GRADE 10

Standards
Clarification and Examples
Number Sense - Grade 10  
1 Students understand and represent imaginary and complex numbers symbolically and use them in basic operations.  
1.1 Students use i notation (i = square-root of -1) for representing imaginary and complex numbers, and compute the sum, difference, product, and quotient of two complex numbers  
2 Students understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems using logarithms.  
2.1 Students use the definition of logarithms to translate between logarithmic and exponential representations of numbers Note: Logarithms using any base should be included
2.2 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values  
Symbols and Algebra - Grade 10  
1 Students compute with and simplify rational expressions and those containing radicals and fractional exponents.  
1.1 Students add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and polynomial denominators, and simplify complex fractions including fractions with negative exponents in the denominator  
1.2 Students add, subtract, multiply, divide, and simplify expressions containing radicals and fractional and exponents  
2 Students apply and solve quadratic equations.  
2.1 Students select, justify, and apply algebraic techniques (factoring, completing the square, using the quadratic formula) to solve a quadratic equation and identify the real and imaginary roots  
2.2 Students use the discriminant of the quadratic formula to determine the nature of the roots (rational, irrational, or imaginary) of quadratic equations with real coefficients  
2.3 Students determine the minimum, maximum, and roots (if any) of a quadratic function, locate these points on a graph a function, and apply the solution to minimum and maximum value application problems  
2.4 Students demonstrate and explain the effect that changing a coefficient has on the graph of a quadratic function describe the changes in the graph of y = ax2 + bx + c as the coefficients a, b, and c change
3 Students solve factorable polynomial equations in one variable and identify the properties of graphs of polynomial functions  
3.1 Students use factoring techniques (common factor, sums and differences of cubes, grouping) and synthetic division and the rational root theorem to solve polynomial equations of degree four or less  
3.2 Students sketch graphs of polynomial functions locating integral roots, locating irrational roots between two appropriate integers, and estimating maximum and minimum points  
4 Students solve, graph, and interpret non-linear systems of equations in two variables and solve systems of first-order equations in three variables  
4.1 Students solve linear-quadratic and quadratic-quadratic systems of equations and inequalities algebraically, represent them graphically, verify solutions, and relate the solutions and graphs to the situations modeled by the equations Systems of equations involve conic sections whose axes are parallel to the x and y axes.
4.2 Students solve systems of three first-order equations in three variables by substitution or linear combinations and identify whether or not such systems have unique solutions  
Functions - Grade 10  
1 Students identify features of and perform algebraic operations on standard functions Standard functions here include linear, quadratic, exponential, rational, radical, absolute value, and factorable polynomial functions.
1.1 Students graph linear, exponential, polynomial, square-root, cube-root, absolute value, step and piecewise functions and identify the key characteristics of these functions  
1.2 Students determine which type of equation models a function based on the information provided about the situation  
1.3 Students find the value of a function for a given element in its domain and determine the natural domain of a given function  
1.4 Students identify the domain, range, zeros, and express the inverse (for 1-to-1 functions) of functions  
1.5 Students determine the composition of two or more functions and determine the domain and range of the original and the resultant function and the value of each for a given element in its domain  
1.6 Students use the properties of logarithms and the inverse relationship between logarithms and exponents to simplify logarithmic expressions and to solve equations with variables in the exponents. Logarithms will be used to solve exponential growth and decay problems.
2 Students demonstrate an understanding of arithmetic and geometric sequences and series. (  
2.1 Students identify, describe, extend, and find the nth term of arithmetic and geometric sequences  
2.2 Students find the sum of finite arithmetic and geometric sequences and infinite geometric sequences with |r| < 1, use and interpret summation [sigma] notation  
Measurement and Geometry - Grade 10  
1 Students identify and give information about equations for circles, parabolas, ellipses, and hyperbolas whose axes are parallel to the x- and y-axes.  
1.1 Students recognize the standard form of equations for a circle, ellipse, hyperbola, and parabola, use this information to identify key features and sketch a graph the figure Features include the center and radius of a circle, the center, vertices, foci, and axes of an ellipse or hyperbola, and the vertex, axis of symmetry, focus, and directrix of a parabola
Statistics, Data Analysis and Probability - Grade 10  
1 Students use fundamental counting principles to compute combinations and permutations, and relate these to the determination of theoretical probabilities Probability computations and their relation to binomial expansion need to be the focus this year, and are therefore given more attention in the following
1.1 Students multiply to find the number of permutations and use numbers of permutations to compute theoretical probabilities  
1.2 Students relate the coefficients of binomial powers to the computation of theoretical probability, express these terms in factorial notation, and use this to find probabilities of combinations of events  
1.3 Students differentiate between permutations and combinations in problem situations, and use this distinction to correctly formulate probability computations  
Mathematical Reasoning - Grade 10  
1 Students expand their mathematical reasoning in justifying steps and in proof.  
1.1 Students produce a proof of the quadratic formula using completing the square, and explain the reasoning for each step  
1.2 Students judge the validity of an argument based on whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step  
1.3 Students determine if a specific algebraic statement involving rational expressions, radical expressions, logarithmic or exponential functions, is sometimes true, always true, or never true  
1.4 Students prove simple laws of logarithms If M and N are positive numbers, prove that Log MN = Log M + Log N

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