Michigan Edition correlated to the Michigan Merit Curriculum Course / Credit Requirements Geometry
McDougal Littell Geometry 2008 (Michigan Edition) correlated to the Michigan Merit Curriuclum Course / Credit Requirements Geometry STRAND 1: QUANTITATIVE LITERACY AND LOGIC (L) STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason about numbers, number systems, and the relationships between them. They represent quantitative relationships using mathematical symbols, and interpret relationships from those representations. L1.1 Number Systems and Number Sense L1.1.6 Explain the importance of the irrational numbers 2 and 3 in basic right triangle trigonometry and the importance of π because of its role in circle relationships. Geom: Lesson 1.7: Find Perimeter, Circumference, and Area, 49-56 Lesson 7.4: Special Right Triangles, 457-464 Lesson 11.4: Circumference and Arc Length, 746-752 Lesson 11.5: Areas of Circles and Sectors, 755-761 L1.2 Representations and Relationships L1.2.3 Use vectors to represent quantities that have magnitude and direction, interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors. Geom: Lesson 9.1: Vectors, 574-579 1
STANDARD L2: CALCULATION, ALGORITHMS AND ESTIMATION Students calculate fluently, estimate proficiently and describe and use algorithms in appropriate situations (e.g., approximating solutions to equations.) They understand the basic ideas of iteration and algorithms. L2.3 Measurement Units, Calculations and Scales L2.3.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly. Geom: Lesson 1.7: Find Perimeter, Circumference, and Area, 49, 51, 53-54 Lesson 6.1: Unit Analysis, 356, 359, 362-363 Lesson 6.2: Unit Analysis, 366 Lesson 7.4: Unit Analysis, 460 Lesson 11.1: Unit Analysis, 722 Lesson 11.3: Unit Analysis, 739 Lesson 12.4: Problem Solving Workshop: Using Alternative Methods, 826-827 Skills Review Handbook: Converting Measurements and Rates, 886 Tables: Measures, 921 2
STANDARD L3: MATHEMATICAL REASONING, LOGIC AND PROOF Students understand mathematical reasoning as being grounded in logic and proof and can distinguish mathematical arguments from other types of arguments. They can interpret arguments made about quantitative situations in the popular media. Students know the language and laws of logic and can apply them in both mathematical and everyday settings. They write proofs using direct and indirect methods and use counterexamples appropriately to show that statements are false. L3.1 Mathematical Reasoning L3.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each. Geom: Lesson 2.1: Use Inductive Reasoning, 72-78 Lesson 2.3: Apply Deductive Reasoning, 87-93 L3.1.2 L3.1.3 Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic. Geom: (related topics) Lesson 2.1: Use Inductive Reasoning, 72-78 Lesson 2.3: Apply Deductive Reasoning, 87-93 Lesson 6.1: Sampling and Surveys, 359, 362, 369 Skills Review Handbook: Sampling and Surveys, 890 Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples and proofs in the logical structure of mathematics. Identify and give examples of each. Geom: Lesson 1.1: Identify Points, Lines, and Planes, 2-8 Lesson 2.1: Use Inductive Reasoning, 72-78 Lesson 2.6: Prove Statements about Segments and Angles, 112-119 Postulates: 9, 10, 24, 25, 96, 126, 148, 154, 161, 172, 234, 240, 249, 381, 660, 720, 819, 926 Proofs: Throughout text Theorems: Throughout text Tables: Postulates, 926; Theorems, 927-931 3
L3.2 Language and Laws of Logic L3.2.1 Know and use the terms of basic logic. Geom: Lesson 2.1: Use Inductive Reasoning, 72-78 Lesson 2.2: Analyze Conditional Statements, 79-85 Lesson 2.3: Apply Deductive Reasoning, 87-93 Lesson 2.3: Extension: Symbolic Notation and Truth Tables, 94-95 L3.2.2 L3.2.3 L3.2.4 Use the connectives not, and, or, and if, then, in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives. Geom: Lesson 2.2: Analyze Conditional Statements, 79-85 Lesson 2.3: Apply Deductive Reasoning, 87-93 Lesson 2.3: Extension: Symbolic Notation and Truth Tables, 94-95 Use the quantifiers there exists and all in mathematical and everyday settings and know how to logically negate statements involving them. Geom: (related sections) Lesson 2.2: Analyze Conditional Statements, 79-85 Lesson 2.3: Apply Deductive Reasoning, 87-93 Lesson 2.3: Extension: Symbolic Notation and Truth Tables, 94-95 Write the converse, inverse and contrapositive of an If, then statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original while the inverse and converse are not. Geom: Lesson 2.2: Analyze Conditional Statements, 79-85 Lesson 2.3: Extension: Symbolic Notation and Truth Tables, 94-95 Lesson 5.6: Using a Contrapositive, 341 Lesson 10.3: Apply Properties of Chords, 668 4
L3.3 Proof L3.3.1 L3.3.2 L3.3.3 Know the basic structure for the proof of an If, then statement (assuming the hypothesis and ending with the conclusion) and that proving the contrapositive is equivalent. Geom: Lesson 2.6: Prove Statements about Segments and Angles, 112-119, 120-121 Lesson 2.7: Prove Angle Pair Relationships, 124-131 Lesson 3.2: Parallel and Perpendicular Lines: 156, 159-160 Lesson 3.3: Prove Lines Are Parallel, 163-169 Lesson 4.5: Congruent Triangles, 254 Lesson 4.7: Using Isosceles and Equilateral Triangles, 269 Lesson 5.3: Use Angle Bisectors of Triangles, 315 Lesson 5.6: Using a Contrapositive, 341 Lesson 6.3: Use Similar Polygons, 379 Lesson 6.4: Prove Triangles Similar by AA, 386 Lesson 8.2: Use Properties of Parallelograms, 521 Lesson 10.1: Use Properties of Tangents, 658 Lesson 10.4: Use Inscribed Angles and Polygons, 678 Lesson 10.5: Apply Other Angle Relationships in Circles, 685 Lesson 10.6: Find Segment Lengths in Circles, 694 Construct proofs by contradiction. Use counterexamples, when appropriate, to disprove a statement. Geom: Lesson 2.1: Counterexamples, 74, 76, 78 Lesson 5.6: Indirect Reasoning, 337-338, 339-340 Lesson 8.5: Use Properties of Trapezoids and Kites, 549 Lesson 10.1: Use Properties of Tangents, 658 Lesson 10.3: Apply Properties of Chords, 670 Explain the difference between a necessary and a sufficient condition within the statement of a theorem. Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied. Geom: (related topics) Lesson 2.2: Biconditional Statements, 82 Lesson 2.3: Extension: Symbolic Notation and Truth Tables, 94-95 Lesson 8.4: Biconditionals, 536 5
STRAND 3: GEOMETRY AND TRIGONOMETRY (G) STANDARD G1: FIGURES AND THEIR PROPERTIES Students represent basic geometric figures, polygons and conic sections and apply their definitions and properties in solving problems and justifying arguments, including constructions and representations in the coordinate plane. Students represent 3-dimensional figures, understand the concepts of volume and surface area and use them to solve problems. They know and apply properties of common 3-dimensional figures. G1.1 Lines and Angles; Basic Euclidean and Coordinate Geometry G1.1.1 Solve multi-step problems and construct proofs involving vertical angles, linear pairs of angles, supplementary angles, complementary angles and right angles. Geom: Lesson 1.5: Describe Angle Pair Relationships, 35-41 Lesson 2.6: Prove Statements about Segments and Angles, 117-118 Lesson 2.7: Prove Angle Pair Relationships, 122-123, 124-131 G1.1.2 G1.1.3 G1.1.4 G1.1.5 G1.1.6 Solve multi-step problems and construct proofs involving corresponding angles, alternate interior angles, alternate exterior angles and same-side (consecutive) interior angles. Geom: Lesson 3.1: Identify Pairs of Lines and Angles, 147-152 Lesson 3.2: Use Parallel Lines and Transversals, 153, 154-160 Lesson 3.3: Prove Lines Are Parallel, 161-169 Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass. Geom: Lesson 1.4: Construction: Copy and Bisect Segments and Angles, 33-34 Lesson 3.3: Challenge, 169 Lesson 4.6: Proving a Construction, 261 Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass. Given a line and a point, construct a line through the point that is perpendicular to the original line. Justify the steps of the constructions. Geom: Lesson 3.1: Construction, 152 Lesson 3.3: Challenge, 169 Lesson 3.6: Construction, 195 Lesson 5.4: Construction, 323 Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint. Geom: Lesson 1.3: Use Distance and Midpoint Formulas, 15-22 Lesson 5.1: Midsegment Theorem and Coordinate Proof, 297-299 Recognize Euclidean Geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms, axioms, definitions and theorems. Geom: Lesson 1.1: Undefined and Defined Terms, 2-3 Lesson 2.4: Postulates, 96 Lesson 2.6: Theorems, 113 Tables: Postulates, 926; Theorems, 927-931 Additional Lesson A: Euclidean and Non-Euclidean Geometry, A2-A3 6
G1.2 Triangles and Their Properties G1.2.1 Prove that the angle sum of a triangle is 180 and that an exterior angle of a triangle is the sum of the two remote interior angles. Geom: Lesson 4.1: Apply Triangle Sum Properties, 216, 218-219, 223 G1.2.2 G1.2.3 G1.2.4 G1.2.5 Construct and justify arguments and solve multi-step problems involving angle measure, side length, perimeter and area of all types of triangles. Geom: Lesson 1.7: Find Perimeter, Circumference, and Area, 49-56 Lesson 4.1: Apply Triangle Sum Properties, 216, 217-224 Lesson 4.2: Apply Congruence and Triangles, 227, 229-231 Lesson 4.3: Prove Triangles Congruent by SSS, 234-239 Lesson 4.4: Prove Triangles Congruent by SAS and HL, 240-246 Lesson 4.5: Prove Triangles Congruent by ASA and AAS, 249-255 Lesson 4.6: Use Congruent Triangles, 256-263 Lesson 4.7: Use Isosceles and Equilateral Triangles, 265-270 Lesson 5.1: Midsegment Theorem and Coordinate Proof, 297-301 Lesson 5.2: Use Perpendicular Bisectors, 303-309 Lesson 5.3: Use Angle Bisectors of Triangles, 310-316 Lesson 5.5: Use Inequalities in a Triangle, 328-334 Lesson 5.6: Inequalities in Two Triangles and Indirect Proof, 335-341 Lesson 6.3: Use Similar Polygons, 373, 376-378 Lesson 6.4: Prove Triangles Similar by AA, 381-387 Lesson 6.5: Prove Triangles Similar by SSS and SAS, 388-395 Lesson 7.1: Apply the Pythagorean Theorem, 433-439 Lesson 7.2: Use the Converse of the Pythagorean Theorem, 440, 441-447 Lesson 7.3: Use Similar Right Triangles, 449-456 Lesson 7.4: Special Right Triangles, 457-464 Lesson 7.5: Apply the Tangent Ratio, 466-472 Lesson 7.6: Apply the Sine and Cosine Ratios, 473-480, 481-482 Lesson 7.7: Solve Right Triangles, 483-489 Lesson 11.1: Areas of Triangles, 721-726 Know a proof of the Pythagorean Theorem and use the Pythagorean Theorem and its converse to solve multi-step problems. Geom: Lesson 7.1: Apply the Pythagorean Theorem, 432-439 Lesson 7.2: Use the Converse of the Pythagorean Theorem, 441-447 Lesson 7.3: Use Similar Right Triangles, 455 Lesson 7.6: Problem-Solving Workshop: Using Alternative Methods, 481-482 Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º triangles and 45º- 45º- 90º triangles. Geom: Lesson 7.4: Special Right Triangles, 457-464 Solve multi-step problems and construct proofs about the properties of medians, altitudes and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a triangle. Using a straightedge and compass, construct these lines. Geom: Lesson 5.2: Use Perpendicular Bisectors, 303-309 Lesson 5.3: Use Angle Bisectors of Triangles, 310-316 Lesson 5.4: Use Median and Altitudes, 318, 319-325, 326-327 7
G1.3 Triangles and Trigonometry G1.3.1 Define the sine, cosine and tangent of acute angles in a right triangle as ratios of sides. Solve problems about angles, side lengths, or areas using trigonometric ratios in right triangles. Geom: Lesson 7.5: Apply the Tangent Ratio, 466-472 Lesson 7.6: Apply the Sine and Cosine Ratios, 473-480, 481-482 Lesson 7.7: Solve Right Triangles, 483-489 G1.3.2 G1.3.3 Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle ø using the formula Area = (1/2) ab sin ø. Geom: Lesson 7.7: Extension: Law of Sines and Law of Cosines, 490-491 Determine the exact values of sine, cosine and tangent for 0, 30, 45, 60 and their integer multiples and apply in various contexts. Geom: Lesson 7.5: Apply the Tangent Ratio, 469 Lesson 7.6: Apply the Sine and Cosine Ratios, 476-477, 480 G1.4 Quadrilaterals and Their Properties G1.4.1 Solve multi-step problems and construct proofs involving angle measure, side length, diagonal length, perimeter and area of squares, rectangles, parallelograms, kites and trapezoids. Geom: Lesson 1.7: Find Perimeter and Area, 49, 51, 52-56, 57 Lesson 8.2: Use Properties of Parallelograms, 515-521 Lesson 8.3: Show that a Quadrilateral is a Parallelogram, 522-529 Lesson 8.4: Properties of Rhombuses, Rectangles, and Squares, 533-540, 541 Lesson 8.5: Use Properties of Trapezoids and Kits, 542-549 Lesson 8.6: Identify Special Quadrilaterals, 557 Lesson 11.1: Areas of Triangles and Parallelograms, 720-726 Lesson 11.2: Areas of Trapezoids, Rhombuses, and Kites, 729, 730-736 G1.4.2 G1.4.3 G1.4.4 Solve multi-step problems and construct proofs involving quadrilaterals using Euclidean methods or coordinate geometry. Geom: Lesson 8.2: Use Properties of Parallelograms, 515-521 Lesson 8.3: Show that a Quadrilateral Is a Parallelogram, 522-530, 530-531 Lesson 8.4: Properties of Rhombuses, Rectangles, and Squares, 533-540, 541 Lesson 8.5: Use Properties of Trapezoids and Kits, 542-549 Lesson 8.6: Identify Special Quadrilaterals, 555 Describe and justify hierarchical relationships among quadrilaterals. Geom: Lesson 8.3: Show that a Quadrilateral Is a Parallelogram, 522-530, 530-531 Lesson 8.6: Identify Special Quadrilaterals, 552-557 Prove theorems about the interior and exterior angle sums of a quadrilateral. Geom: Lesson 8.1: Find Angle Measures in Polygons, 507-515 8
G1.5 Other Polygons and Their Properties G1.5.1 Know and use subdivision or circumscription methods to find areas of polygons. Geom: Lesson 11.1: Areas of Triangles and Parallelograms, 722, 724-725 Lesson 11.2: Areas of Trapezoids, Rhombuses, and Kites, 734-735 Lesson 11.6: Areas of Regular Polygons, 762-768, 769 G1.5.2 Know, justify and use formulas for the perimeter and area of a regular n-gon and formulas to find interior and exterior angles of a regular n-gon and their sums. Geom: Lesson 8.1: Find Angle Measures in Polygons, 510-513 Lesson 11.6: Areas of Regular Polygons, 762-768, 769 G1.6 Circles and Their Properties G1.6.1 Solve multi-step problems involving circumference and area of circles. Geom: Lesson 1.7: Find Circumference and Area, 49-50, 53-55 Lesson 11.4: Circumference, 746-747, 749, 750, 751 Lesson 11.5: Areas of Circles, 755, 758, 760, 761 G1.6.2 G1.6.3 G1.6.4 Solve problems and justify arguments about chords and lines tangent to circles. Geom: Lesson 10.1: Use Properties of Tangents, 650, 651-658 Lesson 10.3: Apply Properties of Chords, 664-670 Lesson 10.6: Find Segment Lengths in Circles, 689-695 Solve problems and justify arguments about central angles, inscribed angles and triangles in circles. Geom: Lesson 10.4: Use Inscribed Angles, 671, 672-679 Know and use properties of arcs and sectors and find lengths of arcs and areas of sectors. Geom: Lesson 10.2: Find Arc Measures, 659-663 Lesson 11.4: Arc Length, 747-752 Lesson 11.5: Areas of Sectors, 756-761 G1.8 3-Dimensional Figures G1.8.1 Solve multi-step problems involving surface area and volume of pyramids, prisms, cones, cylinders, hemispheres and spheres. Geom: Lesson 12.2: Surface Area of Prisms and Cylinders, 802, 803-809 Lesson 12.3: Surface Area of Pyramids and Cones, 810-817 Lesson 12.4: Volume of Prisms and Cylinders, 819-825, 826-827 Lesson 12.5: Volume of Pyramids and Cones, 828, 829-836, 837 Lesson 12.6: Surface Area and Volume of Spheres, 838-845 Lesson 12.7: Explore Similar Solids, 846, 847-854 G1.8.2 Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres and spheres. Geom: (related topics) Lesson 8.5: Extension: Draw Three-Dimensional Figures, 550 Lesson 12.1: Explore Solids, 794-801 9
STANDARD G2: RELATIONSHIPS BETWEEN FIGURES Students use and justify relationships between lines, angles, area and volume formulas and 2- and 3-dimensional representations. They solve problems and provide proofs about congruence and similarity. G2.1 Relationships Between Area and Volume Formulas G2.1.1 Know and demonstrate the relationships between the area formula of a triangle, the area formula of a parallelogram and the area formula of a trapezoid. Geom: Lesson 11.1: Areas of Triangles and Parallelograms, 720-726 Lesson 11.2: Areas of Trapezoids, Rhombuses, and Kites, 729, 730-736 G2.1.2 G2.1.3 Know and demonstrate the relationships between the area formulas of various quadrilaterals. Geom: Lesson 11.1: Areas of Triangles and Parallelograms, 720-726 Lesson 11.2: Areas of Trapezoids, Rhombuses, and Kites, 729, 730-736 Know and use the relationship between the volumes of pyramids and prisms. Geom: Lesson 12.5: Volume of Pyramids and Cones, 828, 829, 834 G2.2 Relationships Between 2-dimensional and 3-dimensional Representations G2.2.1 Identify or sketch a possible 3-dimensional figure, given 2-dimensional views. Create a 2- dimensional representation of a 3-dimensional figure. Geom: Lesson 8.5: Extension: Draw Three-Dimensional Figures, 550-551 Lesson 12.1: Explore Solids, 794-801 G2.2.2 Identify or sketch cross-sections of 3-dimensional figures. Identify or sketch solids formed by revolving 2-dimensional figures around lines. Geom: Lesson 12.1: Explore Solids, 794-801 10
G2.3 Congruence and Similarity G2.3.1 Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria and that right triangles are congruent using the hypotenuse-leg criterion. Geom: Lesson 4.3: Prove Triangles Congruent by SSS, 234-239 Lesson 4.4: Prove Triangles Congruent by SAS and HL, 240-246, 247 Lesson 4.5: Prove Triangles Congruent by ASA and AAS, 249-255 G2.3.2 G2.3.3 G2.3.4 G2.3.5 Use theorems about congruent triangles to prove additional theorems and solve problems, with and without use of coordinates. Geom: Lesson 4.6: Use Congruent Triangles, 256-263 Lesson 4.7: Use Isosceles and Equilateral Triangles, 264-270 Prove that triangles are similar by using SSS, SAS and AA conditions for similarity. Geom: Lesson 6.4: Prove Triangles Similar by AA, 381-387 Lesson 6.5: Prove Triangles Similar by SSS and SAS, 388-395 Use theorems about similar triangles to solve problems with and without use of coordinates. Geom: Lesson 6.4: Prove Triangles Similar by AA, 383, 386-387 Lesson 6.5: Prove Triangles Similar by SSS and SAS, 390, 393-395 Know and apply the theorem stating that the effect of a scale factor of k relating one twodimensional figure to another or one three-dimensional figure to another, on the length, area and volume of the figures is to multiply each by k, k 2, and k 3, respectively. Geom: Lesson 6.3: Use Similar Polygons, 372-379 Lesson 11.3: Perimeter and Area of Similar Figures, 737-743, 744 Lesson 12.7: Explore Similar Solids, 846, 847-854 11
STANDARD G3: TRANSFORMATIONS OF FIGURES IN THE PLANE Students will solve problems about distance-preserving transformations and shape-preserving transformations. The transformations will be described synthetically and in simple cases, by analytic expressions in coordinates. G3.1 Distance-preserving Transformations: Isometries G3.1.1 Define reflection, rotation, translation and glide reflection and find the image of a figure under a given isometry. Geom: Lesson 4.8: Perform Congruence Transformations, 271, 272-279 Lesson 9.1: Translate Figures and Use Vectors, 572-579 Lesson 9.2: Use Properties of Matrices, 581, 585 Lesson 9.3: Perform Reflections, 588, 589-596 Lesson 9.4: Perform Rotations, 598-606 G3.1.2 G3.1.3 Given two figures that are images of each other under an isometry, find the isometry and describe it completely. Geom: Lesson 4.8: Perform Congruence Transformations, 271, 272-279 Lesson 9.1: Translate Figures and Use Vectors, 572-579 Lesson 9.3: Perform Reflections, 588, 589-596 Lesson 9.4: Perform Rotations, 598-606 Find the image of a figure under the composition of two or more isometries and determine whether the resulting figure is a reflection, rotation, translation or glide reflection image of the original figure. Geom: Lesson 9.5: Apply Compositions of Transformations, 607, 608-615 G3.2 Shape-preserving Transformations: Dilations and Isometries G3.2.1 Know the definition of dilation and find the image of a figure under a given dilation. Geom: Lesson 6.7: Perform Similarity Transformations, 408, 409-415 Lesson 9.7: Identify and Perform Dilations, 625, 626-632 G3.2.2 Given two figures that are images of each other under some dilation, identify the center and magnitude of the dilation. Geom: Lesson 6.7: Perform Similarity Transformations, 411-414 Lesson 9.7: Identify and Perform Dilations, 626-632 RECOMMENDED: *G1.4.5 Understand the definition of a cyclic quadrilateral, and know and use the basic properties of cyclic quadrilaterals. Geom: Lesson 10.4: Inscribed Quadrilateral, 675-678 *G3.2.3 Find the image of a figure under the composition of a dilation and an isometry. Geom: Lesson 9.7: Find Image of a Composition, 628, 630, 633 12
MI 265 12/2007 2008 CC2