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Geometry Curriculum Updated 2017

North Bergen School District Benchmarks

Grade: 9, 10, 11, 12

Subject: Geometry

First Marking Period

Students will master the following topics:

1. Understanding the Essentials of Geometry • Identify points, Lines, and Planes HSG-MG.A.3, HSG-CO.A.1 • Use Segments and Congruency HSG-MG.A.3, HSG-CO.A.1 • Use Midpoint And Distance Formulas HSG-MG.A.3, HSG-CO.A.1 • Measure and Classify Angles HSG.MG.A.3, HSGCO.A.1 • Describe Angle Pair Relations HSG-CO.1, HSG-CO.9

2. Use Reasoning and Proofs • Use Inductive Reasoning. Apply Deductive Reasoning

HSG-CO.1, HSG-CO.9, HSG-CO.10, HSG-CO.11 • Analyze Conditional Statements HSG-CO.A.1 • Use Postulates and Diagrams (Postulates) HSG-CO.9, HSG-CO.10, HSG-CO.11 • Reason Using Properties from Algebra HSG-MG.A.3 • Prove Statements about Segments and Angles HSG-CO.A.9 • Prove Angle Pair Relationships HSG-CO.A.9

3. Investigate Parallel and Perpendicular Lines

• Identify Pairs of Lines and Angles HSG-MG.A.3, HSG-CO.A.1 • Use Parallel Lines and Transversals HSG-MG.A.3, HSG-CO.A.9 • Prove Lines are Parallel HSG-CO.A.9, HSG-CO.A.12 • Find and Use Slopes of Lines HSG-CO.A.9, HSG-GPE.B.5 • Write and Graph Equations of Lines HSG-GPE.B.5 • Prove Theorems about Perpendicular Lines HSG-CO.A.9, HSG-GPE.B.5


Second Marking Period

Students will master the following topics:

4. Prove Congruent Triangles • Apply Triangle Sum Properties HSG-MG.A.3, HSG-CO.A.10 • Apply Congruence and Triangles HSG-MG.A.3, HSG-SRT.B.5 • Prove Triangles Congruent by SSS HSG-MG.A.3, HSG-CO.A.12, HSG-SRT.B.5 • Prove Triangles Congruent by SAS and HL HSG-MG.A.3, HSG-SRT.B.5 • Prove Triangles Congruent by ASA and AAS HSG-SRT.B.5 • Use Congruent Triangles HSG-CO.A.12, HSG-SRT.B.5 • Use Isosceles and Equilateral Triangles HSG-MG.A.3, HSG-CO.A.10 • Midsegment Theorem and Coordinate Proof

HSG-MG.A.3, HSG-CO.A.10, HSG-GPE.B.4 • Use Perpendicular Bisectors HSG-C.A.3, HSG-CO.A.9, HSG-CO.A.10,

HSG-CO.A.12 • Use Angle Bisectors of Triangle

HSG-C.A.3, HSG-CO.A.10, HSG-CO.A.12, HSG-GPE.B.4

5. Investigate the Relationships Within Triangles • Use Medians and Altitudes HSG-CO.A.10, HSG-CO.A.12, HSG-GPE.B.4 • Use Inequalities in a Triangle HSG-MG.A.3, HSG-CO.A.10 • Inequalities in Two Triangles and Indirect Proof HSG-CO.A.10

6. Apply the Properties and Theorems of Similarity • Classify Polygons HSG-GMD.4, HSG-MG.1 • Use Similar Polygons HSG-SRT.B.5 • Prove Triangles Similar by AA HSG-SRT.A.3, HSG-SRT.B.5 • Prove Triangles Similar by SSS and SAS

HSG-MG.A.3, HSG-SRT.B.4, HSG-SRT.B.5


Third Marking Period

Students will be able to:

7. Use Proportionality Theorems HSG-CO.A.12, HSG-SRT.B.4, HSG-SRT.B.5

8. Apply Properties of Right Triangles and Trigonometry • Apply the Pythagorean Theorem HSG-SRT.C.8 • Use the Converse of the Pythagorean Theorem HSG-SRT.C.8 • Use Similar Right Triangles HSG-MG.A.3, HSG-SRT.B.4, HSG-SRT.B.5 • Special Right Triangles HSG-SRT.B.4, HSG-SRT.B.5, HSG-SRT.C.8 • Apply the Tangent Ratio HSG-SRT.C.6, HSG-SRT.C.8 • Apply the Sine and Cosine Ratios

HSG-SRT.C.6, HSG-SRT.C.7, HSG-SRT.C.8, HSG-SRT.D.9 • Solve Right Triangles HSG-MG.A.3, HSG-SRT.C.8

9. Understand and Apply Properties of Quadrilaterals

• Find Angle Measures in Polygons HSG-MG.A.1 • Use Properties of Parallelograms HSG-MG.A.1, HSG-CO.A.11, HSG-GPE.B.4 • Show that a Quadrilateral is a Parallelogram

HSG-MG.A.1, HSG-CO.A.11, HSG-CO.A.12, HSG-GPE.B.4 • Properties of Rhombuses, Rectangles, and Squares

HSG-MG.A.1, HSG-MG.A.3, HSG-CO.A.11, HSG-GPE.B.4 • Use Properties of Trapezoids and Kites HSG-MG.A.1, HSG-GPE.B.4 • Identify Special Quadrilaterals HSG-CO.A.11, HSG-GPE.B.4


Fourth Marking Period

Students will be able to:

10. Apply Properties of Transformations • Relate Transformation and Congruence HSG-CO.A.2, HSG-CO.A.6, HSG-CO.A.7 • Perform Congruence Transformations

HSG-MG.A.3, HSG-CO.A.2, HSG-CO.A.5, HSG-CO.A.6 • Relate Transformations and Similarity HSG-SRT.1, HSG-SRT.1, HSG-C.1 • Perform Similarity Transformation HSG-CO.2, HSG-SRT.1, HSG-GPE.4 • Translate Figures and Use Vectors

HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6 • Perform Reflections HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6 • Perform Rotations HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6 • Apply Compositions of Transformations

HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.A.6 • Identify Symmetry HSG-SO.3 • Identify and Perform Dilation HSG-SRT.1

11. Understand and Apply Properties of Circles

• Use Properties of Tangents HSG-CO.1, HSG-C.1, HSG-C.4(+) • Find Arc Measures HSG-CO.1 • Apply Properties of Chords HSG-CO.12, HSG-C.2, HSG-C.3 • Use Inscribed Angles and Polygons HSG-C.2, HSG-C.3, HSG-C.4(+), HSG-C.5 • Apply Other Angle Relationships in Circles HSG-C.2, HSG-C.5 • Find Segment lengths in Circles HSG-C.2

Domain: Congruence

Cluster: Experiment with transformations in the plane

Standards: (G.CO.1-5) 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • Why do you need to

know, understand, and speak in geometric terms to function in everyday life?

• How does a

transformation change the image of a figure?

• Why do we need to know that?

• Comprehending and applying basic geometric terms are important for future theorems and postulates

• An understanding of transformations can be applied in real-life situations.

• Create vocabulary flash cards • Students will draw corresponding examples with proper

notation for given terms. • Given a figure, such as a rectangle, on the coordinate

plane, students will model each transformation individually.

• Students will model basic transformations’ compositions.

• Students will investigate transformations using tessellations.

• Interactive Example for distance. • Sample Lessons, Examples, Class work, Homework,

Worksheet


• Sample Lessons and Examples • SMART board lessons and examples.

Content Statements 21st Century Life and Careers Differentiation

• Students should define 9.1 Personal Financial Literacy 9.2 (ELLs, Special Education, Gifted and Talented) Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries basic geometric terms

Preparation • Students should • Preferential seating

• Learning Centers • Cooperative learning grouping • Independent Study • Teacher & Peer tutoring • Modified assignments • Differentiated instruction

identify and draw the three basic transformations

• Students can represent a figure’s size without changing its shape.

• Students will change a figure’s position without changing its size and shape.

Assessments Interdisciplinary Connections


http://www.state.nj.us/education/cccs/2014/career/91.pdf




Task 1: Each figure below is part of a capital letter in the English alphabet. To find the whole letter, combine the figure with its image for the appropriate rotation or reflection. What letter corresponds to each figure? What transformation produces each letter?

Answer:

Social Studies: Map reading Parkway exits are also ride markers Parking/Driving (parallel lines and parking) Career and Life Skills: Carpentry (perpendicular objects)


Task 2:

Copy the graph shown below. On the same set of axes, graph the image of MNOP for a dilation with center (0, 0) and scale factor 2. Use coordinate geometry and the definition of similar polygons to prove that MNOP is similar to its image.

Answer:


Teacher Resources 2012 • Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (See Attached) • Homework

Equipment Needed: SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Compass Protractor Ruler Straightedge Graph Paper Isometric Dot Paper Hands-on and virtual two- and three-dimensional

manipulatives (i.e. prisms) Geo-boards Geometer’s Sketchpad

Teacher Resources:

Math Open Reference Website: http://www.mathopenref.com/coorddist.html

Pennsylvania Department of Education Standards Aligned System: http://www.pdesas.org/module/content/resources/18110/ view.ashx

Click here for Resource Folder. http://my.hrw.com/index.jsp http://www.khanacademy.com http://prc.parcconline.org Math Series


http://www.mathopenref.com/coorddist.html

http://www.pdesas.org/module/content/resources/18110/view.ashx



http://www.hcstonline.org/hcccmath/math%209-12/geometry/g.co.1-5/resources/

http://my.hrw.com/index.jsp

http://www.khanacademy.com/

http://prc.parcconline.org/

Domain: Congruence

Cluster: Understand congruence in terms of rigid motions

Standards: (G.CO.6-8) 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. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • What types of motion

in the plane maintain the congruence of a figure?

• Students will know that rigid transformations preserve size and shape or distance and angles, within the concept of congruency.

• Based on rigid motion, students will be able to develop and explain the triangle congruency based on the following criteria: ASA, SSS, and SAS.

• Given two triangles, prove they are congruent based on congruency definitions and theorems.

• Given three points, students will prove that these are the vertices of an isosceles triangle, translate this figure 5 units to the right, and prove that the final image is congruent to the pre-image.

• Each student to make a quilt block using congruent right triangles with specific measure. Then combine all students work to create a class quilt.

• Sample Lessons, Examples, software for triangle congruence.

• Discovery Activity - Angle Sum in Triangles Proof using Rotation and a Parallel Line

• Interactive Congruent Triangles Activity • SMART board lessons and examples



• Review identification and definitions of three basic transformations including isometries.

• Students will apply the criteria for figure congruence, i.e. corresponding parts of congruent triangles.

(ELLs, Special Education, Gifted and Talented) • Extended time • Supplemental Materials, i. e. translation dictionaries • Preferential seating • Learning Centers • Cooperative learning grouping • Independent Study • Teacher & Peer tutoring • Modified assignments • Differentiated instruction



• Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (See Attached) • Homework

Writing: Math Journal



Teacher Resources:

Pennsylvania Department of Education Standards Aligned System: http://www.pdesas.org/module/content/resources/4890/vi ew.ashx

Texas Instruments: http://education.ti.com/calculators/downloads/US/Activit ies/Detail?id=7381&ref=%2fcalculators%2fdownloads% 2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022 %26sa%3d5024%26t%3d5056

IES Inc. ( International Education Software ): http://www.ies.co.jp/math/products/geo1/applets/cong/co ng.html

http://my.hrw.com/index.jsp http://www.khanacademy.com http://prc.parcconline.org Math Series





http://education.ti.com/calculators/downloads/US/Activities/Detail?id=7381&amp%3Bref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fSubject%3fs%3d5022%26sa%3d5024%26t%3d5056






http://www.ies.co.jp/math/products/geo1/applets/cong/cong.html







make sure your students know what the point of a proof

Domain: Congruence

Cluster: Prove geometric theorems

Standards: (G.CO.9-11) 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • What is a geometric

proof? • Why do we have to

prove statements that have already been proven by mathematicians of the past?

• Student will understand that practicing geometric proofs teaches the logic of deductive reasoning.

• Students should know that a theorem is different from a postulate because it is a formula or statement deduced from a chain of reasoning, or a series of other proofs already accepted. A postulate, on the other hand, is an assumed truth based on rational geometric principles.

• Different students prefer different types of proofs. Some take a liking to paragraph proofs so they can provide a stream-of-consciousness rant and have it be an acceptable answer. Others prefer the two-column proof so that all their arguments are clearly laid out in front of them. Be lenient as to the format of the proof at first, but

Content Statements 21st Century Life and Careers

• Students will be able to apply definitions, theorems, postulates and properties about vertical angles, parallel lines, perpendicular

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and actually is and what their goal is in writing one. Preparation • Knowing basic information and definitions about lines

and angles is crucial, since they'll be the building blocks of your students' arguments. For instance, many






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lines, bisectors, triangles, and parallelograms to develop and justify a geometric proof.

theorems concerning lines and angles will make use of the fact that a straight line is 180°. Vertical angles come to mind.

• You can let students know that once one proof is worked out, they can use the result in future proofs. Proofs are meant to be collected and used when tackling difficult problems. So we can use the vertical angle proof to prove another theorem about alternate interior angles.

(Taken from: http://www.shmoop.com/common-core- standards/ccss-hs-g-co-9.html)

• Students should know the basic definitions that come

with triangles and how to classify them based on angles and sides. When they can use the words "equiangular" and "isosceles" in everyday conversation, you'll know you've done a good job.

• Students should also be comfortable with the angles of a triangle, both interior and exterior. They should know that all the interior angles of a triangle add up to 180°, and they should know how to prove it. It's better to introduce these concepts to them by using concepts they should already know, like parallel lines and transversals.

• But that's just the tip of the triangular iceberg. There's way more inside triangles than just three interior angles. For instance, we can fill a triangle with medians, line segments that join the vertices of a triangle to the midpoints of its opposite sides. We can also connect the midpoints of each side in the triangle to form a similar triangle that's half the size of the original one.

• Finally, students shouldn't get lost with all the theorems and postulates. They all build on each other, and it's best to keep track of these proofs and postulates so that

Assessments • Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (Click here for Resources

Folder.) • Homework

http://www.shmoop.com/common-core-standards/ccss-hs-g-co-9.html






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students don't get confused. Also, students should know that using proofs and theorems they've already learned isn't cheating; it's applying the skills they've learned, and it's highly encouraged.


• Students should learn how to link what they know in a

logical string to prove or disprove an argument. They should have already done this with lines and triangles, so the next logical step would be to head right into the world of parallelograms.

• In order to prove theorems about parallelograms, students might want to know what a parallelogram is. (Spoiler alert: it's a quadrilateral with opposite sides that are parallel.) From there, students will want to come up with an argument to prove, whether it's that opposite sides of a parallelogram are congruent, opposite angles are congruent, or that the diagonals bisect each other.

• You wouldn't order a grilled cheese just to sit there and stare at it. (Marveling at the beauty of the perfect grilled cheese is excusable and even understandable, but not tasting its delicious cheesy goodness is near blasphemous.) Just the same, students didn't learn about parallel lines, transversals, congruent triangles, and complementary angles for nothing. They should use the knowledge they already have and apply it to parallelograms.

• If students are struggling, tell them that pictures are always an excellent way to start proofs. That way, they should at least be able to get off the ground to start with.




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Refreshing their memory about the theorems and definitions they'll be using might be helpful as well. Also, using the two-column proof format might help students organize their thoughts better, at least in the beginning. Those paragraph proofs can get messy to the point of uselessness.


Differentiation (ELLs, Special Education, Gifted and Talented)

• Extended time • Supplemental Materials, i. e. translation dictionaries • Preferential seating • Learning Centers • Cooperative learning grouping • Independent Study • Teacher & Peer tutoring • Modified assignments • Differentiated instruction

Interdisciplinary Connections Writing: Math Journal




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Teacher Resources: Math Warehouse Website:

http://www.mathwarehouse.com/geometry/angle/interact ive-transveral-angles.php

Click here for Resources Folder. http://my.hrw.com/index.jsp http://www.khanacademy.com http://prc.parcconline.org Math Series

http://www.mathwarehouse.com/geometry/angle/interactive-transveral-angles.php







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Domain: Congruence

Cluster: Make geometric constructions

Standards: (G.CO.12-13) 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences What can geometric

constructions teach you?

How are constructions different from sketches and drawings?

Do you need geometric constructions to erect a building or other structure?

Students will understand that geometric constructions are evident and necessary in the surrounding environment.

• The standard itself lists a few examples of both the tools students might be presented with and the tasks they should be able to perform. However, it takes more than fancy tools to make these constructions. Students should keep definitions, properties, and theorems about line segments, rays, angles, and parallel and perpendicular lines safe in their back pockets in order to support their drawings.

• Students will benefit from plenty of opportunities to practice each skill. Even more helpful will be different contexts for each one. For example, students might first construct an angle bisector on a single angle, then the angle bisectors for all three angles of a triangle to show


Students will be able to use tools, e.g. ruler, compass, protractor, straightedge, etc. to copy, bisect, and construct line segments, angles, circles and polygons.

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and that they are concurrent, and finally explore whether the

angle bisectors for other polygons are also concurrent. Preparation • Once they have mastered using a compass and

straightedge for a particular construction, students will benefit from exploring using other tools and methods, such as paper folding or using a mirror, to complete the same task. This will reinforce their understanding of the rules of the shape they're working with.






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Assessments • Students are expected to construct equilateral triangles, squares, and circle-inscribed regular hexagons. This means they need to know the theorems and rules about these shapes, specifically that each shape is composed of congruent sides and congruent angles.

• Students also need to know how to construct congruent line segments and congruent angles. They'll need to be able to draw perpendicular lines, too.

• It will be beneficial for students to see demonstrations of how to construct each shape, particularly noting that the side length of the hexagon-in-a-circle is equal to the radius of the circle.

(Taken from: http://www.shmoop.com/common-core- standards/ccss-hs-g-co-12.html and http://www.shmoop.com/common-core-standards/ccss-hs-g-co- 13.html)

• Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (See Attached) • Homework • Valentine Construction Assessment (with

Rubric); Click for Resource Folder.

Equipment Needed: SMART board Projector Paper and pencil TI-84 (plus software) Compass Protractor Ruler Straightedge Graph Paper Isometric Dot Paper Hands-on and virtual two- and three-dimensional

manipulatives (i.e. prisms) Geo-boards

Teacher Resources: Texas Instruments:

http://education.ti.com/calculators/downloads/US/Activit ies/Detail?id=13208&ref=%2fcalculators%2fdownloads %2fUS%2fActivities%2fSearch%2fSubject%3fs%3d502 2%26sa%3d5024%26t%3d5049

Click for Resource Folder. Illuminations - National Council of Teachers of

Mathematics: http://illuminations.nctm.org/ActivityDetail.aspx?ID=22













http://illuminations.nctm.org/ActivityDetail.aspx?ID=22





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Domain: Similarity, Right Triangles and Trigonometry

Cluster: Understand similarity in terms of similarity transformations

Standards: (G.SRT.1-3) 1. Verify experimentally the properties of dilations given by a center and a scale factor.

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How do you recognize

similarity in figures? • What are the effects of

performing dilations on geometric figures?

• How are transformations and similar figures related?

• Students will be able to

perform dilations and predict the effects of the dilation

• Students will use

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

• Complete the interactive activities in the following site and write a reflection based on what you learned. http://bit.ly/xa7x4d


http://bit.ly/xa7x4d


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• Students will be able to 9.1 Personal Financial Literacy; 9.2 (ELLs, Special Education, Gifted and Talented) Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries perform dilations given

Preparation a center and a scale • Preferential seating


factor • Students will identify

similarity transformations and






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verify properties of similarity.


• Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test • Homework

Photo Editing Architecture – Blue Prints on actual buildings

Equipment Needed: Teacher Resources:

SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Compass Protractor Ruler Straightedge Graph Paper Isometric Dot Paper Hands-on and virtual two- and three-dimensional


http://www.hcstonline.org/hcccmath/math%209- 12/geometry/g.srt.1-3/resources/G.SRT.1- 3.Similarity%20and%20Dilations.pdf

http://www.shmoop.com/common-core-standards/ccss- hs-g-srt-1.html




http://www.hcstonline.org/hcccmath/math%209-12/geometry/g.srt.1-3/resources/G.SRT.1-3.Similarity%20and%20Dilations.pdf





http://www.shmoop.com/common-core-standards/ccss-hs-g-srt-1.html













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Cluster: Understand similarity in terms of similarity transformations

Standards: (G.SRT.4-5) 4. Prove theorems about triangles. 5. Use congruence and similarity criteria for triangles to solve problems to prove relationships in geometric figures.

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences How do you use

proportions to find side lengths in similar figures?

How do you show that two triangles are similar?

How do you identify corresponding parts of similar triangles?

Students will be able to

show that two triangles are similar knowing the relationship between only two or three pairs of corresponding parts.

Students will be able to

apply similar triangles to real-life situations.

• Students should know that in triangles and in life, fair doesn't necessarily mean equal. The pieces of the triangle are proportional, but that doesn't always mean congruent. They each get the right amount according to their own proportional needs.

• As for the Pythagorean Theorem, well, that's not implicitly about splitting things up, but if we intend to prove it using similar triangles (which we do), then we will be explicitly splitting things up. We're talking about the fact that to prove similarity, we should look at ratios of side lengths rather than the lengths themselves.

• Students will probably want to keep in mind the importance of being similar. Not to each other—though they will certainly be on top of the mismatched socks trend anyway—but the similarity between triangles that


Students will form proportions based on know lengths of corresponding sides.

Students will use the AA similarity postulate and the

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and are, by definition, similar.

• If they forget the basics of triangle similarity, give them Preparation a quick review of corresponding angles, corresponding

sides, congruence transformations, and similarity transformations. These are pretty much the fundamental blocks with which students can build proofs about triangles.






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SAS and SSS similarity theorems.

Students will be able to prove that a line parallel to one side of a triangle divides the other two proportionally.

• Furthermore, students should definitely be able to name the three tests for identifying similar triangles: AA, SSS, and SAS. Remind them that when we're talking about similarity, those S's are more about the right proportions than they are about equality.

(Taken from: http://www.shmoop.com/common-core- standards/ccss-hs-g-srt-4.html)

• The criteria to determine whether triangles are congruent

or similar include proportionality or congruency of side lengths and congruency of angles. Fortunately, students don't have to check all three sides and all three angles every time they want to test for congruency or similarity. That would make the process rather inefficient.

• There are several ways to combine the side length proportionality or congruency and angle congruency to check for triangle congruency or similarity. They're various combinations and permutations of the letters S (for side) and A (for angle) and students should already be familiar with them. In fact, they should have mastered these congruency versus similarity tests for triangles.

(Taken from: http://www.shmoop.com/common-core- standards/ccss-hs-g-srt-5.html)

Assessments Student Participation Questioning Self-Quizzes Lesson Quizzes (See attached) Unit Test







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Computer Lab / Projector SMART board

http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_1. html




http://www.khanacademy.org/video/similarity- postulates?playlist=Geometry


http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_1.html












http://www.khanacademy.org/video/similarity-postulates?playlist=Geometry







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Cluster: Define trigonometric ratios and solve problems involving right triangles

Standards: (G.SRT.6-8) 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences

• How do you find a side

length or angle measure in a right triangle?

• How do trigonometric ratios relate to similar right triangles?

• Students will understand

how to use trigonometric ratios to form proportions

• Students will understand how to apply sine, cosine and tangent ratios and Pythagorean theorem to solve problems with right triangles

Interactive examples, PowerPoint lessons, regular level

and challenge level problems, sketchpad activities, and hands on activities

TI-84 Activity Videos – Trig. Ratio Introduction, Proof of Pythagorean

Theorem with Similarity, Special Right Triangles SMART Notebook File – Examples and Problems Hypsometer Activity PowerPoint lessons, Interactive Examples, Worksheets –

Sine, Cosine, Tangent



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9.1 Personal Financial Literacy 9.2 (ELLs, Special Education, Gifted and Talented) • Students will be able to Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries Preparation explore the sine of an • Preferential seating


acute angle and the cosine of its complement and determine their relationship.

• Students will examine sine, cosine and tangent ratios.

• The student will solve






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real-world problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry.


• Student Participation • Questioning • Self-Quizzes

o http://teachers.henrico.k12.va.us/math/igo/07 RightTriangles/7-2PythagoreanThm/7-2t.htm

o http://teachers.henrico.k12.va.us/math/igo/07 RightTriangles/7-3SpecialRtTriangles/7-3.htm

o http://teachers.henrico.k12.va.us/math/igo/07 RightTriangles/7-4Trigonometry/7-4t.htm

• Lesson Quizzes (Click here for Resource Folder.) • Unit Test

Indirect measurement as related to various fields of student Astronomy


http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7-2PythagoreanThm/7-2t.htm



http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7-3SpecialRtTriangles/7-3.htm



http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7-4Trigonometry/7-4t.htm



http://www.hcstonline.org/hcccmath/math%209-12/geometry/g.srt.6-8/resources/


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SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Compass Protractor Straws Paperclips Ruler Straightedge Graph Paper Isometric Dot Paper Hands-on and virtual two- and three-dimensional


http://www.shmoop.com/common-core-standards/ccss-hs- g-srt-6.html



Henrico County Public Schools: o http://teachers.henrico.k12.va.us/math/igo/07

RightTriangles/7_2.html o http://teachers.henrico.k12.va.us/math/igo/07

RightTriangles/7_3.html o http://teachers.henrico.k12.va.us/math/igo/07

RightTriangles/7_4.html Texas Instruments:

o http://education.ti.com/calculators/downloads /US/Activities/Detail?ID=1855&MICROSIT E=ACTIVITYEXCHANGE

Khan Academy video archive: o http://www.khanacademy.org/video/basic-

trigonometry?playlist=Trigonometry o http://www.khanacademy.org/video/pythagor

ean-theorem-proof-using- similarity?playlist=Geometry

o http://www.khanacademy.org/video/30-60- 90-triangle-side-ratios- proof?playlist=Geometry

o http://www.khanacademy.org/video/45-45- 90-triangle-side-ratios?playlist=Geometry

Math Warehouse Website: o http://www.mathwarehouse.com/trigonometr

y/sine-cosine-tangent-home.php










http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7_2.html









http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=1855&amp%3BMICROSITE=ACTIVITYEXCHANGE



http://www.khanacademy.org/video/basic-trigonometry?playlist=Trigonometry



http://www.khanacademy.org/video/pythagorean-theorem-proof-using-similarity?playlist=Geometry





http://www.khanacademy.org/video/30-60-90-triangle-side-ratios-proof?playlist=Geometry





http://www.khanacademy.org/video/45-45-90-triangle-side-ratios?playlist=Geometry



http://www.mathwarehouse.com/trigonometry/sine-cosine-tangent-home.php




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Domain: Similarity, Right Triangles and Trigonometry Cluster: Apply Trigonometry to General Triangles Standards: (G.SRT.9-11) 9. (+) Derive the formula A=½ ab sin (C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • Can trigonometry be

used on non-right triangles?

• How would you find the missing measurements of a non-right triangle given two measurements?

• Students will be able to use the Law of Sines and Cosines to find missing angles and side lengths in non-right triangles.

• Use the Laws of Sines and Cosines solve real-life problems.

• Guided Activities – Students will use right triangle trigonometry and Pythagorean theorem to develop the Law of Sines and Law of Cosines

• Hands on Activity – Students will explore the ambiguous SSA case by building a model

• Interactive examples, PowerPoint lessons, and regular level and challenge level problems

• Geometer’s Sketchpad Activity - Law of Sines Investigation • TI-84/TI-Nspire Discovery Activities • Videos - Proof of Law of Sines and Cosines • SMART Notebook File – Examples and Problems • Active Inspire File – Flowchart of Law of Sines and Cosines

Procedure



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• In a non-right triangle, students will be able to draw an auxiliary line from a vertex, perpendicular to the opposite side and derive the formula, A=½ ab sin (C), for the area of a triangle, using the fact that the height of the triangle is, h=a sin(C).

9.1 Personal Financial Literacy 9.2 (ELLs, Special Education, Gifted and Talented) Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries Preparation • Preferential seating







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• Students will be able to use trigonometry and the relationship among sides and angles of any triangle, such as sin(C) = (h/a), prove the Law of Sines.

• Students will be able to trigonometry and the relationship among sides and angles of any triangle and the Pythagorean Theorem to prove the Law of Cosines.

• Students will be able to use the Laws of Sines and Cosines solve problems.


• Student Participation • Questioning • Self Quizzes: o http://teachers.henrico.k12.va.us/math/ito_08/10Addition

alTrig/10les1/lawofsinessq.htm o http://teachers.henrico.k12.va.us/math/ito_08/10Addition

alTrig/10les2/lawofcosinessq.htm • Lesson Quiz (Click here for Resource Folder.) • Benchmark Assessment (Click here for Resource Folder.) • Unit Test

Career and Life Skills: Geometry at work, explore how using trigonometry; a surveyor can find the latitude, longitude and elevation of each by point in the survey.

http://teachers.henrico.k12.va.us/math/ito_08/10AdditionalTrig/10les1/lawofsinessq.htm



http://teachers.henrico.k12.va.us/math/ito_08/10AdditionalTrig/10les2/lawofcosinessq.htm






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Promethean Board / Smart Board

Computer lab access with Geometer’s Sketchpad

Projector for interactive examples

Activity Sheets

TI-84/TI-Nspire Calculators & Activity files

Rulers

Scissors

Markers

Card Stock

Paper fasteners

• http://www.shmoop.com/common-core-standards/ccss-hs-g- srt-9.html



• Illuminations - National Council of Teachers of Mathematics: o http://illuminations.nctm.org/LessonDetail.aspx?ID=

U177 • Henrico County Public Schools:

o http://teachers.henrico.k12.va.us/math/ito_08/10Addit ionalTrig/10-1LawSines.html

o http://teachers.henrico.k12.va.us/math/ito_08/10Addit ionalTrig/10-2LawCosines.html

• Texas Instruments:

o http://education.ti.com/calculators/downloads/US/Acti vities/Detail?id=5367

o http://education.ti.com/calculators/downloads/US/Acti vities/Detail?id=9849

• Khan Academy Video Archive:

o http://www.khanacademy.org/video/proof--law-of- sines?playlist=Trigonometry

o http://www.khanacademy.org/video/law-of- cosines?playlist=Trigonometry










http://illuminations.nctm.org/LessonDetail.aspx?ID=U177



http://teachers.henrico.k12.va.us/math/ito_08/10AdditionalTrig/10-1LawSines.html



http://teachers.henrico.k12.va.us/math/ito_08/10AdditionalTrig/10-2LawCosines.html



http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5367






http://www.khanacademy.org/video/proof--law-of-sines?playlist=Trigonometry



http://www.khanacademy.org/video/law-of-cosines?playlist=Trigonometry




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Domain: Circles

Cluster: Understand and apply theorems about circles

Standards: (G.C.1-4) 1. Prove that all circles are similar. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4. (+) Construct a tangent line from a point outside a given circle to the circle. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • Why are all circles

similar? • How can you prove

relationships between angles in a circle?

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

• What is the relationship between inscribed and circumscribed figures?

• Students will understand the different components of circles and how they relate to angle measurements.

• Students will understand that inscribed angles on a diameter are right angles.

• Students will understand that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

• Students should understand that a circle is a closed curve on a plane. What's special about circles is that all the points on a circle are equidistant from the center. That means circles are defined by two details: position (the center of the circle) and size (the distance from the center to a point on the circle).

• Students should know that position isn't a problem when we're talking about similarity or congruence transformations. If we have two congruent figures at different positions, translating them will easily map one on top of the other. But what about size?

• Students should know that unlike polygons that have dimensions independent of one another (base and height, for instance), a circle's size depends only on one measurement: the radius r. Sure, we can look at the diameter d or the circumference C or even the area A, but they all depend on r as well. (Now would be a good time to tell them that d = 2r and C = 2πr and A = πr2.)

• Since all aspects of a circle's size depend on r, we can


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change the size of any circle simply by dilating the radius by a constant scale factor. Students should already know that dilations, whether they're expansions or contractions, are similarity transformations. We're


• Students will be able to use the fact that the ratio of diameter to circumference is the same for circles to prove that all circles are similar.

• Students will be able to use definitions, properties, and theorems, to identify and describe relations among inscribed angles, radii, and chords (including central, inscribed, and circumscribed angles.)

• Students will be able to construct inscribed circles of a triangle.

• Students will be able to construct circumscribed circles of a triangle.

• Students will be able to use definitions, properties, and theorems to prove properties of angles for a quadrilateral inscribed

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and changing the size of the circle, but not its shape.

• If dilation changes a circle's size and translation can Preparation change its position, we can easily map one circle onto

another using only those two transformations. Since both dilation and translation are similarity transformations, you can safely tell students that all circles are similar. In fact, we don't even need reflections or rotations to help us out. (Try rotating and reflecting circles and see how they change. Hint: they don't.)

• Students can also use two circles and proportions to determine similarity. Two similar shapes will have a constant ratio when corresponding sides are compared. Circles don't have sides, but they do have radii, diameters, and circumferences that we can compare. If students do that, they'll quicky realize that a small circle with radius r and a larger circle with radius r' are similar by a constant scale factor.

• Also, we could show in the same way that the area ratio equals the square of the radius ratio.

• The circumferences, diameters, and radii of our two






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in a circle. • Students will be able to

find the measure of an inscribed angle.

• Students will be able to find the measure of an angle formed by a tangent and a chord.

• Students will be able to construct a tangent line from a point outside a given circle to the circle.

circles are all in proportion, which means they're similar. Essentially, this states that circles come in a variety of sizes, but they'll always be the exact same shape: a circle.

(Taken from: http://www.shmoop.com/common-core- standards/ccss-hs-g-c-1.html)

• Before students can identify and describe the various

angles in a circle, they should be familiar with what these angles are. Students should also be familiar with the concept of arc measurement and how it relates to the measure of the different kinds of angles. Throw some chords in there, too, and we aren't talking about your guitar skills (though we're sure you can rock and roll with the best of 'em).

• Students should know that a central angle is formed by two radii (where the vertex of the angle is the center of the circle), an inscribed angle is an angle formed by two chords (where the vertex of the angle is some point on the circle), and a circumscribed angle has a vertex outside the circle and sides that intersect with the circle.

• To solidify the relationships between arc measure and angle measure, we suggest bringing a pizza to class. Explain that usually, we cut pizzas so that each slice of pizza has a particular central angle. The amount of crust (arc measure) depends on the central angle of the pizza slice.

• If you have the whole pizza to yourself, you could even cut a huge slice from one side to the other, resulting in an inscribed angle. The measure of that angle will also affect how much crust you get on your slice.

Assessments • Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes):

o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-1Terminology/8-1t.htm

o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-3Tangents/8-3t.htm

o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-4ArcsChords/8-4t.htm

o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-5AngleFormulas/8-5t.htm

o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8-6SegmentFormulas/8-6t.htm

• Benchmark/Test (Click here for resource folder) http://sites.google.com/site/ppshighschoolmath/ho

http://www.shmoop.com/common-core-standards/ccss-hs-g-c-1.html


http://teachers.henrico.k12.va.us/math/igo/08Circles/8-1Terminology/8-1t.htm



http://teachers.henrico.k12.va.us/math/igo/08Circles/8-3Tangents/8-3t.htm



http://teachers.henrico.k12.va.us/math/igo/08Circles/8-4ArcsChords/8-4t.htm



http://teachers.henrico.k12.va.us/math/igo/08Circles/8-5AngleFormulas/8-5t.htm



http://teachers.henrico.k12.va.us/math/igo/08Circles/8-6SegmentFormulas/8-6t.htm



http://www.hcstonline.org/hcccmath/math%209-12/geometry/g.c.1-4/resources/

http://sites.google.com/site/ppshighschoolmath/home/benchmark-assessments/assessment

http://sites.google.com/site/ppshighschoolmath/home/benchmark-assessments/assessment


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me/benchmark-assessments/assessment • Homework

• Somewhere in between the discussions of angles and pizza toppings, throw in the formulas for finding the measures of inscribed and circumscribed angles when given the central angle or arc measure. Most important of these is that an inscribed angle's measure is equal to half the measure of its intercepted arc.

• After students have mastered these angles and arc measures, talk about tangents and secants of circles, lines that intersect a circle at one and two points, respectively. It's also important for students to know that a point on a circle can only have one tangent and that tangents are always perpendicular to the radius of a circle.


• Students should know that an inscribed circle is the

largest circle that can fit on the inside of a triangle, with the three sides of the triangle tangent to the circle. A circumscribed circle is one that contains the three vertices of the triangle. Students should know the difference between and be able to construct both of these circles.




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• Students should understand that while circles have one

defined center, triangles try to outdo them by having four different centers: the incenter, the circumcenter, the centroid, and the orthocenter. Imitation is the sincerest form of flattery and all, but do you think maybe they're compensating for something?

• Students should know the definitions of these centers and how each one differs from the others. But how do we get to the true center of the triangle? The world may never know.

• We can draw any triangle by plotting three points on a circle. Students can take this one point further, plot four points on a circle, connect them, and make a cyclic quadrilateral. (They're called that because the vertices are all on the circle, in kind of a, well, cycle.)

• Students be able to prove theorems about cyclic quadrilaterals, the most important of which is that opposite angles in a cyclic quadrilateral add to 180°, or opposite angles in a cyclic quadrilateral are supplementary.


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• If students don't already know the properties of circles

and tangents before this construction, they should take away a few main points from it:

o Tangents drawn to a circle are perpendicular to the circle's radius at the point of tangency.

o Two tangents drawn to a circle from the same point outside the circle are equal. You can have students do this construction and measure the segments from the point of tangency to the shared point.

o Two tangents drawn to a circle from the same point outside the circle make an angle that, when bisected, includes the circles center. Students can construct the angle bisector and see for themselves.

o Tangents to a circle at either end of a diameter are parallel.

• Given a circle with center B and a point A outside the circle, the construction of a tangent to ⊙B that goes through A is relatively simple. What's important is that students also understand the properties of circles and their tangents in order to make these constructions.


Equipment Needed: SMART board

Teacher Resources: • Henrico County Public Schools:

http://teachers.henrico.k12.va.us/math/igo/08Circ les/8_1.html



http://www.youtube.com/watch?v=WHym_d8QrpU



http://teachers.henrico.k12.va.us/math/igo/08Circles/8_1.html




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Projector Paper and pencil Calculator Compass Ruler Protractor Geometer’s Sketchpad





Math Warehouse: http://www.mathwarehouse.com/geometry/circle/

interactive-central-angle-of-circle.php http://www.mathwarehouse.com/geometry/circle/

inscribed-angle.html#inscribedAngleDemo http://www.mathwarehouse.com/geometry/circle/

tangent-to-circle.php Click here for Resource Folder. http://my.hrw.com/index.jsp http://www.khanacademy.com http://prc.parcconline.org Math Series













http://www.mathwarehouse.com/geometry/circle/interactive-central-angle-of-circle.php



http://www.mathwarehouse.com/geometry/circle/inscribed-angle.html#inscribedAngleDemo



http://www.mathwarehouse.com/geometry/circle/tangent-to-circle.php



http://www.hcstonline.org/hcccmath/math%209-12/geometry/g.c.1-4/resources/





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(We have 2π as our factor but it must come from 360°

Domain: Circles

Cluster: Find arc lengths and areas of sectors of circles

Standards: (G.C.5) 5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How is arc length

related to the radius and central angle of a circle?

• How is the area of a sector related to the radius and central angle of a circle?

• Students will understand the relationship between the arc length and the radius and central angle of a circle.

• Students will understand the relationship between the area of a sector and the radius and central angle of a circle.

• Students should first draw a circle with radius r and a central angle of θ. Visually, it should be clear that the arc length s depends on both the central angle and the radius. If we look at the circumference C as one bit arc length, we can see that its central angle is 360° (or 2π radians). If C = 2πr, then the arc length should equal the central angle (in radians) times the radius, or s = θr.

• If students aren't as familiar with radians, make sure to take some time and explain it to them. It can be tricky going from degrees to radians, but you can use the circumference equation to translate from one to another.


• Students will be able to use similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the radius and identify the constant of proportionality as the radian measure of the angle.

• Students will be able to find the arc length of a

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and somehow. If we say 360k = 2π and solve for k, we have Preparation our conversion factor from degrees to radians. Yippee.)

• Students should also be able to derive the formula for the sector of a circle. Just like we used the C = 2πr to find arc length, we can use A = πr2 to find the area of a sector. This time, the 360 degrees translates to π, not 2π. That means we're dividing the angle by 2, so our formula should be A = ½θr2.

• The key thing to note here is that θ must be in radians. You can explain to them that a radian is about 57.3° and that it's the angle at which the ratio of the arc length to






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circle. • Students will be able to

use similarity, derive the formula for the area of a sector.

• Students will be able to find the area of a sector in a circle.

the radius is 1:1.


Assessments Differentiation

• Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes)

o http://teachers.henrico.k12.va.us/math/igo/ 08Circles/8- 2AreaCircumference/8_2_circles_arcs_cir c_area.htm

• Benchmark/Test • Homework


Interdisciplinary Connections




http://teachers.henrico.k12.va.us/math/igo/08Circles/8-2AreaCircumference/8_2_circles_arcs_circ_area.htm








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Equipment Needed: SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Compass Protractor Ruler Straightedge Graph Paper

Teacher Resources:

• Henrico County Public Schools: http://teachers.henrico.k12.va.us/math/igo/08Circles/ 8_2.html





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Isometric Dot Paper Hands-on and virtual two- and three-dimensional



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Domain: Expressing Geometric Properties with Equations

Cluster: Translate between the geometric description and the equation for a conic section

Standards: (G.GPE.1-3) 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How do you find the

equation of a circle in the coordinate plane?

• How do you find the equation of a parabola in the coordinate plane?

• How do you find the equation of an ellipse in the coordinate plane?

• How do you find the equation of a hyperbola in the coordinate plane?

• Students will understand the differences between the equations of conic sections.

• Students will understand how the center and the radius of a circle effect the equation of a circle.

• Students will understand how the focus and the directrix of a parabola effect the equation of a parabola.

• Students will understand how the distances to the foci of an ellipse effect the

• Find the equation of a circle centered on point F that has line segment FG as a radius.

Content Statements


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• Students will be able to use the Pythagorean Theorem to derive the equation of a circle, given the center and the radius.

• Students will be able to complete the square to find the center and radius of a circle when given the general conic form of a circle.

• Students will be able to derive the equation of a parabola when given a focus and directrix of a parabola.

• Students will be able to identify the vertex, focus, directrix, and axis of symmetry and note that every point on the parabola is the same distance from the focus and the directrix.

equation of an ellipse. • Students will understand

how the distances to the foci of a hyperbola effect the equation of a hyperbola.

(A)(x – 5)2 + (y – 8)2 = 144

(B)(x + 5)2 + (y + 8)2 = 144

(C)(x + 5)2 + (y + 8)2 = 169

(D)(x – 5)2 + (y – 8)2 = 169

o Explanation: Here, we have a circle with a center at (-5, -8) and radius 13. The right side of the equation must be 13 2 = 169, so (A) and (B) are incorrect. Since the center is negative for both x and y variables, the equation should have x + 5 and y + 8 on the left side.

(Taken from: http://www.shmoop.com/common-core- standards/ccss-hs-g-gpe-1.html)

• Students should know that a focus is a point near a

parabola and a directrix is a line near a parabola. So what's so special about them? In Mathese, the parabola is the set of all the points that are the same distance between two things: a given point and a given line. In picture form, that would be something like this.

21st Century Life and Careers

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and Preparation

http://www.shmoop.com/common-core-standards/ccss-hs-g-gpe-1.html








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• Students will be able to identify the major and minor axis of an ellipse.

• Students will be able to derive the equation of a hyperbola when given the foci, and note that the absolute value of the differences of the distances from the foci to a point on the hyperbola is constant.

• Students will be able to identify the vertices, center, transverse axis, conjugate axis, and asymptotes of a hyperbola.

• Students should know that because the distances are equal, they find the equation of a parabola by applying the bulky equation below, where xf and yf are the coordinates of the focus and either xd or yd can be substituted with the value of the directrix. The other value becomes its respective variable (so either xd

becomes x, or yd becomes y).

• To help students become more efficient, tell them to square both sides and then expand them. Then, they can solve for y to get the equation of the parabola. No sweat (unless they happen to be on a treadmill or something).

• Students can also use the (x – h)2 = 4p(y – k) formula for a horizontal directrix and (y – k)2 = 4p(x – h) for a vertical directrix. They should know that (h, k) are the coordinates of the parabola's vertex and p is the distance from the focus to the vertex. Remind students that p will be negative when the focus is either under or to the left

Assessments • Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (See Attached) • Homework


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of the directrix.


• Focus, vertex, asymptote, transverse, conjugate, standard

position, Pythagorean Theorem—all important geometric terms that students need to know. On the other hand, square, simplify, solve for x, and complete the square are crucial algebraic skills they need to be able to perform.

• They should know the equations that describe both ellipses (which look like elongated circles) and hyperbolas (which look like a pair of parabolas). For ellipses or hyperbolas with center (h, k), a major axis of a, and a minor axis of b, the formulas are:

• Students should know that when the major axes are vertical rather than horizontal, the a2 and b2 values are switched. Whichever axis is elongated gets the larger a underneath it.

• Chances are students will also want to know the distance formula. After all, the standard itself features the word "distances." Lucky for them, the distance formula is still the same as it always was.

• Students should understand that while parabolas have a single focus, ellipses and hyperbolas have two of them.




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(The plural of focus is "foci" so don't let that trip them up.) The distance from one focus to any point on the ellipse and back to the other focus will always be constant. The same applies to hyperbolas, only instead of adding the distances, we subtract them.

• Given a set of two foci, students should be able to find the equation of an ellipse. They can do this by setting up the equation in the form of two distances calculations (the distance from one focus to (x, y) and the other focus to that same point) that equal a constant, or by finding the values of h, k, a, and b.

• It also helps to imagine a right triangle inside the ellipse with vertices at the center, one of the foci, and at the vertical vertex. The length of the hypotenuse is half the constant distance c (the other half is the hypotenuse of the triangle on the other side), the length of the base helps find the location of the foci, and the length of the vertical leg is b (the same b as in the equations above). Students can also find a by dividing the constant distance by 2.

• You might consider representing the constant distance as


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a string loosely connected to two foci. Using the string to guide your marker, draw the ellipse and explain the relationship between the values and distances. Bringing in triangles will help mathematically cement these relationships. Students should familiarize themselves with all the intricacies of Ellipses so that they can actually get around.

• The itinerary for Hyperbola City should look about the same. Students should be able to perform essentially the same calculations, finding the equation when given the foci. The only difference is that instead of a sum of two distances, we're talking about a difference.

• If students are struggling, they might benefit from: o reviewing the algebra used to simplify big

equations, especially completing the square o considering how the Pythagorean Theorem is

applied here o focusing on when and where to use addition and

subtraction o referring to a labeled drawing of the shape in

question


Equipment Needed: SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Compass Protractor

Teacher Resources: 1. Illuminations - National Council of Teachers of

Mathematics: http://illuminations.nctm.org/LessonDetail.aspx?ID=L792, http://illuminations.nctm.org/ActivityDetail.aspx?ID=195

2. Math Warehouse Website: http://www.mathwarehouse.com/conic-sections/equations- formula-of-conic-sections.php#



http://illuminations.nctm.org/LessonDetail.aspx?ID=L792

http://illuminations.nctm.org/LessonDetail.aspx?ID=L792


http://www.mathwarehouse.com/conic-sections/equations-formula-of-conic-sections.php




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Ruler Straightedge Graph Paper Isometric Dot Paper Hands-on and virtual two- and three-dimensional



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Domain: Expressing Geometric Properties with Equations

Cluster: Use coordinates to prove simple geometric theorems algebraically

Standards: (G.GPE.4-7) 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How do you use

coordinate geometry to prove theorems?

• What properties of geometric figures can be proven using coordinate geometry?

• How can you derive the distance and midpoint formulas using coordinate geometry?

• How can you calculate areas using coordinate geometry?

• Students will understand the relationship between algebra and coordinate geometry.

• Students will understand the concept of slope and how it relates to geometric figures.

• Students will need to know several major formulas and equations in order to prove they can hold their own against Descartes. Here are a few of the most important equations:

o the distance formula:

o the slope formula: o the midpoint

formula:

o the equation of a circle: (x – h)2 + (y – k)2 = r2

o the equation of an

ellipse: Content Statements 21st Century Life and Careers


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• Students will be able to use coordinate geometry to prove geometric theorems algebraically.

• Students will be able to use slope to prove lines are parallel or perpendicular.

• Students will be able to find equations of lines based on certain slope criteria such as finding the equation of a line parallel or perpendicular to a given line that passes through a given point.

• Students will be able to find a point on a line segment that divides the segment into a given ratio when given two points.

• Students will be able to use coordinate geometry and the distance formula to find the perimeters of polygons and the areas of triangles and rectangles.

9.1 Personal Financial Literacy 9.2 o the equations of all the rest of the Career Awareness, Exploration, and shapes/functions that can be plotted on the

coordinate plane (parabolas, hyperbolas, etc.) Preparation

• They'll also need to be pretty fluent in the properties of geometric shapes. If they don't know that the sum of any two sides of a triangle is always greater than the remaining side, or that all four sides of a rhombus are congruent, they may or may not be in some serious trouble.

• To prove a geometric theorem about a shape on the coordinate plane, it's not enough for students to know these properties and formulas. They'll need to be able to apply them to the coordinate plane as well.

• Most statements students need to prove will somehow involve angles and lengths. For instance, to prove that a given figure is a square, students could prove that the slopes of adjacent sides are perpendicular (slope formula), and that all sides are the exact same length (distance formula).

• There are a great number of possibilities for algebraically proving the many properties of the many shapes in the coordinate plane. While students don't have to know all the ways to prove a particular geometric theorem, they should be able to come up with at least one.


• Students are expected to know that lines in the same

plane can fall into one of three categories: parallel,








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Assessments perpendicular, or neither. Given two lines, they should be able to prove which of these three categories is applicable. They should also know how to find the equation of a line that is parallel or perpendicular to a given line.

• Your students may need a quick refresher on some slope basics. You know, point your toes together, watch out for trees, and don't eat yellow snow. That kind of stuff.

• A line's slope, of course, is the ratio of rise over run. In the slope-intercept form of a line's equation, it is the coefficient on the x term. Positively sloped lines point up, while negatively sloped ones point down. The greater the magnitude of the slope, the steeper the line.

• Parallel lines, or what second graders would refer to as "train track lines," have the same slope and never cross each other. Perpendicular lines, on the other hand, cross at a right angle and feature slopes that are opposite reciprocals of each other.


• The three main methods that students can use to find the

partition point are as distinct from each other as sitcom character archetypes. Students should be able to use the midpoint formula (if the ratio of the parts is 1:1), the section formula, and the distance formula.

• If we're being honest, though, the midpoint formula is just a special case of the section formula. But it comes first in the list because it's the easiest (unlike the game show Pyramid). Just find the averages of the x and y coordinates to find the midpoint, which gives the




http://www.youtube.com/watch?v=vxCpg4tkz_8


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partition point these coordinates.

• The section formula is just a fancier version of the midpoint formula. If a line segment has endpoints (x1, y1) and (x2, y2), and a partition point P will separate the line segment into a ratio of m:n, then students should plug the numbers into the section formula to find the coordinates of P.

• Essentially, the midpoint formula is to finding averages as the section formula is to finding weighted averages. Given the endpoints of a line segment, students should be able to use both formulas to find the midpoint M and the partition point P at a specified ratio.

• Students should also be able to determine the ratio of a partition using the distance formula. They need to remember that we are talking about directed line segments here. So it does matter on which side of the partition point the bigger segment lies. Remind them to keep track of which segments they're looking at.


• Students are expected to be able to use the distance

formula to find the distance between two coordinate points and then apply that information and know-how to calculate the perimeter and area of various polygons.

• They should note that the distance formula is derived from the Pythagorean theorem. We suggest squaring




both sides of the distance formula so it's a little easier to see that both equations set the sum of two squared terms equal to a third squared term.

• Basically, the distance formula assumes that the distance we're measuring is the hypotenuse of a right triangle. The base of the triangle is denoted as a, or (x2 – x1). The height is b, or (y2 – y1). And the hypotenuse, c, is the distance, D.

• As one might expect, the distance formula is pretty handy for computing the distance between two coordinate points. Of course, it's best saved for when those points are not located along the same vertical or horizontal, because then simple subtraction would be the most efficient way to find the distance between them. No, the distance formula is more aptly used when the points are spaced diagonally.

• Students should know that if we compute the distances between the points around a polygon, then the distances can be added to find the polygon's perimeter. Students should know that this works for all polygons.

• Area is a little different, since it depends on the actual shape. Students should know that pretty much all polygons on the coordinate plane can be split into rectangles and triangles. As such, students should know how to calculate the areas of rectangles and triangles. Adding the areas of the individual pieces should give them the area of the entire shape.

• We recommend giving students a variety of shapes so they can apply their knowledge and problem-solving skills rather than automatically plugging points into a formula. If students are ever confused, they can always

plot points on the coordinate plane, too.




Teacher Resources: Math Warehouse Website:

http://www.mathwarehouse.com/algebra/linear_equation /parallel-perpendicular-lines.php





http://www.mathwarehouse.com/algebra/linear_equation/parallel-perpendicular-lines.php

http://www.mathwarehouse.com/algebra/linear_equation/parallel-perpendicular-lines.php




Domain: Geometric Measurement and Dimension

Cluster: Explain volume formulas and use them to solve problems

Standards: (G.GMD.1-3) 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences Would we be able to

function in everyday life without these measurements?

How would we be able to cook if did not have volume?

Students will understand that consistency in the units of measure is essential in describing and solving real- world situations.

Students will understand that the formula and unit used to calculate the measurement is based upon the reason for which the measurement is calculated.

Students will understand and apply the formulas for two- and three-dimensional figures.

• “Popcorn Cylinders” activity. Calculate and compare volumes.

• Interactive examples, PowerPoint lessons, regular level and challenge level problems, sketchpad activities, and hands on activities

• Video - Cylinder Volume and Surface Area • Video - Solid Geometry Volume • Video – Volume of a Sphere • Sample Lesson Plans and Examples



Students will be able to 9.1 Personal Financial Literacy 9.2 (ELLs, Special Education, Gifted and Talented) calculate the following Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries Preparation formulas: area, volume, • Preferential seating


perimeter, and circumference.

Students will be able to informally justify the formulas for two- and three-dimensional figures.


1. Self-quizzes Career and Life Skills:

Architecture (Historical) Marketing Art and Design






Equipment Needed: SMART board Projector Paper and pencil Calculator Hands-on and virtual two- and three-dimensional

manipulatives (i.e. prisms) Geometer’s Sketchpad

Teacher Resources: • Henrico County Public Schools:

o http://teachers.henrico.k12.va.us/math/igo/09Are aVolume/9_1.html



• Khan Academy video archive: o http://www.khanacademy.org/video/cylinder-

volume-and-surface-area?playlist=Geometry • Khan Academy video archive:

o http://www.khanacademy.org/video/solid- geometry-volume?playlist=Geometry

• Khan Academy video archive: o http://www.khanacademy.org/video/volume-of-a-

sphere?playlist=Developmental+Math+2 • http://www.shmoop.com/common-core-standards/ccss-hs-

g-gmd-1.html • http://www.shmoop.com/common-core-standards/ccss-hs-

g-gmd-2.html • http://www.shmoop.com/common-core-standards/ccss-hs-

g-gmd-3.html


http://teachers.henrico.k12.va.us/math/igo/09AreaVolume/9_1.html









http://www.khanacademy.org/video/cylinder-volume-and-surface-area?playlist=Geometry



http://www.khanacademy.org/video/solid-geometry-volume?playlist=Geometry



http://www.khanacademy.org/video/volume-of-a-sphere?playlist=Developmental%2BMath%2B2



http://www.shmoop.com/common-core-standards/ccss-hs-g-gmd-1.html









Domain: Geometric Measurement and Dimension

Cluster: Visualize relationships between two-dimensional and three-dimensional objects

Standards: (G.GMD.4) 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. Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How can we use two-

and three-dimensional figures and their characteristics to describe real-world solids and solve problems?

• Students will understand that there is application in the relationships of two- dimensional and three- dimensional objects in present careers, e.g. architecture, construction trades.

• Students will understand how two-dimensional shapes are associated with three- dimensional forms.

• Jared is scheduled for some tests at his doctor’s office tomorrow. His doctor has instructed him to drink 3 liters of water today to clear out his system before the tests. Jared forgot to bring his water bottle to work and was left in the unfortunate position of having to use the annoying paper cone cups that are provided by the water dispenser at his workplace. He measures one of these cones and finds it to have a diameter of 7cm and a slant height (measured from the bottom vertex of the cup to any point on the opening) of 9.1cm. Note: 1 cm 3 =1 ml

o How many of these cones of water must Jared drink if he typically fills the cone to within 1cm of the top and he wants to complete his drinking during the work day?

o Suppose that Jared drinks 25 cones of water during the day. When he gets home he measures one of his cylindrical drinking glasses and finds it to have a diameter of 7cm and a height of 15cm. If he typically fills his glasses to 2cm from the top, about how many glasses of water must he drink before going to bed?



• Students will identify 9.1 Personal Financial Literacy 9.2 (ELLs, Special Education, Gifted and Talented) Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries attributes of two- and

Preparation three-dimensional • Preferential seating


objects and relate them to each other.

• Students will be able construct three- dimensional objects using two-dimensional nets and rotations.







• Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (Click here for Resource Folder.) • Homework



manipulatives (i.e. models, prisms) Math journal for each student Glue or tape Geo-boards Geometer’s Sketchpad

Teacher Resources:

• http://map.mathshell.org/materials/download.php?fileid= 828

• http://www.illustrativemathematics.org/standards/hs • three-dimensional solid shapes (actual models, or

classroom/home objects that resemble solid shapes) or create paper models using free printables from the following Web site (http://www.senteacher.org/worksheet/12/nets.xhtml)

• variety of nets copied for students to construct prisms and pyramids (http://www.senteacher.org/worksheet/12/nets.xhtml or http://mathematics.hellam.net/nets.html)

• Illuminations – NCTM National Council of Teachers of Mathematics: http://illuminations.nctm.org/ActivityDetail.aspx?ID=20 5


http://www.hcstonline.org/hcccmath/math%209-12/geometry/g.gmd.4/resources/

http://map.mathshell.org/materials/download.php?fileid=828



http://www.illustrativemathematics.org/standards/hs

http://www.senteacher.org/Worksheet/12/Nets.xhtml

http://www.senteacher.org/Worksheet/12/Nets.xhtml

http://mathematics.hellam.net/nets.html

http://mathematics.hellam.net/nets.html




Domain: Modeling with Geometry

Cluster: Apply geometric concepts in modeling situations

Standards: (G.MG.1-3) 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* 3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How do geometric

shapes relate to everyday objects?

• How do we use area and volume to explore the concepts of density?

• How can geometric properties be used in design?

• Students will understand the relationship between geometry and real life situations.

• Find a picture of an object of interest to you. Use a program such as Paint or PowerPoint to create the object using only geometric shapes

• Research the U.S. at night over time. Create a visual with no less than four graphics showing the changes in population density based on images from the evening sky

• Use a computer program to investigate fractals and discuss the apparent limitations of the objects: http://www.fractovia.org/



http://www.fractovia.org/

• Students will be able to 9.1 Personal Financial Literacy 9.2 (ELLs, Special Education, Gifted and Talented) Career Awareness, Exploration, and • Extended time

• Supplemental Materials, i. e. translation dictionaries use geometric shapes,

Preparation their measures, and • Preferential seating


their properties to describe objects.

• Students will be able to use the concept of density when referring to situations involving area and volume






models. • Students will be able to

solve design problems by designing an object or structure that satisfies certain constraints.


• Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (See Attached) • Homework

Career and Life Skills: Occupancy Regulations Fire safety Engineering Code Officer/ Building Inspector




Teacher Resources: http://www.yummymath.com/tag/g-mg-1/ Illuminations - National Council of Teachers of

Mathematics: http://illuminations.nctm.org/LessonDetail.aspx?id=L79 3

http://www.shmoop.com/common-core-standards/ccss- hs-g-mg-1.html





http://www.yummymath.com/tag/g-mg-1/

http://illuminations.nctm.org/LessonDetail.aspx?id=L793



http://www.shmoop.com/common-core-standards/ccss-hs-g-mg-1.html












Domain: Statistics and Probability

Cluster: Understand independence and conditional probability and use them to interpret data

Standards: (S-CP.1-5) 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”). 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. 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.

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. 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.

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.

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • How can probability be

used to predict the outcome of an event?

• How can the concepts of probability be applied to real life situations?

• Students will understand how to organize data and identify sample space using frequency tables.

• Students will understand how to measure the likelihood that an event will occur.

Geometric Probability Problem • Have students determine the probability that a point

randomly chosen in a partially shaded square lands in each of the shaded regions.

• Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.

o In your teams, make conjectures about what it


Content Statements • Students will understand how the outcome of one event can affect the probability of another event.

means to find the area of the shaded region and how you might determine the shaded area for each square.

o When ready, in your teams determine the probability that a randomly chosen point in the square lies in the shaded region.

o In your teams, try multiple approaches and be sure to ask the question, does our conjecture make sense?”

Conditional Probability Problem

• In the following questions, identify two ways to group a sample (labels for a two-way frequency table) that could answer the question you are given.

o Do third grade girls prefer chocolate or vanilla pudding when compared to boys?

o Do kiwi birds live longer than sea turtles? o Do guavas prefer to have rubber bands, paper

clips, or paint balls shot at them? o Is there a relationship between high school

students that score well in math and those who brush their teeth twice daily?

o Is there a relationship between people who snore at night and people who enjoy strawberry shortcake?

• Students will be able to define a sample space and events within the sample space.

• Students will be able to identify subsets from sample space when given defined events, including unions, intersections and complements of events.

• Students will be able to identify two events as independent or not.

• Students will be able to explain properties of Independence and Conditional Probabilities in context and simple English.

• Students will be able to define and calculate conditional probabilities.

• Students will be able to use the Multiplication Principal to decide if two events are independent and to calculate conditional



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probabilities. • Students will be able to

construct and interpret two-way frequency tables of data for two categorical variables and calculate probabilities from the table. They will also be able to use probabilities from the table to evaluate independence of two variables.

• Students will be able to recognize and explain the concepts of independence and conditional probability in everyday situations.

21st Century Life and Careers Differentiation





History: Political party comparison English: Venn Diagrams describing character traits and literature Technology: Computers and programs such as Excelto create Venn Diagrams






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Equipment Needed: SMART board Projector Paper and pencil

Teacher Resources: • http://learnzillion.com/lessons/2323-determine-the-

probability-of-an-event-and-its-complement • http://www.mathsisfun.com/pascals-triangle.html

http://learnzillion.com/lessons/2323-determine-the-probability-of-an-event-and-its-complement



http://www.mathsisfun.com/pascals-triangle.html


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Calculator (Graphing and Scientific) Ruler

http://www.shmoop.com/common-core-standards/ccss- hs-s-cp-1.html






http://www.shmoop.com/common-core-standards/ccss-hs-s-cp-1.html



















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Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model

Standards: (S.CP.6-9) 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. 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. 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. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • What is the difference

between combinations and permutations?

• How do you decide when to use combinations and permutations?

• In what type of situations would compound probability be used?

• Students will understand that the probability of compound events can be found by using the probability of each part of the compound event.

• Students will understand how to use tables, tree diagrams, and formulas to find conditional probability.

• Students will understand when to use permutations or combinations.

• Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

• Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

• Modeling Conditional probabilities: o http://map.mathshell.org/materials/download.php

?fileid=1215 • http://www.youtube.com/watch?v=LIM_PSq0LeQ • https://www.khanacademy.org/math/trigonometry/prob_

comb/dependent_events_precalc/v/independent-events-1

Content Statements • Students will be able to

calculate conditional probabilities using the definition: ―the conditional probability of A given B as the

http://www.shmoop.com/common-core-standards/ccss-hs-s-md-5a.html





http://www.shmoop.com/common-core-standards/ccss-hs-s-md-5b.html









http://map.mathshell.org/materials/download.php

http://www.youtube.com/watch?v=LIM_PSq0LeQ

https://www.khanacademy.org/math/trigonometry/prob_comb/dependent_events_precalc/v/independent-events-1




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fraction of B’s outcomes that also belong to A and interpret the probability in context.

• Students will be able to identify two events as disjoint (mutually exclusive).

• Students will be able to calculate probabilities using the Addition Rule and interpret the probability in context.

• Students will be able to calculate probabilities using the General Multiplication Rule and interpret in context.

• Students will be able to identify situations as appropriate for use of a permutation or combination to calculate probabilities.

• Students will be able to use permutations and

21st Century Life and Careers Differentiation (ELLs, Special Education, Gifted and Talented)

• Extended time • Supplemental Materials, i. e. translation dictionaries • Preferential seating • Learning Centers • Cooperative learning grouping • Independent Study • Teacher & Peer tutoring • Modified assignments • Differentiated instruction

Interdisciplinary Connections Physical Education and Health: Probabilities in sports









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and to solve problems.

Assessments • Student Participation • Questioning • Quizzes (Teacher Given and Self Quizzes) • Benchmark/Test (Click here for Resource Folder.) • Homework

http://www.hcstonline.org/hcccmath/math%209-12/geometry/s.cp.6-9/resources/


6262

Equipment Needed: SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Ruler

Teacher Resources: http://geometrycommoncore.com/index.html http://www.mathsisfun.com/pascals-triangle.html http://www.shmoop.com/common-core-

standards/ccss-hs-s-cp-6.html http://learnzillion.com/lessons/2576-understand-

conditional-probability http://mathforum.org/library/drmath/sets/high_perms

_combs.html http://www.shmoop.com/common-core-

standards/ccss-hs-s-cp-6.html http://www.shmoop.com/common-core-



standards/ccss-hs-s-cp-9.html http://my.hrw.com/index.jsp http://www.khanacademy.com http://prc.parcconline.org Math Series

http://geometrycommoncore.com/index.html

http://www.mathsisfun.com/pascals-triangle.html




http://learnzillion.com/lessons/2576-understand-conditional-probability



http://mathforum.org/library/drmath/sets/high_perms_combs.html

http://mathforum.org/library/drmath/sets/high_perms_combs.html

















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o Select a randomly chosen card from a "fresh"


Cluster: Use probability to evaluate outcomes of decisions

Standards: (S.MD.6-7) 5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast- food restaurant.

b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low- deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

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

Essential Questions Enduring Understandings Activities, Investigation, and Student Experiences • What does it mean for

an event to be random? • How can probability be

used to make choices and to help make decisions based on prior experience?

• Students will understand the concept of randomness.

• Students will understand how to use probabilities in decision making.

• Come up with an example or a decision that needs to be made, and then propose ways by which it can be made. Are the decisions fair from a probability standpoint? Let the students decide after numerous tries to ensure randomness. Are the decisions fair from a moral standpoint (for instance, randomly deciding who fails the class based on a coin toss)?

o Toss a fair die o Drop a fair die o Toss an unfair die (but don't tell them about it)


• Students will be able to set up a probability distribution for a random variable representing payoff values in a game of chance.

• Students will be able to make decisions based

9.1 Personal Financial Literacy 9.2 Career Awareness, Exploration, and deck of 52 Preparation o Flip a coin

o Use lots or straws o Use a spinning top o Use a random number generator

• Discuss the strengths and weaknesses, randomness or bias, fairness or unfairness of each method from a






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on expected values • Students will be able to

use expected values to compare long term benefits of several situations.

mathematical standpoint. Discuss methods to undermine or enhance the fairness of each method, and when such methods will be useful in making decisions.

Assessments Differentiation



Interdisciplinary Connections Physical Education: Game decisions Technology: Excel/graphing calculators to create/ analyze number generators


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Equipment Needed: SMART board Projector Paper and pencil Calculator (Graphing and Scientific) Ruler

Teacher Resources: http://ccssmath.org/?page_id=2400 http://www.shodor.org/interactivate/activities/Adjustable

Spinner/ http://www.shodor.org/interactivate/activities/Advanced

MontyHall/ http://www.shmoop.com/common-core-standards/ccss-

hs-s-md-6.html http://www.shmoop.com/common-core-standards/ccss-

hs-s-md-7.html http://my.hrw.com/index.jsp http://www.khanacademy.com http://prc.parcconline.org Math Series

http://ccssmath.org/?page_id=2400

http://www.shodor.org/interactivate/activities/AdjustableSpinner/



http://www.shodor.org/interactivate/activities/AdvancedMontyHall/



http://www.shmoop.com/common-core-standards/ccss-hs-s-md-6.html









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