Geometry Curriculum Map

Transcription

1 Quadrilaterals 7.1 Interior Angle Sum Theorem 7.2 Exterior Angle Sum Theorem 7.3 Using Interior and Exterior Angles to Solve Problems Define the Angle Sum Theorem. Illustrate interior angles with the Angle Sum Theorem with examples. Apply the Angle Sum Theorem to convex polygons. Define the Angle Sum Theorem. Illustrate exterior angles with the Angle Sum Theorem with examples. Apply the Angle Sum Theorem to convex polygons. Apply the Angle Sum Theorem to convex polygons. Combine interior and exterior angles to solve problems. Solve problems using the Angle Sum Theorem. 7.4 Quadrilaterals Define and identify quadrilaterals. Distinguish between types of quadrilaterals. 8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 2.G.A.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle

2 7.5 The Parallelogram Identify and label the parts of a parallelogram. Use midpoints to construct parallelograms. Prove that opposite sides of a parallelogram are congruent. 7.6 Parallelogram Proofs Prove that opposite angles of a parallelogram are congruent. Prove that rectangles are parallelograms with congruent diagonals. Prove that a parallelogram is a rectangle if and only if its diagonals are congruent. criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. G-CO.11 Prove theorems about parallelograms. G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G-CO.11 Prove theorems about parallelograms.

3 7.7 Rhombus Proofs Prove that if a parallelogram has two consecutive sides congruent, it is a rhombus. Prove that a parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G-CO.11 Prove theorems about parallelograms. 7.8 Algebraic Proofs involving Quadrilaterals 7.9 Applications Involving Quadrilaterals 7.10 Modeling Real-Life Situations with Quadrilaterals 7.11 Module Review Prove that a parallelogram is a rhombus if and only if the diagonals are perpendicular. 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, sqrt 3) lies on the circle centered at the origin and containing the point (0,2). Apply algebra to solve problems involving quadrilaterals. Model real-life situations using quadrilaterals. Solve design problems using quadrilaterals. G-GPE-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). 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).

4 Transformations 8.1 Rigid Motion in a Plane 8.2 Identifying Transformations between Two Figures Define and name transformations. Examine how to preserving angles and lengths in transformations. Identifying transformations in real life. Identify types of transformations between figures. Examine examples of multiple transformations of figures. Determine transformations between two figures. G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-CO.B.6 Use geometric descriptions of rigid

5 8.3 Constructing Multiple Transformations 8.4 Rotational and Reflectional Symmetry Review types of transformations. Analyze multiple transformations on figures. Construct multiple transformations on figures using tracing paper, graphing paper or software. Define rotational and reflectional symmetry. Illustrate rotational and reflectional symmetry through examples. Create examples that distinguish between rotational and reflectional symmetry. motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 4.G.A.3 Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

6 Solve problems involving rotational and reflectional symmetry. 8.5 Translations Define translations in the coordinate plane. Identify properties of translations. Transformations using vectors. Solving problems involving translations. G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

7 8.6 Transformation Problem Solving Review rotations, reflections and translations in the coordinate plane. Distinguish between different transformations. Solve problems using transformations. G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 8.7 Tessellations Define tessellations and their use in Geometry. Generate tessellations using tools such as paper and software.

8 8.8 Using Tessellations to Model Real-Life Problems 8.9 Applications of Transformations 8.10 Creating Frieze Patterns 8.11 Module Review Review the definition of tessellations and how they are made. Analyze tessellations in reallife scenarios. Review different types of transformations. Distinguish between the types of transformations. Solve real-life problems involving transformations. Define frieze patterns Create different frieze patterns using transformations. Research uses of frieze patterns in art, architecture, etc. MP.3 Construct viable arguments and critique the reasoning of others. MP.5 Use appropriate tools strategically. G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

9 Parts of a Circle 9.1 Parts of a Circle Identify the parts of a circle. 9.2 Circumference and Area of a Circle Examine the relationship between the parts of a circle. Define the area and circumference of a circle. Distinguish between the area and circumference of circle problems. Solve problems involving the area and circumference of a circle. 9.3 Arcs and Sectors Define arc, minor arc, major arc, semi-circle, and chord. Name and identify arcs and chords and state their relationship. Use the arc addition postulate to solve problems. Solve problems using the properties of chords and minor arcs in congruent circles. 9.4 Circumference and Arc Length Calculate circumference of a circle and the length of a circular arc. 4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. MP.5 Use appropriate tools strategically. 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 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. 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

10 Use circumference and arc length to solve real life problems. 9.5 Tangent to a Circle Identify the tangent of a circle. Construct a tangent line from a point outside of a circle using tools such as compass, straightedge and software. measure of the angle as the constant of proportionality; derive the formula for the area of a sector. G-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle. 9.6 Measuring Angles with Radians and Degrees 9.7 Inscribed and Circumscribed Angles 9.8 Inscribed Figures in Circles Examine the properties of tangent lines. Solve problems involving the tangent of a circle. Define angle measurements with radians vs degrees. Convert angles in radian measure to degree measure. Convert angles in degree measure to radian measure. Define inscribed and circumscribed angles in circles. Describe the relationships among inscribed angles, radii and chords. Use the inscribed angle theorem to solve problems algebraically. Describe inscribed figures inside of a circle. Draw inscribed figures such as 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. 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. G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G-C.A.3 Construct the inscribed and circumscribed

11 9.9 Finding Angles Involving Tangents and Circles triangles, squares and hexagons in a circle using tools such as straightedge, compass or software. Apply tangents in relation to circles. Illustrate angles formed from tangents on circles. Solve problems involving angles formed from tangent lines on circles Equation of a Circle Write the equation of a circle in the coordinate plane. Use the equation of a circle to graph the circle and solve related problems Module Review circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 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. G-GPE-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). 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. Area 10.1 Perimeter of Polygons Find the perimeter of various polygons. Find the perimeter of polygons in the coordinate plane. G.MG.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). G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,

12 10.2 Area of Polygons Find the area of various polygons. Find the area of polygons in the coordinate plane. e.g., using the distance formula.* G.MG.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) Sector Area Find the area of a sector. Find the area of a segment Calculating Area Find the area of unknown figures. G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* G-HSG.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. G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot) Discovering Solids Use density to calculate other quantities related to it and interpret these answers in terms of their contexts Cubes & Spheres Define and classify solids. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Pyramids & Cones Find surface area of a cone. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Cylinders & Prisms Find surface area of a pyramid. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects

13 composed of triangles, quadrilaterals, polygons, cubes, and right prisms Unit Conversions Find surface area of a prism. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Area Applications Find surface area of a cylinder. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Module Review Volume 11.1 Introduction to Volume Define volume. Analyze volume of objects problems through examples Volume of Cubes Define the volume of a cube formula. Analyze volume examples with cube. Solve problems involving the volume of cube Volume of Prisms "Define the volume of a prism formula. Analyze volume examples with prisms. CCSS.Math.Content.6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Solve problems involving the

14 11.4 Volume of Rectangular Prisms volume of prisms. Define the volume of a rectangular prism formula. Distinguish volume examples with prisms vs rectangular prisms. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Solve problems involving the volume of rectangular prisms Volume of Cylinders Define the volume of cylinders formula. Examine problems that involve the volume of cylinder. Apply dissection arguments, Cavalieri s Principle and informal limits to solve volume problems. Solve volume problems involving the volume of cylinders Volume of Spheres Define the volume of sphere formula. Examine problems that involve the volume of sphere. Solve volume problems involving the volume of sphere. G-HSG.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. G-HSG.GMD.A.2 (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. G-HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* G-HSG.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. G-HSG.GMD.A.2 (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. G-HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve

15 problems.* 11.7 Volume of Cones Define the volume of cones formula. Analyze problems involving the volume of cones. Solve problems that involve finding the volume of cones Volume of Pyramids Define the volume of pyramids formula. Analyze problems involving the volume of pyramids. Solve volume problems involving pyramids. G-HSG.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. G-HSG.GMD.A.2 (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. G-HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* G-HSG.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. G-HSG.GMD.A.2 (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures Changing Dimensions Identify how changing dimensions effect the resulting figure. G-HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* G-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

16 11.1O Solving Real-Life Volume Problems Module Review Solve real-life problems involving the concept of volume. G-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems Simple Events Review probability vocabulary. Calculate the probabilities of simple events. S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). Probability 12.2 Using an Area Model Use an area model to solve real-life problems and predict outcomes. S-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong

17 to A, and interpret the answer in terms of the model. S- CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model Using a Tree Diagram Use a tree diagram to represent probability situations and solve problems. Compare theoretical and experimental probability. S-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. S- CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. S-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer

18 12.4 Probability Models Determine which tool (a tree diagram, a systematic list, or an area model) is better for modeling certain situations. Calculate some expected values of a "fair" game. in terms of the model. S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. S- CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model Unions, Intersections, and Complements Use the language for calculating probabilities of unions, intersections, and complements of events. Use precise calculations to figure out probabilities as well as communicate findings Expected Value Solve problems involving chance. S-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics

19 Analyze and make conjectures about outcomes. Look for patterns around expected value Counting Use the Fundamental Principle of Counting to count permutations and other outcomes when there are too many to list Permutations Identify permutations. Develop two formulas for calculating the number of permutations Combinations Identify combinations. Compare permutations and combinations. Develop a method for counting Categorizing Counting Problems Module Review combinations. Determine the counting methods for situations that involve order and repetition, order and no repetition, no order with repetition, and no order without repetition. (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.