Download Geometry - Grade 10 - Rahway Public Schools

CURRICULUM
FOR
GEOMETRY
GRADE 10
This curriculum is part of the Educational Program of Studies of the Rahway Public Schools.
ACKNOWLEDGMENTS
Christine H. Salcito, Director of Curriculum and Instruction
Kevin Robinson, Program Supervisor of STEM
The Board acknowledges the following who contributed to the preparation of this curriculum.
Rachael Gulley
Subject/Course Title:
Geometry
Grade 10
Date of Board Adoptions:
August 30, 2011
RAHWAY PUBLIC SCHOOLS CURRICULUM
UNIT ONE - Congruence, Proofs (*with special emphasis on coordinate proofs) and Constructions
Content Area: Geometry
Unit Title: Congruence, Proofs (*with special emphasis on coordinate proofs) and Constructions
Target Course/Grade Level: Geometry/Grade 10
Unit Summary:
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Experiment with transformations in the plane.
Understand congruence in terms of rigid motions.
Prove geometric theorems.
Prove or disprove whether the figure is a certain type of quadrilateral or triangle*
Make geometric constructions.
Approximate Length of Unit:
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12 weeks
Primary interdisciplinary connections:
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Art, Social Studies, Language Arts, and Science
LEARNING TARGETS
Content Area Domain
Content Area Cluster
Standard
Congruence
Experiment with transformations in the plane
G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-CO.5
Congruence
Understand congruence in terms of rigid motion
G-CO.6, G-CO.7, G-CO.8
Congruence
Prove geometric theorems
G-CO.9, G-CO.10, G-CO.11
Congruence
Make geometric constructions
G-CO.12, G-CO.13
Expressing Geometric Properties With Equations
Use coordinates to prove simple geometric theorems algebraically
G-GPE.4, G-GPE.5
Unit Understandings
Students will understand that…
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Geometric properties can be used to construct geometric figures.
Geometric relationships provide a means to make sense of a variety of phenomena.
Shape and area can be conserved during mathematical transformations.
Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry.
Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Unit Essential Questions
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Describe the result of applying each rule to a figure in the coordinate plane:
o A(x,y) = ( x-6 , y+7 )
o B(x,y) = ( x , y – 14 )
o C(x,y) = ( x + 5 , y)
o D(x,y) = ( x , - y )
o E(x,y) = ( - x , y )
o F(x,y) = ( - x , - y )
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Write the transformation that would translate a figure 5 unit to the left and 12 units down
Given four coordinates, prove mathematically which type of quadrilateral is formed.
In detail, describe the symmetries found in a piece of artwork
Determine whether two given triangles are congruent and state which postulate justifies your answer.
Identify the properties of quadrilaterals and the relationships among the properties.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Knowledge and Skills
Students will know…
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Vocabulary –
o Point, line, ray, segment, plane, angle, diagonal, endpoint, polygon, center of a polygon, central angle, median, circle, arc
o transformation, translation, reflection, axis of symmetry, rotation
o congruent, congruence, corresponding parts, equilateral, equiangular, regular polygon, equidistant
o hypothesis, conclusion, conditional, converse, counterexample, biconditional, logical chain, proof, theorem
o inductive reasoning, deductive reasoning, conjecture, theorem, postulate, proof, two-column proof, paragraph proof
o vertical angles, adjacent angles, consecutive angles, complementary, supplementary, linear pair
o right angel, acute angle, obtuse angle
o transversal, alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles
o parallel, perpendicular, bisect, perpendicular bisector
o inscribed, circumscribed
o CPCTC- Corresponding Parts of Congruent Triangles are Congruent
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Formulas –
o distance, midpoint, slope
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Segment Addition and Angle Addition postulates
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Overlapping Segments and Overlapping Angles Theorems
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Linear Pair property
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Vertical Angles Theorem
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Transitive, Reflexive and Symmetric Properties
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Corresponding Angles Postulate and it’s converse
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Alternate Interior, Alternate Exterior, Same-Side Interior Theorems and their converses
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Triangle Sum Theorem
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Sum of Interior Angles of a Polygon
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The Measure of an Interior Angle of a regular polygon
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Sum of Exterior Angle of a Polygon
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the slopes of parallel and perpendicular lines are opposite reciprocals
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the criteria for triangle congruence (ASA, SAS, SSS, and the special case of ASS (HL))
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CPCTC- Corresponding Parts of Congruent Triangles are Congruent
Students will be able to…
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use mathematical vocabulary fluently
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make connections between function transformations and geometric transformations
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use appropriate vocabulary to describe rotations and reflections
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use the characteristics of a figure to determine and then describe what happens to the figure as it is rotated (such as axis of symmetry,
congruent angles or sides…)
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interpret and perform a given sequence of transformations and draw the result
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accurately use geometric vocabulary to describe the sequence of transformations that will carry a given figure onto another
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recognize the effects of rigid motion on orientation and location of a figure
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use rigid motions to map one figure onto another
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use the definition of congruence as a test to see if two figures are congruent
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use the vocabulary of corresponding parts and the connection to the given triangles
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identify the corresponding parts of two triangles
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recognize why particular combinations of corresponding parts establish congruence and why others do not
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show why congruence of particular combinations of corresponding parts do not establish congruence of the triangles
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prove theorems about lines and angles (e.g., vertical angles are congruent; when a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.)
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construct proofs using a variety of methods: Two-column, paragraph, flowchart, coordinate, table
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use understanding of geometric concepts to establish a rationale for the steps/procedures used in completing a construction
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use distance, slope and midpoint formulas then use the information to solve geometric problems
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calculate slopes of lines and use the information to determine whether two lines are parallel, perpendicular or neither
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make sense of problems and persevere in solving them
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reason abstractly and quantitatively
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construct viable arguments and critique the reasoning of others
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model with mathematics
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use appropriate tools strategically
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attend to precision
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look for and make use of structure
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look for and express regularity in repeated reasoning
EVIDENCE OF LEARNING
Assessment
What evidence will be collected and deemed acceptable to show that students truly “understand”?
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Unit tests, quizzes, quarterly formative assessments
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Open-ended problems that involve written responses
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Daily student work
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Student/group presentations
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Daily Homework
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Long term projects
Learning Activities
What differentiated learning experiences and instruction will enable all students to achieve the desired results?
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Mathematical investigations
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Construct and analyze geometric figures
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Make conjectures from geometric figures and data and then prove or disprove them
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Work with tools of geometry
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Use geometric properties to solve real-world problems
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(HRW) Text, Pg.68 Chapter 1 Project – Origami Paper Folding
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“What Am I?” - Translations Instruction Sheet
o Create an instruction sheet that guides a person to recreate a drawing by connecting points found by using translation functions.
o Complete another students’ instruction sheet.
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Design a piece of art that illustrates reflectional and/or rotational symmetry. Describe the symmetries in detail.
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Create a translation or rotation tessellation. (examples by M.C. Esher)
RESOURCES
Teacher Resources:
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Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing
assignments
Teacher developed worksheets and activities
Literature: Flatland: A Romance of Many Dimensions by E.A.Abbott
Movie: Flatland (starring Martin Sheen)
Geometers Sketchpad
Visual aids (suggestions) e.g.
o Shape sorter for symmetry
o Dominos for logical chains
o AngLegs for triangle congruence
Equipment Needed:
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Rulers/straight edge
Protractors
Compasses
Chalk/Pencil & String
Reflective devices
Patty paper
Graph paper
Geometers Sketchpad
RAHWAY PUBLIC SCHOOLS CURRICULUM
UNIT TWO - Similarity, Right Triangles and Trigonometry
Content Area: Geometry
Unit Title: Similarity, Right Triangles and Trigonometry
Target Course/Grade Level: Geometry/Grade 10
Unit Summary:
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Understand similarity in terms of similarity transformations.
Prove theorems involving similarity.
Define trigonometric ratios and solve problems involving right triangles
Apply geometric concepts in modeling situations.
Approximate Length of Unit:
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10 weeks
Primary interdisciplinary connections:
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Art, Social Studies, Language Arts, and Science
LEARNING TARGETS
Content Area Domain
Content Area Cluster
Standard
Similarity, Right Triangles & Trigonometry
Understand similarity in terms of similarity transformations
G-SRT.1, G-SRT.2, G-SRT.3
Similarity, Right Triangles & Trigonometry
Prove theorems involving similarity
G-SRT.4, G-SRT.5
Similarity, Right Triangles & Trigonometry
Define trigonometric ratios and solve problems involving right triangles
G-SRT.6, G-SRT.7, G-SRT.8
Expressing Geometric Properties With Equations
Use coordinates to prove simple geometric theorems algebraically
G.GPE.6, G.GPE.7
Unit Understandings
Students will understand that…
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Geometric properties can be used to establish similarity.
Comparing similar figures is useful when there is a need for indirect measurement.
Smaller scale models can be created by using similarity
Geometric relationships provide a means to make sense of a variety of phenomena.
Measurements can be used to describe, compare, and make sense of phenomena.
Unit Essential Questions
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Show how to find the measure of a tree using indirect measurement.
What properties do all triangles share? How are triangles classified?
How are similarity and congruence established? Why is this important?
Prove that two given triangles are similar.
Given a side and an angle of a right triangle, use the trigonometric ratios to find the remaining angles and sides of the triangle.
Knowledge and Skills
Students will know…
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Vocabulary –
o Dilation, center of dilation, contraction, expansion, scale factor, similar, proportionate, corresponding parts
o Isosceles triangle, vertex angle, base angle, base and legs of an isosceles triangle
o Corollary
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o Altitude, base and height of a parallelogram, trapezoid and triangle
o Legs of a trapezoid
o Apothem
Formulas –
o Trigonometric ratios: SohCahToa
o Area of a triangle, parallelogram, trapezoid and a regular polygon
Triangle similarity theorems and postulate (ASA, SAS, SSS, the special case of ASS (HL)and AA)
Side-splitting theorem
Proportional Altitudes, Medians, Angle Bisectors and Segments Theorems
Polygon similarity Postulate
Pythagorean Theorem and it’s converse
Pythagorean Inequalities
Pythagorean triples
45-45-90 Triangle Theorem
30-60-90 Triangle Theorem
Isosceles Triangle Theorem and it’s converse
Triangle Midsegment Theorem
Triangle Inequality Theorem
Students will be able to…
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connect experiences with dilations and orientation to experiences with lines
develop a hypothesis based on observations
make connections between the definition of similarity and the attributes of two given figures
set up and use appropriate ratios and proportions
recognize why particular combinations of corresponding parts establish similarity and why others do not
construct a proof using one of a variety of methods
use information given in verbal or pictorial form about geometric figures to set up a proportion that accurately models the situation
generalize that side ratios from similar triangles are equal and that these relationships lead to the definition of the six trigonometric ratios
explain and use the relationship between the sine and cosine of complementary angles.
use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
use geometric shapes, their measures, and their properties to describe objects. (e.g., modeling a tree trunk or a human torso as a cylinder).
apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
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).
make sense of problems and persevere in solving them
reason abstractly and quantitatively
construct viable arguments and critique the reasoning of others
model with mathematics
use appropriate tools strategically
attend to precision
look for and make use of structure
look for and express regularity in repeated reasoning
EVIDENCE OF LEARNING
Assessment
What evidence will be collected and deemed acceptable to show that students truly “understand”?
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Unit tests, quizzes, quarterly formative assessments
Open-ended problems that involve written responses
Daily student work
Student/group presentations
Daily Homework
Long term projects
Learning Activities
What differentiated learning experiences and instruction will enable all students to achieve the desired results?
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Mathematical investigations
Construct and analyze geometric figures
Make conjectures from geometric figures and data and then prove or disprove them
Work with tools of geometry
Use geometric properties to solve real-world problems
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(HRW) Text, Pg.542 Portfolio Activity – Techniques for Indirect Measurement
Use proportions and at least two of the methods listed below to find the dimensions of a building or other structure at your school or in your
neighborhood.
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Measure the shadow of the building and the shadow of a person or object with a known height
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Use a mirror to create similar triangles
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Take a photograph of the building with a person or object of known height standing in front of it. Measure the building and
person in the photograph.
(HRW) Text, Pg.552 Chapter Eight Project – Indirect Measurement
o Build a scale model of your school and possibly the area around it
RESOURCES
Teacher Resources:
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Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments
Teacher developed worksheets and activities
Literature: Flatland: A Romance of Many Dimensions by E.A.Abbott
Movie: Flatland (starring Martin Sheen)
Geometers Sketchpad
Visual aids (suggestions) e.g.
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Examples of scale models
Equipment Needed:
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Tape measure
Ruler
Protractor
Reflective devices
Poster board
Scissors
Tape
Geometers Sketchpad
RAHWAY PUBLIC SCHOOLS CURRICULUM
UNIT THREE - Extending to Three Dimensions
Content Area: Geometry
Unit Title: Extending to Three Dimensions
Target Course/Grade Level: Geometry/Grade 10
Unit Summary:
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Explain volume formulas and use them to solve problems.
Visualize the relation between two‐dimensional and three‐dimensional objects.
Apply geometric concepts in modeling situations.
Approximate Length of Unit:
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8 weeks
Primary interdisciplinary connections:
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Art, Social Studies, Language Arts, Science and Business
LEARNING TARGETS
Content Area Domain
Content Area Cluster
Standard
Geometric Measurement & Dimension
Explain volume formulas and use them to solve problems
G-GMD.1, G-GMD.2, G-GMD.3
Geometric Measurement & Dimension
Visualize relationships between two-dimensional and three-dimensional objects
G-GMD.4
Modeling With Geometry
Apply geometric concepts in modeling situations
G-MG.1, G-MG.2, G-MG.3
Expressing Geometric Properties With Equations
Use coordinates to prove simple geometric theorems algebraically
G-GPE.7
Unit Understandings
Students will understand that…
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Three dimensional figures can be represented in two dimensions by using the orthographic projections
Three dimensional figures can be drawn/created by referring to the orthographic projections of the figure.
Geometric properties can be used to construct geometric figures.
Geometric relationships provide a means to make sense of a variety of phenomena.
Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Unit Essential Questions
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Given a choice of two box designs with different dimensions, which of the two is better from the manufacturer’s point of view. Justify your
answer.
Give examples of when it is better to maximize the surface area to volume ratio and when it is better to minimize the surface area to volume
ratio. Explain what makes it better. (e.g. Tums v. Tylenol, Plants in hot climates v. cold climates)
Calculate the volume, surface area or specific dimensions of a variety of polyhedral.
Knowledge and Skills
Students will know…
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Vocabulary –
o Orthographic projection, isometric drawing, parallel planes
o Polyhedron, faces, edges, vertices, dihedral angle, cross section,
o Prism, base, height, slant height, lateral height, lateral edge, lateral face, right prism, oblique prism, altitude
o Cylinder, pyramid, cone, sphere
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o Surface area, volume, density
o Surface area to volume ratio
Formulas –
o Area of a regular polygon
o Volume of prism, cylinder, pyramid, cone, sphere
o Surface Area of a right prism, right cylinder, right pyramid and right cone
o Diagonal of a right rectangular prism
o Distance Formula in Three Dimensions
Cavalieri’s Principle
Students will be able to…
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give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by
rotations of two-dimensional objects.
use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
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).
make connections between two-dimensional figures such as rectangles, squares, circles, and triangles and three-dimensional figures such as
cylinders, spheres, pyramids and cones.
connect experiences with this standard as it related to the two-dimensional shapes studied in Unit 2 to three-dimensional shapes.
make sense of problems and persevere in solving them
reason abstractly and quantitatively
construct viable arguments and critique the reasoning of others
model with mathematics
use appropriate tools strategically
attend to precision
look for and make use of structure
look for and express regularity in repeated reasoning
EVIDENCE OF LEARNING
Assessment
What evidence will be collected and deemed acceptable to show that students truly “understand”?
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Unit tests, quizzes, quarterly formative assessments
Open-ended problems that involve written responses
Daily student work
Student/group presentations
Daily Homework
Long term projects
Learning Activities
What differentiated learning experiences and instruction will enable all students to achieve the desired results?
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Mathematical investigations
Construct and analyze geometric figures
Make conjectures from geometric figures and data and then prove or disprove them
Work with tools of geometry
Use geometric properties to solve real-world problems
(HRW) Text, Pg.485 Portfolio Activity Creating Solids of Revolution
(HRW) Alternative Assessment Chapter 7 Form A – Product Packaging
“Building a Castle” – Find the volume and surface area of a ‘castle’ built with a variety of children’s blocks
RESOURCES
Teacher Resources:
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Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments
Teacher developed worksheets and activities
Geometers Sketchpad
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Literature: Flatland: A Romance of Many Dimensions by E.A.Abbott
Movie: Flatland (starring Martin Sheen)
Visual aids (suggestions) e.g.
o Unit blocks, children’s blocks
o Deck of cards to illustrate Cavalieri’s Principle
o Tums, pestle and mortar, vinegar to show an application of surface area to volume ratio
o Set of Geometric Solids, everyday solids
o Nets of solids
o Real life examples of solids
o Assortment of packaging options
o Cutouts made from Honeycomb balls to identify three-dimensional objects generated by rotations of two-dimensional objects.
Equipment Needed:
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Isometric Dot paper
Graph paper
Rulers
Calculators
Student set of Geometric Solids
Nets of solids
Poster board
Scissors
Tape/Glue
Geometers Sketchpad
RAHWAY PUBLIC SCHOOLS CURRICULUM
UNIT FOUR - Circles With and Without Coordinates
Content Area: Geometry
Unit Title: Circles With and Without Coordinates
Target Course/Grade Level: Geometry/Grade 10
Unit Summary:
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Understand and apply theorems about circles.
Find arc lengths and areas of sectors of circles.
Translate between the geometric description and the equation for a conic section.
Use coordinates to prove simple geometric theorems algebraically.
Apply geometric concepts in modeling situations.
Approximate Length of Unit:
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4 weeks
Primary interdisciplinary connections:
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Art, Social Studies, Language Arts, Science
LEARNING TARGETS
Content Area Domain
Content Area Cluster
Standard
Circles
Understand and apply theorems about circles
G-C.1, G-C.2, G-C.3, G-C.4
Circles
Find arc lengths and areas of sectors of circles
G-C.5
Modeling With Geometry
Apply geometric concepts in modeling situations
G-MG.1
Expressing Geometric Properties With Equations
Translate between the geometric description and the equation for a conic section
G-GPE.1, G-GPE.2
Expressing Geometric Properties With Equations
Use coordinates to prove simple geometric theorems algebraically
G-GPE.4
Unit Understandings
Students will understand that…
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Circles have unique properties and applications which are different from those of other geometric figures.
Measures of line segments and angles associated with a circle are found by using the properties of a circle
Geometric relationships provide a means to make sense of a variety of phenomena.
Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Unit Essential Questions
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What is a circle?
What is a parabola?
Given the center and the radius, find the equation of the circle in the coordinate plane or given the equation of a circle in center-radius form,
state the center and the radius of the circle.
Given three non-collinear points construct the circle that passes through them.
How do you find the measure of an arc?
(HRW) Text, Pg.593, Q.9 Navigation
o Lighthouses are located at points A and B on the circle of danger. If a ship is located at point X and the measure of angle BXA = 27
degrees, is the ship inside or outside the circle of danger? (Diagram is given in the text)
Find the center of a large circle by using a carpenters’ square. Justify why this works.
Knowledge and Skills
Students will know…
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Vocabulary –
o Circle, arc, radii, diameter, semicircle, minor arc, major arc. chord, sector, tangent, point of tangency, tangent segment, perpendicular,
perpendicular bisector, secant, secant segment, external secant segment
o Central angle, arc length, arc measure, intercepted arc, inscribed angle
o Inscribed and circumscribed polygons
o Radian measure
o Parabola, focus, directrix
Formulas –
o Equation of a circle
o Circumference, Area
o Central angle, intercepted arc
o Length of an Arc, Area of a Sector
o Pythagorean Theorem
o Distance
That the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Chords and Arcs theorem and it’s converse
Tangent theorem and it’s converse
Radius and Chord Theorem
Inscribed Angle theorem, Right Angle Corollary, Arc Intercept Corollary
Students will be able to…
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prove that all circles are similar.
identify and describe relationships among inscribed angles, radii, and chords.
include the relationship between central, inscribed, and circumscribed angles
use concurrence of perpendicular bisectors and angle bisectors for the basis of the construction
construct a tangent line from a point outside a given circle to the circle.
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.
derive the equation of a circle of given centre and radius using the Pythagorean Theorem; complete the square to find the centre and radius of a
circle given by an equation.
use coordinates to prove simple geometric theorems algebraically.; for example prove or disprove that the point (1, √3) lies on the circle
centered at the origin and containing the point (0, 2).
connect experiences from Unit 2 and Unit 3 with two-dimensional and three-dimensional shapes to circles
connect the distance formula and the definition of a parabola
connect the algebraic and geometric definitions of a parabola
make sense of problems and persevere in solving them
reason abstractly and quantitatively
construct viable arguments and critique the reasoning of others
model with mathematics
use appropriate tools strategically
attend to precision
look for and make use of structure
look for and express regularity in repeated reasoning
EVIDENCE OF LEARNING
Assessment
What evidence will be collected and deemed acceptable to show that students truly “understand”?
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Unit tests, quizzes, quarterly formative assessments
Open-ended problems that involve written responses
Daily student work
Student/group presentations
Daily Homework
Long term projects
Learning Activities
What differentiated learning experiences and instruction will enable all students to achieve the desired results?
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Mathematical investigations using a circular geoboard
Construct and analyze geometric figures
Make conjectures from geometric figures and data and then prove or disprove them
Work with tools of geometry
Use geometric properties to solve real-world problems
Design window art using a compass and straightedge
RESOURCES
Teacher Resources:
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Geometry Textbook: Teachers’ Edition & accompanying resources, e.g. Transparencies, practice worksheets, assessments, writing assignments
Teacher developed worksheets and activities
Geometers Sketchpad
Visual aids (suggestions) e.g.
o Circle Art
o Circular Geoboard for the overhead projector
o Rubber bands
o Maps of the coast and lighthouses, model boat
o Compass (for drawing circles)
o Compass (for directions)
Equipment Needed:
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Compass (for drawing circles)
Compass (for directions)
Straight edge
Circular Geoboard & rubber bands
Graph paper
Spaghetti (for parabolas)
(Cheap) Transparencies (for circle window art)
Colored permanent markers
Geometers Sketchpad