trenton public schools curricul… · web view2017. 8. 31. · demonstrate comprehension by...

Unit Title: Geometry Unit 2Grade Level: HS Geometry

Timeframe: Marking Period 2

Unit Focus and Essential Questions

Unit Focus Use coordinates to prove simple geometric theorems Define trigonometric ratios and solve problems involving right triangles Translate between the geometric description and the equation for a conic section Understand and apply theorems about circles Find arc lengths and areas of sectors of circles Explain volume formulas and use them to solve problems. Visualize relationships between two dimensional and three-dimensional objects Apply geometric concepts in modeling situations

Essential QuestionsHow are real-world figures, formulas, and coordinates related?

What are the relationships between two-dimensional shapes and three-dimensional shapes in terms of properties, coordinates, and formulas and how can we prove these?

New Jersey Student Learning Standards

Standards/Cumulative Progress Indicators (Taught and Assessed): G.MG.A.1 G.MG.A.2G.MG.A.3G.GPE.B.4G.GPE.B.5G.GPE.B.6G.GPE.B.7

G.GMD.A.3G.GMD.B.4G.GMD.A.1G.GPE.A.1G.C.A.1G.C.A.2G.C.A.3G.C.B.5

Key: Green = Major Clusters; Blue = Supporting; Yellow = Additional Clusters

Standard/SWBAT and Pacing Student Strategies Based on Instructional Framework

Formative Assessment

Activities and Resources Standards Based Assessment

CAR © 2009

G.MG.A.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.

SWBAT

identify cross-sections of three dimensional objects.

identify three-dimensional objects generated by rotation of two-dimensional objects.

solve problems using volume formulas for cylinders, pyramids, cones, and spheres.

model real-world objects with geometric shapes.

describe the measures and properties of geometric shapes that best represent a real-world object.

Math Journal: Identify 10 real-world objects and compare them with the geometric shapes they most closely resemble. For example, a tree trunk resembles a cylinder.

Direct Instruction Option 1 https://www.youtube.com/watch?v=hlD_j3AtxGshttps://www.youtube.com/watch?v=28-MznX-xxU

Option 2 –https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-6

https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-7

Option 3 – https://njctl.org/courses/math/geometry/3d-geometry/attachments/3d-geometry-3/

Slides: 192-275

Option 4 – https://www.illustrativemathematics.org/HSG-MG.A

Option 5 – https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-solids-intro/e/volume-word-problems-with-cones--cylinders--and-spheres

https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-2d-vs-3d/v/vertical-slice-of-rectangular-pyramid

Option 6 –Geometry Textbook: 11-2, 11-3, 11-4, 11-5, 11-6, 11-7

CentersTeacher Center – The teacher works in a small group with 1-4 students. [The engage NY lessons are written with teacher guidance and should also be implemented in the teacher center]

Teachers will agree on common classwork problems in their professional learning communities or grade level meetings. Problems should be selected that most closely match the assessments in column 5.

Martin’s Swimming Pool

Martin is planning to construct a swimming pool behind his house. The architect shows him a plan for the swimming pool. The pool, if viewed from the top looks like a 40 ft long and 20 ft wide rectangle. The pool is divided into two equal sections along its length—the shallow section and the deep section. The shallow section has a constant depth of 5 ft. Once the shallow section ends, the floor of the pool starts sloping until it reaches a maximum depth of 20 ft at the other end of the pool.

Part A. What is the length of the slope of the deep section of the pool? Draw a two-dimensional side view of the swimming pool to show the shallow and deep sections. Remember to write the measures of all sides.

EngageNYhttps://www.engageny.org/resource/geometry-module-3-

topic-b-lesson-6


Pearson Geometry Common Core 11-7

YouTube

https://www.youtube.com/watch?v=hlD_j3AtxGshttps://www.youtube.com/watch?v=28-MznX-xxU

PMI/NJCTLhttps://njctl.org/courses/math/geometry/3d-geometry/attachments/3d-geometry-3/

Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-solids-intro/e/volume-word-problems-with-cones--cylinders--and-spheres

https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-2d-vs-3d/v/vertical-slice-of-rectangular-pyramid

Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-MG.A

CPalms

http://www.cpalms.org/Public/PreviewStandard/Preview/5639

Type 2-3 Question Bank

G-MG.1- Type 2-3 Question Bank

Quarterly Assessment

Geometry Touchpoint - G.MG.1

Geometry OCR - G.MG.1

CAR © 2009



https://www.illustrativemathematics.org/HSG-MG.A

https://www.youtube.com/watch?v=28-MznX-xxU

https://www.youtube.com/watch?v=hlD_j3AtxGs





https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-solids-intro/e/volume-word-problems-with-cones--cylinders--and-spheres




https://njctl.org/courses/math/geometry/3d-geometry/attachments/3d-geometry-3/









Standards Based Problems Center – Students work in a group to solve the style of problems the assessments (Quarterly) will use to measure that standard. [Engage NY has excellent problems for this… as well as EdConnect Type 2 and Type 3 problems

Individual Center – Students work on the individual skill that they need based on their pre-test data.

Manipulative Center – Students work on creating transformations using protractor, compass, and straight-edge.

Interdisciplinary Center – Students work on transformation problems relating to video game graphics and programming.

Review Classwork

Exit Ticket

Part B. Martin contracts a construction company to dig up earth for the swimming pool. The company charges $15 per hour and estimates that it will be able to complete the job within 8 hours. How much earth will be dug up? How much earth is the company digging out for every dollar charged? Round your answer to the hundredths place.

Part C. Once the earth has been excavated, the next step is to paint all four lateral sides and to tile the floor of the swimming pool. What area needs to be painted? What area needs to be tiled? Martin has the option to purchase square tiles of length 1 ft or 2 ft or 3 ft, and so on. What is the largest tile he can purchase? The cost of each type of tile, in dollars, is twice the length of its diagonal. If Martin purchases the biggest possible tile, how much would they cost? Round your answers to the hundredths place.

Part D. The pool is filled with water one foot below the top. The water is pumped into the pool through an 18-inch wide pipe with velocity of 1.5 ft/second. Find the area of the cross section of

CAR © 2009

the pump and multiply it by the velocity to calculate the rate, in cubic feet per second, at which the water will be pumped into the pool? Use 3.14 as the value of pi and round your answers to the hundredths place.

Part E. How much water does the shallow section hold? How much more water does the deep section hold as compared to the shallow section?

Part F. Another pipe is attached at the bottom of the pool, which is used to drain water from the pool. The pipe has a diameter of 36 inches and the pump at its end sucks water out with velocity of 0.25 ft/s. One day, when the pool was empty, both the inlet and outlet pipes were opened simultaneously. How many hours will it now take to fill the pool? Use 3.14 as the value of pi and round your answers to the hundredths place.

CAR © 2009

ELL/Bilingual Modifications

Student Learning Objective (SLO) Language Objective Language NeededSLO: 4CCSS:G.MG.1 WIDA ELDS: 3ReadingSpeakingWriting

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★

Describe objects using geometric shapes, their measures and their properties using models, a word wall and a partner.

VU: Right circular cylinder, rectangular, base radius

LFC: Passive voice

LC: Varies by ELP level

ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives

Describe objects using geometric shapes, their measures and their properties in L1 and/or use gestures, examples and selected technical words.

Describe objects using geometric shapes, their measures and their properties in L1 and/or use selected technical vocabulary in phrases and short sentences.

Describe objects using geometric shapes, their measures and their properties using key, technical vocabulary in simple sentences.

Describe objects using geometric shapes, their measures and their properties using key technical vocabulary in expanded sentences.

Describe objects using geometric shapes, their measures and their properties using technical vocabulary in complex sentences.

Learning Supports

ModelingDemonstrationPartner workWord/picture wallL1 text and/or supportPictures /illustrations Cloze Sentences

ModelingPartner workWord/picture wallL1 text and/or supportSentence Frame

ModelingPartner work Word wall

ModelingPartner work

ModelingPartner work

CAR © 2009

http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/InteractionSupports/PartnerWork.pdf

http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/InteractionSupports/TeacherModeling.pdf



http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/ClassroomDisplays/WordWall.pdf



http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/GraphicOrganizersWorksheets/SentenceFrames.pdf

http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/L1Supports.pdf




http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/Cloze.pdf

http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/Visualizations/PicturesPhotographs.pdf





G.MG.A.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

SWBAT

model real-world situations, applying density concepts based on area.

model real-world situations, applying density concepts based on volume.

Math Journal: According to Wikipedia, NJ has a population density of 1218 people/mi2 , while AZ has a population density of 60 people/mi2. Describe the differences that you might expect to find in these two states based on this data.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-8

Option 2 – https://www.illustrativemathematics.org/HSG-MG.A

Option 3 – https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-density/v/density-example-blimp

Option 4 – Geometry Textbook: 11-7







A population density map of a county is given below. One square unit represents one square kilometer.

Part A. How many square kilometers is the county?

Part B. Based on the map, what is the smallest possible number of people who live in the county? What is the largest possible number?

Part C. To keep housing affordable and


topic-b-lesson-8


Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-

solids/hs-geo-density/v/density-example-blimp

Illustrative Mathematics:https://www.illustrativemathematics.org/HSG-MG.A


G-MG.2 - Type 2-3 Question Bank




CAR © 2009

Review Classwork

Exit Ticket

safe, it is recommended that counties have a ratio of 2 units of housing for every 5 residents. Below is a housing density map.

The county gives out grants for housing developments. Which area should be the top priority for the county officials to encourage the building of additional housing units? Explain. Create a table to show how many units they should build in each area. Be sure to take the range of population into account and support your answer with numbers obtained from the information above.

Use words, numbers, and/or pictures to show your work.

CAR © 2009

Student Learning Objective (SLO) Language Objective Language NeededSLO: 5CCSS:G.MG.2 WIDA ELDS: 3ReadingSpeakingWriting

Use density concepts in modeling situations based on area and volume. (e.g., persons per square mile, BTUs per cubic foot). ★

Demonstrate comprehension of how to use density concepts in modeling situations based on area and volume in real world problems using models, linguistic supports and drawings.

VU: Density, disk, population density

LFC: Prepositional clauses



Demonstrate comprehension of how to use density concepts in modeling situations based on area and volume in real world problems in L1 and/or use gestures, examples and selected technical words.

Demonstrate comprehension of how to use density concepts in modeling situations based on area and volume in real world problems in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension of how to use density concepts in modeling situations based on area and volume in real world problems using key technical vocabulary in simple sentences.

Demonstrate comprehension of how to use density concepts in modeling situations based on area and volume in real world problems using key, technical vocabulary in expanded and some complex sentences.

Demonstrate comprehension of how to use density concepts in modeling situations based on area and volume in real world problems using technical vocabulary in complex sentences.

Learning Supports

ModelingMath JournalSmall group/ triadsWord/Picture WallL1 text and/or supportPictures / Illustrations/diagrams/drawings Cloze Sentences

ModelingSmall group/ triadsWord/Picture WallL1 text and/or supportSentence Frame

ModelingSmall group/ triads Sentence StarterWord Wall

ModelingSmall group/ triads

Modeling

CAR © 2009


http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/InteractionSupports/InTriadsOrSmallGroups.pdf



http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/GraphicOrganizersWorksheets/SentenceStarters.pdf






http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/InteractionSupports/PartnerWork.pdfhttp:/www.nj.gov/education/modelcurriculum/ela/ellscaffolding/InteractionSupports/PartnerWork.pdf



http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/Visualizations/IllustrationsDiagramsDrawings.pdf





http://www.nj.gov/education/modelcurriculum/math/ellscaffolding/MathJournal.pdf


G.MG.A.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).

SWBAT

design objects or structures satisfying physical constraints

design objects or structures to minimize cost.

solve design problems.

Math Journal: You have been hired by a local company to design packaging for an ice cream cone. Your cone needs to have a plastic disc over the opening (to keep it from being crushed while shipping) as well as paper wrapping. Describe (in general terms) what you would need to calculate to determine the cost of this process. How would the dimensions of the cone affect the costs?

Direct Instruction Option 1: http://www.shmoop.com/common-core-standards/ccss-hs-g-mg-3.html

Option 2 –https://www.illustrativemathematics.org/HSG-MG.A







Review Classwork


Pet Fence

Dana is planning to build an enclosure in her yard so that her dogs can play in a secure area. She is planning to use fencing that comes in rigid 6-foot-long sections. She cannot bend the individual sections, but she can join them at any angle to form different polygons. Dana has enough money to buy 24 sections of fencing, including one with a gate. Dana plans to use all 24 sections of fencing when building the enclosure for her dogs.

Part A. Dana first considers making a rectangular enclosure. In the table below, list all possible ways Dana could use the fencing to make an enclosure that has an area of at least 900 square feet. What is the greatest

Shmoop.comhttp://www.shmoop.com/common-core-standards/ccss-hs-

g-mg-3.html


Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-MG.A


G-MG.1- Type 2-3 Question Bank




CAR © 2009


Exit Ticket rectangular area Dana could enclose with the 24 sections of fencing? Explain your answer.

Part B. Dana decides to sketch models of the rectangular enclosures. She uses tick-marks to show each section of fencing on the models, and she labels what will be the actual length and width of the enclosures. If represents two pieces of fencing placed next to each other, use a ruler or graph paper to sketch models of all of the possible enclosures

CAR © 2009

that have an area of at least 1,000 square feet. Label the models with what will be the actual lengths and widths of the enclosures. How does the area of each enclosure, in square feet, relate to the area of each enclosure in fence section by fence section? Use the models you drew to help explain your answer.

Part C. Dana is also considering making the enclosure in the shape of a regular hexagon. Use a ruler or graph paper to sketch a model of a regular hexagon with tick-marks to show how many fence sections would be needed for each side. Include the length of each side, in feet. Then, divide the hexagon into sections so that you can compute its area in square feet. Show how you chose to divide the hexagon and show your work for computing the area. When appropriate, leave side lengths in radical form. For your final answer, round the area to the nearest square foot.

Part D. Dana’s sister suggested she make the enclosure in the shape of a regular

CAR © 2009

octagon. Use a ruler or graph paper to sketch a model of a regular octagon with tick-marks to show how many fence sections would be needed for each side. Include the length of each side, in feet. Then, divide the octagon into sections so that you can compute its area in square feet, and sketch your divisions on your model. Show your work and label the lengths you used in your calculations. When appropriate, leave side lengths in radical form. For your final answer, round the area to the nearest square foot.

Part E. If Dana uses all 24 pieces of fencing as the sides of the enclosure, how could Dana construct the enclosure in order to maximize the area? Describe the configuration and explain your answer.

Student Learning Objective (SLO) Language Objective Language NeededSLO: 6CCSS:G.MG.3

Solve design problems using geometric methods. (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★

Explain how to solve design problems using geometric methods using online math glossary, visuals, models and a partner.

VU: Scaled down, thumbnail

CAR © 2009

WIDA ELDS: 3Listening ReadingWriting

LFC: Mathematical statements, cause/effect



Explain how to solve design problems using geometric methods in L1 and/or with gestures, examples and selected technical words.

Explain how to solve design problems using geometric methods in L1 and/or with selected technical vocabulary in phrases and short sentences.

Explain how to solve design problems using geometric methods using key, technical vocabulary in simple sentences.

Explain how to solve design problems using geometric methods using key, technical vocabulary in expanded complex sentences.

Explain how to solve design problems using geometric methods using technical vocabulary in complex sentences.

Learning Supports

ModelingMath JournalDemonstrationOnline math glossaryVisualsWord/Picture WallL1 text and/or supportPictures / Illustrations/diagrams/ drawings Cloze Sentences

ModelingMath JournalOnline math glossaryVisualsWord/Picture WallL1 text and/or supportSentence Frame

ModelingMath JournalWord WallOnline math glossaryVisualsSentence Starter

ModelingMath Journal

ModelingMath Journal

CAR © 2009






















G.GPE.B.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).

SWBAT use coordinates to prove geometric theorems including:

prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle (or other quadrilateral);

and prove or disprove that a given point lies on a circle of a given center and radius or point on the circle.

Math Journal: How can you tell if the figure formed by four coordinates on a graph create a square or rectangle? List as many different ways as you can.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-4-topic-d-lesson-13

Option 2 – https://njctl.org/courses/math/geometry/quadrilaterals/attachments/quadrilaterals-2/Slides 309-343

Option 3 – https://www.illustrativemathematics.org/HSG-GPE.B

Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-dist-problems/v/area-of-trapezoid-on-coordinate-plane







The vertices of parallelogram PQRS are

and

Part A. PQRS shares a side with squarePSTU. What are the coordinates of T and U? Show your work.

Part B. Prove that PSTU is a square.



topic-d-lesson-13


PMI/NJCTLhttps://njctl.org/courses/math/geometry/quadrilaterals/

attachments/quadrilaterals-2/


analytic-geometry/hs-geo-dist-problems/v/area-of-trapezoid-on-coordinate-plane

Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-GPE.B


G-GPE.4 - Type 2-3 Questions


Geometry Touchpoint - G-GPE.4

Geometry OCR - G-GPE.4


CAR © 2009

https://www.illustrativemathematics.org/HSG-GPE.B


Review Classwork

Exit Ticket

Student Learning Objective (SLO) Language Objective Language NeededSLO: 7CCSS:G.GPE.4 WIDA ELDS: 3ListeningReadingWriting

Use coordinates to prove simple geometric theorems algebraically. Demonstrate comprehension of how to use coordinates to prove simple geometric theorems algebraically by explaining the process using a Teacher Modeling, Charts/Posters and Partner work.

VU: Theorems, rhombus, parallelogram, quadrilateral

LFC: Embedded clauses

LC: Varies by ELP levelELP 1 ELP 2 ELP 3 ELP 4 ELP 5

Language Objectives

Demonstrate comprehension of how to use coordinates to prove simple geometric theorems algebraically by explaining the process in L1 and/or use gestures, examples and selected technical words.

Demonstrate comprehension of how to use coordinates to prove simple geometric theorems algebraically by explaining the process in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension of how to use coordinates to prove simple geometric theorems algebraically by explaining the process using key vocabulary in simple sentences.

Demonstrate comprehension of how to use coordinates to prove simple geometric theorems algebraically by explaining the process using key, technical vocabulary in expanded sentences.

Demonstrate comprehension of how to coordinates to prove simple geometric theorems algebraically by explaining the process using technical vocabulary in complex sentences.

Learning Supports

Teacher ModelingMath JournalDemonstrationCharts/PostersWord/Picture WallL1 text and/or supportPictures /illustrations

Teacher ModelingMath JournalCharts/PostersWord/Picture WallL1 text and/or supportSentence Frame

Teacher ModelingMath JournalCharts/PostersSentence StarterWord Wall

Teacher ModelingMath JournalCharts/Posters

Teacher Modeling

CAR © 2009

http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/GraphicOrganizersWorksheets.pdf

http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/GraphicOrganizersWorksheets/ChartsPosters.pdf






















G.GPE.B.5Prove 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).

SWBAT

prove the slope criteria for parallel lines (parallel lines have equivalent slopes).

prove the slope criteria for perpendicular lines (the product of the slopes of perpendicular lines equals -1).

solve problems using the slope criteria for parallel and perpendicular lines.

Math Journal: How can you tell if two lines are parallel? How can you tell if two lines are perpendicular? How can you tell this without graphing them?

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-4-topic-b-lesson-8

Option 2 – https://njctl.org/courses/math/geometry/analytic-geometry/Slides 73-132

Option 3 – https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5

Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-parallel-perpendicular-lines/v/parallel-and-perpendicular-lines-intro

https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-parallel-perpendicular-eq/v/parallel-lines

Option 5– Pearson Geometry Common Core: 3-8, 7-3, 7-4




Read "Designing a Hotel" and answer the questions.

From her study of architecture, Yasmin knows that diagonal bracing makes buildings more stable. She wants to make the building stronger. She likes the pattern the diagonal supports add to the front of the building, so she strategically places the diagonal supports on the front face of the hotel.

Part A. What is the slope of the support passing through the points labeled A andC? What is the slope of the support passing through the points labeled B and D? Explain and show your work.

Part B. Compare the slopes of these two supports. What does the slope tell you about


topic-b-lesson-8

Pearson Geometry Common Core 3-8, 7-3, 7-4

PMI/NJCTLhttps://njctl.org/courses/math/geometry/analytic-geometry/


analytic-geometry/hs-geo-parallel-perpendicular-lines/v/parallel-and-perpendicular-lines-intro

https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-parallel-perpendicular-eq/v/

parallel-lines

Illustrative Mathematics: https://www.illustrativemathematics.org/content-

standards/HSG/GPE/B/5







CAR © 2009

https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/5


https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-parallel-perpendicular-lines/v/parallel-and-perpendicular-lines-intro






https://njctl.org/courses/math/geometry/analytic-geometry/







Review Classwork

Exit Ticket

the relationship between these two supports? Using slope, explain why this relationship must be true. How would the relationship between the supports change if the location of point B was moved up 1 unit on the grid? Explain.


Student Learning Objective (SLO) Language Objective Language NeededSLO: 5CCSS:G.GPE.5 WIDA ELDS: 3SpeakingWriting

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.)

Explain orally and in writing the proof of the slope criteria for parallel and perpendicular lines and use criteria to solve geometric problems using a Math Journal, Teacher Modeling and a Template.

VU: Complementary, congruent

LFC: Mathematical sentences, cause/effect


Language Objectives

Explain orally and in writing the slope criteria for parallel and perpendicular lines and use criteria to solve geometric problems in L1 and/or use gestures, examples and selected technical words.

Explain orally and in writing the slope criteria for parallel and perpendicular lines and use criteria to solve geometric problems in L1 and/or use selected technical vocabulary in phrases and short sentences.

Explain orally and in writing the slope criteria for parallel and perpendicular lines and use criteria to solve geometric problems using key technical vocabulary in simple sentences.

Explain orally and in writing the slope criteria for parallel and perpendicular lines and use criteria to solve geometric problems using key, technical vocabulary in expanded and some complex sentences.

Explain orally and in writing the slope criteria for parallel and perpendicular lines and use criteria to solve geometric problems using technical vocabulary in complex sentences.

Learning Supports

Teacher ModelingDemonstrationTemplateMath Journal

Teacher ModelingTemplateMath JournalSmall group

Teacher ModelingTemplateMath JournalSmall group

Teacher ModelingSmall group

Teacher Modeling

CAR © 2009






http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/GraphicOrganizersWorksheets/Template.pdf










Small groupWord/Picture WallL1 text and/or supportPictures /illustrations Cloze Sentences

Word/Picture WallL1 text and/or supportSentence Frame

Sentence StarterWord Wall

G.GPE.B.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

SWBAT

locate the point on a directed line segment that creates two segments of a given ratio.

Math Journal: Explain how you could cut a board x units long so that the ratio between the two pieces is 3 to 2.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-4-topic-d-lesson-12

Option 2 – https://njctl.org/courses/math/geometry/analytic-geometry/Slides: 49-72


Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-dividing-segments/v/finding-a-point-part-way-between-two-points

Option 5 – Geometry Textbook: 1-3, 1-7




A highway connecting two cities is represented on the coordinate plane below. The highway has two rest areas along the route such that they divide the distance between the cities into three equal parts.


topic-d-lesson-12

Pearson Geometry Common Core 1-3, 1-7



analytic-geometry/hs-geo-dividing-segments/v/finding-a-point-part-way-between-two-points

Illustrative Mathematics:https://www.illustrativemathematics.org/content-



G-GPE.6 - Type 2-3 Question Bank




CAR © 2009
















Review Classwork

Exit Ticket

Part A. In what ratio does the rest area closest to city A divide the distance from city A to city B?

Part B. What are the coordinates of the point representing the first rest area?

Part C. What are the coordinates of the point representing the second rest area?

Part D. If a service station is built halfway between the rest areas, what are the coordinates of the point representing the service station?


G.GPE.B.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,

Math Journal: Explain how to find the area of a regular pentagon with a length of 12 cm on each side.

Direct Instruction

Teachers will agree on common classwork problems in their professional


topic-c-lesson-9https://www.engageny.org/resource/geometry-module-4-



CAR © 2009

https://www.engageny.org/resource/geometry-module-4-topic-c-lesson-9


using the distance formula. find perimeters of polygons

using coordinates, the Pythagorean theorem and the distance formula.

find areas of triangle and rectangles using coordinates.

Option 1 https://www.engageny.org/resource/geometry-module-4-topic-c-lesson-9https://www.engageny.org/resource/geometry-module-4-topic-c-lesson-10


Option 3 – https://njctl.org/courses/math/geometry/analytic-geometry/

Slides 133-146

Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-coordinate-plane-proofs/v/classfying-a-quadrilateral-on-the-coordinate-plane






learning communities or grade level meetings. Problems should be selected that most closely match the assessments in column 5.

Read “College Apartments” and answer the questions.

Part A. Find the area of the Campus East Apartment complex.

Part B. Find the area of the Campus West Apartment complex. Explain how you divided the complex into shapes you knew how to find the area of.

Part C. The footprint of each of the smaller apartment buildings Nancy is designing is 100 meters by 150 meters. The footprints of Michael’s buildings are both 620 meters long by 120 meters wide. Based on this information and the information in the passage, which apartment complex should have Nancy’s smaller apartments and which apartment complex should have Michael’s apartment buildings? Use calculations and

topic-c-lesson-10



Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-coordinate-plane-proofs/v/classfying-a-quadrilateral-on-the-coordinate-plane

Illustrative Mathematicshttps://www.illustrativemathematics.org/content-





CAR © 2009



https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-coordinate-plane-proofs/v/classfying-a-quadrilateral-on-the-coordinate-plane












Review Classwork

Exit Ticket

information from the passage to defend your answer.


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

Students are able to:

solve problems using volume formulas for cylinders, pyramids, cones, and spheres.

model real-world objects with geometric shapes.

describe the measures and properties of geometric shapes that best represent a real-world object.

Math Journal: Describe at least two ways to find the volume of a right rectangular prism.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-8https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-11

Option 2 –https://njctl.org/courses/math/geometry/3d-geometry/attachments/3d-geometry-3/

Slides: 192-275

Option 3 – https://www.illustrativemathematics.org/content-standards/HSG/GMD/A/3

Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-solids-intro/e/solid_geometry

Option 5 – http://www.shmoop.com/common-core-standards/ccss-hs-g-gmd-3.html


The Creative Cup Company has designed two new glass drinking cups. Design #1 is a hemisphere hollowed out of a cylinder, and design #2 is a cone hollowed out of a cylinder, as shown below.

EngageNYhttps://www.engageny.org/resource/geometry-module-3-topic-b-lesson-8https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-11

Pearson Geometry Common Core 11-4, 11-5, 11-6

PMI/NJCTLhttps://njctl.org/courses/math/geometry/3d-geometry/attachments/3d-geometry-3/


solids/hs-geo-solids-intro/e/solid_geometry


standards/HSG/GMD/A/3


G-GMD.1 - Type 2-3 Question Bank


Geometry Touchpoint - G-GMD.3

Geometry OCR - G-GMD.3

CAR © 2009

https://www.illustrativemathematics.org/content-standards/HSG/GMD/A/3

https://www.illustrativemathematics.org/content-standards/HSG/GMD/A/3





Option 6: Geometry Textbook: 11-4, 11-5, 11-6






Review Classwork

Exit Ticket

Part A. If design #1 has a diameter of 8 cm and a height of 10 cm, determine how much glass is needed to create the cup. Show your work and round your answer to the nearest tenth of a centimeter.

Part B. If design #2 has a radius of 4 cm and a height of 8 cm and the height of the cone is the same as the height of the cylinder, how much glass is needed to create the cup? Show your work and round your answer to the nearest tenth of a centimeter.

CAR © 2009

Part C. A customer is deciding between these two designs and wants to purchase the cup that can hold the most liquid. The customer decides to purchase the cup based on design #1 because it is taller than the cup based on design #2. Did the customer correctly choose the cup that can hold the most liquid? Explain your answer.

Part D. If a cone and a hemisphere have the same radius and the same volume, what is the height of the cone in terms of the radius? Use volume formulas to determine your answer algebraically. Show your work.


Student Learning Objective (SLO) Language Objective Language NeededSLO: 2CCSS:G. GMD.3

Solve problems using volume formulas for cylinders, pyramids, cones, and spheres.★

Demonstrate comprehension of volume formulas for cylinders, pyramids, cones and spheres by solving written problems using a chart, a word wall, and prompts.

VU: Hemisphere, radius, silo, beaker

CAR © 2009

WIDA ELDS: 3ListeningReadingWriting

LFC: Complex, mathematical statements


Language Objectives

Demonstrate comprehension of volume formulas for cylinders, pyramids, cones and spheres by solving written problems in L1 and/or use gestures, examples and selected technical words.

Demonstrate comprehension of volume formulas for cylinders, pyramids, cones and spheres by solving written problems in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension of volume formulas for cylinders, pyramids, cones and spheres by solving written problems using key technical vocabulary in simple sentences.

Demonstrate comprehension of volume formulas for cylinders, pyramids, cones and spheres by solving written problems using key, technical vocabulary in expanded sentences.

Demonstrate comprehension of volume formulas for cylinders, pyramids, cones and spheres by solving written problems using technical vocabulary in complex sentences.

Learning Supports

ChartsPromptsWord/Picture WallL1 text and/or supportCloze Sentences

ChartsPromptsWord/Picture WallL1 text and/or supportSentence Frame

ChartsPromptsSentence StarterWord Wall

ChartsPrompts

Charts

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

SWBAT:

identify cross-sections of three dimensional objects.

identify three-dimensional objects generated by rotation of two-dimensional objects.

Math Journal: A 3-D printer creates an object by layering plastic –essentially by placing layers of two dimensional objects (with minimal thickness) on top of each other to create a 3-D object. Describe the process that a 3-D printer would use to create a solid cube, a solid cone, a hollow cube, and a hollow cone.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-13https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-6

Option 2 – https://www.youtube.com/watch?v=hlD_j3AtxGshttps://www.youtube.com/watch?v=28-MznX-xxU

Option 3 – https://www.illustrativemathematics.org/content-standards/HSG/GMD/B/4

Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-2d-vs-3d/v/vertical-slice-of-rectangular-pyramid

https://www.khanacademy.org/math/geometry/hs-geo-


Jimmy wants to observe the change in the shape of the surface of water in a container as it is being filled with water. He uses the two right cylindrical containers shown below.


topic-b-lesson-13https://www.engageny.org/resource/geometry-module-3-

topic-b-lesson-6


YouTube:https://www.youtube.com/watch?v=hlD_j3AtxGshttps://www.youtube.com/watch?v=28-MznX-xxU

Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-2d-vs-3d/v/vertical-slice-of-rectangular-pyramid

https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-2d-vs-3d/v/rotating-2d-shapes-in-3d


standards/HSG/GMD/B/4




CAR © 2009

https://www.illustrativemathematics.org/content-standards/HSG/GMD/B/4

https://www.illustrativemathematics.org/content-standards/HSG/GMD/B/4










http://www.nj.gov/education/modelcurriculum/ela/ellscaffolding/InteractionSupports/Prompts.pdf
















solids/hs-geo-2d-vs-3d/v/rotating-2d-shapes-in-3d

Option 5:– Geometry Textbook: 11-1, 12-6






Review Classwork

Exit Ticket

Part A. What is the shape of the surface of the water in each container when it is half-filled with water?

Part B. How will the shape of the surface of the water change in each container as he continues to increase the water level?




Student Learning Objective (SLO) Language Objective Language NeededSLO: 3CCSS:G.GMD.4 WIDA ELDS: 3SpeakingReadingWriting

Identify the shape of a two-dimensional cross-section of a three-dimensional figure and identify three-dimensional objects created by the rotation of two-dimensional objects.

Demonstrate comprehension by identifying the shape of a two-dimensional cross section of a three dimensional figure and identify three dimensional objects created by the rotation of two-dimensional objects using demonstrations, diagrams, word wall and linguistic supports.

VU: Cylindrical, two and three dimension, rotation

LFC: Passive voice


CAR © 2009



Demonstrate comprehension by identifying the shape of a two-dimensional cross section of a three dimensional figure and identify three dimensional objects created by the rotation of two-dimensional objects in L1 and/or use drawings, examples and selected technical words.

Demonstrate comprehension by identifying the shape of a two-dimensional cross section of a three dimensional figure and identify three dimensional objects created by the rotation of two-dimensional objects in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension by identifying the shape of a two-dimensional cross section of a three dimensional figure and identify three dimensional objects created by the rotation of two-dimensional objects using key, technical vocabulary in simple sentences.

Demonstrate comprehension by identifying the shape of a two-dimensional cross section of a three dimensional figure and identify three dimensional objects created by the rotation of two-dimensional objects using key, technical vocabulary in expanded sentences.

Demonstrate comprehension by identifying the shape of a two-dimensional cross section of a three dimensional figure and identify three dimensional objects created by the rotation of two-dimensional objects using technical vocabulary in complex sentences.

Learning Supports

DemonstrationPartner workWord/Picture WallL1 text and/or supportPictures / Illustrations/diagrams Cloze Sentences

DemonstrationDiagramsPartner workWord/Picture WallL1 text and/or supportSentence Frame

DemonstrationDiagramsPartner workSentence StarterWord Wall

DemonstrationModelingPartner work

Demonstration

G.GMD.A.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.

SWBAT construct viable

dissection arguments and informal limit arguments.

apply Cavalieri’s principle.

construct an informal argument for the formula for the circumference of a circle.

construct an informal argument for the formula for the area of a circle.

construct an informal argument for the formula for the volume of a cylinder, pyramid, and cone.

Math Journal: Suppose you take a circular pizza (8 slices) and arrange the slices so that the crust alternates from the top to the bottom of the figure. Sketch this figure. If you were to assume that the resulting figure were a rectangle, tell how long each side is. Then explain how to find the area.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-10https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-11https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-12

Option 2 –https://njctl.org/courses/math/geometry/3d-geometry/attachments/3d-geometry-3/

Slides: 276-286

Option 3 – https://www.illustrativemathematics.org/HSG-GMD.A.1


Jenny is given the task of carving out a triangular pyramid and a square pyramid of the same height from two identical wooden blocks. She first needs to determine the dimensions of each pyramid before starting to carve out the blocks.

Part 1She decides that the triangular pyramid will have a right triangle as


topic-b-lesson-11https://www.engageny.org/resource/geometry-module-3-

topic-b-lesson-12

Pearson Geometry Common Core cb 10-7, 11-4

PMI/NJCTLhttps://njctl.org/courses/math/geometry/3d-geometry/

attachments/3d-geometry-3/

Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-GMD.A.1






CAR © 2009

https://www.illustrativemathematics.org/HSG-GMD.A.1































Option 4 – Geometry Textbook: cb 10-7, 11-4






Review Classwork

Exit Ticket

its base with legs of

lengths a and and its height will be h. Use Cavalieri’s principle to determine the formula for the volume of the triangular pyramid.

Part 2She decides to carve out a square pyramid with a base of

side Use Cavalieri’s principle to determine the formula for the volume of the square pyramid.

Part 3What is the relation between a and b?

Part 4If Jenny finally decides to carve out the triangular pyramid such that its longer leg measures 12 inches (in.) and its height is 25 in., what will be the area of the base of the square pyramid? What will be the volume of the square pyramid?


Student Learning Objective (SLO) Language Objective Language Needed

CAR © 2009

SLO: 1CCSS:G.GMD.1WIDA ELDS: 3SpeakingReadingWriting

Develop informal arguments to justify 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).

Explain ,by developing , informal arguments the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone using personal whiteboards, charts, models and a partner.

VU: Square base, pyramid, apex, vertices

LFC: If, then…questions, complex sentences



Explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone using informal arguments in L1 and/or using gestures and selected technical words.

Explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone using informal arguments in L1 and/or selected technical vocabulary in phrases and short sentences.

Explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone using informal arguments with technical vocabulary in simple sentences.

Explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone using informal arguments with key, technical vocabulary in expanded sentences.

Explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid and cone using informal arguments with technical vocabulary in complex sentences.

Learning Supports

ModelingWhite BoardChartsMath JournalPartner workWord/Picture WallL1 text and/or supportPictures / Illustrations/diagrams/drawings

White BoardChartsModelingMath JournalPartner workWord/Picture WallL1 text and/or support

White BoardChartsModelingMath JournalPartner work

White BoardMath JournalPartner work

White BoardMath Journal

G.GPE.A.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.

SWBAT given the center and radius, derive the equation of a circle (using the Pythagorean Theorem).

SWBAT given an equation of a circle in any form, use the method of completing the square to determine the center and radius of the circle.

Math Journal: Explain how to solve the following problems by completing the square:x2-2x-15=0 and x2+2x=35

Direct Instruction Option 1: https://www.engageny.org/resource/geometry-module-5-topic-d-lesson-17https://www.engageny.org/resource/geometry-module-5-topic-d-lesson-18

Option 2 – https://www.illustrativemathematics.org/HSG-GPE.A

Option 3 – https://njctl.org/courses/math/geometry/analytic-geometry/


Come Full Circle

In this task, you will be deriving equations for circles that are graphed on the coordinate plane.


topic-d-lesson-17https://www.engageny.org/resource/geometry-module-5-

topic-d-lesson-18



Khanacademy.orghttps://www.khanacademy.org/math/algebra2/intro-to-

conics-alg2/expanded-equation-circle-alg2/v/completing-the-square-to-write-equation-in-standard-form-of-a-circle

Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-GPE.A

Shmoop.comhttp://www.shmoop.com/common-core-standards/ccss-hs-




CAR © 2009

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

https://www.illustrativemathematics.org/HSG-GPE.A

https://www.khanacademy.org/math/algebra2/intro-to-conics-alg2/expanded-equation-circle-alg2/v/completing-the-square-to-write-equation-in-standard-form-of-a-circle



https://www.engageny.org/resource/geometry-module-5-topic-d-lesson-17







http://www.nj.gov/education/modelcurriculum/math/ellscaffolding/Whiteboards.pdf

























Slides 147-191

Option 4 – https://www.khanacademy.org/math/algebra2/intro-to-conics-alg2/expanded-equation-circle-alg2/v/completing-the-square-to-write-equation-in-standard-form-of-a-circle

Option 5: http://www.shmoop.com/common-core-standards/ccss-hs-g-gpe-1.html

Option 6: Geometry Textbook: 12-5






Review Classwork

Exit Ticket

Part A.

Using the

Pythagorean

Theorem, write an

equation that

gives the set of all

points at

a distance r units

from the origin.

Make a sketch of

the figure on the

coordinate plane

below that

satisfies this

equation.

In your sketch

above, label

g-gpe-1.html



CAR © 2009

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

the distances you

used in your

equation.

Explain your

answer.

Part B. Often circles are not centered at the origin. The center of the circle shown below is located at the

point

In the diagram

above, sketch a

right triangle that

can be used to

determine the

equation for the

CAR © 2009

circle with a

center

at

Find and label the

lengths of the legs

of the right

triangle in terms

of the values

shown in the

figure.

Use the

Pythagorean

Theorem to write

the equation of

this circle.

Part C. The center of the circle shown below is located at the

point

CAR © 2009

What is the

equation of this

circle with

center

and radius r? Does the equation

of a circle with

center

change based on

which quadrant

the center of the

circle is in?

Explain your

answer.

Part D. The equation below represents a circle. Complete the square to rewrite this equation in the form

CAR © 2009

that you derived in part C.

What are the

coordinates of the

center of the

circle?

What is the length

of the radius?

Sketch a graph

of the circle on the

coordinate grid

provided below.

CAR © 2009

Part E. Consider the general form for the equation of a circle, shown below. Complete the square to rewrite this equation in the form that you derived in part C.

Find the center

and the radius in

terms of C, D,

and E.

Part F. Look again at the equation from part D, shown again below.

Identify the values

of C, D, and E in

the general form

for the equation of

a circle.

Use these values

in the expressions

for the center and

the radius that

you found in part

CAR © 2009

E. Show your

work.

Did you find the

same center and

radius that you

found in part D?

Student Learning Objective (SLO) Language Objective Language NeededSLO: 4CCSS:G.GPE.1WIDA ELDS: 3ReadingSpeakingWriting

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.

Demonstrate comprehension by deriving the equation of a circle of a given center and radius using the Pythagorean Theorem, and finding the center and radius of a circle by completing the square given by an equation using Teacher Modeling, a Word Wall and Partner work.

VU: Distance formula, standard form

LFC: Explanations, passive voice



Demonstrate comprehension of the Pythagorean Theorem, by deriving and explaining the equation of a circle of given center and radius in L1 and/or use gestures, examples and selected technical words.

Demonstrate comprehension of the Pythagorean Theorem, by deriving and explaining the equation of a circle of given center and radius in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension of the Pythagorean Theorem, by deriving and explaining the equation of a circle of given center and radius using key, technical vocabulary in simple sentences.

Demonstrate comprehension of the Pythagorean Theorem, by deriving and explaining the equation of a circle of given center and radius the using key technical vocabulary in expanded sentences.

Demonstrate comprehension of the Pythagorean Theorem, by deriving and explaining the equation of a circle of given center and radius using technical vocabulary in complex sentences.

Learning Supports

Teacher ModelingDemonstrationPartner workWord/Picture WallL1 text and/or supportPictures /illustrations Cloze Sentences

Teacher ModelingPartner workWord/Picture WallL1 text and/or supportSentence Frame

Teacher ModelingPartner workSentence StarterWord Wall

Teacher ModelingPartner work


G.C.A.1 Prove that all circles are similar.

Math Journal: Which statement best explains why Teachers will agree on common



CAR © 2009























SWBAT construct a formal proof of the similarity of all circles.

all circles are similar?

A All circles have exactly one center point.

BThe diameter of all circles is twice the length of the radius.

CAll circles can be mapped onto any other circle using only translations.

DAll circles can be mapped onto any other circle using a translation and dilation.

Direct Instruction Option 1 : https://www.engageny.org/resource/geometry-module-2-topic-c-lesson-14 -- examples 1-3

Option 2 – https://www.illustrativemathematics.org/content-standards/HSG/C/A/1

Option 3 – https://www.khanacademy.org/math/geometry-home/cc-geometry-circles/circle-basics/v/seeing-circle-similarity-through-translation-and-dilation



Standards Based Problems Center – Students work in a group to solve the style of problems the assessments (Quarterly) will use to measure that standard. [Engage NY has excellent problems for

classwork problems in their professional learning communities or grade level meetings. Problems should be selected that most closely match the assessments in column 5.

Similarity in Circles

Geometric similarity is an extremely useful concept. Similar figures are alike except for their size; their corresponding angles are congruent, and their corresponding parts are proportional. On the coordinate plane, one figure can be mapped to the other by a series of transformations.

Part A. Consider these two equilateral triangles. Are they similar? How do you know? Write a proportion showing the relationship of their sides.

topic-c-lesson-14


Khanacademy.orghttps://www.khanacademy.org/math/geometry-home/cc-geometry-circles/circle-basics/v/seeing-circle-similarity-

through-translation-and-dilation


standards/HSG/C/A/1


G-C.1- Type 2-3 Question Bank

Geometry Touchpoint - G.C.1

Geometry OCR - G.C.1

CAR © 2009

https://www.illustrativemathematics.org/content-standards/HSG/C/A/1


https://www.khanacademy.org/math/geometry-home/cc-geometry-circles/circle-basics/v/seeing-circle-similarity-through-translation-and-dilation






this… as well as EdConnect Type 2 and Type 3 problems




Review Classwork

Exit Ticket

Part B. Are any two squares similar? Tell how you know.

Remember that the measure of each angle of a regular polygon

is where nis the number of sides. Can you make a general statement about the similarity of two regular polygons (n-gons) with the same number of sides? Explain your answer.

Part C. As the number of sides of a regular polygon increases, what figure does it begin to look like?

CAR © 2009

What is a reasonable conclusion about the similarity of figures of this kind of different sizes?

Part D. Consider these two circles on the coordinate plane. What is the radius of circle A? Of circle B? Write the ratio. Write the ratios for the diameters and circumferences of the two circles. Are the circles proportional?

CAR © 2009

Part E. You can also prove that two figures are similar by showing that a series of transformations will map one figure to the other. What is the equation for circle A?

Part F. What series of transformations will

CAR © 2009

map circle A to circle B? Write the equation for circle B. Are these two circles similar?

Part G. Any two circles can be centered at the origin through translations. If both circles are centered at the origin, what one transformation will map one to the other, proving their similarity? If the equation of one circle

is and the radius of the other circle is f times the radius of the first, what is the equation of the second circle?

What is the equation of the second circle if the center is NOT

In either case, no matter what the size or position of the circles, are all circles similar?

SLO: 1CCSS:G.C.1WIDA ELDS: 3SpeakingWriting

Generate proofs that demonstrate that all circles are similar. Sequence the steps and explain proofs that demonstrate that all circles are similar orally and in writing using a Teacher Modeling, a word wall, and Partner work.

VU: Proof, radius/radii, translate, dilation, congruent

LFC: Embedded clauses

CAR © 2009





Sequence the steps and explain proofs that demonstrate that all circles are similar orally and in writing in L1 and/or use gestures, Illustrations/diagrams/drawingsand selected technical words.

Sequence the steps and explain proofs that demonstrate that all circles are similar orally and in writing in L1 and/or use selected technical vocabulary in phrases and short sentences.

Sequence the steps and explain proofs that demonstrate that all circles are similar orally and in writing using key, technical vocabulary in simple sentences.

Sequence the steps and explain proofs that demonstrate that all circles are similar orally and in writing using key, technical vocabulary in expanded sentences.

Sequence the steps and explain proofs that demonstrate that all circles are similar orally and in writing using technical vocabulary in complex sentences.

Learning Supports

Teacher ModelingMath Journal/dictionaryPartner workWord/Picture WallL1 text and/or supportIllustrations Cloze sentences

Teacher ModelingMath Journal/dictionaryPartner workWord/Picture WallL1 text and/or supportSentence frames

Teacher ModelingMath Journal/dictionaryPartner workSentence StarterWord wall


Teacher Modeling

G.C.A.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.

SWBAT use the relationship between inscribed angles, radii and chords to solve problems.

SWBAT use the relationship between central, inscribed, and circumscribed angles to solve problems.SWBAT identify inscribed angles on a diameter as right angles.

SWBAT identify the radius of a circle as perpendicular to the tangent where the radius intersects the circle.

Math Journal: Have students define the following terms: central angle, inscribed angle, minor arc, major arc, and intercepted arc of an angle. Draw and label a circle with each object.

Direct Instruction Option 1: https://www.engageny.org/resource/geometry-module-5-topic-a-lesson-4


Option 3 – https://njctl.org/courses/math/geometry/circles/attachments/circles-3/

Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circle-basics/v/language-and-notation-of-the-circle

Option 5 - Geometry Textbook: 10-6, CB 10-6, 12-2, 12-3


A city planner is designing a new park, as shown in the figure below. The circular park will have a fountain located at the center, represented by the black dot in the figure below. There will also be five different walking paths within the park, represented by the line segments shown in the figure below. One of these paths will form the diameter of the circle, which is 100 meters long. The other paths are


topic-a-lesson-4

Pearson Geometry Common Core 10-6, CB 10-6, 12-2, 12-3

PMI/NJCTLhttps://njctl.org/courses/math/geometry/circles/attachments/circles-3/

Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circle-basics/v/language-and-notation-of-

the-circle


standards/HSG/C/A/2


G-C.2 - Type 2-3 Question Bank




CAR © 2009



https://www.engageny.org/resource/geometry-module-5-topic-a-lesson-4

https://www.engageny.org/resource/geometry-module-5-topic-a-lesson-4


























Review Classwork

Exit Ticket

labeled a, b, c, and d.

Part A. Based on the location of paths aand b within the park, at what angle must those two paths intersect? Explain your reasoning.

Part B. If path d is 40 meters in length and perpendicular to path c, what is the length of path c? Show your work.

Part C. The architect wants to design path a

so that its endpoints intercept an arc on the circle that is 116°. What is the measure of the angle formed by the diameter and path b? Explain your answer.

Part D. What is the measure of the arc intercepted by the endpoints of segment b? Explain

CAR © 2009

your reasoning.


Student Learning Objective (SLO) Language Objective Language NeededSLO: 1CCSS:G.C.2.6WIDA ELDS: 3ListeningReadingWriting

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.

After listening to an oral explanation and reading the directions, demonstrate comprehension by identifying and describing relationships among inscribed angles, radii and chords using White Boards, Charts/Posters, Teacher Modeling and Partner work.

VU: Inscribed angles, circumscribed, tangent, radians

LFC: Wh-questions



After listening to oral explanation and reading the directions, demonstrate comprehension by identifying and describing relationships among inscribed angles, radii and chords in L1 and/or using gestures and selected technical words.

After listening to oral explanation and reading the directions, demonstrate comprehension by identifying and describing relationships among inscribed angles, radii and chords using L1 and/or selected technical vocabulary in phrases and short sentences.

After listening to oral explanation and reading the directions, demonstrate comprehension by identifying and describing relationships among inscribed angles, radii and chords using technical vocabulary in simple sentences.

After listening to oral explanation and reading the directions, demonstrate comprehension by identifying and describing relationships among inscribed angles, radii and chords using key, technical vocabulary in expanded sentences.

After listening to oral explanation and reading the directions demonstrate comprehension by identifying and describing relationships among inscribed angles, radii and chords using technical vocabulary in complex sentences.

Learning Supports

Teacher ModelingWhite BoardCharts/PostersMath Journal/dictionaryDemonstrationPartner workWord/Picture WallL1 text and/or supportPictures /illustrations

White BoardCharts/PostersTeacher ModelingMath Journal/dictionaryPartner workWord/Picture WallL1 text and/or support

White BoardCharts/PostersTeacher ModelingMath Journal/dictionaryPartner work

White BoardMath journal/dictionary Partner work

White BoardMath journal/dictionary

G.C.A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties

Math Journal: List all radii shown in Circle C below.

Teachers will agree on common classwork problems in their professional


topic-a-lesson-4



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of angles for a quadrilateral inscribed in a circle.

SWBAT construct the inscribed circle of a triangle.

SWBAT construct the circumscribed circle of a triangle.

SWBAT prove properties of the angles of a quadrilateral that is inscribed in a circle.

Define inscribed and circumscribed circles of a triangle. Identify any that are drawn in the above figure. IF there aren’t any, draw them in.

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-5-topic-a-lesson-4



Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-inscribed-angles/v/inscribed-angles-exercise-example

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circum-in-circles/v/constructing-circle-inscribing-triangle

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circum-in-circles/v/constructing-circumscribing-circle


CentersTeacher Center – The teacher works in a small group with 1-4 students. [The engage NY lessons are written with teacher guidance and should also be

learning communities or grade level meetings. Problems should be selected that most closely match the assessments in column 5.

Amy is designing a piece of jewelry to sell in her craft store. She begins with the triangular piece of silver, as shown below.

Part A. Amy wants to add a circular piece of gold that will be inscribed inside the triangular piece of silver. Use a compass and straightedge to show how she can add the circular piece to the triangle above. Explain the steps you used to perform the construction.

Part B. She needs to know the radius of the inscribed circle so that


PMI/NJCTLhttps://njctl.org/courses/math/geometry/circles/

attachments/circles-3/https://njctl.org/courses/math/geometry/circles/attachments/circles-3/

Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-inscribed-angles/v/inscribed-angles-exercise-example


https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circum-in-circles/v/constructing-circumscribing-circle


standards/HSG/C/A/3




CAR © 2009






https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-inscribed-angles/v/inscribed-angles-exercise-example









implemented in the teacher center]





Review Classwork

Exit Ticket

she can calculate the circumference and area of the circular gold piece she needs to make for the jewelry. Given that the silver triangle is a right triangle with side lengths a, b, and c, find the equation Amy can use to determine the radius of the circle, r. Explain your answer and draw a diagram or use your construction in part A to support your reasoning.

Part C. Amy then decides to inscribe another similar silver triangle inside a circular piece of copper so that each vertex of the triangle touches the edge of the copper circle. Use a compass and straightedge to construct her design below. Explain the steps you used to perform the construction.

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Part D. A couple of months ago, Amy designed a piece of jewelry with a gold quadrilateral inscribed on a circular piece of silver. She found the sketch of her design in her desk drawer, as shown below.

Now Amy wants to produce an identical piece of jewelry but needs to know the exact angle measures for the gold quadrilateral. What geometric property about quadrilaterals can Amy use to find the measures of the angles

CAR © 2009

of her jewelry design? Use a paragraph proof to justify your response.

Part E. What are the measures of the three missing angles in Amy’s sketch of the piece of jewelry in part D? Explain how you know.

Student Learning Objective (SLO) Language Objective Language NeededSLO: 2CCSS:G.C.3, G.C.4 WIDA ELDS: 3ListeningReadingWriting

Prove the properties of angles for a quadrilateral inscribed in a circle and construct inscribed and circumscribed circles of a triangle, and a tangent line to a circle from a point outside a circle, using geometric tools and geometric software.

Demonstrate comprehension by proving the properties of angles for a quadrilateral inscribed in a circle and constructing inscribed and circumscribed circles of a triangle and a tangent line to a circle from a point outside the circle using a Charts/Posters, a Word Wall, drawings and Prompts.

VU: Inscribed, circumscribed, tangent

LFC: Mathematical statements



Demonstrate comprehension by proving the properties of angles for a quadrilateral inscribed in a circle and constructing inscribed and circumscribed circles of a triangle in L1 and/or use gestures, examples and selected technical words.

Demonstrate comprehension by proving the properties of angles for a quadrilateral inscribed in a circle and constructing inscribed and circumscribed circles of a triangle in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension by proving the properties of angles for a quadrilateral inscribed in a circle and constructing inscribed and circumscribed circles of a triangle using key technical vocabulary in simple sentences.

Demonstrate comprehension by proving the properties of angles for a quadrilateral inscribed in a circle and constructing inscribed and circumscribed circles of a triangle using key, technical vocabulary in expanded sentences.

Demonstrate comprehension by proving the properties of angles for a quadrilateral inscribed in a circle and constructing inscribed and circumscribed circles of a triangle using technical vocabulary in complex sentences.

Learning Supports

Charts/PostersStudent-generated dictionaryPartially completed proof

Charts/PostersStudent-created dictionaryPartially completed proof

Charts/PostersPromptsSentence Starter

Charts/Posters Charts/Posters

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PromptsWord/Picture WallL1 text and/or supportCloze Sentences

PromptsWord/Picture WallL1 text and/or supportSentence Frame

Word Wall

G.C.B.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.

SWBAT use similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the radius.

SWBAT define radian measure of an angle as the constant of proportionality when the length of the arc intercepted by an angle is proportional to the radius.

SWBAT derive the formula for the area of a sector.

SWBAT compute arc lengths and areas of sectors of circles.

Math Journal: See opening exercise: https://www.engageny.org/resource/geometry-module-5-topic-c-lesson-15

Direct Instruction Option 1 https://www.engageny.org/resource/geometry-module-5-topic-c-lesson-15



Option 4 – https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-measures/v/intro-arc-measure

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-length-deg/v/length-of-an-arc-that-subtends-a-central-angle




Jacob is working on a design for a company's logo. The logo is shown below. The radius of the small circle is 1 inch and the radius of the large circle is 7.5 inches.

The measure of the arc of the unit circle cut by the angle is 0.785 inches. We define this to be the radian


topic-c-lesson-15


PMI/NJCTLhttps://njctl.org/courses/math/geometry/circles/attachments/circles-3/

Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-measures/v/intro-arc-measure

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-length-deg/v/length-of-an-arc-that-subtends-a-central-angle


standards/HSG/C/A/3




Geometry Touchpoint - G.C.B.5

Geometry OCR - G.C.B.5

CAR © 2009



https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-measures/v/intro-arc-measure



















Review Classwork

Exit Ticket

measure of the angle.

Part A

Set up a proportion to determine the area of a sector for any circle.

Let represent the radian measure of the central angle and A represent the area of the sector SPR.

Part B

Solve the proportion

for

Part C

Looking at the logo, show or explain how you could find the area of the unshaded part of the sector created by the central angle.

Part D

Find the area, in square inches, of the unshaded part of the sector

CAR © 2009

created by the central angle. Round your answer to the nearest tenth of a square inch.

Student Learning Objective (SLO) Language Objective Language NeededSLO: 3CCSS:G.C.5 WIDA ELDS: 3SpeakingReadingWriting

Use similarity to show 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.

Demonstrate comprehension by using similarity to show and explain that the length of the arc intercepted by an angle is proportional to the radius and define radian measure of the angle as the constant of proportionality using Teacher Modeling(The word through is misspelled in the description), diagrams, Word Wall and Multilingual Math Glossary.

VU: Proportional, arc, intercept, radian measure

LFC: Cause/effect statements



Demonstrate comprehension by using similarity to show and explain that the length of the arc intercepted by an angle is proportional to the radius in L1 and/or use drawings, examples and selected technical words.

Demonstrate comprehension by using similarity to show and explain that the length of the arc intercepted by an angle is proportional to the radius in L1 and/or use selected technical vocabulary in phrases and short sentences.

Demonstrate comprehension by using similarity to show and explain that the length of the arc intercepted by an angle is proportional to the radius using key, technical vocabulary in simple sentences.

Demonstrate comprehension by using similarity to show and explain that the length of the arc intercepted by an angle is proportional to the radius using key, technical vocabulary in expanded and some complex sentences.

Demonstrate comprehension by using similarity to show and explain that the length of the arc intercepted by an angle is proportional to the radius using technical vocabulary in complex sentences.

Learning Supports

Multilingual Math GlossaryTeacher ModelingPartner workWord/Picture WallL1 text and/or supportPictures /illustrations Cloze Sentences

Multilingual Math GlossaryTeacher ModelingPartner workWord/Picture WallL1 text and/or supportSentence Frame

Multilingual Math GlossaryTeacher ModelingPartner workSentence StarterWord Wall

Multilingual Math GlossaryTeacher ModelingPartner work

Multilingual Math Glossary

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http://www.glencoe.com/sec/math/mlg/
























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