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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
https://www.engageny.org/resource/geometry-module-3-topic-b-lesson-7
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
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
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
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]
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.
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.
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
EngageNYhttps://www.engageny.org/resource/geometry-module-3-
topic-b-lesson-8
Pearson Geometry Common Core 11-7
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
Type 2-3 Question Bank
G-MG.2 - Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G.MG.2
Geometry OCR - G.MG.2
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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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
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
Option 3 – Geometry Textbook: 3-4
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]
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
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.
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
Pearson Geometry Common Core 3-4
Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-MG.A
Type 2-3 Question Bank
G-MG.1- Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G.MG.3
Geometry OCR - G.MG.3
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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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
Option 5 – Geometry Textbook: 6-9
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]
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.
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.
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.
Use words, numbers, and/or pictures to show your work.
EngageNYhttps://www.engageny.org/resource/geometry-module-4-
topic-d-lesson-13
Pearson Geometry Common Core 6-9
PMI/NJCTLhttps://njctl.org/courses/math/geometry/quadrilaterals/
attachments/quadrilaterals-2/
Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-
analytic-geometry/hs-geo-dist-problems/v/area-of-trapezoid-on-coordinate-plane
Illustrative Mathematicshttps://www.illustrativemathematics.org/HSG-GPE.B
Type 2-3 Question Bank
G-GPE.4 - Type 2-3 Questions
Quarterly Assessment
Geometry Touchpoint - G-GPE.4
Geometry OCR - G-GPE.4
G-GPE.4 - Type 2-3 Questions
CAR © 2009
Interdisciplinary Center – Students work on transformation problems relating to video game graphics and programming.
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
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
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]
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
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.
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
EngageNYhttps://www.engageny.org/resource/geometry-module-4-
topic-b-lesson-8
Pearson Geometry Common Core 3-8, 7-3, 7-4
PMI/NJCTLhttps://njctl.org/courses/math/geometry/analytic-geometry/
Khanacademy.orghttps://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
Illustrative Mathematics: https://www.illustrativemathematics.org/content-
standards/HSG/GPE/B/5
Type 2-3 Question Bank
G-GPE.5 - Type 2-3 Questions
Quarterly Assessment
Geometry Touchpoint - G-GPE.5
Geometry OCR - G-GPE.5
G-GPE.5 - Type 2-3 Questions
CAR © 2009
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
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.
Use words, numbers, and/or pictures to show your work.
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
LC: Varies by ELP levelELP 1 ELP 2 ELP 3 ELP 4 ELP 5
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
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 3 – https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/6
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
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]
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
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.
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.
EngageNYhttps://www.engageny.org/resource/geometry-module-4-
topic-d-lesson-12
Pearson Geometry Common Core 1-3, 1-7
PMI/NJCTLhttps://njctl.org/courses/math/geometry/analytic-geometry/
Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-
analytic-geometry/hs-geo-dividing-segments/v/finding-a-point-part-way-between-two-points
Illustrative Mathematics:https://www.illustrativemathematics.org/content-
standards/HSG/GPE/B/6
Type 2-3 Question Bank
G-GPE.6 - Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G-GPE.6
Geometry OCR - G-GPE.6
CAR © 2009
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 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?
Use words, numbers, and/or pictures to show your work.
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
EngageNYhttps://www.engageny.org/resource/geometry-module-4-
topic-c-lesson-9https://www.engageny.org/resource/geometry-module-4-
Quarterly Assessment
Geometry Touchpoint - G-GPE.7
CAR © 2009
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 2 – https://www.illustrativemathematics.org/content-standards/HSG/GPE/B/7
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
Option 5 – Geometry Textbook: 6-7, 10-1
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]
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.
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
Pearson Geometry Common Core 6-7, 10-1
PMI/NJCTLhttps://njctl.org/courses/math/geometry/analytic-geometry/
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-
standards/HSG/GPE/B/7
Type 2-3 Question Bank
G-GPE.7 - Type 2-3 Question Bank
Geometry OCR - G-GPE.7
CAR © 2009
Interdisciplinary Center – Students work on transformation problems relating to video game graphics and programming.
Review Classwork
Exit Ticket
information from the passage to defend your answer.
Use words, numbers, and/or pictures to show your work.
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
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.
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/
Khanacademy.orghttps://www.khanacademy.org/math/geometry/hs-geo-
solids/hs-geo-solids-intro/e/solid_geometry
Illustrative Mathematicshttps://www.illustrativemathematics.org/content-
standards/HSG/GMD/A/3
Type 2-3 Question Bank
G-GMD.1 - Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G-GMD.3
Geometry OCR - G-GMD.3
CAR © 2009
Option 6: Geometry Textbook: 11-4, 11-5, 11-6
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]
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 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.
Use words, numbers, and/or pictures to 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
LC: Varies by ELP levelELP 1 ELP 2 ELP 3 ELP 4 ELP 5
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-
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.
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.
EngageNYhttps://www.engageny.org/resource/geometry-module-3-
topic-b-lesson-13https://www.engageny.org/resource/geometry-module-3-
topic-b-lesson-6
Pearson Geometry Common Core 11-1, 12-6
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
Illustrative Mathematicshttps://www.illustrativemathematics.org/content-
standards/HSG/GMD/B/4
Quarterly Assessment
Geometry Touchpoint - G-GMD.4
Geometry OCR - G-GMD.4
CAR © 2009
solids/hs-geo-2d-vs-3d/v/rotating-2d-shapes-in-3d
Option 5:– Geometry Textbook: 11-1, 12-6
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]
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 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?
Use words, numbers, and/or pictures to show your work.
Type 2-3 Question Bank
G-GMD.1 - Type 2-3 Question Bank
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
LC: Varies by ELP level
CAR © 2009
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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
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.
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
EngageNYhttps://www.engageny.org/resource/geometry-module-3-
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
Type 2-3 Question Bank
G-GMD.1 - Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G-GMD.1
Geometry OCR - G-GMD.1
CAR © 2009
Option 4 – Geometry Textbook: cb 10-7, 11-4
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]
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
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?
Use words, numbers, and/or pictures to show your work.
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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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/
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.
Come Full Circle
In this task, you will be deriving equations for circles that are graphed on the coordinate plane.
EngageNYhttps://www.engageny.org/resource/geometry-module-5-
topic-d-lesson-17https://www.engageny.org/resource/geometry-module-5-
topic-d-lesson-18
Pearson Geometry Common Core 12-5
PMI/NJCTLhttps://njctl.org/courses/math/geometry/analytic-geometry/
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-
Quarterly Assessment
Geometry Touchpoint - G-GPE.1
Geometry OCR - G-GPE.1
CAR © 2009
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
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]
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 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
Type 2-3 Question Bank
G-GPE.1 - Type 2-3 Question Bank
CAR © 2009
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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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
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
EngageNYhttps://www.engageny.org/resource/geometry-module-2-
Quarterly Assessment
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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
Option 4 – Geometry Textbook: 10-6
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]
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
Pearson Geometry Common Core 10-6
Khanacademy.orghttps://www.khanacademy.org/math/geometry-home/cc-geometry-circles/circle-basics/v/seeing-circle-similarity-
through-translation-and-dilation
Illustrative Mathematicshttps://www.illustrativemathematics.org/content-
standards/HSG/C/A/1
Type 2-3 Question Bank
G-C.1- Type 2-3 Question Bank
Geometry Touchpoint - G.C.1
Geometry OCR - G.C.1
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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. 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?
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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?
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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
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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
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LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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 ModelingPartner work
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 2 – https://www.illustrativemathematics.org/content-standards/HSG/C/A/2
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
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.
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
EngageNYhttps://www.engageny.org/resource/geometry-module-5-
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
Illustrative Mathematicshttps://www.illustrativemathematics.org/content-
standards/HSG/C/A/2
Type 2-3 Question Bank
G-C.2 - Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G.C.2
Geometry OCR - G.C.2
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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]
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
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
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your reasoning.
Use words, numbers, and/or pictures to show your work.
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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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
EngageNYhttps://www.engageny.org/resource/geometry-module-5-
topic-a-lesson-4
Quarterly Assessment
Geometry Touchpoint - G.C.3
CAR © 2009
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 2 – https://www.illustrativemathematics.org/content-standards/HSG/C/A/3
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-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
Option 5 – Geometry Textbook: 5-3, 12-3
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
Pearson Geometry Common Core 5-3, 12-3
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-circle-inscribing-triangle
https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circum-in-circles/v/constructing-circumscribing-circle
Illustrative Mathematicshttps://www.illustrativemathematics.org/content-
standards/HSG/C/A/3
Type 2-3 Question Bank
G-C.3 - Type 2-3 Question Bank
Geometry OCR - G.C.3
CAR © 2009
implemented in the teacher center]
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
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
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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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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 2 – https://www.illustrativemathematics.org/content-standards/HSG/C/A/3
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-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
Option 5 – Geometry Textbook: 10-6, 10-7
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.
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
EngageNYhttps://www.engageny.org/resource/geometry-module-5-
topic-c-lesson-15
Pearson Geometry Common Core 10-6, 10-7
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
Illustrative Mathematicshttps://www.illustrativemathematics.org/content-
standards/HSG/C/A/3
Type 2-3 Question Bank
G-C.5 - Type 2-3 Question Bank
Quarterly Assessment
Geometry Touchpoint - G.C.B.5
Geometry OCR - G.C.B.5
CAR © 2009
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
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
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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
LC: Varies by ELP level
ELP 1 ELP 2 ELP 3 ELP 4 ELP 5Language Objectives
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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