Gizmos for West Virginia - Mathematics: Mathematics II (College- and Career-Readiness Standards adopted 2023)

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.

EE: : Expressions and Equations

1.1: : Interpret the structure of expressions and equations in terms of the context they model.

EE.M.A1HS.1: : Interpret linear, exponential, and quadratic expressions that represent a quantity in terms of its context.

EE.M.A1HS.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

EE.M.A1HS.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

EE.M.A1HS.2: : Use the structure of quadratic and exponential expressions to identify ways to rewrite them.

1.2: : Write expressions in equivalent forms to solve problems.

EE.M.A1HS.5: : Choose and produce an equivalent form of linear, exponential, and quadratic expressions to reveal and explain properties of the quantity represented by the expression through connections to a graphical representation of the function.

EE.M.A1HS.5.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

EE.M.A1HS.5.b: : Complete the square in a quadratic expression, when a = 1 only, to reveal the maximum or minimum value of the function it defines.

1.3: : Perform arithmetic operations on polynomials.

EE.M.A1HS.6: : Recognize that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Focus on linear or quadratic terms.

1.4: : Solve equations and inequalities in one variable.

EE.M.A1HS.11: : Solve quadratic equations in one variable by inspection (e.g., for x^2 = 49), taking square roots, factoring, completing the square when a = 1 only, and the quadratic formula, as appropriate for the initial form of the equation.

EE.M.A1HS.11.a: : Recognize the concept of complex solutions when the quadratic formula gives complex solutions.

EE.M.A1HS.11.b: : Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)^2 = q. Derive the quadratic formula from this method of completing the square.

F: : Functions

2.1: : Interpret functions that arise in applications in terms of a context.

F.M.A1HS.22: : For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship. Relate the domain of a function to its linear, exponential, and quadratic graphs and, where applicable, to the quantitative relationship it describes.

F.M.A1HS.22.a: : Key features of linear and exponential graphs include: intercepts; and intervals where the function is increasing, decreasing, positive, or negative.

F.M.A1HS.22.b: : Key features of quadratic graphs include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximum or minimum; symmetry; and end behavior.

2.2: : Analyze functions using different representations.

F.M.A1HS.23: : Graph linear, exponential, and quadratic functions expressed symbolically and show key features of the graph.

F.M.A1HS.23.a: : For linear functions, focus on intercepts.

F.M.A1HS.23.b: : For exponential functions, focus on intercepts and end behavior.

F.M.A1HS.23.c: : For quadratic functions, focus on intercepts, maxima, minima, end behavior, and the relationship between coefficients and roots to represent in factored form.

F.M.A1HS.25: : Write a function defined by a linear, exponential, or quadratic expression in different but equivalent forms to reveal and explain different properties of the function.

F.M.A1HS.25.a: : Use the process of factoring and completing the square for a = 1 only in a quadratic function to show zeros, extreme values, symmetry of the graph, the relationship between coefficients and roots represented in factored form and interpret these in terms of a context.

F.M.A1HS.25.b: : Use the properties of exponents to interpret expressions in exponential functions.

2.3: : Build a function that models a relationship between two quantities.

F.M.A1HS.26: : Write linear, exponential, and quadratic functions that describe a relationship between two quantities.

F.M.A1HS.26.a: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

F.M.A1HS.26.b: : Combine standard function types using arithmetic operations.

2.4: : Build new functions from existing functions.

F.M.A1HS.28: : Identify the effect on the graphs of linear and exponential functions, f(x), with f(x) + k, and the graphs of quadratic functions, g(x), with g(x) + k, kg(x), g(kx), and g(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

2.5: : Construct and compare linear, quadratic, and exponential models and solve problems.

F.M.A1HS.29: : Distinguish between situations that can be modeled with linear functions, with exponential functions, and with quadratic functions.

F.M.A1HS.29.a: : Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.

F.M.A1HS.29.b: : Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F.M.A1HS.29.c: : Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F.M.A1HS.29.d: : Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. Extend the comparison of linear and exponential growth to quadratic growth.

TC: : Transformations and Congruence

3.1: : Prove geometric theorems.

TC.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 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; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

TC.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.

3.2: : Use coordinates to prove simple geometric theorems algebraically.

TC.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).

ST: : Similarity and Trigonometry

4.1: : Understand similarity in terms of similarity transformations.

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

ST.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.

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

ST.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.

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

4.2: : Prove theorems involving similarity.

ST.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.

ST.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.

4.3: : Define trigonometric ratios and solve problems involving right triangles.

ST.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.

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

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

C: : Circles

5.1: : Understand and apply theorems about circles.

C.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.

5.2: : Find arc lengths and areas of sectors of circles.

C.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.

5.3: : Make geometric constructions.

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

ETM: : Extending to Three Dimensions and Modeling

6.1: : Explain volume formulas and use them to solve problems.

ETM.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.

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

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

ETM.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.

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

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

SP: : Statistics and Probability

7.1: : Understand independence and conditional probability and use them to interpret data.

SP.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.

SP.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.

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

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

SP.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.

SP.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.

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

7.3: : Use probability to evaluate outcomes of decisions.

SP.M.GHS.47: : Use probabilities to make fair decisions.

SP.M.GHS.48: : Analyze decisions and strategies using probability concepts.

Correlation last revised: 10/13/2023

West Virginia - Mathematics: Mathematics II

College- and Career-Readiness Standards | Adopted: 2023