curriculum management system - monroe township … · 2015-01-20 · goals/essential...

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

mailto:[email protected]



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

http://www.corestandards.org/

http://www.corestandards.org/the-standards

http://www.corestandards.org/the-standards

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

http://parcconline.org/

Quarter 1

Unit Topics(s)

I. Points, Lines, Planes 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

h. Distance Between Two Parallel Lines

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 a. Perfect Squares b. Radical Simplification c. Geometric Mean d. Pythagorean Theorem and its Converse e. Proof of Pythagorean Theorem f. Special Right Triangles 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

b. Proving Quadrilaterals to be Parallelograms c. Rectangles, Rhombi, and Squares d. Trapezoids e. Properties of Trapezoids f. Medians of Trapezoids (Midsegments) g. Coordinate Geometry h. Other Quadrilaterals (Kites)

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

CC9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

CC9-12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

CC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

CC9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Transfer Students will be able to independently use their learning to… -Understand Geometry is a mathematical system built on accepted facts, basic terms, and definitions.

-Show segments, rays, and lines are very similar but each have their own properties and can be combined to form larger figures in the geometric world.

- Use formulas to find the midpoint and length of any segment on a coordinate plane.

-Apply special angle pairs to identify geometric relationships and to find angle measures.

Meaning UNDERSTANDINGS Students will understand that… -A line is made up of an infinite amount of points. Two lines can intersect a point, two planes can intersect to form a line, and three planes intersect at a point in space. - Linear measure is the distance between 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.

ESSENTIAL QUESTIONS -What are the building blocks of geometry? -How can you describe the attributes of a segment or angle? -Why are units of measure important?

- 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… -Definitions: Point Line Ray Segment Plane Collinear Coplanar Space Right Angles Obtuse Angles Acute Angles Adjacent Angles Vertical Angles Linear Pair Complementary Angles Supplementary Angles Perpendicular Polygon -The Distance Formula -The Midpoint Formula -Segment Addition Postulate -Angle Addition Postulate

Students will be skilled at… _ -Modeling each concept using coordinate geometry.

-Comparing and contrasting the similarities and differences between each undefined term.

-Generating a philosophical conversation on the idea of space and how it applies to that around them.

-Writing and solving algebraic expressions using segment addition and angle addition.

- Using the distance formula or Pythagorean Theorem to determine the length of a segment on a coordinate plane.

-Using the midpoint formula to find the exact middle of a line segment with two endpoints given. Be able to find the endpoint of a segment when the midpoint and one other endpoint are given.

-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.

4 – Student demonstrates a complete understanding of the concept with a correct solution.

3 – Solution contains one minor arithmetic error but demonstrates complete 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.

0 – Incorrect answer demonstrating no understanding of the concept.

Students will be assessed through a formative assessment that may contain but is not limited to the following tasks:

• Identifying a point, line, plane, ray, or segment when given a diagram. They are responsible for using the proper symbolic notation as outlined by the instructor.

• Understanding points that are coplanar and comparing them to points that are collinear.

• Distinguishing the difference between space and planes. • Using angle pair relationships such as linear pairs, vertical angles, complementary

angles, and supplementary angles, to solve real life application problems. • Using the Angle Addition Postulate or Segment Addition Postulate in algebraic

problem solving. • Applying the Pythagorean Theorem, Distance Formula and Midpoint Formula as

needed on a coordinate plane. • Comparing and contrasting polygons that are regular or irregular, convex or

concave.

The following is a student rubric to assess individual understanding during class activities.

4 – I understand completely and I can teach it to a classmate. 3 – I understand the concept but I do not

OTHER EVIDENCE:

• Homework/Classwork • Written response to one of the essential questions using vocabulary, mathematical

properties, and insights • Formative assessments such as individual or pair quizzes • Peer and self-assessment • Collaborative group work

think I can explain it to a classmate.

2 – I can complete the task with assistance.

1 – I need help.

• Geometer’s Sketchpad Explorations

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

http://www.illuminations.nctm.org/

http://www.khanacademy.org/

http://www.learn360.com/

http://www.regentsprep.org/

http://www.jeopardylabs.com/

http://www.parcconline.org/

http://www.desmos.com/calculator

Unit 2: Proof and Reasoning Stage 1 Desired Results

ESTABLISHED GOALS

CC9-12.G.CO.9 Prove theorems about lines and angles.

CC9-12.G.CO.10 Prove theorems about triangles.

Transfer Students will be able to independently use their learning to… -Decipher patterns and determine the next logical term requires the analysis and synthesis of numbers, figures, and various objects. Finding the truths in these can lead to the proof of the hypothesis for the next term. -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 Students will understand that… -If-then statements are often referred to as conditional statements and can be transformed into the inverse, converse, and contrapositive by moving the hypothesis 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

ESSENTIAL QUESTIONS -How can you make a conjecture and prove that it’s true? -Is there a “best practice” to proving an answer is correct?

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… -Definitions: Conjecture Counterexample Hypothesis Conclusion Conditional Statement Converse Inverse Contrapositive Algebraic Proof Flow Proof Two-Column Proof Paragraph Proof -Properties of Equalities

Students will be skilled at… -Making conjectures about the next term in a pattern.

-Finding counterexamples to prove statements to be false.

-Writing the converse, inverse, and contrapositive of a sentence when given the conditional statement and determine whether each of these are true or false.

-Using postulates involving segments and lines as well as the Midpoint Theorem to complete several paragraph proofs.-

-Segment Congruence Properties -Angle Congruence Properties -Supplement Congruence Theorem -Complement Congruence Theorem -Midpoint Theorem -Angle Bisector Theorem

-Using Equality Properties to set up and complete a two-column algebraic proof.

-Showing angle congruency on a clock by using equality properties and definitions of 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.





• Identifying conjectures and stating whether they are true or false. If false a counterexample is required.

• 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 call the converse, inverse, and contrapositive.

• Writing algebraic proofs by using the two-column format and the properties of equalities. • Writing proofs (two-column or flow proofs) using the Segment Congruence Properties as

well as the Segment Addition Postulate and Midpoint Theorem. • Writing proofs (two-column or flow proofs) using the Angle Congruence Properties,

Supplement Congruence Theorem, Complement Congruence Theorem, and Angle Bisector Theorem.




4 – I understand completely and I can teach it to a classmate. 3 – I understand the concept but I do not think I can explain it to a classmate.


1 – I need help.

OTHER EVIDENCE:


properties, and insights • Formative assessments such as individual or pair quizzes • Peer and self-assessment • Collaborative group work • Geometer’s Sketchpad Explorations



• 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.


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

CC9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.


CC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

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 perpendicular to a given line that passes through a given point)

Transfer Students will be able to independently use their learning to… -Understand not all lines and all planes intersect. When a line intersects two or more lines, the angles formed at the intersection points create special angle pairs.

-Use properties including special angle pairs formed by parallel lines and a transversal in proof.

-Prove angle pairs can be used to decide whether two lines are parallel.

-Show the relationships of two lines to a third line can be used to decide whether two lines are parallel or perpendicular to each other.

-Derive the theorem that states the sum of the angles of a triangle is always the same.

- Create the equation of a line when certain facts about the line, such as its slope and a point on the line are known.

-Compare the slopes of two lines can show whether the lines are parallel or perpendicular.

Meaning UNDERSTANDINGS Students will understand that… -Parallel lines are coplanar and skew lines are non-coplanar. Planes can also be parallel. -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

ESSENTIAL QUESTIONS -How do you write the equation of a line in a coordinate plane? -What information is necessary prove that two lines are parallel or perpendicular?

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… -Definitions: Parallel Lines Skew Lines Parallel Planes

Students will be skilled at… Naming segments that are skew and segments that are parallel when given a prism. Identifying transversals if lines of

Transversal Auxiliary Line Alternate Interior Angles Alternate Exterior Angles Same-Side (Consecutive) Interior Angles Corresponding Angles -Corresponding Angles Postulate and it’s Converse -Alternate Interior Angle Theorem and it’s Converse -Alternate Exterior Angle Theorem and it’s Converse -Same Side Interior Angle Theorem and it’s Converse -Slope Formula -Slope-Intercept Formula -Point Slope Formula -Perpendicular Transversal Theorem -Parallel Postulate -Perpendicular Postulate

prisms are extended.

- Completing error analysis when identifying special angle pair relationships.

-Proving lines parallel using various converse’s of special angle pair theorems and postulates.

- Writing and using algebraic expressions to find the value of angle measures including those in which an auxiliary line must be created.

-Using the slope formula to determine if two lines are parallel, perpendicular, or neither. - Deciphering information and using linear equations to solve application problems in the everyday world. -Finding the distance between two parallel lines by using the understanding that the shortest distance between two lines is the perpendicular between them.




• Identifying parallel lines, skew lines, and parallel planes when given a diagram of a prism. • Using the transversal of two lines to identify alternate interior angles, alternate exterior

angles, corresponding angles, and same side interior angles. • Compare the measure s of these special angle pairs when the lines are parallel and non-





parallel. • Proving two or more lines parallel by using the Converse of the Alternate Interior Angle

Theorem, Alternate Exterior Angle Theorem, Corresponding Angle Postulate, and Same-Side Interior Angle Theorem.

• Using the slope formula to determine whether two lines are parallel, perpendicular, or have no relationship to each other.

• Writing equations of lines that are parallel or perpendicular to each other. • Using the Perpendicular Postulate to solve algebraic problem solving real world application

questions.




1 – I need help.

OTHER EVIDENCE:





• Develop conditional statements that are tautologies.

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


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

Graphing Calculator Activity: Parallel Lines l and m are cut by a transversal t. The equations of l, m, and t are 1 14, 6,2 2

y x y x= − = −

and 2 1y x= − + , 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.

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/e/line_relationships

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:



Resources:









Unit 4: Triangle Congruence Stage 1 Desired Results

ESTABLISHED GOALS

CC9-12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

CC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion.


CC9-12.G.CO.10 Prove theorems about triangles

Transfer Students will be able to independently use their learning to… -Understand that classification is used in all facets of mathematics and can be used as a first step to dissecting a problem.

Prove the sums of the angles of a triangle are always 180 degrees.

-Use visualization, tick marks, and arc marks to show corresponding sides and angles in congruent triangles.

-Prove two triangles congruent using the Congruence Postulates (SSS, SAS, ASA, AAS, HL).

-Distinguish the various differences between isosceles and equilateral triangles.

-Derive the relationships involving perpendicular bisectors and angle bisectors in triangles.

Meaning UNDERSTANDINGS Students will understand that… -Triangles are classified by their side lengths and angle measures. - The Triangle Angle Sum Theorem states that any triangle has three angles that will add up to 180 degrees. -The Exterior Angle Theorem states the exterior angle of a triangle is congruent to the sum of the remote interior angles inside the triangle. - Congruent triangles have corresponding

ESSENTIAL QUESTIONS -Where do we see classification used in concepts involving mathematics? -How do you identify corresponding parts of congruent triangles? -How do you show that two triangles are congruent? -Can an equilateral triangle be classified isosceles? Vice-versa? -How do you use coordinate geometry to find relationships within triangles?

congruent sides and angles. Congruence is preserved through transformations such as slides, flips, and turns. - 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

-How do you solve problems that involve measurements of triangles?

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… Definitions: Legs Bases Congruence Congruence Statement Corresponding Included Angle Included Side Vertex Angle Isosceles Equilateral Equiangular -SAS Postulate -SSS Postulate -ASA Postulate -AAS Theorem -HL Theorem -Isosceles Triangle Theorem -Properties and Corollaries of Equilateral Triangles

Students will be skilled at… -Classifying triangles by using the side classification and angle classifications.

-Using the properties of an equilateral or isosceles triangle to set up algebraic equations and solve for missing side measures.

- Recalling the distance formula is used to find the lengths of the sides of a triangle on a coordinate plane and can assist in classifying the triangle by its side lengths.

-Writing and solving algebraic equations using the Angle Sum Theorem to find the measures of the angles in a triangle.

-Writing and solving algebraic equations using the Exterior Angle Theorem to find 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.


4 – Student demonstrates a complete understanding of the concept with a correct


• Classifying triangles by their sides and angles. • Using the Interior Angle Sum Theorem and Triangle Exterior Angle Theorem in algebraic

solution.





problem solving. • Identifying congruent triangles using SSS, SAS, ASA, AAS, and HL. • Writing congruence statements based off the congruent triangles found. • Using the Isosceles Triangle Theorem and the properties of equilateral triangles in real life

application questions.




1 – I need help.

OTHER EVIDENCE:





• 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°.

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Unit 5: Relationships in Triangles Stage 1 Desired Results

ESTABLISHED GOALS



CC9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Transfer Students will be able to independently use their learning to… Understand that triangles play a key role in relationships involving perpendicular bisectors and angle bisectors.

-Use special parts of a triangle that are always concurrent. A triangle’s three perpendicular bisectors are always concurrent, as are a triangles three angle bisectors, its three medians, and its three altitudes.

-Find that the measures of angles of a triangle are related to the lengths of the opposite sides.

Meaning UNDERSTANDINGS Students will understand that… -When three or more lines intersect a common point these lines are called concurrent line and they meet at a point of concurrency. -Three perpendicular bisectors meet at the circumcenter . The circumcenter is 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

ESSENTIAL QUESTIONS -How do you use coordinate geometry to find relationships within triangles? -How do you solve problems that involve measurements of triangles? -When can we apply points of concurrencies in real life?

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… Definitions: Midsegment Perpendicular Bisector Angle Bisector Median Altitude Concurrent Lines Point of Concurrency Circumcenter Incenter Centroid Orthocenter Inequality -Triangle Midsegment Theorem -Perpendicular Bisector Theorem and it’s Converse -Angle Bisector Theorem and it’s Converse -Circumcenter Theorem -Incenter Theorem -Centroid Theorem -Orthocenter Theorem -The Exterior Angle Inequality Theorem -The Triangle Inequality Theorem

Students will be skilled at… -Identifying different concurrent lines and their points of concurrency.

-Using the properties of the points of concurrency to solve algebraic problems.

-Using systems of equations, distance formula, slope formula, and midpoint formula to find points of concurrency on a coordinate plane.

-Forming tables in which they special segments in a triangle are listed, the type of segment is identified, and their point of concurrency is determined.

Writing and using basic inequalities.

-Performing algebraic operations on inequalities.

-Using deductive reasoning to write inequalities involving angle measures and 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.







PERFORMANCE TASK(S): Students will be assessed through a formative assessment that may contain but is not limited to the following tasks:

• Identifying points of concurrency in diagrams and using their properties in algebraic problem solving.

• Completing PARCC like application problems involving the Circumcenter Theorem. • Using the Triangle Midsegment Theorem to develop a two-column proof. • Writing inequalities using the Triangle Inequality Theorem and the Exterior Angle

Inequality Theorem.




1 – I need help.

OTHER EVIDENCE:





• 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.


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 Bisectors

Angle Bisectors Lines Containing the Altitudes

Medians

Acute Triangle

Right Triangle

Obtuse Triangle

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

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Unit 6: Similarity Stage 1 Desired Results

ESTABLISHED GOALS



CC9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.


CC9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Transfer Students will be able to independently use their learning to… -An equation can be written stating that two ratios are equal, and if the equation contains a variable, it can be solved to find the value of the variable.

- Ratios and proportions can be used to decide whether two polygons are similar and the find unknown side lengths of similar figures.

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

-Triangles can be similar based on the relationship of two or three corresponding parts.

Meaning UNDERSTANDINGS Students will understand that… -A ratio is a comparison of two quantities. A proportion is an equation stating that two ratios are equal. - Polygons are similar if their corresponding angles are congruent and corresponding sides proportional.

-Similarity statements can be written to show congruent angles and corresponding 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.

ESSENTIAL QUESTIONS -How do you use proportions to find side lengths in similar polygons? -How do you show two triangles are similar? -How do you identify corresponding parts of similar triangles?

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… Definitions: Ratio Extended Ratio Proportion Means Extremes Similar Figures Similar Polygons Scale Factor Indirect Measurement -Means and Extremes Property -The Hinge Theorem (SAS Inequality) and it’s Converse (SSS Inequality) -AA Similarity Postulate - Triangle Proportionality Theorem -Triangle Angle Bisector Theorem

Students will be skilled at… -Using ratios to find angle measures in a triangle and to find the missing side lengths when given the perimeter of a triangle.

-Cross multiplying the means and the extremes to determine if the cross product of a proportion is equal.

-Setting up and solving proportions in which a variable needs to be found.

-Writing proportions of corresponding parts to determine if two figures are similar.

-Using scale factor in application problems 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.







PERFORMANCE TASK(S): Students will be assessed through a formative assessment that may contain but is not limited to the following tasks:

• Writing ratios and solving proportions when given application questions. • Determining whether two figures are similar . • Using the SAS Inequality, SSS Inequality, and AA Similarity to determine whether two

triangles are similar. • Finding and using scale factor to compare real life buildings to their scale models. • Identifying parallel lines within figures to apply to Triangle Proportionality Theorem and

solve algebraic problems.




1 – I need help.

OTHER EVIDENCE:





• 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.


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:



Resources:









Unit 7: Right Triangles and Trigonometry Stage 1 Desired Results

ESTABLISHED GOALS




CC9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

CC9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

CC9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorems to solve right triangles in applied problems.

CC9-12.G.SRT.10 Prove the Law of Sine’s and the Law of Cosines and use them to solve problems.

CC9-12.G.SRT.11 Understand and apply the Law of Sine’s and the Law of Cosines to find

Transfer Students will be able to independently use their learning to… -Understand geometric mean describes the relationship between the altitude and hypotenuse of a right triangle.

-Use the Pythagorean Theorem when given lengths of any two sides of a right triangle to find the length of the third side.

-Derive ratios when certain combinations of side lengths and angle measures of a right triangle are known.

-Understand ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of side lengths and angle measures are known.

-Use angles of elevation and depression, which are the acute angles of right triangles formed by a horizontal distance and a vertical height.

-Solve a triangle using the Law of Sine’s or Law of Cosines when given a specific combination of sides and angles.

Meaning UNDERSTANDINGS Students will understand that… - When an altitude is drawn to the hypotenuse of one triangle three similar right triangles are formed.

-The Pythagorean Theorem can be used on right triangles to determine the missing side lengths.

-If three lengths are given the converse of the Pythagorean Theorem can be used to

ESSENTIAL QUESTIONS -How do you find a side length or angle measure in a right triangle? -How do trigonometric ratios relate to similar right triangles?

unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces)

determine if the triangle is right.

-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… Definitions: Geometric Mean Hypotenuse Legs

Students will be skilled at… -Identifying and find the geometric mean when given a diagram and writing a similarity statement for the three right

Special Right Triangles Trigonometric Ratio Angle of Elevation Angle of Depression -Pythagorean Theorem and it’s Converse -Geometric Mean Theorem and its’ Corollaries -45-45-90 Theorem -30-60-90 Theorem -Law of Sines -Law of Cosines

triangles formed.

-Using the Pythagorean Theorem to determine the missing side lengths.

-Determining the classification of a triangle using the Pythagorean Theorem.

-Verifying a triangle is right on the coordinate plane.

-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.








• Using similar triangles to find the geometric mean of a right triangle by writing proportions. • Determining whether a triangle is right, acute, or obtuse. • Completing PARCC like application problems involving special right triangles and the

extended ratio of their sides. • Identifying which trigonometric ratios to use when finding a missing angle or side of a right

triangle. • Applying trigonometric ratios to application questions involving the angle of elevation or

angle of depression. • Solving any kind of triangle by using the Law of Sines and Law of Cosines.




1 – I need help.

OTHER EVIDENCE:





• 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.


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.

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Unit 8: Circles Stage 1 Desired Results

ESTABLISHED GOALS

CC9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords.

CC9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CC9-12.G.C.4 Construct a tangent line from a 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 square to find the center and radius of a circle given by an equation.

Transfer Students will be able to independently use their learning to… -A radius of a circle and the tangent that intersects the endpoint of the radius on the circle has a special relationship.

-Information about congruent parts of a circle (or congruent circles) can be used to find information about other parts of the circle (or circles)

-Angles formed by intersecting lines have a special relationship to the arcs the intersecting 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, intersecting secants, or a secant that intersects a tangent.

Meaning UNDERSTANDINGS Students will understand that… -A circle has many parts that can be measured and used in various ways. The radius is needed to find the circumference and area of a circle.

-An angle formed by two radii that meet at the center is called a central angle. The central angles of a circle can be added to 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 semi-circle. Arcs can be added together by using

ESSENTIAL QUESTIONS -How can you prove relationships between angles and arcs in a circle? -When lines intersect a circle or within a circle, how do you find the measures of the resulting angles, arcs, and segments? -How do you find the equation of a circle in the coordinate plane?

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… Definitions: Circle Center Radius Chord Diameter Tangent Point of Tangency Secant Circumference Area Minor Arc Major Arc Semi-Circle Arc Measure Arc Length -Tangency Theorem and its Converse -Congruent Circles Theorem and its Converse -Congruent Arcs Theorem and its Converse -Inscribed Angles Theorem and its Corollaries -Angles and Arcs Theorem and its Converse

Students will be skilled at… -Identifying parts of a circle.

-Finding the radius and diameter of a circle when given specific information. Problem solving for the circumference, radius, and diameter of a figure when given a specific equation.

-Writing and using algebraic equations to find the measures of central angles and their intercepted arcs.

-Identifying various types of arcs in a given diagram and find the value of their measure.

-Finding the length of the arcs of circle when given the central angle measure and circumference.

-Using proofs to show the Congruent Arcs and Chords Theorem are valid.

-Using circumscribed circles to determine 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..








• Identifying parts of a circle. • Using the Tangency Theorem and its Converse to find the radius of a circle. • Solving algebraic problems involving the Congruent Circles Theorem and its Converse. • Developing two-column proofs using the properties of arc measure and the Inscribed

Angles Theorem and its Corollaries. • Comparing and contrasting arc length vs. arc measure, and knowing when to use each. • Developing proofs using circle theorems, properties of similarity, and trigonometric ratios.


4 – I understand completely and I can teach it to a classmate. 3 – I understand the concept but I do not

OTHER EVIDENCE:


properties, and insights • Formative assessments such as individual or pair quizzes • Peer and self-assessment • Collaborative group work

think I can explain it to a classmate.


1 – I need help.

• Geometer’s Sketchpad Explorations



• 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.



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?

http://whistleralley.com/construction/c1.htm

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?

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Unit 9: Transformations and Symmetry Stage 1 Desired Results

ESTABLISHED GOALS G.CO.2 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not. G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections, translations and dilations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another. G.CO.6 Use geometric descriptions of rigid motions to transform figures and predict the effect of a given rigid motion on a given figure: two given figures use the definition of congruence in term of rigid motions to determine if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two

Transfer Students will be able to independently use their learning to… -Change the position of geometric figures so the angle measure and distance between any two points of a figure stay the same. -A reflection of a figure across a line is a mapping of each point to another point the same distance from the line but on the other side. The orientation of the figure reverses. -Show rotations preserve distance, angles measures, and orientation of figures. -Isometries can be expressed as compositions of reflections. -Congruence can be understood by using compositions of rigid motion. -Scale factor can be used to make a larger or smaller copy of a figure that is similar to the original. -Compositions of rigid motions and dilations help understand similarity and its properties.

Meaning UNDERSTANDINGS Students will understand that… -The image and preimage must preserve distance and angle measures. -Corresponding points have the same position in the names of the preimage and image. -A translation maps all points of a figure the same distance in the same direction. -The composition of any two translations is another translation. -Two figures reflect is they corresponding vertices of the preimage and image read in opposite directions.

ESSENTIAL QUESTIONS -How can you change a figure’s position without changing its size and shape? -How can you change a figure’s size without changing its shape? -How can you represent a transformation in the coordinate plane? -How do you recognize congruence and similarity in figures?

triangles are congruent if and only if corresponding pairs of sides and angles are congruent. G.CO.8 Explain how criteria for triangle congruence follow from the definition of congruence in terms of rigid motion. G.SRT.1a The dilation of a line segment is longer or shorter in the ratio given by the scale factor. .

-A reflection across a line is called line reflection. -Reflections preserve distance and angle measures. They map each point of the preimage to one and only one corresponding point of its image. -A rotation about a point is a rigid motion. 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… Definitions: Transformation Preimage Image Rigid Motion Translation Composition of Transformations Reflection Line of Reflection Rotation Center of Rotation Angle of Rotations Symmetry Line of Symmetry Rotational Symmetry Point Symmetry Glide Reflection Isometry Congruence Transformation Dilation Center of Dilation Enlargement Reduction -Isometry Theorem -Reflections Across Parallel Lines Theorem -Reflections Across Intersecting Lines Theorem

Students will be skilled at… -Identifying transformations when given a coordinate plane. -Making shifts to a preimage based on the notation given to them. -Comparing and contrasting rigid motions. -Graphing images and labeling the vertices in prime notation. -Rotating a preimage about a point in any direction. -Proving a transformation is an isometry on and off a coordinate plane. -Understanding glide reflections and their orientations. -Proving transformational congruence and similarity. -Creating tessellations based on rigid motions. -Enlarging or shrinking figures by using their scale factor on or off a coordinate plane.








• Identifying transformations when given a image and preimage. • Graphing images and using prime notation. • Shifting a preimage based on the rotation, translation, reflections, and dilation given. • Proving a transformation is an isometry. • Showing a transformation preserves congruence on a coordinate plane. • Using scale factor to dilate a figure. • Determining the transformation based on coordinate notation. • Using the Reflections Across Parallel Lines Theorem in PARCC like application questions.




1 – I need help.

OTHER EVIDENCE:





• 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:



Resources:









Unit 10: Quadrilaterals Stage 1 Desired Results

ESTABLISHED GOALS

CC9-12.G.CO.11 Prove theorems about parallelograms.

CC9-12.G.CO.13 Construct and equilateral triangle, a square, and a regular hexagon inscribed in a circle.


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 perpendicular to a given line that passes through a given point)

CC9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios.)

Transfer Students will be able to independently use their learning to… -The sum of the angle measures of a polygon depends on the number of sides the polygons has.

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

-If a quadrilaterals sides, angles, and diagonals have certain properties; it can be shown that the quadrilateral is a parallelogram.

-The special parallelograms, rhombus, rectangle, and square have basic properties of their sides, angles, and diagonals that help identify them.

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

Meaning UNDERSTANDINGS Students will understand that… -A diagonal of a polygon is a segment that connects any two nonconsecutive vertices. You can find the sum of the interior angles by using the Interior Angle Sum Theorem. -The Exterior Angle Sum Theorem states that all the exterior angles of a polygon will 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

ESSENTIAL QUESTIONS -How can you find the sum of the measures of polygon angles? -How can we prove quadrilaterals to be parallelograms? -How can you classify quadrilaterals on and off a coordinate plane?

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… Definitions: Regular Polygons Parallelogram Opposite Angles Opposite Sides Consecutive Angles Rhombus Rectangle Square Trapezoid Base Leg Base Angle Isosceles Trapezoid Midsegment Kite Coordinate Proof -Polygon Angle Sum Theorem -Polygon Exterior Angle Sum Theorem -Properties of Parallelograms Theorems and Corollaries -Converses of Parallelograms Theorems -Rectangle Theorem -Rhombus Theorems -Isosceles Trapezoid Theorem

Students will be skilled at… -Deriving the Interior Angle Sum Theorem through an interactive activity.

-Using the Interior Angle Sum Theorem to find the sum of the angles of regular polygons.

-Completing application problems in which the Interior Angle Sum Theorem is used.

-Finding the number of sides a polygon has when given the sum of the interior angles.

-Deriving the Exterior Angle Theorem by using the Geometer’s Sketchpad activity to investigate its properties.

-Using the Exterior Angle Theorem to determine the amount of sides a polygon has when the angle measure is known.

-Writing a two-column proof in which they prove that the opposite angles of a parallelogram are congruent.

-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.





1 – Solution contains multiple arithmetic errors while demonstrating minimal


• Using the properties and theorems of various parallelograms to classify a quadrilateral. • Proving the Converses of the Parallelogram Theorems. • Using the Rectangle Theorem, Rhombus Theorem, and properties of a parallelogram in

various problem solving application questions. • Using the Isosceles trapezoid Theorem and Midsegment Theorem in algebraic problem

solving. • Determining what exact polygon a quadrilateral is on a coordinate plane. • Deriving the Polygon Angle Sum Theorem.

understanding of the concept.

0 – Incorrect answer demonstrating no understanding of the concept. The following is a student rubric to assess individual understanding during class activities.



1 – I need help.

OTHER EVIDENCE:





• 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.


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:










Resources:


Unit 11: Area of Polygons Stage 1 Desired Results

ESTABLISHED GOALS

CC9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

CC9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

CC9-12.G.MG.3 Apply geometric methods to solve design problems.

.

Transfer Students will be able to independently use their learning to… -The area of a parallelogram or a triangle can be found when the length of its base and height are known.

-The area of a trapezoid can be found when the height and its lengths of its bases are known.

-The area of a rhombus or kite can be found when lengths of the diagonals are known.

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

-Trigonometry can be used to find the area of a regular polygon when the length of a side, radius, or apothem, is known or to find the area of a triangle when the length of two sides 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 Students will understand that… -Any side of a parallelogram can be called a base. For each base, there is a corresponding altitude that is perpendicular to the base. The area of a parallelogram is found by

ESSENTIAL QUESTIONS -How do you find the area of a polygon or find the circumference or area of a circle? -Compare and contract the ratios of the perimeters and areas of similar polygons.

multiplying its base and its height.

-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

-What methods can be used to find the area of any regular polygon?

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… Definitions: Base of a Parallelogram Altitude of a Parallelogram Height of a Parallelogram Base of a Triangle Height of a Triangle Height of a Trapezoid Apothem Circumference Pi -Area of a Rectangle Theorem -Area of a Parallelogram Theorem -Area of a Triangle Theorem -Area of a Trapezoid Theorem -Area of a Rhombus or Kite Theorem -Area of a Regular Polygon Theorem -Perimeter and Areas of Similar Figures Theorem -Area of a Circle Theorem -Circumference of a Circle Theorem

Students will be skilled at… -Finding the area and perimeter of a parallelogram.

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

-Finding the area of a parallelogram on a coordinate plane.

-Finding the areas of different triangles including those that are equilateral, right, scalene, and isosceles.

-Finding the area of a trapezoid and a rhombus.

-Using coordinate geometry to find the area of a trapezoid and rhombus on a coordinate plane.

-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.




• Finding the area of parallelograms by identifying the height and base of the quadrilateral. • Deriving the area of a triangle formula from the area of a parallelogram formula. • Using the area of a trapezoid to solve PARCC application questions. • Comparing and contrasting the Area of a Rectangle, Area of a Parallelogram, Area of a

Triangle, Area of a Trapezoid, and Area of a Rhombus Theorems.





• Using previous topics such as inscribed polygons, Polygon Angle Sum Theorem, special right triangles and trigonometric ratios to find the area of any regular polygon.

• Determining the area and perimeter of similar figures. • Finding the area and circumference of a circle and leaving the answer in terms of Pi. • Using complex reasoning to find the area of the shaded region of two overlapping polygons.




1 – I need help.

OTHER EVIDENCE:





• 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.


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:


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:



Unit 12: Surface Area and Volume Stage 1 Desired Results

ESTABLISHED GOALS

CC9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

CC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

CC9-12.G.GMD.4 Identify the shapes of two-dimensional cross sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Transfer Students will be able to independently use their learning to… -A three dimensional figure can be analyzed by describing the relationship among its vertices, edges, and faces.

-The area of three-dimensional figure is equal to the sum of the areas of each surface of the figure.

-The volume of a prism and cylinder can be found when its height and area of its base are known.

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

-The surface area and the volume of a sphere can be found when the radius is known.

Meaning UNDERSTANDINGS Students will understand that… -A net is a pattern for a three-dimensional figure if it was laid flat.

- Lateral area is the sum of the area of the 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

ESSENTIAL QUESTIONS -How can you determine the intersection of a solid and a plane? -How is the surface area of a solid related to the volume?

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… Definitions: Polyhedra Face Edge Vertex Cross Section Lateral Face Lateral Edge Lateral Area Surface Area Volume Prism Right Prism Oblique Prism Cylinder Right Cylinder Oblique Cylinder Pyramid Regular Pyramid Cone Right Cone Sphere -Lateral Area and Surface Area of a Prism Theorem -LA and SA of a Cylinder Theorem -LA and SA of a Pyramid Theorem -LA and SA of a Cone Theorem -SA and Volume of a Sphere Theorem

Students will be skilled at… -Identifying and name three-dimensional figures.

-Drawing nets for any solid.

-Using net to determine surface area.

-Finding the lateral area of all different types of prisms (ex. triangular, pentagonal, etc.)

-Finding the surface area of various types of prisms.

-Using the apothem to find the surface area of prisms with bases that are regular polygons.

-Using surface area of a prism to solve real world problems.

-Finding the lateral area of a cylinder.

-Using the lateral area of a cylinder to find the surface area.

-Finding missing dimensions of a cylinder when given the surface area.

-Finding the lateral area of a regular

-Volume of a Prism Theorem -Volume of a Cylinder Theorem -Volume of a Pyramid Theorem -Volume of a Cone Theorem

pyramid.

-Finding the surface area of a regular 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.





1 – Solution contains multiple arithmetic


• Finding the surface area and volume of any right prism. • Deriving the surface area and volume formulas for a cylinder based off the formulas for a

right prism. • 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,

trigonometric ratios, and special right triangles. • Finding the surface area and volume of any right pyramid. • Deriving the surface area and volume formulas for a cone based off the formulas for a right

pyramid. • Using the cone formulas to find the surface area and volume in terms of Pi. • Finding the surface area and volume of a sphere in terms of Pi.

errors while demonstrating minimal understanding of the concept.




1 – I need help.

OTHER EVIDENCE:





• 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:










Resources:


Unit 13: Probability and Measurement Stage 1 Desired Results

ESTABLISHED GOALS S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), 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.

S.CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English.

Transfer Students will be able to independently use their learning to… -Various counting methods can help you analyze situations and develop theoretical probabilities.

-You can use multiplication to quickly count the number of ways certain things can happen.

-The probability of an impossible event is 0. The probability of a certain event is 1. Otherwise the probability of an event is a number between 0 and 1.

-To find the probability of two events occurring you have to decide whether one event occurring affects the other event.

-Conditional probability exists when two events are dependent.

- In geometric probability, numbers of favorable and possible outcomes are geometric measures such as lengths of segments or areas of regions.

Meaning UNDERSTANDINGS Students will understand that… The Fundamental Counting Principle describes the method of using multiplication to count. A permutation is an arrangement of items in a particular order. Using factorial notation allows you to write 3x2x1 as 3!. A selection in which order does not matter is called a combination.

When you gather data from observations, you calculate an experimental probability.

ESSENTIAL QUESTIONS -What is the difference between experimental and theoretical probability? -How are the laws of probability used to predict outcomes in the real world? -How is statistics used to analyze data in real world situations?

Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

S.CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

S.CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

S.CP.9 Use permutations and combinations to compute probabilities of compound

Each observation is an experiment or a trial. A simulation is a model of the event. The set of all possible outcomes to an experiment or activity is a sample space. If the outcomes in a sample space have the same chance of occurring, the outcomes are called equally likely outcomes. Theoretical probability is when a sample space has equally likely outcomes to an event occurring. When the occurrence of one event affects how a second event can occur, the events are dependents events. Otherwise the events are independent events. Two events that cannot happen at the same time are mutually exclusive events. 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 two different categories. Using the formula for conditional probability you can calculate the conditional probability from other probabilities.

In geometric probability, numbers of favorable and possible outcomes are geometric measures such as lengths of segments or areas of regions. The probability of a point being located on a specific part of a segment is the ratio of the specific length of the segment to the length of the whole segment. Probability can also be used when finding the chances of a point

events and solve problems.

S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of the game).

being in an inscribed figure.

Acquisition Students will know… Definitions: Outcome Event Sample Space Probability Experimental Probability Theoretical Probability Complement of an Event Frequency Table Relative Frequency Probability Distribution Permutation N-Factorial Combination Compound Event Independent Event Dependent Event Mutually Exclusive Events Overlapping Events Conditional Probability Geometric Probability Expected Value -Fundamental Counting Principle

Students will be skilled at… -Using the Fundamental Counting Principle.

-Finding the number of permutations of n items.

-Using the permutation formula.

-Using the combination formula.

-Identifying whether the order matters in a event.

-Finding experimental probability.

-Using a simulation.

-Finding theoretical probability.

-Finding probability using combinations.

-Classifying events as independent or dependent.

-Finding the probability of independent 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.





1 – Solution contains multiple arithmetic errors while demonstrating minimal


• Using the Fundamental Counting Principle in PARCC application questions. • Finding the permutation or combinations of n items. • Determining the probability of independent events, dependent events, mutually exclusive

events. • Comparing and contrasting experimental probability and theoretical probability. • Developing contingency tables and tree diagrams to find conditional probability of an event

occurring. • Applying probability principles to find the geometric probability of an event occurring

through segment length or area.

understanding of the concept.




1 – I need help.

OTHER EVIDENCE:





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


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:



Resources:









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.


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.


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.


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.

curriculum management system - monroe township … · 2015-01-20 · goals/essential...

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