West Virginia

Basics of Geometry

• Experiment with transformations in the plane.

  • M.GHS.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.

    • Angle vocabulary (G-C.1)

  • Checkpoint opportunity

    • Checkpoint: Definitions of geometric objects (G-D.9)

• Identify and utilize inductive and deductive reasoning.

  • M.GHS.2 Construct and justify the validity of a logical argument.

    • M.GHS.2.a Identify the converse, inverse, and contrapositive of a conditional statement.

      • Converses, inverses, and contrapositives (G-A.5)

    • M.GHS.2.b Translate a short, verbal argument into symbolic form.

      • Truth tables (G-A.7)
      • Truth values (G-A.8)

    • M.GHS.2.c Use Venn diagrams to represent set relationships.

      • Use Venn diagrams to solve problems (G)

    • M.GHS.2.d Use inductive and deductive reasoning.

      • Counterexamples (G-A.2)
      • Conditionals (G-A.3)
      • Negations (G-A.4)
      • Biconditionals (G-A.6)

• Prove geometric theorems.

  • M.GHS.3 Use appropriate methods of proof to 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.

    • Perpendicular Bisector Theorem (G-B.9)
    • Angle bisectors (G-C.6)
    • Proofs involving angles (G-C.9)
    • Transversals of parallel lines: solve for x (G-D.5)
    • Proofs involving parallel lines I (G-D.7)
    • Proofs involving parallel lines II (G-D.8)

  • Checkpoint opportunity

    • Checkpoint: Line and angle theorems (G-D.10)

• Use coordinates to prove simple geometric theorems algebraically.

  • M.GHS.4 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

    • Midpoints (G-B.6)
    • Midpoint formula: find the midpoint (G-B.10)
    • Partition a line segment in a given ratio (G-B.11)
    • Midpoint formula: find the endpoint (G-B.12)

  • Checkpoint opportunity

    • Checkpoint: Partition a line segment (G-B.16)

• Make geometric constructions.

  • M.GHS.5 Make formal geometric constructions with a variety of tools and methods, such as a compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.:

    • M.GHS.5.a copying a segment;

      • Construct a congruent segment (G-B.8)

    • M.GHS.5.b copying an angle;

      • Construct a congruent angle (G-C.8)

    • M.GHS.5.c bisecting a segment;

      • Construct the midpoint or perpendicular bisector of a segment (G-B.15)

    • M.GHS.5.d bisecting an angle;

      • Construct an angle bisector (G-C.7)

    • M.GHS.5.e constructing perpendicular lines, including the perpendicular bisector of a line segment; and

      • Construct the midpoint or perpendicular bisector of a segment (G-B.15)
      • Construct a perpendicular line (G-D.2)

    • M.GHS.5.f constructing a line parallel to a given line through a point not on the line.

      • Construct parallel lines (G-D.6)

Transformations and Congruence

• Experiment with transformations in the plane.

  • M.GHS.6 Build on prior knowledge from rigid motions to:

    • M.GHS.6.a represent transformations using geometric concepts in the plane.

      • Translations: graph the image (G-H.2)
      • Reflections: graph the image (G-H.5)
      • Rotations: graph the image (G-H.8)

    • M.GHS.6.b describe transformations as functions that take points in the plane as inputs and give other points as outputs.

      • Classify congruence transformations (G-H.1)
      • Translations: find the coordinates (G-H.3)
      • Translations: write the rule (G-H.4)
      • Reflections: find the coordinates (G-H.6)
      • Rotations: find the coordinates (G-H.9)
      • Reflections and rotations: write the rule (G-H.10)

    • M.GHS.6.c compare transformations that preserve distance and angle to those that do not.

  • M.GHS.7 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

    • Transformations that carry a polygon onto itself (G-H.15)

  • M.GHS.8 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

    • Rotate polygons about a point (G-H.7)

  • M.GHS.9 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Describe a sequence of transformations that will carry a given figure onto another.

    • Translations: graph the image (G-H.2)
    • Reflections: graph the image (G-H.5)
    • Rotations: graph the image (G-H.8)
    • Sequences of congruence transformations: graph the image (G-H.12)
    • Sequences of congruence transformations: choose the sequence (G-H.13)
    • Sequences of congruence transformations: find the rules (G-H.14)

  • Checkpoint opportunity

    • Checkpoint: Congruence transformations (G-H.17)

• Understand congruence in terms of rigid motions.

  • M.GHS.10 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.

    • Sequences of congruence transformations: graph the image (G-H.12)
    • Sequences of congruence transformations: choose the sequence (G-H.13)
    • Transformations that carry a polygon onto itself (G-H.15)
    • Congruence transformations: mixed review (G-H.16)
    • Determine if two figures are congruent: justify your answer (G-K.4)

  • M.GHS.11 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 congruent.

    • Prove triangles are congruent using rigid motions (G-L.1)

  • M.GHS.12 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

  • M.GHS.13 Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures.

    • SSS and SAS Theorems (G-L.2)
    • Proving triangles congruent by SSS and SAS (G-L.3)
    • ASA and AAS Theorems (G-L.4)
    • Proving triangles congruent by ASA and AAS (G-L.5)
    • SSS, SAS, ASA, and AAS Theorems (G-L.6)
    • SSS Theorem in the coordinate plane (G-L.7)
    • Proving triangles congruent by SSS, SAS, ASA, and AAS (G-L.8)
    • Proofs involving corresponding parts of congruent triangles (G-L.9)
    • Congruency in isosceles and equilateral triangles (G-L.10)
    • Proofs involving isosceles triangles (G-L.11)
    • Hypotenuse-Leg Theorem (G-L.12)
    • Proofs involving triangles and quadrilaterals (G-O.14)

  • Checkpoint opportunity

    • Checkpoint: Rigid motion and congruence (G-L.13)

• Prove geometric theorems.

  • M.GHS.14 Use appropriate methods of proof to prove theorems about triangles and lines. 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; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

    • Perpendicular Bisector Theorem (G-B.9)
    • Triangle Angle-Sum Theorem (G-F.2)
    • Proofs involving isosceles triangles (G-L.11)
    • Midsegments of triangles (G-N.1)
    • Construct the circumcenter or incenter of a triangle (G-N.6)
    • Construct the centroid or orthocenter of a triangle (G-N.7)
    • Find the centroid of a triangle (G-N.8)
    • Proofs involving triangles I (G-N.9)
    • Proofs involving triangles II (G-N.10)

  • M.GHS.15 Use appropriate methods of proof to prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

    • Properties of parallelograms (G-O.6)
    • Proving a quadrilateral is a parallelogram (G-O.7)
    • Properties of rhombuses (G-O.8)
    • Properties of squares and rectangles (G-O.9)

  • Checkpoint opportunity

    • Checkpoint: Triangle theorems (G-N.11)
    • Checkpoint: Parallelogram theorems (G-O.16)

• Use coordinates to prove simple geometric theorems algebraically.

  • M.GHS.16 Use coordinates to prove simple geometric theorems about right triangles, quadrilaterals, and circles algebraically (e.g., derive the equation of a circle of given center and radius using the Pythagorean Theorem).

    • Classify shapes on the coordinate plane: justify your answer (G-O.13)
    • Determine if a point lies on a circle (G-Y.3)
    • Derive equations of circles using the Pythagorean theorem (G-Y.4)

  • Checkpoint opportunity

    • Checkpoint: Coordinate proofs (G-E.8)

Similarity and Trigonometry

• Understand similarity in terms of similarity transformations.

  • M.GHS.17 Verify experimentally the properties of dilations given by a center and a scale factor.

    • M.GHS.17.a 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.

      • Dilations: graph the image (G-I.1)
      • Dilations and parallel lines (G-I.6)

    • M.GHS.17.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

      • Dilations: find length, perimeter, and area (G-I.3)
      • Dilations: find the scale factor (G-I.4)
      • Dilations: find the scale factor and center of the dilation (G-I.5)

  • M.GHS.18 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.

    • Identify similar figures (G-M.3)
    • Side lengths and angle measures in similar figures (G-M.4)
    • Similar triangles and similarity transformations (G-M.9)
    • Determine if two figures are similar: justify your answer (G-M.10)

  • M.GHS.19 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

    • Similarity rules for triangles (G-M.8)

  • Checkpoint opportunity

    • Checkpoint: Transformations in the plane (G-I.7)
    • Checkpoint: Dilations (G-I.8)
    • Checkpoint: Similarity transformations (G-M.18)

• Prove theorems involving similarity.

  • M.GHS.20 Use appropriate methods of proof to prove theorems about triangles involving 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.

    • Proofs involving similarity in right triangles (G-M.17)
    • Proofs involving triangles II (G-N.10)
    • Prove the Pythagorean theorem (G-P.2)

  • M.GHS.21 Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use the Pythagorean Theorem and similarity criteria to derive and apply special right triangles to solve problems.

    • Similar triangles and indirect measurement (G-M.5)
    • Similarity rules for triangles (G-M.8)
    • Triangle Proportionality Theorem (G-M.12)
    • Similarity and altitudes in right triangles (G-M.13)
    • Prove similarity statements (G-M.15)
    • Prove proportions or angle congruences using similarity (G-M.16)

  • Checkpoint opportunity

    • Checkpoint: Triangle similarity and congruence (G-M.19)

• Define trigonometric ratios and solve problems involving right triangles.

  • M.GHS.22 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

    • Trigonometric ratios: sin, cos, and tan (G-Q.1)
    • Trigonometric ratios with radicals: sin, cos, and tan (G-Q.2)
    • Trigonometric ratios: csc, sec, and cot (G-Q.3)
    • Trigonometric ratios in similar right triangles (G-Q.4)

  • M.GHS.23 Explain and use the relationship between the sine and cosine of complementary angles.

    • Sine and cosine of complementary angles (G-Q.9)

  • M.GHS.24 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

    • Pythagorean theorem (G-P.1)
    • Converse of the Pythagorean theorem (G-P.3)
    • Pythagorean Inequality Theorems (G-P.4)
    • Trigonometric ratios: find a side length (G-Q.11)
    • Trigonometric ratios: find an angle measure (G-Q.12)
    • Solve a right triangle (G-Q.13)

  • Checkpoint opportunity

    • Checkpoint: Right triangle trigonometry (G-Q.14)

• Apply trigonometry to general triangles.

  • M.GHS.25 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

    • Area of a triangle: sine formula (G-R.4)

  • M.GHS.26 Prove the Laws of Sines and Cosines extending the definitions of sine and cosine to obtuse angles.

    • Law of Sines (G-R.1)
    • Law of Cosines (G-R.2)
    • Area of a triangle: Law of Sines (G-R.5)

  • M.GHS.27 Understand and apply the Law of Sines and the Law of Cosines to solve problems and to find unknown measurements in right and non-right triangles.

    • Solve a triangle (G-R.3)

  • Checkpoint opportunity

    • Checkpoint: Laws of Sines and Cosines (G-R.6)

Circles

• Understand and apply theorems about circles.

  • M.GHS.28 Prove that all circles are similar.

    • Similarity of circles (G-M.11)

  • M.GHS.29 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.

    • Central angles and arc measures (G-W.2)
    • Arcs and chords (G-W.9)
    • Tangent lines (G-W.10)
    • Inscribed angles (G-W.12)
    • Angles in inscribed right triangles (G-W.13)
    • Angles formed by chords, secants, and tangents (G-W.16)
    • Segments formed by chords, secants, and tangents (G-W.17)

  • Checkpoint opportunity

    • Checkpoint: Prove circles are similar (G-M.20)
    • Checkpoint: Angles and lines in circles (G-W.18)

• Find arc lengths and areas of sectors of circles.

  • M.GHS.30 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.

    • Understand arc length and sector area of a circle (G-W.3)
    • Arc length (G-W.4)
    • Convert between radians and degrees (G-W.5)
    • Radians and arc length (G-W.6)
    • Area of sectors (G-W.7)

  • Checkpoint opportunity

    • Checkpoint: Arc length and area of sectors (G-W.20)

• Make geometric constructions.

  • M.GHS.31 Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.

    • Construct the circumcenter or incenter of a triangle (G-N.6)
    • Angles in inscribed quadrilaterals I (G-W.14)
    • Angles in inscribed quadrilaterals II (G-W.15)
    • Construct the inscribed or circumscribed circle of a triangle (G-X.5)

  • M.GHS.32 Construct a tangent line from a point outside a given circle to the circle.

    • Construct a tangent line to a circle (G-X.1)

  • M.GHS.33 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    • Construct an equilateral triangle inscribed in a circle (G-X.2)
    • Construct a square inscribed in a circle (G-X.3)
    • Construct a regular hexagon inscribed in a circle (G-X.4)

  • Checkpoint opportunity

    • Checkpoint: Inscribed and circumscribed circles (G-W.19)
    • Checkpoint: Geometric constructions (G-X.6)

Extending to Three Dimensions and Modeling

• Explain volume formulas and use them to solve problems.

  • M.GHS.34 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.

  • M.GHS.35 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems, including how area and volume scale under similarity transformations.

    • Volume of prisms and cylinders (G-U.7)
    • Volume of pyramids and cones (G-U.8)
    • Volume of spheres (G-U.9)
    • Perimeter, area, and volume: changes in scale (G-U.14)

  • Checkpoint opportunity

    • Checkpoint: Volume (G-U.17)

• Visualize the relation between two-dimensional and three-dimensional objects and apply geometric concepts in modeling situations.

  • M.GHS.36 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

    • Cross sections of three-dimensional figures (G-T.4)
    • Solids of revolution (G-T.5)

  • M.GHS.37 Use two- and three-dimensional shapes and circles, their measures, and their properties to describe objects.

    • Pythagorean theorem (G-P.1)

    • M.GHS.37.a Apply concepts of density based on area and volume in modeling situations.

      • Calculate density, mass, and volume (G-V.9)

    • M.GHS.37.b Apply geometric methods to solve design problems to satisfy given constraints.

  • Checkpoint opportunity

    • Checkpoint: Cross sections and solids of revolution (G-T.6)
    • Checkpoint: Geometric modeling and design (G-V.10)
    • Checkpoint: Density (G-V.11)

Statistics and Probability

• Understand independence and conditional probability and use them to interpret data.

  • M.GHS.38 Describe events as subsets of a sample space using characteristics of the outcomes or as unions, intersections, or complements of other events.

    • Theoretical and experimental probability (G-AA.1)
    • Outcomes of compound events (G-AA.2)

  • M.GHS.39 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities. Use this characterization to determine if they are independent.

    • Identify independent and dependent events (G-AA.3)
    • Probability of independent and dependent events (G-AA.4)
    • Identify independent events (G-AA.10)

  • M.GHS.40 Recognize 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.

    • Find conditional probabilities (G-AA.11)
    • Independence and conditional probability (G-AA.13)

  • M.GHS.41 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

    • Find probabilities using two-way frequency tables (G-AA.9)
    • Find conditional probabilities using two-way frequency tables (G-AA.12)

  • M.GHS.42 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

  • Checkpoint opportunity

    • Checkpoint: Understand independence and conditional probability (G-AA.16)

• Use the rules of probability to compute probabilities of compound events in a uniform probability model.

  • M.GHS.43 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.

    • Find conditional probabilities using two-way frequency tables (G-AA.12)
    • Geometric probability (G-AA.14)

  • M.GHS.44 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.

    • Find probabilities using the addition rule (G-AA.15)

  • M.GHS.45 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.

    • Find conditional probabilities (G-AA.11)

  • M.GHS.46 Use permutations and combinations to compute probabilities of compound events and solve problems.

    • Counting principle (G-AA.5)
    • Permutations (G-AA.6)
    • Permutation and combination notation (G-AA.7)
    • Find probabilities using combinations and permutations (G-AA.8)

  • Checkpoint opportunity

    • Checkpoint: Probabilities of compound events (G-AA.17)

• Use probability to evaluate outcomes of decisions.

  • M.GHS.47 Use probabilities to make fair decisions.

  • M.GHS.48 Analyze decisions and strategies using probability concepts.