Content Standards
| Geometry | Algebra II |
Content Standards
©2002
Lori Killpatrick
1. Identify, give examples
and use definitions, undefined terms, axioms, postulates, theorems,
inductive and deductive reasoning.
2. Construct and judge the
validity of a logical argument including giving counterexamples.
3. Select and use
deductive, indirect, algebraic or coordinate methods of proof in a variety
of formal proof formats (paragraph, flow chart, two-column).
4. Describe the
relationships between angles (e.g. vertical, right, supplementary,
complementary, adjacent, linear pairs, angle bisectors, angles formed when
parallel lines are cut by a transversal) and solve problems concerning
them.
5. Find and use measures
of sides, interior and exterior angles of triangles and polygons to
classify figures (e.g. isosceles, obtuse, convex, regular) and solve
problems concerning the lengths of sides and/or measures of angles.
6. Identify similar and
congruent triangles and other polygons and their corresponding parts and
prove basic theorems and solve problems about them.
7. Know and prove theorems
about angles (e.g. vertical, right, corresponding), triangles (e.g.
similar, congruent) and polygons (e.g. regular, convex).
8. Compare, contrast,
classify and solve problems involving quadrilaterals (square, rhombus,
rectangle, parallelogram, trapezoid) on the basis of their definitions and
properties (e.g. opposite sides, consecutive angles, diagonals).
9. Know the definitions of
the basic trigonometric functions defined by the angles of a right
triangle.
10. Understand algebraic
and geometric proofs of the Pythagorean Theorem.
11. Use the Pythagorean
Theorem, its converse, properties of special right triangles (e.g.
30-60-90 and 45-45-90) and right triangle trigonometry to find missing
information about triangles.
12. Apply knowledge of
angles, arcs, chords, radii, tangents and secants to solve problems
involving circles.
13. Know and prove
theorems about the properties of angles, arcs, chord, radii, tangents and
secants of circles.
14. Perform standard
straightedge and compass constructions (e.g. bisect angles, draw parallel
lines, divide a line segment into proportional parts, draw perpendicular
segment bisectors).
15. Identify the
structural parts (e.g. angles, shape of sides, orientation of sides) and
characteristics (e.g. symmetry, shape of cross-sections) of
three-dimensional objects (e.g. cylinders, prisms, cones, pyramids) and
use these to classify the objects and answer questions about them.
16. Know, use and/or
derive formulas for and solve problems involving perimeter, circumference,
area, volume, lateral area and surface area of common geometric figures.
17. Describe how changes
in the dimensions of an object affect the perimeter, area and volume.
18. Verify geometric
statements using coordinate geometry, including the midpoint of a line
segment, distance formula, various forms of equations of lines and
circles.
19. Use transformations in the coordinate plane or in space (translations, rotations and reflections) (e.g. identify the type of transformation underlying symmetry in figures, show how one object in a pair of congruent objects can be translated, reflected and/or rotated to “sit on top” of its counterpart).
1. Add, subtract, multiply, divide and simplify expressions containing radicals, fractional exponents and negative exponents, including expressions in scientific notation.
2. Use, interpret and relate radicals, fractional exponents and negative exponents.
3. Demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. Graph complex numbers as points in the plane.
4. Add, subtract, multiply and divide complex numbers.
5. Compare and contrast different forms of linear equations (e.g. slope-intercept, point-slope, standard), describe the advantages of each form and convert from one form to another.
6. Write equations of parallel and perpendicular lines to solve problems.
7. Write and solve linear and quadratic equations and inequalities and absolute value inequalities in one variable, verify the solutions and interpret the results graphically, relating the solutions and graphs to the situations modeled.
8. Add, subtract, multiply, divide and simplify polynomials, rational and radical expressions and solve equations involving them.
9. When given a polynomial of degree three or higher in factored form, sketch its graph using characteristics of the function (e.g. end behavior, intercepts).
10. Factor polynomial representing the difference of squares, perfect squares, trinomials and the sum and difference of cubes.
11. Write and solve systems of linear and quadratic equations (including linear-quadratic and quadratic-quadratic systems) and inequalities, verify solutions and interpret the results in terms of the graph and the situation modeled by the equations.
12. Demonstrate understanding that systems of equations are either inconsistent (no solutions), have a finite number of solution or have infinitely many solutions.
13. Given a relation in the form of a table of values, an equation or a graph, determine if it is a function and explain its key characteristics (domain, range, intercepts, one-to-one).
14. Determine the inverse of a one-to-one function, the composition of two or more functions and the composition of a function with itself when the function is given as an equation, a graph or a set of ordered pairs and determine the domain and range of the resultant function.
15. Given an equation (linear, quadratic, exponential, absolute value, polynomial, rational or radical), sketch its graph and determine its significant features (intercepts, slope, vertex, asymptotes, holes).
16. Demonstrate and explain the effect that transformations (translations, reflections, dilations) have on both the equation and graph of a function.
17. Understand the relationships among the factors, roots, zeros and x-intercepts of a quadratic function (and related quadratic equation) and use these to solve factorable quadratic equations.
18. Use the method of completing the square to develop the quadratic formula and use the quadratic formula to solve any quadratic equation in one variable, including those with irrational or complex solutions and state solutions in simplified radical form.
19. Explain the effect changing a coefficient and/or constant has on the graph of the quadratic function y = a(x - b)2 + c.
20. Find and interpret the maximum or minimum value of a quadratic function.
21. Use the discriminant to determine the nature of the roots of a quadratic equation, the zeros of its related function and the x-intercepts of its graph.
22. Convert between exponential and logarithmic expressions and equations.
23. Translate between logarithms in any bases.
24. Evaluate common logarithms.
25. Solve simple logarithmic equations.
26. Identify, describe, extend and find the nth term of arithmetic, geometric and other sequences.
27. Find the sum of finite arithmetic and geometric sequences and find the sum of infinite geometric sequences when |r| < 1.
28. Relate arithmetic and geometric sequences to linear and exponential functions and describe them in explicit and recursive form.
29. Apply the Binomial Theorem and use it to expand binomial expressions which are raised to positive integer powers.
30. Use combinations and permutations to count the number of arrangements of a set of elements and distinguish between the two and relate to the determination of theoretical probabilities.
31. From the equation of a conic section, sketch the graph noting its key features (e.g. center, vertices, directrix, asymptotes, foci, intercepts, axis of symmetry).
32. Given a graph or characteristics of a conic, determine its equation.
33. Identify situations and phenomena that are cyclic in nature, identify the length of the cycle and use the values in that cycle to determine values everywhere.
34. Relate central angles of a circle centered at the origin on a coordinate grid with the sine, cosine and tangent of their reference triangles and explain why these can be considered periodic functions.
35. Use Laws of Sines and Cosines to find missing sides of triangles.