Geometry Curriculum Framework

Geometry Curriculum Framework Mapping Knowledge and Skills to Cognitive Demands

Knowledge and Skills	Cognitive Demands
	Generalize	Make Connections	Understand Concepts	Perform Procedures	Memorize
Basic Understandings Ga1 Foundation concepts for high school mathematics... Ga2 Geometric thinking and spatial reasoning… Ga3 Geometric figures and their properties… Ga4 The relationship between geometry, other mathematics and other disciplines… Ga5 Tools for geometric thinking… Ga6 Underlying Mathematical Processes	Ga1 Expand understanding of K-8 foundation concepts, e.g. generalization of the Pythagorean theorem Ga2 Ex press generalizations about space and spatial relationships using geometric thinking and spatial reasoning Ga5 Use a variety of representations including tools and techno logy to analyze and prove relationships between transformed geometric figures Ga6 Use geometry to justify and prove conjectures	Ga1 Make vertical connections between K-8 concepts & main geometric ideas e.g. connect the concept of ratio with similarity Ga2 Use geometric figures to connect mathematical concepts & their relationships and use these relationships to solve problems Ga3 Connect relationships among geometric figures of 0, 1, 2 or 3- dimensions, their size, shape, location, direction, and orientation to algebraic representations of these figures Ga4 Use geometry to model and represent mathematics and real world situations Ga4 Make connections between geometry and the real and mathematical worlds Ga4 Use geometric ideas, relationships and properties to solve real-world problems Ga5 Use a variety of representations to solve meaningful problems by representing and transforming figures and analyzing geometric relationships Ga6 Connect geometry to other domains within mathematics and other disciplines	Ga1 Build on K-8 mathematics concepts to strengthen the understanding of geometry through other mathematical experiences Ga2 Use geometric figures, to understand mathematical concepts and to represent mathematical situations Ga3 Investigate properties of relationships with size, shape, location, direction, and orientation among 0, 1, 2, and 3-dimensional figures to solve problems Ga4 Recognize that geometry can be used to make connections between geometry and the real and mathematical worlds Ga5 Use a variety of representations (concrete, pictorial, numerical, symbolic, graphical and verbal ), tools, and technology to solve meaningful problems. Ga6 Use appropriate language to communicate understanding of geometry	Ga1 Use procedures based on K-8 mathematics to solve routine problems Ga3 Use properties to de termine size , shape, location, and direction and orientation of 0, 1, 2 and 3-dimensional figures Ga5 Draw and represent 0, 1, 2 and 3- dimensional figures needed to solve problems Ga5 Perform basic constructions using appropriate tools e.g. ruler, protractor, compass, & technology Ga6 Use numerical fluency to solve problems in geometry Ga6 Use and manage the language of structure of geometry	Ga1 Foundation concepts presented in K-8 Ga3 Properties & relationships of 1, 2 and 3 dimensional figures previously investigated
Timeline	Textbook and Materials		NCTM Standards
			Analyze characteristics and properties of 2 and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning and geometric modeling to solve problems
Knowledge and Skills	Cognitive Demands
	Generalize	Make Connections	Understand Concepts	Perform Procedures	Memorize
Geometric Structure bG.1 The student understands the structure of and relationships within, an axiomatic system. bG.2 The student analyzes geometric relationships in order to make and verify conjectures. bG.3 The student applies logical reasoning to justify and prove mathematical statements. bG.4 The student uses a variety of representations to describe geometric relationships and solve problems.	bG.1C Compare and contrast the structure and implications of Euclidean and non-Euclidean geometries Ga2 bG.2A Use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships Ga5 bG.2B Make conjectures about angles, lines, polygons, circles, and three –dimensional figures, and determine the validity of conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic Ga5, Ga6 bG.3D Use inductive reasoning to formulate a conjecture Ga6 bG.3E Use deductive reasoning to prove a statement Ga6	bG.1A Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems Ga2 bG.3A Determine the validity of a conditional statement, its converse, inverse, and contra positive Ga6 bG.3B Construct and justify statements about geometric figures and their properties Ga3	bG.1B Recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes Ga4 bG.3C Use logical reasoning to prove statements are true and find counter examples to disprove statements that are false Ga6 bG.4 Select an appropriate representation (concrete, pictorial, graphical, verbal or symbolic ) to describe geometric relationships and solve problems Ga2, Ga4, Ga5, Ga6	conditional statement Ga1, Ga6 Form the converse of a conditional statement Ga1, Ga6	Mathematical system Geometric system Euclidean geometry Non-Euclidean geometry Axiom Postulate Theorem Converse Conditional statement Ga1
Timeline	Textbook and Materials		NCTM Standards
			geometric shapes and develop mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning and geometric modeling to solve problems
Knowledge and Skills	Cognitive Demands
	Generalize	Make Connections	Understand Concepts	Perform Procedures	Memorize
Geometric Patterns bG.5 The student uses a variety of representations to describe geometric relationships and solve problems	bG.5B Use numeric and geometric patterns to make generalizations, justify, and prove conjectures about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles Ga2, Ga5, Ga6	bG.5C Use properties of transformations and their compositions to make connections between mathematics and the real world such as tessellations Ga2, Ga3, Ga4, Ga5, Ga6 bG.5D Identify and apply patterns from right triangles to solve novel meaningful problems, including special right triangles (45 – 45 – 90 and 30 – 60 – 90) and triangles whose sides are Pythagorean triples. Ga2, Ga3, Ga4, Ga6	bG.5D Identify and apply patterns from right triangles to solve problems, including special right triangles (45 – 45 – 90 and 30 – 60 - 90 ) and triangles whose sides are Pythagorean triples Ga1, Ga2, Ga3, Ga5, Ga6 bG.5A Use numeric and geometric patterns to develop algebraic expressions representing geometric properties	Gc3 Draw and represent special right triangles (45 - 45 - 90) and (30 – 60 - 90) and triangles whose sides are Pythagorean triples Ga1, Ga3, Ga5	Definitions: Right triangles Pythagorean triples Transformation Compositions of transformations 2-D figure 3-D figure Polygon Solid Tessellation Ratios Circumference Diameter Loci Π Ga 1
Timeline	Textbook and Materials		NCTM Standards
			develop mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning and geometric modeling to solve problems
Knowledge and Skills	Cognitive Demands
	Generalize	Make Connections	Understand Concepts	Perform Procedures	Memorize
Dimensionality and the Geometry of Location bG.6 The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. bG.7 The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.	bG.7C Derive and use formulas involving length, slope, and midpoint Ga2, Ga5	G.6C Use orthographic and isometric views of 3-dimensional figures to represent and construct three-dimensional geometric figures and solve problems Ga2, Ga3, Ga5 bG.7B Use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons in solving non-routine problems Ga2, Ga3, Ga4, Ga5	bG.6A Describe and draw the intersection of a given plane with various three-dimensional geometric figures Ga2, Ga5 bG.7B Use slope and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons Ga2, Ga3, Ga5 bG.6B Use nets to represent and construct three-dimensional geometric figures Ga2, Ga3, Ga5	bG.7A Use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments and figures Ga1, Ga3, Ga5 bG.7C Use given formulas, to determine distance between two points and slope and midpoint of a line Ga1, Ga6	Distance Formula Midpoint Pythagorean Theorem Point Line Line segment Figure Parallel lines Perpendicular lines Rectangular coordinate system Ga1
Timeline	Textbook and Materials		NCTM Standards
			Analyze characteristics and properties of 2 and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning and geometric modeling to solve problems
Knowledge and Skills	Cognitive Demands
	Generalize	Make Connections	Understand Concepts	Perform Procedures	Memorize
Congruence and the Geometry of Size bG.8 The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. bG.9 The student analyzes properties and describes relationships in geometric figures. bG.10 The student applies the concept of congruence to justify properties of figures and solve problems.	bG.9A Formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and using concrete models Ga2 bG.9B Formulate and test conjectures about properties and attributes of polygons and their comp onent parts based on explorations and using concrete models Ga2 bG.9C Formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and using concrete models Ga2 bG.10A Use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane Ga5, Ga6 bG.10B Justify and apply triangle congruence relationships Ga6 bG.8C Derive, extend, and use the Pythagorean theorem Ga1, Ga5, Ga6	bG.10B Apply triangle congruence relationships to solve problems Ga3, Ga4 bG.9D Analyze the characteristics of polyhedra and other three dimensional figures and their component parts based on explorations and concrete models Ga2, Ga3, Ga5, Ga6 bG.8C Apply the Pythagorean theorem to solve problems Ga1, Ga2, Ga4	bG.8A Explain how to use simple shapes to find the area of composite figures Ga2, Ga3, Ga5 bG.8B Describe how proportional reasoning is used to find areas of sectors and arc lengths of circles Ga1, Ga2, Ga6	bG.8A Find areas of regular polygons, circles, and composite figures Ga1, Ga3, Ga5 bG.8B Find area of sectors and arc length of circles using proportional reasoning Ga1, Ga3, Ga5 bG.8D Find surface area and volume of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations Ga1, Ga3, Ga5 bG.8C Use the Pythagorean theorem to solve problems to find missing information from right triangles Ga1, Ga3, Ga5	Pythagorean theorem Definitions: Parallel Perpendicular Congruent figures Congruent transformations Reflection of an object Rotation of an object Translation of an object Circumference Diameter Loci Π Formulas: area, perimeter, volume, and circumference of 2- dimensional and 3-dimensional figures Ga1, Ga3
Timeline	Textbook and Materials		NCTM Standards
			Analyze characteristics and properties of 2 and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning and geometric modeling to solve problems
Knowledge and Skills	Cognitive Demands
	Generalize	Make Connections	Understand Concepts	Perform Procedures	Memorize
Similarity and the Geometry of Shape bG.11 The student applies the concepts of similarity to justify properties of figures and solve problems	bG.11A Use and extend similarity properties and transformations to explore and justify conjectures about geometric figures Ga2, Ga5 bG.11B Make conjectures from patterns of similar figures Ga2, Ga5 bG.11C Justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods Ga1, Ga5, Ga6	bG.11C Apply triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples to solve problems using a variety of methods Ga1, Ga2, Ga3, Ga4, Ga6	bG.11C Develop triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples to solve problems using a variety of methods Ga2, Ga3, Ga6 bG.11D Describe the effect on perimeter, area, and volume when one of more dimensions of a figure are changed and apply this idea in solving problems Ga2, Ga3, Ga5	bG.11B Use ratios to solve problems involving similar figures Ga1, Ga3, Ga5	Pythagorean triples Ratio Right triangle ratios Similarity Transformation Trigonometric ratios Ga1
Timeline	Textbook and Materials		NCTM Standards
			Use visualization, spatial reasoning, and geometric modeling to solve problems Specify locations and describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning, and geometric modeling to solve problems

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