geometry(curriculum(map( - amazon s3 · 2015-02-13 · 4.g.a.1 draw points, lines, line segments,...

Geometry Curriculum Map

Module LO# Lesson Title Objective(s) Common Core State Standard(s)

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1.1 Notation 1) Define points, lines, segments, rays and planes. 2) Draw and label points, lines, segments, rays and planes.

G-CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MP.5 Use appropriate tools strategically.

1.2 Patterns and Conjectures

1) Complete patterns from given sequences of numbers and objects. 2) Draw general conclusions (conjectures) from given examples. 3) Verify the correctness of conclusions from given examples.

MP.2 Reason abstractly and quantitatively MP.7 Look for and make use of structure

1.3 Definitions and Postulates

1) Identify how definitions, postulates, and axioms will help you justify your answers. 2) Use the Segment Addition Postulate to solve problems algebraically.

MP.6 Attend to precision. MP.1 Make sense of problems and persevere in solving them.

1.4

Basic Geometric Shapes

1) Identify basic geometric shapes such as triangles, squares and circles. 2) Define properties of triangles, squares and circles. 3) Distinguish between shapes based on their properties.

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. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.


1.5 Congruency 1) Define congruence. 2) Construct congruent shapes using tools such as a ruler. 3) Define similarity.

G-CO.A.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.7 Use the definition of congruent in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

1.6 Perimeter, Circumference and Area

1) Define perimeter, circumference and area of rectangles, triangles and circles. 2) Solve problems involving the perimeter, circumference and area of rectangles and triangles. 3) Solve problems involving the circumference and area of a circle.

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. 4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.


1.7 Transformations 1) Define rotations, reflections and translations for objects and shapes. 2) Draw transformed figures given the shape and rotation, reflection or translation. 3) Determine the transformations that will carry a given figure onto another.

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.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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.

1.8 Geometric Constructions

1) Bisect a segment. 2) Bisect an angle. 3) Construct a perpendicular bisector of a line segment.

G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

1.9 Congruent Constructions

1) Construct congruent angles and segments. 2) Construct a figure congruent to a given figure.

7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. MP.5 Use appropriate tools strategically.


1.10 Reasoning through Logic Puzzles

1) Decide whether information drawn from a passage is true, false, or cannot be determined. 2) Draw correct conclusions from given statements. 3) Use logic to solve puzzles.

MP.3 Construct viable arguments and critique the reasoning of others. MP.5 Use appropriate tools strategically.

1.11 Module Review

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2.1 Geometric Proofs 1) Create informal proofs based on given conjectures. 2) Distinguish between an informal proof and a formal proof. 3) Describe the process for constructing a formal proof.


2.2 Conditional and Bi-Conditional Statements

1) Explain the concept of the conditional statement. 2) Distinguish between conditional and bi-conditional statements used in specific examples. 3) Create conditional and bi-conditional statements given situations.

MP.6 Attend to precision. MP.3 Construct viable arguments and critique the reasoning of others.

2.3 Conjectures 1) Use Venn Diagrams to verify statements are true. 2) Distinguish between conjectures and proofs.

MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. MP.5 Use appropriate tools strategically.

2.4 Conditional Forms 1) Define converse, inverse and contrapositive statements. 2) Write converse, inverse and



contrapositive statements. 3) Verify conclusions from a given conditional statement.

2.5 Syllogism 1) Define and identify syllogism. 2) Use (fill in the blank) syllogisms to validate conclusions. 3) Explain why a set of statements cannot be a syllogism.


2.6 Reasoning & Logic 1) Distinguish between inductive and deductive reasoning. 2) Illustrate the difference between inductive and deductive reasoning. 3) Create conjectures using inductive and deductive reasoning.


2.7 Counterexamples 1) Explain the nature of counterexamples. 2) Provide counterexamples for false conditional statements.


2.8 Paragraph Proofs 1) Use sentence explanations to initiate proof format.


2.9 Understanding Formal Proofs

1) Define the structure of formal proofs. 2) Complete fill in the blank proofs.


2.10 Writing Proofs 1) Construct proofs using logic statements.


Geometry Curriculum Map 2.11 Module Review

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3.1 Drawing and Measuring Angles

1) Define measures of angles. 2) Measure given angles using a protractor. 3) Draw angles using a protractor.

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.

3.2 Classifying Angles 1) Define the types of angles and triangles that exist. 2) Distinguish between types of angles. 3) Draw various types of angles given descriptions.


3.3 Angle Relationships 1) Describe the relationships between various angles. 2) Define complementary and supplementary angles using lines. 3) Distinguish relationships between angles in context.

7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

3.4 Proofs Involving Angles

1) Use angle properties (such as addition of angles) to make conjectures about angles. 2) Verify angle congruence from a given image. 3) Develop informal proofs verifying the conjectures made about angles.

G-CO.C.9 Prove theorems about lines and angles. Theorems include: 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.

3.5 Relationships Between Lines

1) Identify examples of the relationships between lines: intersecting, parallel, perpendicular, and skew. 2) Identify and compare the

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,


angles formed by transversals of parallel lines.

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.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

3.6 Parallel and Perpendicular Lines

1) Construct parallel and perpendicular lines using geometric tools. 2) Identify congruencies formed from perpendicular bisectors of segments.

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. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

3.7 Properties of Parallel Lines

1) Draw transversals of parallel lines. 2) Develop line and angle properties formed from transversals of parallel lines.

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. MP.5 Use appropriate tools strategically.

3.8 Proofs Involving Lines and Angles

1) Generate theorems that relate angle pairs in parallel lines.

G-CO.C.9 Prove theorems about lines and angles. Theorems include: 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.

3.9 Lines in the Coordinate Plane

1) Graph lines on a coordinate plane using tools such as

G-GPE.B.6 Find the point on a directed line segment between two given points that partitions


graph paper, straightedge or software. 2) Find the distance between two points on a line using the distance formula. 3) Find the midpoint on a line segment using the midpoint formula.

the segment in a given ratio. 6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. HSN.CN.B.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

3.10 Applications Involving Lines and Angles

1) Solve problems involving lines, angles and their properties.

7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

3.11 Module Review

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4.1 Angles and Triangles

1) Classify different types of angles and triangles. 2) Identify properties of triangles. 3) Define congruence for triangles.

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. 4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

4.2 Showing Congruence in Triangles

1) Explain CPCTC and its use. 2) Explain the criteria for triangle congruence (ASA, SAS, SSS).

G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are


3) Create congruence statements about congruent parts of pairs of triangles.

congruent. G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

4.3 Congruence in Right Triangles

1) Analyze side relationships between congruent and similar right triangles. 2) Informally prove that right triangles are congruent through HL. 3) Solve problems involving HL criteria for congruence in right triangles.

G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MP.6 Attend to precision.

4.4 Ratio and Proportion 1) Define ratio and proportion. 2) Calculate ratios and proportion in examples. 3) Solve problems involving ratios and proportions of figures.

G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

4.5 Solving Problems with Ratios and Proportions

1) Solve real-life problems involving ratios and proportions of figures.

7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

4.6 Similarity 1) Define similarity for figures in terms of transformations. 2) Explain how similar shapes are created using dilation of line segments. 3) Distinguish between congruence and similarity, using knowledge of transformations.

G-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for


triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

4.7 Similar Triangles 1) Define similarity for triangles in terms of transformations. 2) Decide between congruence and similarity for given triangles. 3) Create similarity statements for parts of similar triangles.

G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

4.8 Triangle Sum Theorem

1) Create conjectures about angle relationships such as interior and exterior angles in triangles. 2) Develop an informal proof for triangle similarity using AA criterion.

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. G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

4.9 Proofs Involving Congruence and Similarity

1) Use angle and line relationships to make conjectures about congruent and similar triangles. 2) Illustrate angle and line relationships in congruent and similar triangles. 3) Develop informal proofs verifying the conjectures made about triangles.

G-SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a


triangle meet at a point.

4.10 Applications Involving Congruence and Similarity

1) Differentiate between congruent and similar triangles. 2) Explain the angle-side relationships in congruent and similar triangles. 3) Solve problems involving congruent and similar triangles.

G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

4.11 Module Review

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5.1 Triangle Properties 1) Identify specific triangle types, including right, isosceles, scalene, and equilateral. 2) Identify the parts of specific triangles types. 3) Construct triangles. 4) Prove that the measures of the interior angles of a triangle have a sum of 180 degrees.

4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

5.2 Pythagorean Theorem

1) Apply the Pythagorean Theorem to find an unknown side length of a right triangle. 2) Construct a diagram and use the Pythagorean Theorem to solve real world problems involving right triangles.

8.G.B.7 Understand and apply the Pythagorean Theorem. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. MP.6 Attend to precision.

5.3 Proving the 1) Prove that a line parallel to G-SRT.B.4 Prove theorems about triangles.


Pythagorean Theorem

one side of a triangle divides the other two sides proportionally and conversely. 2) Prove the Pythagorean theorem using triangle similarity.

Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

5.4 Isosceles Triangles 1) Prove base angles of isosceles triangles are congruent.

G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

5.5 Medians and Altitudes of Triangles

1) Prove the medians of a triangle meet at a point. 2) Prove that the altitude drawn to the hypotenuse of a right triangle creates two similar right triangles, each similar to the original right triangle and similar to each other.

G-SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

5.6 The Midsegment Theorem

1) Solve problems using the midsegment theorem. 2) State and extend the midsegment theorem to trapezoids.

MP.1 Make sense of problems and persevere in solving them. MP.6 Attend to precision.

5.7 Inequalities in One Triangle

1) Explore that any side of a triangle must be shorter than the other two sides added together. 2) Use this rule to understand and solve problems.

MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision.


5.8 Inequalities in Two Triangles

1) Prove the Hinge Theorem. 2) Prove the converse of the Hinge Theorem.

MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others.

5.9 Proportional Triangles

1) Prove the leg rule. 2) Prove the altitude rule.

G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MP.6 Attend to precision.

5.1O Triangle Applications 1) Apply various triangle postulates and theorems to solving problems involving triangles.

G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MP.6 Attend to precision.

5.11 Module Review

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6.1 Similarity in Right Triangles

1) List parts and properties of a right triangle. 2) Establish relationships between sides and acute angles in right triangles. 3) Use the geometric mean to find segment lengths in right triangles. 4) Apply similarity relationships in right triangles to solve problems.

8.G.B.7 Understand and apply the Pythagorean Theorem. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

6.2 Using Special Right Triangles

1) State the relationship between the sides of a 30-60-90 and 45-45-90 triangle. 2) Use the relationships

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques


between the sides of a 30-60-90 and 45-45-90 triangle to find missing lengths of the sides of triangles. 3) Calculate the area of a right triangle.

in the context of solving real-world and mathematical problems. MP.6 Attend to precision.

6.3 The Tangent Ratio 1) Define the tangent ratio. 2) Determine the tangent ratio for a right triangle with known leg lengths.

G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MP.6 Attend to precision.

6.4 Applying the Tangent Ratio

1) Use the tangent ratio to measure distance.

G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MP.6 Attend to precision.

6.5 Sine, Cosine & Applications

1) Define the sine and cosine ratios. 2) Determine the sine and cosine ratios for a right triangle with known leg lengths.

G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

6.6 Law of Sines 1) Define the law of sines. 2) Use the law of sines to

G-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.


solve for triangles. G-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

6.7 Law of Cosines 1) Define the law of cosines. 2) Use the law of cosines to solve for triangles.

G-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

6.8 Applications of all Right Triangles

1) Use definitions, properties, and theorems of triangles to solve right triangle problems.

G-SRT.D.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

6.9 Right Triangles in Rectangles

1) Find the lengths of sides in a rectangular solid using right triangle properties. 2) Find the missing parts of a right triangle within a rectangular solid.

G-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

6.10 Distance Formula in 2 & 3 Dimensions

1) Apply the Pythagorean theorem to calculate distances in two dimensions. 2) Apply the Pythagorean theorem to calculate distances in three dimensions.

6.11 Module Review

Geometry Curriculum Map Q

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7.1 Interior Angle Sum Theorem

Define the Angle Sum Theorem. Illustrate interior angles with the Angle Sum Theorem with examples. Apply the Angle Sum Theorem to convex polygons.

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.

7.2 Exterior Angle Sum Theorem

Define the Angle Sum Theorem. Illustrate exterior angles with the Angle Sum Theorem with examples. Apply the Angle Sum Theorem to convex polygons.


7.3 Using Interior and Exterior Angles to Solve Problems

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.

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


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.

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.

G-CO.11 Prove theorems about parallelograms.

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.

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.


7.7 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. Prove that a parallelogram is a rhombus if and only if the diagonals are perpendicular.

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

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).

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).

7.9 Applications Involving Quadrilaterals

Apply algebra to solve problems involving quadrilaterals.

7.10 Modeling Real-Life Situations with Quadrilaterals

Model real-life situations using quadrilaterals. Solve design problems using quadrilaterals.

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).

7.11 Module Review


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8.1 Rigid Motion in a Plane

Define and name transformations. Examine how to preserving angles and lengths in transformations. Identifying transformations in real life.

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.

8.2 Identifying Transformations between Two Figures

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.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.

8.3 Constructing Multiple Transformations

Review types of transformations. Analyze multiple transformations on figures. Construct multiple transformations on figures using tracing paper, graphing paper or software.

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.

8.4 Rotational and Reflectional Symmetry

Define rotational and reflectional symmetry. Illustrate rotational and reflectional symmetry through examples. Create examples that distinguish between rotational and reflectional symmetry.

4.G.A.3 Recognize a line of symmetry for a two-dimensional 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.


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.


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 Using Tessellations to Model Real-Life Problems

Review the definition of tessellations and how they are made. Analyze tessellations in real-life scenarios.


8.9 Applications of Transformations

Review different types of transformations. Distinguish between the types of transformations. Solve real-life problems involving 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.

8.10 Creating Frieze Patterns

Define frieze patterns Create different frieze patterns using transformations. Research uses of frieze patterns in art, architecture, etc.

8.11 Module Review


Part

s of

a C

ircle

9.1 Parts of a Circle Identify the parts of a circle. Examine the relationship between the parts of a circle.


9.2 Circumference and Area 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.

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.

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.

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


Use circumference and arc length to solve real life problems.

measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

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. Examine the properties of tangent lines. Solve problems involving the tangent of a circle.

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

Define angle measurements with radians vs degrees. Convert angles in radian measure to degree measure. Convert angles in degree measure to radian measure.

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.

9.7 Inscribed and Circumscribed Angles

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.

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.

9.8 Inscribed Figures in Circles

Describe inscribed figures inside of a circle. Draw inscribed figures such as

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


triangles, squares and hexagons in a circle using tools such as straightedge, compass or software.

circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

9.9 Finding Angles Involving Tangents and Circles

Apply tangents in relation to circles. Illustrate angles formed from tangents on circles. Solve problems involving angles formed from tangent lines on circles.

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).

9.10 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.

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.

9.11 Module Review

Are

a

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,


e.g., using the distance formula.* 10.2 Area of Polygons Find the area of various

polygons. Find the area 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, e.g., using the distance formula.*

10.3 Sector Area Find the area of a sector. Find the area of a segment.

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.

10.4 Calculating Area Find the area of unknown figures.

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).★

10.5 Discovering Solids Use density to calculate other quantities related to it and interpret these answers in terms of their contexts.

10.6 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.

10.7 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.

10.8 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


composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

10.9 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.

10.10 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.

10.11 Module Review

Volu

me

11.1 Introduction to Volume

Define volume. Analyze volume of objects problems through examples.

11.2 Volume of Cubes Define the volume of a cube formula. Analyze volume examples with cube. Solve problems involving the volume of cube.

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.

11.3 Volume of Prisms "Define the volume of a prism formula. Analyze volume examples with prisms. Solve problems involving the

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.


volume of prisms. 11.4 Volume of

Rectangular Prisms Define the volume of a rectangular prism formula. Distinguish volume examples with prisms vs rectangular prisms. Solve problems involving the volume of 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.

11.5 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.

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.*

11.6 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.*

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.


11.8 Volume of Pyramids Define the volume of pyramids formula. Analyze problems involving the volume of pyramids. Solve volume problems involving pyramids.


11.9 Changing Dimensions

Identify how changing dimensions effect the resulting figure.

G-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★


11.1O Solving Real-Life Volume Problems

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.★

11.11 Module Review

Prob

abili

ty

12.1 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”). 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.

12.2 Using an Area Model Use an area model to solve real-life problems and predict outcomes.

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 in terms of the model.

12.3 Using a Tree Diagram

Use a tree diagram to represent probability situations and solve problems. Compare theoretical and experimental probability.

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


in terms of the model. 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.

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 in terms of the model.

12.5 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.

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”).

12.6 Expected Value Solve problems involving chance.

S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics


Analyze and make conjectures about outcomes. Look for patterns around expected value.

(or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

12.7 Counting Use the Fundamental Principle of Counting to count permutations and other outcomes when there are too many to list.

S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

12.8 Permutations Identify permutations. Develop two formulas for calculating the number of permutations.


12.9 Combinations Identify combinations. Compare permutations and combinations. Develop a method for counting combinations.


12.10 Categorizing Counting Problems

Determine the counting methods for situations that involve order and repetition, order and no repetition, no order with repetition, and no order without repetition.


12.11 Module Review

geometry(curriculum(map( - amazon s3 · 2015-02-13 · 4.g.a.1 draw points, lines, line segments,...

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