresponding sides of the triangles.

-Using the Angle Bisector Theorem in proofs

of proportional parts.

Stage 2 – Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each PERFORMANCE TASK(S): problem on any summative assessment. Students will be assessed through a formative assessment that may contain but is not limited to the following tasks: 4 – Student demonstrates a complete • Writing ratios and solving proportions when given application questions. understanding of the concept with a correct • Determining whether two figures are similar . solution. • Using the SAS Inequality, SSS Inequality, and AA Similarity to determine whether two triangles are similar. 3 – Solution contains one minor arithmetic • Finding and using scale factor to compare real life buildings to their scale models. error but demonstrates complete • Identifying parallel lines within figures to apply to Triangle Proportionality Theorem and understanding of the concept. solve algebraic problems.

2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Use scale factor to find the size of American monuments when given scale models of them. • Solve proportions to develop and solve a system of equations. • Derive the SSS Inequality after proving the SAS Inequality. • Use the properties of parallel lines to prove the Triangle Proportionality Theorem.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 7 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Foresters, environmentalists, and other professionals in the timber industry take many measurements in their field work for purposes including forest management planning and forest health monitoring. Workers in these industries also need to calculate the height of trees in order to determine the “merchantable height” of the tree to find out how much lumber will be available to sell. Two tools commonly used are a clinometer, to measure angles of elevation, and a Biltmore Stick, to measure diameter and height of trees. These tools are based on similar triangles, since a tree’s height usually cannot be measured directly. Use Geometer’s Sketchpad on your iPad to create the following: ƒ

• Construct a segment that appears vertical and label it AB . • Measure the segment’s length. • Drag either endpoint so that the length is 1. • Construct a point not on AB . • Construct a line parallel to AB through the new point. • Hide the point. • Construct a segment whose endpoints are on the parallel line and label it TS . • Hide the parallel line. • Construct a line from T (the top of the tree) through A (the top of the stick).

• Construct line SB • Find the point of intersection of TA and SB and label it O (for observer). • Measure the lengths of OB and OS • Calculate the height of the tree by setting up and solving a proportion involving the lengths of OB , OS , AB and TS . • Confirm that your calculation is correct by measuring TS in your figure.

How will your work change if the original segment AB is known to be 3 feet long? Or 25 inches long?

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on ratio, scale factor, proportions, similar figures, similar triangles, and proportionality in triangle with parallel lines.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 7: Right Triangles and Trigonometry Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.9 Prove theorems about lines -Understand geometric mean describes the relationship between the altitude and and angles. hypotenuse of a right triangle.

CC9-12.G.CO.10 Prove theorems about -Use the Pythagorean Theorem when given lengths of any two sides of a right triangle to triangles find the length of the third side.

CC9-12.G.SRT.5 Use congruence and -Derive ratios when certain combinations of side lengths and angle measures of a right similarity criteria for triangles to solve triangle are known. problems and to prove relationships in -Understand ratios can be used to find side lengths and angle measures of a right triangle geometric figures. when certain combinations of side lengths and angle measures are known. CC9-12.G.SRT.6 Understand that by -Use angles of elevation and depression, which are the acute angles of right triangles similarity, side ratios in right triangles are formed by a horizontal distance and a vertical height. properties of the angles in the triangle, leading to definitions of trigonometric ratios -Solve a triangle using the Law of Sine’s or Law of Cosines when given a specific for acute angles. combination of sides and angles.

CC9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of Meaning complementary angles. UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… -How do you find a side length or angle CC9-12.G.SRT.8 Use trigonometric ratios - When an altitude is drawn to the measure in a right triangle? and the Pythagorean Theorems to solve hypotenuse of one triangle three similar right triangles in applied problems. right triangles are formed. -How do trigonometric ratios relate to similar right triangles? CC9-12.G.SRT.10 Prove the Law of Sine’s -The Pythagorean Theorem can be used on and the Law of Cosines and use them to right triangles to determine the missing side solve problems. lengths.

CC9-12.G.SRT.11 Understand and apply the -If three lengths are given the converse of Law of Sine’s and the Law of Cosines to find the Pythagorean Theorem can be used to

unknown measurements in right and non- determine if the triangle is right. right triangles (e.g., surveying problems, resultant forces) -There are certain right triangles in which a pattern can be formed to find their side lengths. These triangles have angle measures of 45-45-90 or 30-60-90.

-Trigonometry is used to find missing side lengths and angles measures when given a certain combination of sides and angles.

-The angle of elevation is the angle between the horizontal and the line of sight when an observer is looking up. The angle of depression is the angle formed with the horizontal and the line of sight when the observer is looking down. The angle of elevation and angle of depression are congruent and can be used when solving application problems.

-The Law of Sine’s and Law of Cosines can be used to find parts of ANY triangle that is not a right triangle. Solving a triangle means finding all the missing side and angle measures.

Acquisition Students will know… Students will be skilled at… Definitions: -Identifying and find the geometric mean  Geometric Mean when given a diagram and writing a  Hypotenuse similarity statement for the three right  Legs

 Special Right Triangles triangles formed.  Trigonometric Ratio  Angle of Elevation -Using the Pythagorean Theorem to  Angle of Depression determine the missing side lengths. -Pythagorean Theorem and it’s Converse -Geometric Mean Theorem and its’ -Determining the classification of a triangle Corollaries using the Pythagorean Theorem. -45-45-90 Theorem -Verifying a triangle is right on the -30-60-90 Theorem -Law of Sines coordinate plane. -Law of Cosines -Identifying and using Pythagorean Triples

to determine right triangles.

-Using properties of special right triangles to find missing side lengths and the missing hypotenuse.

-Determining if a triangle on a coordinate plane is a special right triangle by using its side measures.

-Identifying and using trigonometric ratios to find side lengths or angle measures.

-Finding missing side and angle measures of a triangle on a coordinate plane.

- Solving application questions in which the angle of depression or angle of elevation is needed or must be found.

-Writing and solve ratios with the proper amount of information using the Law of Sines (AAS. ASA, SSA).

-Using the Law of Cosines to solve

application problems when given 3 sides or one angle and two sides.

-Solving triangles for all missing information using the Law of Sine’s and/or the Law of Cosines.

Stage 2 - Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each Students will be assessed through a formative assessment that may contain but is not problem on any summative assessment. limited to the following tasks: • Using similar triangles to find the geometric mean of a right triangle by writing proportions. 4 – Student demonstrates a complete • Determining whether a triangle is right, acute, or obtuse. understanding of the concept with a correct • Completing PARCC like application problems involving special right triangles and the solution. extended ratio of their sides. • Identifying which trigonometric ratios to use when finding a missing angle or side of a right 3 – Solution contains one minor arithmetic triangle. error but demonstrates complete • Applying trigonometric ratios to application questions involving the angle of elevation or understanding of the concept. angle of depression. • Solving any kind of triangle by using the Law of Sines and Law of Cosines. 2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

0 – Incorrect answer demonstrating no understanding of the concept.

OTHER EVIDENCE:

The following is a student rubric to assess • Homework/Classwork individual understanding during class • Written response to one of the essential questions using vocabulary, mathematical activities. properties, and insights • Formative assessments such as individual or pair quizzes 4 – I understand completely and I can teach it to a classmate. • Peer and self-assessment • Collaborative group work 3 – I understand the concept but I do not • Geometer’s Sketchpad Explorations think I can explain it to a classmate.

2 – I can complete the task with assistance.

1 – I need help. Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Simplify radicals that are not perfect squares • Using the Pythagorean Theorem and Geometric Mean to find the missing sides of right triangles. • Derive a proof for 45-45-90 degree triangles when given a square. • Create their own angle of depression and angle of elevation application problems. • Solve triangles using Law of Sines and/or Law of Cosines.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 8 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Students will create an instructional video using the Show Me app. They will be asked to create and record the steps of solving various types of trigonometry questions. Students will upload these videos to the class YouTube account for review. Students will

be required to create a video for each concept learned!

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on adding, subtracting, multiplying, dividing radicals as well as rationalizing the denominator, finding side lengths by using special right triangles, Pythagorean Theorem and trigonometric ratios.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 8: Circles Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.C.2 Identify and describe -A radius of a circle and the tangent that intersects the endpoint of the radius on the circle relationships among inscribed angles, radii, has a special relationship. and chords. -Information about congruent parts of a circle (or congruent circles) can be used to find CC9-12.G.C.3 Construct the inscribed and information about other parts of the circle (or circles) circumscribed circles of a triangle, and prove properties of angles for a -Angles formed by intersecting lines have a special relationship to the arcs the intersecting quadrilateral inscribed in a circle. lines intercept. This includes arcs formed by chords that inscribe angles, angles and arcs formed by lines intersecting either within a circle or outside a circle and intersecting chord, CC9-12.G.C.4 Construct a tangent line from a intersecting secants, or a secant that intersects a tangent. point outside a given circle to the circle.

CC9-12.G.GPE.1 Derive the equation of a circle of given center and radius by using the Pythagorean Theorem; complete the Meaning square to find the center and radius of a UNDERSTANDINGS ESSENTIAL QUESTIONS circle given by an equation. Students will understand that… -How can you prove relationships between -A circle has many parts that can be angles and arcs in a circle?

measured and used in various ways. The radius is needed to find the circumference -When lines intersect a circle or within a and area of a circle. circle, how do you find the measures of the resulting angles, arcs, and segments? -An angle formed by two radii that meet at the center is called a central angle. The -How do you find the equation of a circle in central angles of a circle can be added to the coordinate plane? form 360 degrees.

-An arc smaller than 180 degrees is a minor arc, one greater than 180 degrees is a major arc, and one equal to 180 degrees is a semicircle. Arcs can be added together by using

the Arc Addition Postulate. Ratios can be used to find the length of an arc.

-In circles two arcs are congruent if their intercepting chords are congruent.

-Polygons can be inscribed if all of their vertices lie on the circle. A circle is circumscribed if the polygons vertices lie on its perimeter.

-When a diameter and a chord are perpendicular to each other the chord is bisected as well as the arc it intercepts. Chords are congruent if they are equidistant from the center of the circle.

-Angles are inscribed in circles when their vertex lies on the circumference of the circle. Its intercepted arc is two times the measure of the angle.

-If two inscribed angles intercept the same arc then they are congruent to one another. Every inscribed angle that intercepts a semicircle is a right angle. When a quadrilateral is inscribed in a circle then its opposite angles are supplementary.

- If a tangent and a radius intersect at the point of tangency a right angle is formed. The converse is also true. If two tangents to one circle meet at the same exterior point then they are congruent. Polygons are circumscribed about the circle if each one of

their sides intersects the circle just once at a point of tangency.

- If two secants intersect at the interior of a circle then the measure of the angle formed is one half the sum of the intercepted arcs formed by the angle and its vertical angle.

-When a secant and a tangent intersect at the point of tangency then the angle is half the measure of the intercepted arc.

-When two secants, two tangents, or a secant and a tangent intersect in the exterior of the circle the angle formed is one half the difference of the values of the bigger arc and the smaller arc.

-When two chords intersect the products of their parts are equal to one another.

-When two secants intersect outside of the circle the product of the whole secant and the exterior parts are congruent to one another.

-When a tangent and secant intersect outside of the circle, the square of the exterior tangent is equal to the exterior part of the secant multiplied by the whole secant.

Acquisition Students will know… Students will be skilled at… Definitions: -Identifying parts of a circle.  Circle  Center -Finding the radius and diameter of a circle  Radius when given specific information. Problem  Chord solving for the circumference, radius, and  Diameter diameter of a figure when given a specific  Tangent equation.  Point of Tangency  Secant -Writing and using algebraic equations to  Circumference find the measures of central angles and their  Area intercepted arcs.  Minor Arc  Major Arc -Identifying various types of arcs in a given  Semi-Circle diagram and find the value of their measure.  Arc Measure -Finding the length of the arcs of circle when  Arc Length -Tangency Theorem and its Converse given the central angle measure and -Congruent Circles Theorem and its circumference. Converse -Using proofs to show the Congruent Arcs -Congruent Arcs Theorem and its Converse -Inscribed Angles Theorem and its and Chords Theorem are valid. Corollaries -Using circumscribed circles to determine -Angles and Arcs Theorem and its Converse the values of interior angles of a polygon.

-Writing and solving algebraic equations using the properties of inscribed angles.

-Proving two triangles to be congruent when their angles intercept the same arcs.

-Finding lengths of radii or tangents by using the Pythagorean Theorem.

-Identifying tangents using the Converse of

the Pythagorean Theorem.

-Writing and solving algebraic equations using tangents, secants, and chords..

Stage 2 - Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each Students will be assessed through a formative assessment that may contain but is not problem on any summative assessment. limited to the following tasks: • Identifying parts of a circle. 4 – Student demonstrates a complete • Using the Tangency Theorem and its Converse to find the radius of a circle. understanding of the concept with a correct • Solving algebraic problems involving the Congruent Circles Theorem and its Converse. solution. • Developing two-column proofs using the properties of arc measure and the Inscribed Angles Theorem and its Corollaries. 3 – Solution contains one minor arithmetic • Comparing and contrasting arc length vs. arc measure, and knowing when to use each. error but demonstrates complete • Developing proofs using circle theorems, properties of similarity, and trigonometric ratios. understanding of the concept.

2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

• Peer and self-assessment 3 – I understand the concept but I do not • Collaborative group work

think I can explain it to a classmate. • Geometer’s Sketchpad Explorations

2 – I can complete the task with assistance.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Use Geometer’s Sketchpad to determine the relationship between two chords that intersect and their lengths. • Derive the Inscribed Angles Theorem using similar triangles. • Write the equations of a circle on a coordinate plane. • Use trigonometry to find central angle measure and arc measures.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 12 Solve It! and Lesson Check from the Geometry Common Core textbook.

Constructions: Given three non-collinear points, construct the circle that includes all three points.

1. Begin with points A, B, and C.

2. Draw line segments AB and BC.

3. Construct the perpendicular bisectors of line segments AB and BC. Let point P be the intersection of the perpendicular bisectors.

4. Center the compass on point P, and draw the circle through points A, B, and C.

Foldable:

Part One

1. Use a compass to draw a circle on tracing paper. 2. Use a straightedge to draw two radii. 3. Set your compass to distance shorter than the radii. Place its point at the center of the circle. Mark two congruent segments one on each radius. 4. Fold a line perpendicular to each radius at the point marked on the radius.

a) How do you measure the distance between a point and a line? b) Each perpendicular contains a chord. Compare the lengths of the chords. c) What is the relationship among the lengths of the chords that’s are equidistant from the center of a circle?

Part Two:

1. Use the compass and draw a circle on tracing paper. 2. Use a straightedge to draw two chords that are not diameters. 3. Fold the perpendicular bisector for each chord.

a) Where do the perpendicular bisectors appear to intersect the other now? b) Draw a third chord and fold its perpendicular bisector. Where does it appear to intersect the other two? c) What is true about the perpendicular bisector of a chord?

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 9: Transformations and Symmetry Stage 1 Desired Results ESTABLISHED GOALS Transfer G.CO.2 Represent transformations in the Students will be able to independently use their learning to… plane. Describe transformations as functions -Change the position of geometric figures so the angle measure and distance between any that take points in the plane as inputs and two points of a figure stay the same. give other points as outputs. Compare -A reflection of a figure across a line is a mapping of each point to another point the same transformations that preserve distance and distance from the line but on the other side. The orientation of the figure reverses. angle to those that do not. -Show rotations preserve distance, angles measures, and orientation of figures. -Isometries can be expressed as compositions of reflections. G.CO.3 Given a rectangle, parallelogram, -Congruence can be understood by using compositions of rigid motion. trapezoid, or regular polygon, describe -Scale factor can be used to make a larger or smaller copy of a figure that is similar to the rotations and reflections that carry it onto original. itself. -Compositions of rigid motions and dilations help understand similarity and its properties.

G.CO.4 Develop definitions of rotations, reflections, translations and dilations in Meaning terms of angles, circles, perpendicular lines, UNDERSTANDINGS ESSENTIAL QUESTIONS parallel lines, and line segments. Students will understand that… -How can you change a figure’s position -The image and preimage must preserve without changing its size and shape? G.CO.5 Given a geometric figure and a distance and angle measures. rotation, reflection, or translation, draw the -How can you change a figure’s size without transformed figure. Specify a sequence of -Corresponding points have the same changing its shape? transformations that will carry a given position in the names of the preimage and figure onto another. image. -How can you represent a transformation in the coordinate plane? G.CO.6 Use geometric descriptions of rigid -A translation maps all points of a figure the motions to transform figures and predict the same distance in the same direction. -How do you recognize congruence and effect of a given rigid motion on a given similarity in figures? figure: two given figures use the definition -The composition of any two translations is of congruence in term of rigid motions to another translation. determine if they are congruent. -Two figures reflect is they corresponding G.CO.7 Use the definition of congruence in vertices of the preimage and image read in terms of rigid motions to show that two opposite directions.

triangles are congruent if and only if corresponding pairs of sides and angles are -A reflection across a line is called line congruent. reflection.

G.CO.8 Explain how criteria for triangle -Reflections preserve distance and angle congruence follow from the definition of measures. They map each point of the congruence in terms of rigid motion. preimage to one and only one corresponding point of its image. G.SRT.1a The dilation of a line segment is longer or shorter in the ratio given by the -A rotation about a point is a rigid motion. scale factor. Rotations can be seen in a plane. . -Rotations are assumed to be counterclockwise unless otherwise stated.

-A figure has symmetry if there is a rigid motion that maps the figure onto itself.

-The composition of two or more isometries is an isometry.

-A composition of reflections across two parallel lines is a translation. A composition of reflections across two intersecting lines is a rotation.

-Two figures are congruent if there is a sequence of one or more rigid motions that maps one figure onto the other.

-The scale factor of a dilation is the ratio of a length of the image to the corresponding length in the preimage.

-Two figures are similar if and only if there is a similarity transformation that maps one

figure onto the other.

Acquisition Students will know… Students will be skilled at… Definitions: -Identifying transformations when given a  Transformation coordinate plane.  Preimage  Image -Making shifts to a preimage based on the  Rigid Motion notation given to them.  Translation  Composition of Transformations -Comparing and contrasting rigid motions.  Reflection  Line of Reflection -Graphing images and labeling the vertices  Rotation in prime notation.  Center of Rotation  Angle of Rotations -Rotating a preimage about a point in any  Symmetry direction.  Line of Symmetry  Rotational Symmetry -Proving a transformation is an isometry on  Point Symmetry and off a coordinate plane.  Glide Reflection  Isometry -Understanding glide reflections and their  Congruence Transformation orientations.  Dilation  Center of Dilation -Proving transformational congruence and  Enlargement similarity.  Reduction -Isometry Theorem -Creating tessellations based on rigid -Reflections Across Parallel Lines Theorem motions. -Reflections Across Intersecting Lines Theorem -Enlarging or shrinking figures by using their scale factor on or off a coordinate plane.

Stage 2 - Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each Students will be assessed through a formative assessment that may contain but is not problem on any summative assessment. limited to the following tasks: • Identifying transformations when given a image and preimage. 4 – Student demonstrates a complete • Graphing images and using prime notation. understanding of the concept with a correct • Shifting a preimage based on the rotation, translation, reflections, and dilation given. solution. • Proving a transformation is an isometry. • Showing a transformation preserves congruence on a coordinate plane. 3 – Solution contains one minor arithmetic • Using scale factor to dilate a figure. error but demonstrates complete • Determining the transformation based on coordinate notation. understanding of the concept. • Using the Reflections Across Parallel Lines Theorem in PARCC like application questions.

2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Enlarge figures on a coordinate plane. • Prove figures to be similar through dilations and rules of similarity. • Identify various transformations and create coordinate notation based on the rigid motion. • Create tessellations of images to promote core content understanding. •

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 9 Solve It! and Lesson Check from the Geometry Common Core textbook

iPad Activity: Students will download a Geometer’s Sketchpad activity off of the teacher wikipage. The activity explores translations, reflections, rotations, and dilations as seen in the screenshots below. The students will explore what each of the transformations do and complete an activity sheet summarizing what they have learned.

Coordinate Plane Activities: Please refer to the Concept Bytes sections in the Geometry Common Core textbook for Coordinate Plane Activities relating to transformations.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 10: Quadrilaterals Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.CO.11 Prove theorems about -The sum of the angle measures of a polygon depends on the number of sides the polygons parallelograms. has.

CC9-12.G.CO.13 Construct and equilateral -Parallelograms have special properties regarding sides, angles, and diagonals. triangle, a square, and a regular hexagon inscribed in a circle. -If a quadrilaterals sides, angles, and diagonals have certain properties; it can be shown that the quadrilateral is a parallelogram. CC9-12.G.SRT.5 Use congruence and -The special parallelograms, rhombus, rectangle, and square have basic properties of their similarity criteria for triangles to solve sides, angles, and diagonals that help identify them. problems and to prove relationships in geometric figures. -The angles, sides, and diagonals of a trapezoid have certain properties.

CC9-12.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 Meaning perpendicular to a given line that passes UNDERSTANDINGS ESSENTIAL QUESTIONS through a given point) Students will understand that… -How can you find the sum of the measures -A diagonal of a polygon is a segment that of polygon angles? CC9-12.G.MG.3 Apply geometric methods to connects any two nonconsecutive vertices. solve design problems (e.g., designing an You can find the sum of the interior angles -How can we prove quadrilaterals to be object or structure to satisfy physical by using the Interior Angle Sum Theorem. parallelograms? constraints or minimize cost; working with typographic grid systems based on ratios.) -The Exterior Angle Sum Theorem states -How can you classify quadrilaterals on and that all the exterior angles of a polygon will off a coordinate plane? always add up to 360 degrees.

- All parallelograms have distinctive properties such as: opposite sides congruent and parallel, opposite angles congruent, consecutive interior angles are

supplementary, and if the parallelogram contains one right angle then all four angles are also right.

- The diagonals in a parallelogram bisect one another, and cut the parallelogram into two congruent triangles.

-Properties of parallelograms can be used to prove a quadrilateral to be parallelograms by showing them to be valid.

-A rectangle is a is considered a special type of parallelogram. It has all the properties of a parallelogram except all angles are 90 degrees. The diagonals of a rectangle are congruent. These properties can be used to classify a parallelogram as a rectangle.

-A rhombus is a special type of parallelogram. Rhombi have four congruent sides and all the properties of a parallelogram. Rhombi have diagonals that are perpendicular, and diagonals that bisect the angles they intersect.

-If a parallelogram is both a rhombus and a rectangle then it is considered a square. A square has all the properties of a parallelogram, rectangle, and rhombus.

-Trapezoids are quadrilaterals with exactly one set of parallel sides.

-If a trapezoid is isosceles it has congruent legs and base angles.

-The midsegment of a trapezoid is half the sum of the bases.

Acquisition Students will know… Students will be skilled at… Definitions: -Deriving the Interior Angle Sum Theorem  Regular Polygons through an interactive activity.  Parallelogram  Opposite Angles -Using the Interior Angle Sum Theorem to  Opposite Sides find the sum of the angles of regular  Consecutive Angles polygons.  Rhombus  Rectangle -Completing application problems in which  Square the Interior Angle Sum Theorem is used.  Trapezoid  Base -Finding the number of sides a polygon has  Leg when given the sum of the interior angles.  Base Angle -Deriving the Exterior Angle Theorem by  Isosceles Trapezoid using the Geometer’s Sketchpad activity to  Midsegment  Kite investigate its properties.  Coordinate Proof -Using the Exterior Angle Theorem to -Polygon Angle Sum Theorem determine the amount of sides a polygon has -Polygon Exterior Angle Sum Theorem -Properties of Parallelograms Theorems and when the angle measure is known. Corollaries -Writing a two-column proof in which they -Converses of Parallelograms Theorems prove that the opposite angles of a -Rectangle Theorem parallelogram are congruent. -Rhombus Theorems -Isosceles Trapezoid Theorem -Recognizing and using properties of parallelograms to write algebraic equations and solve for values throughout the

parallelogram.

-Completing standardized test questions involving the use of properties of parallelograms.

-Determining if the given quadrilateral is a parallelogram by showing properties to be true.

-Finding the value of x and y to make the quadrilateral a parallelogram.

-Using the slope and distance formula on a coordinate plane to determine whether the given points form a parallelogram.

-Recognizing and use properties of rectangles to write algebraic equations and find the values of variables.

- Proving parallelograms to be rectangles by using their properties in an application problem.

-Using the slope formula and distance formula on a coordinate plane to prove a parallelogram to be a rectangle.

-Writing a two-column proof in which they must prove the diagonals of a rhombus to be perpendicular.

-Writing and solving algebraic equations to find the measure of sides and angles in a rhombus.

-Using coordinate geometry to determine if the parallelogram is a rectangle, rhombus, or a square.

-Applying the properties of a square to an application problem.

- Finding the median/midsegment of a trapezoid when both bases are given.

-Finding a base of a trapezoid when one base and the midsegment are given.

Stage 2 - Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each Students will be assessed through a formative assessment that may contain but is not problem on any summative assessment. limited to the following tasks: • Using the properties and theorems of various parallelograms to classify a quadrilateral. 4 – Student demonstrates a complete • Proving the Converses of the Parallelogram Theorems. understanding of the concept with a correct • Using the Rectangle Theorem, Rhombus Theorem, and properties of a parallelogram in solution. various problem solving application questions. • Using the Isosceles trapezoid Theorem and Midsegment Theorem in algebraic problem 3 – Solution contains one minor arithmetic solving. error but demonstrates complete • Determining what exact polygon a quadrilateral is on a coordinate plane. understanding of the concept. • Deriving the Polygon Angle Sum Theorem.

2 – Solution contains more than one minor arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal

understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Use properties of parallelograms to prove quadrilateral is in fact a parallelogram, rectangle, rhombus, or square. • Complete coordinate plane activities to discover the properties of each quadrilateral. • Discover trapezoidal midsegments via Geometer’s Sketchpad activities. • Create a graphic organizer to remember which quadrilaterals are related.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 6 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: At the conclusion of this lesson students will use the SAT: Geometry and Measurement Free App to test their knowledge of quadrilaterals, polygons, and circles through practice SAT questions. This free test will serve as an excellent review of current and past topics.

Geometer’s Sketchpad: Exterior Angles of Polygons

Use geometry software. Construct a polygon using extended segments. Mark a point on each ray so you can measure the angles.

-Measure each exterior angle.

-Calculate the sum of the measures of the exterior angles.

-Manipulate the polygon. Observe the sum of the measures of the exterior angles of the new polygon.

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on using various properties of quadrilaterals, proving quadrilaterals, and determining angle measures using trigonometry to prove parallelograms.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 11: Area of Polygons Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.GPE.7 Use coordinates to compute -The area of a parallelogram or a triangle can be found when the length of its base and perimeters of polygons and areas of height are known. triangles and rectangles, e.g., using the distance formula. -The area of a trapezoid can be found when the height and its lengths of its bases are known. CC9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a -The area of a rhombus or kite can be found when lengths of the diagonals are known. circle, area of a circle, volume of a cylinder, -The area of a regular polygon is a function of the distance from the center to a side. pyramid, and cone. -Trigonometry can be used to find the area of a regular polygon when the length of a side, CC9-12.G.MG.3 Apply geometric methods to radius, or apothem, is known or to find the area of a triangle when the length of two sides solve design problems. and the included angle is known. . -The length of part of a circle’s circumference can be found by relating it to an angle in a circle.

-The area of parts of a circle formed by radii and arcs can be found when the circle’s radius is known.

-Ratios can be used to compare the perimeters and area of similar figures.

Meaning UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… -How do you find the area of a polygon or -Any side of a parallelogram can be called a find the circumference or area of a circle? base. For each base, there is a corresponding altitude that is perpendicular to the base. -Compare and contract the ratios of the The area of a parallelogram is found by perimeters and areas of similar polygons.

multiplying its base and its height. -What methods can be used to find the area of any regular polygon? -Multiplying the base and the height then taking the product and dividing by two allows you to find the area of a triangle.

-The formulas for the areas of a trapezoid and a rhombus are related to that of a triangle.

-By taking the sum of the bases multiplying it by the height and dividing by two the area of a trapezoid can be found.

-The rhombi area formula simply requires the product of the diagonals to be divided by two.

-Two figures are congruent if they have congruent areas.

-The area of any regular polygon can be found when it is inscribed in a circle. The apothem is required to find the area of any regular polygon. The area of found when you take the product of the perimeter of the polygon and the apothem then divide by two.

-The area of a circle uses a value of pi and the radius. The area can be found by finding the product of pi and the square of the radius.

-An irregular figure is a figure that cannot be classified into a specific shape. You must

separate these figures into known figures and find the area of each part. Once each part is found add them together to find the area of the entire irregular figure.

-The units for area are always squared.

Acquisition Students will know… Students will be skilled at… Definitions: -Finding the area and perimeter of a  Base of a Parallelogram parallelogram.  Altitude of a Parallelogram  Height of a Parallelogram -Using the area of a parallelogram to solve  Base of a Triangle real world problems.  Height of a Triangle  Height of a Trapezoid -Finding the area of a parallelogram on a  Apothem coordinate plane.  Circumference -Finding the areas of different triangles  Pi -Area of a Rectangle Theorem including those that are equilateral, right, -Area of a Parallelogram Theorem scalene, and isosceles. -Area of a Triangle Theorem -Finding the area of a trapezoid and a -Area of a Trapezoid Theorem -Area of a Rhombus or Kite Theorem rhombus. -Area of a Regular Polygon Theorem -Using coordinate geometry to find the area -Perimeter and Areas of Similar Figures of a trapezoid and rhombus on a coordinate Theorem plane. -Area of a Circle Theorem -Circumference of a Circle Theorem -Using algebra to find the missing measures of a triangle, trapezoid, and rhombus when

the area is given.

-Using area formulas to determine if two figures in real life are congruent.

-Finding the area of a regular polygon when given the apothem.

-Finding the area of a regular polygon when given only the radius.

-Finding the area and circumference of a circle in Pi-Form.

-Determining the radius of a circle when given the area or circumference.

-Using the area of a circle to solve real world problems.

-Using the area of regular polygon formula and area of a circle formula to find the area of the shaded region.

-Finding the area of a irregular figure.

-Using the area of a irregular figure to solve a real life problem.

-Determining the area of a irregular figure on a coordinate plane.

3 – Solution contains one minor arithmetic • Using previous topics such as inscribed polygons, Polygon Angle Sum Theorem, special right error but demonstrates complete triangles and trigonometric ratios to find the area of any regular polygon. understanding of the concept. • Determining the area and perimeter of similar figures. • Finding the area and circumference of a circle and leaving the answer in terms of Pi. 2 – Solution contains more than one minor • Using complex reasoning to find the area of the shaded region of two overlapping polygons. arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal

understanding of the concept. 0 – Incorrect answer demonstrating no understanding of the concept. The following is a student rubric to assess OTHER EVIDENCE: individual understanding during class activities. • Homework/Classwork • Written response to one of the essential questions using vocabulary, mathematical 4 – I understand completely and I can teach properties, and insights it to a classmate. • Formative assessments such as individual or pair quizzes

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Find the area of irregular figures and area of the shaded region. • Use coordinate geometry to find the area of figures. • Apply area and perimeter to real life by having students find the cost of molding and carpeting for a new home.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 10 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Students will be introduced to the idea of area through a brief Geometer’s Sketchpad activity on the iPad. Students will be asked to fill a quadrilateral with circles whose area has already been determined. The circles can be made larger or smaller to fit inside the figure. The app will generate the sum of the areas of the circles to determine an estimated area for the quadrilateral. The pictures below show what the activity can do. Students will then define area in their own words.

Concept Activity: Area of a Circle

Suppose each regular polygon is inscribed in a circle of radius r.

1. Copy and complete the following table. Round to the nearest hundredth. Number of Sides 3 5 8 10 20 50

Measure of a Side 1.73r 1.18r .77r .62r .31r .126r

Measure of Apothem .5r .81r .92r .95r .99r .998r

Area

2. What happens to the appearance of the polygons as the number of sides increases? 3. What happened to the areas as the number of sides increases? 4. Make a conjecture about the area of a circle.

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on area of regular and irregular figures, solving for parts when given the area, and area of the shaded region.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Calculator www.desmos.com/calculator

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 12: Surface Area and Volume Stage 1 Desired Results ESTABLISHED GOALS Transfer Students will be able to independently use their learning to… CC9-12.G.GMD.1 Give an informal argument -A three dimensional figure can be analyzed by describing the relationship among its for the formulas for the circumference of a vertices, edges, and faces. circle, area of a circle, volume of a cylinder, pyramid, and cone. -The area of three-dimensional figure is equal to the sum of the areas of each surface of the figure. CC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to -The volume of a prism and cylinder can be found when its height and area of its base are solve problems. known. -The volume of a pyramid is related to the volume of a prism with the same base and height. CC9-12.G.GMD.4 Identify the shapes of two- dimensional cross sections of three- -The surface area and the volume of a sphere can be found when the radius is known. dimensional objects, and identify three- dimensional objects generated by rotations of two-dimensional objects. Meaning UNDERSTANDINGS ESSENTIAL QUESTIONS Students will understand that… -How can you determine the intersection of -A net is a pattern for a three-dimensional a solid and a plane? figure if it was laid flat. -How is the surface area of a solid related to - Lateral area is the sum of the area of the the volume? lateral faces. Surface area is the sum of the area of each part of the net. Both lateral and surface areas are measured in squared units.

-A prism is a polyhedron with two parallel congruent bases. The rectangular faces that are not bases are called lateral faces. The lateral faces intersect at lateral edges. The

height of the prism is the altitude that connects both bases.

-Lateral area can be found by multiplying the perimeter of the base by the height of the prism.

-The lateral area is used to find the surface area of the prism. The surface area can be found by adding the lateral area to two times the area of the base of the prism.

-The lateral area of a cylinder can be found by taking the product of the circumference of the base and the height of the cylinder. Using the lateral area and adding two times the area of each base to it can find the surface area.

-A pyramid is a polyhedron in which all the lateral faces intersect at one vertex with a base that can be any polygon. The altitude is from the vertex to the center of the base. The slant height is an altitude from the vertex to the edge of the base of the pyramid.

-The lateral area of a pyramid is found by taking half of the perimeter of the base multiplied by the slant height. The surface area is found by adding the lateral area to the area of the base.

-A cone has a circular base and a vertex. The axis (height) goes from the vertex of the center of the circular base. The slant height

is an altitude from the vertex to the circumference of the circular base. The product of pi, the radius of the circular base, and the slant height find the lateral area. The surface area can be found by adding the area of the base to the lateral area.

-A sphere is a set of points in space that are a given distance from a given point. The cross section of the circle through the center is called the “great circle.” Each great circle separates the sphere into two congruent halves or hemispheres. The surface area of a sphere can be found by taking four times the area of the great circle.

-Volume of a figure is the measure of space the figure encloses. It is measured in cubic units.

-Taking 4/3 of pi and cubing the radius finds the volume of a sphere.

-The volume of any prism is found by multiplying the area of the base of the figure times the height of the lateral faces.

-The volume of a cylinder is found by multiplying the area of the base ( ) by the height of the figure.

-The volume of a pyramid is the area of its base multiplied by its height. and divided by 3. The height can be found by using the lateral height.

-The volume of a cone is found by multiplying the area of the base ( ) by the height of the cone.

Acquisition Students will know… Students will be skilled at… Definitions: -Identifying and name three-dimensional  Polyhedra figures.  Face  Edge -Drawing nets for any solid.  Vertex  Cross Section -Using net to determine surface area.  Lateral Face -Finding the lateral area of all different types  Lateral Edge of prisms (ex. triangular, pentagonal, etc.)  Lateral Area  Surface Area -Finding the surface area of various types of  Volume prisms.  Prism  Right Prism -Using the apothem to find the surface area  Oblique Prism of prisms with bases that are regular  Cylinder polygons.  Right Cylinder  Oblique Cylinder -Using surface area of a prism to solve real  Pyramid world problems.  Regular Pyramid  Cone -Finding the lateral area of a cylinder.  Right Cone  Sphere -Using the lateral area of a cylinder to find -Lateral Area and Surface Area of a Prism the surface area. Theorem -LA and SA of a Cylinder Theorem -Finding missing dimensions of a cylinder -LA and SA of a Pyramid Theorem when given the surface area. -LA and SA of a Cone Theorem -SA and Volume of a Sphere Theorem -Finding the lateral area of a regular

-Volume of a Prism Theorem pyramid. -Volume of a Cylinder Theorem -Volume of a Pyramid Theorem -Finding the surface area of a regular -Volume of a Cone Theorem pyramid of any base. (ex. Square, pentagonal, rectangular)

-Finding the lateral area of a right cone.

-Using the lateral area to find the surface area of a right cone.

-Using the surface area of a pyramid and a cone to solve real world problems.

-Finding the area of the great circle.

-Finding the surface area of a sphere.

-Using the surface area formula to solve real world problems involving spheres.

-Finding the volume of a sphere.

-Using the volume of a sphere to solve real world problems.

-Finding the volume of a triangular prism.

-Finding the volume of a rectangular prism.

-Finding the volume of any prism with a regular polygon as its base.

-Finding the volume of a right cylinder.

-Finding the volume of an oblique cylinder.

-Using volume to solve real world problems.

-Finding the volume of a pyramid with a square base.

-Finding the volume of a pyramid with a rectangular base.

-Finding the volume of any pyramid with a regular polygon as its base.

-Finding the volume of a right cone.

-Finding the volume of an oblique cylinder.

-Using volume to solve real world problems.

Stage 2 – Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each Students will be assessed through a formative assessment that may contain but is not problem on any summative assessment. limited to the following tasks: • Finding the surface area and volume of any right prism. 4 – Student demonstrates a complete • Deriving the surface area and volume formulas for a cylinder based off the formulas for a understanding of the concept with a correct right prism. solution. • Using the cylinder formulas to find the surface area and volume in terms of Pi. • Finding the missing parts of a pyramid and cone by using the Pythagorean Theorem, 3 – Solution contains one minor arithmetic trigonometric ratios, and special right triangles. error but demonstrates complete • Finding the surface area and volume of any right pyramid. understanding of the concept. • Deriving the surface area and volume formulas for a cone based off the formulas for a right pyramid. 2 – Solution contains more than one minor • Using the cone formulas to find the surface area and volume in terms of Pi. arithmetic error but demonstrates partial • Finding the surface area and volume of a sphere in terms of Pi. understand of the concept.

1 – Solution contains multiple arithmetic

errors while demonstrating minimal understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Find the surface area and volume of real life solids. • Use the principles of surface area and volume to find the measures of combined polyhedral. • Derive the formulas for cylinders and cones from prisms and pyramids.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 11 Solve It! And Lesson Check from the Geometry Common Core textbook.

Concept Activity: Collect some empty cardboard containers shaped like prisms and cylinder.

1. Measure each container and calculate its surface area. 2. Flatten each container by carefully separating the places where it has been glued together. Find the total area of the packaging material used. 3. For each container, find the percent by which the area of the packing material exceeds the surface area of the container. -How does the unfolded prism-shaped package differ for the net of the prism? -What did you find out about the amount of extra material needed for the prism shaped containers? For the cylindrical? -Why would a manufacturer be concerned about the surface area of a package? About the amount of material used for the package?

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on finding the lateral area, surface area, and volume of prisms, cylinders, pyramids, cones, and spheres.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Unit 13: Probability and Measurement Stage 1 Desired Results ESTABLISHED GOALS Transfer S.CP.1 Describe events as subsets of a Students will be able to independently use their learning to… sample space (the set of outcomes) using -Various counting methods can help you analyze situations and develop theoretical characteristics (or categories) of the probabilities. outcomes, or as unions, intersections, or complements of other events (“or,” “and,” -You can use multiplication to quickly count the number of ways certain things can happen. “not”). -The probability of an impossible event is 0. The probability of a certain event is 1. S.CP.2 Understand that two events A and B Otherwise the probability of an event is a number between 0 and 1. are independent if the probability of A and B -To find the probability of two events occurring you have to decide whether one event occurring together is the product of their occurring affects the other event. probabilities, and use this characterization to determine if they are independent. -Conditional probability exists when two events are dependent.

S.CP.3 Understand the conditional - In geometric probability, numbers of favorable and possible outcomes are geometric probability of A given B as P(A and B)/P(B), measures such as lengths of segments or areas of regions. and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Meaning UNDERSTANDINGS ESSENTIAL QUESTIONS S.CP.4. Construct and interpret two-way Students will understand that… -What is the difference between frequency tables of data when two The Fundamental Counting Principle experimental and theoretical probability? categories are associated with each object describes the method of using being classified. Use the two-way table as a multiplication to count. A permutation is an -How are the laws of probability used to sample space to decide if events are arrangement of items in a particular order. predict outcomes in the real world? independent and to approximate Using factorial notation allows you to write conditional probabilities. For example, 3x2x1 as 3!. A selection in which order does -How is statistics used to analyze data in collect data from a random sample of not matter is called a combination. real world situations? students in your school on their favorite subject among math, science, and English. When you gather data from observations, you calculate an experimental probability.

Estimate the probability that a randomly Each observation is an experiment or a trial. selected student from your school will favor A simulation is a model of the event. The set science given that the student is in tenth of all possible outcomes to an experiment or grade. Do the same for other subjects and activity is a sample space. If the outcomes in compare the results. a sample space have the same chance of occurring, the outcomes are called equally S.CP.5 Recognize and explain the concepts likely outcomes. Theoretical probability is of conditional probability and independence when a sample space has equally likely in everyday language and everyday outcomes to an event occurring. situations. For example, compare the chance of having lung cancer if you are a smoker When the occurrence of one event affects with the chance of being a smoker if you how a second event can occur, the events have lung cancer. are dependents events. Otherwise the events are independent events. Two events S.CP.6 Find the conditional probability of A that cannot happen at the same time are given B as the fraction of B’s outcomes that mutually exclusive events. also belong to A, and interpret the answer in terms of the model. The probability of an event occurring given that another event occurs is called conditional probability. A contingency table is a frequency table that contains data from S.CP.7. Apply the Addition Rule, P(A or B) = two different categories. Using the formula P(A) + P(B) – P(A and B), and interpret the for conditional probability you can calculate answer in terms of the model. the conditional probability from other probabilities.

In geometric probability, numbers of S.CP.8 Apply the general Multiplication Rule favorable and possible outcomes are in a uniform probability model, P(A and B) = geometric measures such as lengths of P(A)P(B|A) = P(B)P(A|B), and interpret the segments or areas of regions. The answer in terms of the model. probability of a point being located on a specific part of a segment is the ratio of the S.CP.9 Use permutations and combinations specific length of the segment to the length to compute probabilities of compound of the whole segment. Probability can also be used when finding the chances of a point

events and solve problems. being in an inscribed figure.

S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Acquisition S.MD.7 Analyze decisions and strategies Students will know… Students will be skilled at… using probability concepts (e.g., product Definitions: -Using the Fundamental Counting Principle. testing, medical testing, pulling a hockey  Outcome -Finding the number of permutations of n goalie at the end of the game).  Event  Sample Space items.  Probability  Experimental Probability -Using the permutation formula.  Theoretical Probability -Using the combination formula.  Complement of an Event  Frequency Table -Identifying whether the order matters in a  Relative Frequency event.  Probability Distribution  Permutation -Finding experimental probability.  N-Factorial  Combination -Using a simulation.  Compound Event  Independent Event -Finding theoretical probability.  Dependent Event  Mutually Exclusive Events -Finding probability using combinations.  Overlapping Events -Classifying events as independent or  Conditional Probability dependent.  Geometric Probability  Expected Value -Finding the probability of independent -Fundamental Counting Principle events.

-Finding the probability of dependent events.

Finding the probability of mutually

exclusive events.

-Finding conditional probability using contingency tables.

-Using conditional probability in statistics.

-Using the conditional probability formula.

-Using a tree diagram to find the conditional probability.

-Using segments to find probability.

-Using area to find probability.

Stage 2 – Evidence Evaluative Criteria Assessment Evidence The following rubric will be applied to each Students will be assessed through a formative assessment that may contain but is not problem on any summative assessment. limited to the following tasks: • Using the Fundamental Counting Principle in PARCC application questions. 4 – Student demonstrates a complete • Finding the permutation or combinations of n items. understanding of the concept with a correct • Determining the probability of independent events, dependent events, mutually exclusive solution. events. • Comparing and contrasting experimental probability and theoretical probability. 3 – Solution contains one minor arithmetic • Developing contingency tables and tree diagrams to find conditional probability of an event error but demonstrates complete occurring. understanding of the concept. • Applying probability principles to find the geometric probability of an event occurring 2 – Solution contains more than one minor through segment length or area. arithmetic error but demonstrates partial understand of the concept.

1 – Solution contains multiple arithmetic errors while demonstrating minimal

understanding of the concept.

1 – I need help.

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

The teacher and students will use class discussion and small group cooperation to accomplish the following tasks:

• Create a lab comparing experimental and theoretical probability. • Create frequency tables via experimentation. • Explore various probability models through random sampling.

Learning Events: Throughout each lesson student understanding will be assessed through an introductory and closing problem set, which assesses vocabulary, problem solving, reasoning, and open-ended questions. Recommended resources are the Chapter 13 Solve It! and Lesson Check from the Geometry Common Core textbook.

iPad Activity: Students will use the Undecided app to generate different probability questions. They can choose between coin flips, rock

paper scissors, drawing straws, spinners, and other random events. The student are to create problems in which probability is dependent, independent, or mutually exclusive using the highlights of the app. They can then switch their problems with a classmate for review!

Concept Activity: You want to find out if your favorite team won this weekend, but forgot that the show was on. You turned it on at 10:14pm. The score will be announced at one random time during the show. What is the probability that you haven’t missed the report about your favorite team?

A point in the figure is chosen at random. In the following figures find the probability that the point lies in the shaded region.

White Board Battle Game: Students will work in groups of four or five. They each need white boards and dry erase markers. As the problems appear on the board through a PowerPoint, the first team to have the correct answer on each team member’s board will receive a point. Each team member must complete each problem alone. If someone finishes early, they can help out their teammates. This activity can focus on finding permutations and combinations, as well as the geometric probability of an event occurring.

Technology:

WolframAlpha APP, Mathway APP, Free GraCalc APP, powerOne SL APP, Geometer’s Sketchpad APP

Resources:

Geometry Common Core; Randall I. Charles, Basia Hall, Dan Kennedy, Laurie E. Bass, Art Johnson, Stuart J. Murphy, Grant Wiggins; Pearson Education Inc., 2012

Benchmark Assessment Quarter 1

1. The student will be able to identify and use parts and types of lines, angles, and planes in problems solving. 2. The student will be able to use logical reasoning and conditional statements to solve problems. 3. The student will be able to use angle relationships with parallel and perpendicular lines to solve problems.

Benchmark Assessment Quarter 2

1. The student will be able to use triangle classifications and congruent triangles to solve problems. 2. The student will be able to use the relationships of sides and angles in triangles to solve problems. 3. The student will be able to use proportions to determine similarity of triangles.

Benchmark Assessment Quarter 3

1. The student will be able to use right triangle trigonometry to solve problems. 2. The student will be able to use and apply properties of lines and angles in circles 3. The student will be able to recognize and apply properties of transformations.

Benchmark Assessment Quarter 4

1. The student will be able to use properties of polygons to solve problems. 2. The student will be able to find the lateral area, surface area, and volume of three-dimensional figures. 3. The students will use geometric probability and statistics to analyze real life situations.