California State Standards for Mathematics: Grade 10
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CA.AI. Algebra I
1.0. Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.
1.1. Students use properties of numbers to demonstrate whether assertions are true or false.
2.0. Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
3.0. Students solve equations and inequalities involving absolute values.
4.0. Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) +4(x-2) = 12.
5.0. Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
6.0. Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x +6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x +6y is less than 4).
7.0. Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.
8.0. Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
9.0. Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
10.0. Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
11.0. Students apply basic factoring techniques to second- and simple third- degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
12.0. Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
13.0. Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
14.0. Students solve a quadratic equation by factoring or completing the square.
15.0. Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
16.0. Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
17.0. Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
18.0. Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.
19.0. Students know the quadratic formula and are familiar with its proof by completing the square.
20.0. Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
21.0. Students graph quadratic functions and know that their roots are the x-intercepts.
22.0. Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
23.0. Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
24.0. Students use and know simple aspects of a logical argument:
24.1. Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
24.2. Students identify the hypothesis and conclusion in logical deduction.
24.3. Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
25.0. Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements.
25.1. Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
25.2. Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.
25.3. Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.
CA.AII. Algebra II
11.1. Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
11.2. Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
3.1. Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).
3.2. Students prove other trigonometric identities and simplify others by using the identity cos2 (x) +sin2 (x) = 1. For example, students use this identity to prove that sec2 (x) = tan2 (x) +1.
CA.MA. Mathematical Analysis
5.1. Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).
5.2. Students can take a geometric description of a conic section - for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6 - and derive a quadratic equation representing it.
CA.LA. Linear Algebra
CA.PS. Probability and Statistics
CA.APPS. Advanced Placement Probability and Statistics
1.2. Students use graphical calculators to verify and estimate limits.
1.3. Students prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.
4.1. Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
4.2. Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.
4.3. Students understand the relation between differentiability and continuity.
4.4. Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
26.0. Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
27.0. Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.