Geometry Core Curriculum Map  Updated:   June 2009

 

 

Core Objective

Vocabulary

Essential Questions

September

Unit 1:  Intro to Geometry (approx 7-8 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 1: Use inductive and deductive reasoning to develop   mathematical arguments.

·         Objective 2: Analyze characteristics and properties of angles.

Standard II: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

·         Objective 2: Describe spatial relationships using coordinate geometry.

1 day – intro to course

1.1a   Write conditional statements, converses, and inverses, and determine the truth value of these statements. (1.5 days)

1.1c   Prove a statement false by using a counterexample. (.5 day)

1.1b   Formulate conjectures using inductive reasoning. (.5 day)

1.2a   Use accepted geometric notation for lines, segments, rays, angles, similarity, and congruence. (2-3 days)

2.2b   Determine whether points in a set are collinear. (.5 day)

 

1 day review

Test 1

 

conditional statements,

converse, inverse, conjecture,

truth value,

counterexample

inductive, deductive

 

 

 

 

collinear

·      What are the established principles of logic in geometry and how do they work? (Explain and give examples for 1) conjectures and counterexamples, 2) conditional statements (if-then),  inverses, converses, negation and contra-positives, 3) inductive and deductive reasoning, 4) the law of detachment, 5) the law of syllogism.

·      How are these principles used in today’s world?

 

·      Why do we use notation for lines, segments, rays, angles, similarity, and congruence?

 

·      How do you determine whether points are collinear?

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - draw a diagram, look for patterns, clarify understanding, “Is this true?, What makes you think so?”, check for reasonableness, guess and check, identify counter examples, consider the thinking of others

Reasoning and proof - investigate mathematical conjectures, formulate counter examples, realize that observing a pattern does not constitute proof

Communication - class and group discussions using precise language and expressing mathematical ideas coherently, journals for essential questions and other questions that may arise

Connections - establish connections among mathematical expressions and physical models, use real-world applications, explore historical and multicultural contributions to math

Representation - use a variety of visual representations and tools (protractor, compass, straight edge, manipulatives, pictures, graph paper, graphing calculators), represent patterns verbally, numerically, geometrically and algebraically

 

Strategies, Materials, and Technology:

Have students bring in ads from magazines or newspapers and have them write conditional, converse, and inverse statements as well as a counter example for each.  Use cooperative groups to create  posters of the symbolic representations of different geometric terms.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions



 October

Unit 2 and 3:  approx 8-.9 days

Unit 2:  Parallel and Perpendicular Lines

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 2: Analyze characteristics and properties of angles.

·         Objective 5: Perform basic geometric constructions, describing and justifying the procedures used.

Standard II: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

·         Objective 1: Describe the properties and attributes of lines and line segments using coordinate geometry.

 

Review as needed:  slope, equations of lines, graphing

1.2e   Prove lines parallel or perpendicular using slope or angle relationships. (1 day)

1.5c   Construct perpendicular and parallel lines. (1 day)

1.5a   Investigate geometric relationships using constructions.

1.5d   Justify procedures used to construct geometric figures.

1.5e   Discover and investigate conjectures about geometric properties using constructions. (1 day)

2.1c   Write an equation of a line perpendicular or a line parallel to a line through a given point. (2 days)

Unit 3:  Angles

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 2: Analyze characteristics and properties of angles.

·         Objective 5: Perform basic geometric constructions, describing and justifying the procedures used.

 

1.5b   Copy and bisect angles and segments. (1 day)

1.2b   Identify and determine relationships in adjacent, complementary, supplementary, or vertical angles and linear pairs. ( 1 day)

Review – 1 day

Test 2, Unit 2 and 3

 

 

 

 

 

 

 

 

 

 

 

 

Unit 4:  Transversals (approx 6 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 2: Analyze characteristics and properties of angles.

Standard IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

 

1.2c   Classify angle pairs formed by two lines and a transversal. ( 1 day)

1.2d   Prove relationships in angle pairs. (2 day)

4.1a   Find linear and angle measures in real-world situations using appropriate tools or technology. ( 1 day)

1.2a   Use accepted geometric notation for lines, segments, rays, angles, similarity, and congruence.

Review – 1 day

Test 3, Unit 4

parallel lines,

perpendicular lines,

skew lines

construct,

perpendicular bisector

 

midpoint

angle bisector

 

slope

 

 

 

adjacent, complementary, supplementary, vertical angles, linear pair

transversal, corresponding, alternate interior, alternate exterior, same-side interior, and consecutive interior angles

acute, right, obtuse, and straight angles

·      How are parallel and skew lines similar?  How are they different?

How are perpendicular and skew lines similar?  How are they different?

·      Why are constructions used to draw parallel lines and perpendicular lines?

·      What is the midpoint of a segment?  How can this help you solve for the length of the two segments created by the midpoint of a line segment?

·      Why are the slopes and equations of parallel and perpendicular lines important in geometry?

·      How can you find an equation when you know an equation of a parallel or perpendicular line and just one data point of the new equation?

·      What are the important relationships of parallel lines cut by transversals and their angles? Why are they important in geometry?

·      Describe different ways you can bisect an angle.  How would you determine that the angle was bisected?

·      How would you create a 15º from a 60º angle?

·      What are angles and how do you measure them?  

·      How do you classify angles?

·      How can identifying angle relationships, such as vertical angles, help us or save us time?

Suggested Processes, Strategies, and Materials

·         Problem solving - make a model or simulation, draw a picture or diagram, eliminate possibilities, extend knowledge by considering the strategies of others, propose and critique alternative approaches Objective 1: Find measurements of plane and solid figures.

 

Reasoning and proof - recognize conclusions as valid or invalid, justify conclusions ,“What makes you think that?”, make and investigate mathematical conjectures

Communication - express mathematical ideas coherently, oral presentations and written communication

Connections - find applications in newspapers, magazines, and other sources, recognize and apply relationships to art and science, extend investigations into real-world situations

Representation - use a variety of visual representations and physical models, use appropriate symbolic notation

Strategies, Materials, and Technology:

Use movement to illustrate parallel, perpendicular, and skew lines; Use Geometer’s Sketchpad or Cabri Geometry to teach constructions; Use graphing calculators to investigate the slopes of parallel and perpendicular lines. Have students do Mix-and-Match to find a partner whose words match their symbol or sketch;  Use compass and protractor, Geometer’s Sketchpad, or Cabri Geometry to construct angles; Use Patty Paper, Geometer’s Sketchpad, Cabri Geometry , Geoboards, or spaghetti to have students create and classify angles formed by two parallel lines cut by a transversal;  Use Patty Paper , Geometer’s Sketchpad, or Cabri Geometry to construct the medians, angle bisectors, altitudes, and perpendicular bisectors.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

November

Unit 5:  Triangles (approx 6-7 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 3: Analyze characteristics and properties of triangles.

·         Objective 5: Perform basic geometric constructions, describing and justifying the procedures used.

 

Review:  triangle classification (.5 day)

1.3e   Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle. (2.5 days)

1.3c   Prove and apply theorems involving isosceles triangles.(1 day)

1.3d   Apply triangle inequality theorems. (1 day)

1.5a   Investigate geometric relationships using constructions.

1.5d   Justify procedures used to construct geometric figures.

1.5e   Discover and investigate conjectures about geometric properties using constructions.

Review, 1 day

Test 4

 

Unit 6 and 7:  Congruent and Similar Triangles

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 2: Analyze characteristics and properties of angles.

·         Objective 3: Analyze characteristics and properties of triangles.

1.3a   Prove congruency ( 4 days)  and similarity (3 days) of triangles using postulates and theorems.  For similar triangles, review ratios and proportions (1 day)

1.2a   Use accepted geometric notation for lines, segments, rays, angles, similarity, and congruence.

Review 1, day /Test 5 – Congruent Triangles (6 days)

Review 1 day,/Test 6 – Similar Triangles (6 days)

median, altitude, point of concurrency, centroid, incenter, orthocenter, circumcenter

 

congruent triangles,

SSS, SAS, and ASA Congruence Postulates;

AAS and HL Congruence Theorems

 

similar triangles,

AA Similarity Postulate, SSS and SAS Similarity Theorems

·      What are the similarities and differences of medians, altitudes, perpendicular bisectors and angle bisectors?

·      How can you determine the classification of a triangle given angle measurements and/or side dimensions?

·      How can knowing the sum of angles in triangle help solve for missing angle measures?

·      Why are constructions used to draw medians, altitudes, angle bisectors of a triangle, and perpendicular bisectors of the sides of a triangle?

·      Explain the similarities and differences between the SSS Congruence Postulate and the SAS Congruence Postulate.

·      How are the ASA Congruence Postulate and the AAS Congruence Theorem alike and how are they different?

·      How can we determine which congruence postulate or theorem is most convenient for proving congruence of a set of triangles?

·      How do the relationships of congruent triangles help solve problems?

·      What does similar mean in geometry?

·      What information must you know in order to solve for one missing side of two similar triangles? 

·      How can the AA Similarity Postulate save time when proving that two triangles are similar?

·      After drawing two similar triangles, what similarity statement can be used to explain why two triangles are similar?

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - select and use appropriate methods for computation ( mental, calculator, pencil and paper), guess and check, choose appropriate operations, make a diagram, extend knowledge by considering the thinking of others, propose alternative approaches

Reasoning and proof - examine pattern for regularities and irregularities, link problem solving to a sequence of steps, use a variety of formal and informal proofs

Communication - organize and consolidate mathematical thinking, express mathematical ideas coherently using correct language

Connections - formulate real-world situations, establish connections among physical models and the mathematical expression or representation

Representation - represent visually, numerically, geometrically and algebraically, use technology

 

Strategies, Materials, and Technology:

Make a table which demonstrates the learning and conclusions to be made related to the segments creating different triangle centers; Use Mira (transparent reflector) to review what it means to be congruent; Use Patty Paper, Geometer’s Sketchpad or Cabri Geometry to investigate the different congruence postulates and theorems about triangles;  Use a compass and straightedge, Geoboards, or dot paper to discover conditions that make triangles similar;  Have students explore the different similarity postulates and theorems by using Patty Paper, Geometer’s Sketchpad or Cabri Geometry;  Create Foldables or graphic organizers after students have investigated the different postulates and theorems about triangles.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

December

Unit 8:  Right Triangles (approx (8-9 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 3: Analyze characteristics and properties of triangles.

Standard II: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

·         Objective 1: Describe the properties and attributes of lines and line segments using coordinate geometry.

Standard III: Students will extend concepts of proportion and similarity to trigonometric ratios.

·         Objective 1: Use triangle relationships to solve problems.

Standard IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

Objective 2: Solve real-world problems using visualization and spatial reasoning.

2.1b   Find the distance between two given points and find the coordinates of the midpoint. (.5 day)

4.2b   Solve problems using the distance formula. (.5 day)

1.3b   Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the Pythagorean Theorem.  (2 days)

4.2a   Solve problems using the Pythagorean Theorem and its converse. (1 day)

3.1a   Solve problems using the properties of special right triangles, e.g., 30°, 60°, 90° or 45°, 45°, 90°. (2 days)

Review, 1 day

Test 7 and Final Exam

Distance Formula

 

Pythagorean Theorem, right triangle, legs, hypotenuse

 

diagonal

 

30-60-90 triangle, 45-45-90 triangle, ratio

·      Describe a real-life situation in which you would use the Distance Formula.

·      What is the relationship between the sides of right triangles and the hypotenuse?

·      Why does the Pythagorean Theorem work? How do you prove it?

·      Describe how to find the diagonal of a rectangle if given the length of the sides.

·      How does the “distance formula” relate to the Pythagorean Theorem?

·      What are special right triangles?

·      How can we use special right triangles to help solve problems?

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - propose and critique alternative approaches to solving problems, make an equation, manipulate formulas, solve a related simpler problem, make a model, ask, “How are these ideas related?”

Reasoning and proof - explain and justify problem solving procedures, identify information as necessary or not, make and investigate mathematical conjectures Communication - express ideas coherently, use portfolios or journals, group discussion

Connections - use real-world applications, use physical models, e.g. Leaning Tower of Pisa, consider historical and multicultural influences

Representation - use visual representations, graph paper, models and technology

 

Strategies, Materials, and Technology:

Use “Cut-apart Proof Puzzles” to illustrate the Pythagorean Theorem; You could also use Geometer’s Sketchpad or Cabri Geometry to explore the Pythagorean Theorem;  To explore the distance formula have students go to Map Quest and select a map, print it, place a coordinate grid over it, and calculate distances between different locations; Program the graphing calculator with the distance formula;  Make a square from balsa wood using no measuring tools;  Use Patty Paper to explore the relationships of the sides of special right triangles;  Use a homemade clinometer or hypsometer (protractor, string and straw) having fixed degrees to solve real life situations with special right triangles.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

January

Unit 9:  Trigonometry (approx 7-8 days)

Standard III: Students will extend concepts of proportion and similarity to trigonometric ratios.

·         Objective 1: Use triangle relationships to solve problems.

·         Objective 2: Use the trigonometric ratios of sine, cosine, and tangent to represent and solve for missing parts of triangles.

Standard IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

·         Objective 2: Solve real-world problems using visualization and spatial reasoning.

3.1b   Identify the trigonometric relationships of sine, cosine, and tangent with the appropriate ratio of sides of a right triangle.  ( 1 day)

3.1c   Express trigonometric relationships using exact values and approximations. (1 day)

3.2a   Find the angle measure in degrees when given the trigonometric ratio. (1 day)

3.2b   Find the trigonometric ratio given the angle measure in degrees, using a calculator. (.5 day)

3.2c   Find unknown measures of right triangles using sine, cosine, and tangent functions and inverse trigonometric functions. (.5 day)

4.2c   Solve problems involving trigonometric ratios. (1 day)

 

Re view, 1 day

Test 8

similar triangles

indirect measurement

right triangle,

adjacent side,

opposite side, hypotenuse

 

trigonometric ratio,

sine, cosine, tangent

 

angle measurement

 

angle of elevation, angle of descent

 

 

·      Give an example of how you can use similar right triangles to solve real-world problems?

·      What is trigonometry? How does it help us?

·      How are sine, cosine and tangent similar?  Different?

·      How can you use sine, cosine and tangent to solve problems?

·      What is the minimum amount of information you need to solve a right triangle?

·      How can you find the measures of the acute angles of a right triangle for which the sides are known?

·      How can you find an angle measure given the trigonometric ratio?  How can you find the trigonometric ratio given the angle measurement?

·      What is the difference between an angle of elevation and an angle of depression?

·      How would you find the height of an object if you know the angle of elevation and your distance from the object?

·      How can you use trigonometry with triangles that aren’t right triangles?

 

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - draw pictures, select and use appropriate methods of computing, write equations, use proportional reasoning, clarify new math concepts and terms

Reasoning and proof - identify information as necessary, sufficient or extraneous, identify conclusions as valid or invalid, examine patterns, note regularities and irregularities

Communication - employ precise language and notation, express ideas clearly, group discussions, portfolios, journals

Connections - formulate real-world situations, solve and justify answers, apply math outside the classroom and in everyday life

Representation - use a variety of visual representations (manipulatives, technology, models, paper), explore and formulate conjectures, represent situations algebraically, verbally, graphically, and geometrically

 

Strategies, Materials, and Technology:

Use a Foldable to introduce the vocabulary for trigonometric ratios; Use the graphing calculator to calculate the angle measure given the trigonometric ratio or the trigonometric ratio when given the angle measure; Have students explore trigonometric ratios using Geometers’ Sketchpad or Cabri Geometry to determine if there is a relation among angle sizes and ratios of  sides of a right triangle; Create a hypsometer  or a clinometer and have students find the height of objects such as a flagpole, a light pole, a building, etc.;  Have students work in cooperative groups and provide each group with a real world scenario that uses angle of elevation or angle of descent.  Have the students solve the problems and then do a Red Robin to have each group present their problem to another group.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

February

Unit 10:  Polygons and Quadrilaterals (approx 8-9 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 4: Analyze characteristics and properties of polygons and circles.

Standard II: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

·         Objective 1: Describe the properties and attributes of lines and line segments using coordinate geometry.

 

Review:  Types of polygons and quadrilaterals ( 1 day)

1.4a   Use examples and counterexamples to classify subsets of quadrilaterals. (1 day)

2.1a   Verify the classifications of geometric figures using coordinate geometry to find lengths and slopes. (1 day)

1.4b   Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems. (1 day)

1.4c   Derive, justify, and use formulas for the number of diagonals, lines of symmetry, angle measures for polygons. (2 days)

Review, 1 day; Test 9

 

Unit 11:  Area (approx 6-7 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 4: Analyze characteristics and properties of polygons and circles.

Standard IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

·         Objective 4: Analyze characteristics and properties of polygons and circles.

 

4.1d   Calculate or estimate the area of an irregular region. (.5 day)

4.2d   Solve problems involving geometric probability. (.5 day)

1.4c   Derive, justify, and use formulas for perimeter and area of regular polygons. (3 days)

 

Review, 1 day; Test 10

quadrilateral, parallelogram, rectangle, square, rhombus, trapezoid, kite

slope of a line, midpoint of a line, distance formula

diagonals, lines of symmetry, angle measures, perimeter, area

 

angles, polygon, interior angles of a polygon, exterior angles of a polygon

regular polygon, irregular regions

 

geometric probability

·      Why is it important to identify characteristics of quadrilaterals? 

·      How does the coordinate system enable us to study the characteristics of quadrilaterals?

·      How are the characteristics of a parallelogram similar to those of a trapezoid and how are they different?

·      Why must a parallelogram with one right angle be a rectangle?

·      How are the characteristics of a rectangle similar to those of a square and how are they different?

·      The diagonals of both kites and rhombi are perpendicular.  Do the diagonals of both bisect each other?  Explain.

·      If at least two opposite angles of a quadrilateral are congruent, what kind of a quadrilateral might it be?

·      How are the formulas for the area of parallelograms, triangles, trapezoids, and regular polygons related?

·      What information do you need to know to find the area of regular polygons?

·      When given an irregular region, how can you find its area?

·      What is the difference between an exterior and an interior angle?

·      How do you find the measure of each interior angle and each exterior angle in a regular polygon?

·      How can you find the sum of the measures of the interior angles of a polygon using triangles?

·      How do you find the probability of a randomly chosen point that lies in a desired part of a segment?

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - look for patterns, make a model, draw a picture or diagram, solve a variety of multi-step problems, extend problems, estimate solutions, extend knowledge by considering the strategies of others

Reasoning and proof - make and investigate conjectures, identify information as necessary or unnecessary

Communication - organize and consolidate thinking, use group discussions, journals or portfolios, employ precise language and notation

Connections - establish connections among math representation and physical models, find applications outside the classroom, use magazines, newspapers, physical models and other sources, consider historical and cultural contributions

Representation - use a variety of visual representations (manipulatives, technology, paper, models), use appropriate symbols

 

Strategies, Materials, and Technology:

Have students make the quadrilaterals listed above and create a graphic organizer that will help them learn characteristics.  The characteristics can be discovered by having students measure angles, side lengths, diagonal lengths, and have them determine parallelism, congruency, etc.;   Give groups of students a long piece of string or rope and assign them a quadrilateral to create.  Have students explain why they know they have created the given quadrilateral; Use Pattern Blocks to create different quadrilaterals and investigate what happens to the area as the linear dimensions are changed; Draw irregular shapes on grid paper and have students investigate the area.  Have them discuss different ways they can find the area e.g., divide the shape into rectangles, triangles, etc.;  Have students use the Geometry Class Reference Sheet;  Use Geometer’s Sketchpad, Cabri Geometry, or Patty Paper to explore the interior and exterior angles of polygons.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

March

Unit 12:  3-D figures, surface area and volume (approx 9-10 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 6: Analyze characteristics and properties of three-dimensional figures.

Standard IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

·         Objective 1: Find measurements of plane and solid figures.

1.6a   Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s). (.5 day)

1.6c   Describe the symmetries of three-dimensional figures. (.5 day)

1.6b   Identify three-dimensional objects from different perspectives using nets, cross-sections, and two-dimensional views. (.5 day)

1.6d   Describe relationships between the faces, edges, and vertices of polyhedra. (.5 day)

4.1b   Develop surface area and volume formulas for polyhedra, cones, and cylinders.

4.1c   Determine perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones, and spheres when given the formulas. (6 days)

 

Review, 1 day

Test 11

 

radius, diameter, chord, secant, (major and minor) arc, semicircle, sector, central angle, inscribed angle, tangent to a circle,

arc length,

intercepted arc,

area of a sector

equation of a circle

 

prism, pyramid, cylinder, cone

 

geometric solid,

 

cross-section

·      What are the similarities and differences between the radius, diameter, and chord of a circle?

·      How do you describe the properties of congruent chords of a circle?

·      What conjectures can you write about chords, tangents, arcs, and angles?

·      What kinds of relationships and properties can we show relating to intersecting chords, intersecting secants and an intersecting tangent and secant?

·      How do you determine whether an arc of a circle is a major or a minor arc?

·      How is the formula for a circle related to the distance formula?

·      What is the relationship between the circumference of a circle and its diameter?  Why is this relationship so important?

·      How does changing the length of the radius of a circle effect the area of the circle?

·      How can we prove congruency of inscribed angles intersecting the same arc?

·      How do you find the area of a sector?

·      What do we use to find geometric probability? 

·      Why is classification of geometric solids helpful?

·      Why might it be valuable to visualize the shapes of cross sections of solids?

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - extend math knowledge by considering the thinking of others, propose alternative approaches, solve a related simpler problem

Reasoning and proof - explain and justify problem solving procedures, “Why do you think so?”,  use informal proof where appropriate , make conjectures, find applications outside classroom

Communication - express math ideas clearly to peers, teachers and others, employ precise notation, organize and consolidate thinking, group discussions, portfolios

Connections - formulate real-world situations, solve and justify them, establish connections among mathematical expressions and physical models

Representation - use a variety of visual representations (manipulatives, technology, paper, models), represent verbally, algebraically and geometrically

Strategies, Materials, and Technology:

Have students use graphing calculators, pattern blocks, Mira, and Geometer’s Sketchpad or Cabri Geometry to investigate transformations; Have students design their own transformation and describe the transformations performed; Use student formations to have students demonstrate their knowledge of the terms associated with a circle; Play the concentration game to help students identify pictures and vocabulary with the definition;  Discuss how the formula for a circle is related to the distance formula; Use Geometer’s Sketchpad or Cabri Geometry to explore the properties and relationships of intersecting chords, inscribed angles, intercepted angles, etc.;  Use the M&M activity to have students explore the relationship between the circumference and diameter of a circle;  Have students throw paper balls at the bulls eye on a target or throw beanbags through the holes of a rectangular surface to explore geometric probability. View cross sections using dynamic software such as NCTM Applets, Sketchpad or Cabri.

 

 

 

 

Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

Fin

Unit 13:  Circles (approx 7-8 days)

Standard I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

·         Objective 4: Analyze characteristics and properties of polygons and circles.

Standard II: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

·         Objective 2: Describe spatial relationships using coordinate geometry.

Standard IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

·         Objective 1: Find measurements of plane and solid figures.

1.4d   Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle. (1 day)

1.4e   Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.  (1 day)

4.1e   Find the length of an arc and the area of a sector when given the angle measure and radius. (1 day)

2.2a   Graph a circle given the equation in the form, and write the equation when given the graph. (1 day)

 

Review, 2 days

Final Exam

 

polyhedron, faces, edges, vertices

 

 

surface area, volume, prisms, spheres, cylinders, cones, pyramids, slant height

 

 

·      What everyday items can be classified as prisms, pyramids, spheres, cylinders, and cones?

·      How can you find the number of vertices of a polyhedron if you know the number of faces and edges?

·      Why is there no polyhedron with six vertices and seven edges?

·      How can the surface area of rectangular prisms, triangular prisms, cylinders, cones, and pyramids be computed?

·      How does the volume of a cone and pyramid compare to the volume of a cylinder and prism with same base and height?

·      How are the formulas for the volume and the surface area of a sphere relate?

·      What everyday situations require finding the surface area and the volume of the various solids?

·      How does the surface area of a prism change if the length doubles but the height remains the same?

·      What happens to the volume of a cylinder if you take half of its height and double its circumference?

·      Cone B has the same height as Cone A, but twice the base diameter.  Cone C has the same base diameter as Cone A but twice the height.  How does the volume of Cone B compare to the volume of Cone C?

·      What effect does doubling the radius of a sphere have on its volume?

·      If the dimensions of a solid are increased by a factor of x, what happens to the surface area of the solid?  To the volume?

·      What shapes are formed by the intersection of a plane and a cube, a plane and a cylinder, a plane and a cone?

 

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - make a model, check for reasonableness, “How does this apply here?”, solve multi-step problems, estimate solutions, work backwards

Reasoning and proof - make and investigate conjectures, identify information as necessary and sufficient and conclusions as valid

Communication - organize and consolidate thinking using group discussions, presentations

Connections - apply to real world situations that require extended investigation, connect math expressions to physical models

Representation - use a variety of visual representations (models, nets, and technology)

 

Strategies, Materials, and Technology:

Have students create nets of the different geometric solids and explore their characteristics; Have students build the platonic solids using nets or straws, toothpicks, marshmallows, and Dots candy.  Have them discover Euler’s formula using their models;  Have students explore the surface area and volume using geometric solids, nets, boxes, cans, ice cream cones, cubes, etc.;  Create a foldable with the formulas for volume and surface area;  Have students use their knowledge of surface area to estimate the amount of skin on their body;  Have students use their knowledge of volume to estimate the volume of a banana; Use food to cut cross sections, fill a clear geometric solid and tip it to a different position to see the cross sections, or make salt dough and have students cut cross sections.


Geometry Core Curriculum Map

 

 

Core Objective

Vocabulary

Essential Questions

May

- Real world projects applying geometric  knowledge

- Review for CRT’s

Review vocabulary taught throughout the year

·      Why is it valuable to learn Geometry?

·      Where do you find Geometry in the world around us?

·      How can we use Geometry to model situations in the real world?

 

·      Review questions from the previous months.

Suggested Processes, Strategies, and Materials

Core Processes:

Problem solving - solve a variety of multi-step problems including puzzles, open-ended or extended problems, ask questions like, “Where have we seen this before?”, propose and critique alternative approaches 

Reasoning and proof - explain and justify procedures, identify information as needed or not, as sufficient or insufficient

Communications - organize and consolidate math thinking through group discussion, journals, portfolios, oral or written reports, employ precise language and notation

Connections - formulate real-world situations or investigations, solve and justify answers, explore multicultural and historical contributions

Representation - use a variety of visual representations, use correct notation

 

Strategies, Materials, and Technology:

Use Jeopardy, Who Wants to Be a Millionaire, or Hollywood Squares to review concepts taught throughout the year; Use the Concentration game to have students match vocabulary taught throughout the year with pictures, formulas, etc.; Use the Benchmark Tests as a review and to practice test taking strategies;  Have students do a picture book or alphabet book on concepts taught in Geometry;  Have students do a group project where they interview someone that uses Geometry in their profession.  Have the groups put together a presentation about their interview and present it to the entire class.