Standards - Mathematics

2019 - Mathematics Course of Study Document
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.15

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems, extending to infinite geometric series.

Examples: calculate mortgage payments; determine the long-term level of medication if a patient takes 50 mg of a medication every 4 hours, while 70% of the medication is filtered out of the patient’s blood.

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Knowledge

Students know:
  • Characteristics of a geometric series.
  • Techniques for performing algebraic manipulations and justifications for the equivalence of the resulting expressions.

Skills

Students are able to:
  • Identify the regularity that exists in a series as being that which defines it as a geometric series.
  • Accurately perform the procedures involved in using geometric series to solve contextual problems,
  • Explain with mathematical reasoning why each step in the derivation of the formula for the sum of a finite geometric series is legitimate, including explaining why the formula does not hold for a common ratio of 1.

Understanding

Students understand that:
  • When each term of a geometric series is multiplied by a value, the result is a new geometric series.
  • When many problems exist with the same mathematical structure, formulas are useful generalizations for efficient solution of problems, (e.g., mortgage payment calculation with geometric series).

Vocabulary

  • Geometric series (finite and infinite)
  • Common ratio
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.16

Derive and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

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Knowledge

Students know:
  • Procedures for dividing a polynomial p(x) by a linear polynomial (x - a).

Skills

Students are able to:
  • Accurately perform procedures for dividing a polynomial p(x) by a linear polynomial (x - a),
  • Evaluate a polynomial p(x) for any value of x.

Understanding

Students understand that:
  • There is a structural relationship between the value of a in (x - a), as well as the remainder when p(x) is divided by (x - a).
  • If p(a)=0, then x-a if a factor of p(x).

Vocabulary

  • If and only if
  • Remainder theorem
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.17

Know and apply the Binomial Theorem for the expansion of $(x + y)^n$ in powers of x and y for a positive integer, n, where x and y are any numbers.

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Knowledge

Students know:
  • Distributive Property of multiplication over addition for polynomials.
  • The generation pattern for Pascal's Triangle and which binomial expansion term has coefficients corresponding to each row.
  • Simplification procedures for expressions involving the number of combinations of n things taken r at a time.
  • The patterns of coefficients and exponents in a binomial expansion.

Skills

Students are able to:
  • Accurately perform algebraic manipulations on polynomial expressions.
  • Generate rows of Pascal's Triangle.
  • Accurately perform simplification procedures for expressions involving the number of combinations of n things taken r at a time.
  • Apply the patterns of coefficients and exponents to expand any binomial raised to a power.

Understanding

Students understand that:
  • Regularities noted in one part of mathematics may also be seen in very different areas of mathematics, (i.e., Pascal's Triangle from counting procedures and the Binomial Theorem). These regularities are useful in computing or manipulating mathematical expressions.
  • The regularities that are seen in exponents and coefficients in a binomial expansion will be generalized to all binomials to aid in identifying specific terms.

Vocabulary

  • Binomial Theorem
  • Pascal's Triangle
  • Combinatorial Argument
  • Mathematical induction

Aligned Learning Resources

Binomial Expansion -Shortcut Please
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.18

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated cases, a computer algebra system.

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Knowledge

Students know:
  • Techniques for long division of polynomials.
  • Techniques for utilizing a computer algebra system.

Skills

Students are able to:
  • Accurately perform polynomial long division.
  • Efficiently and accurately use a computer algebra system to divide polynomials.

Understanding

Students understand that:
  • The role of the remainder in polynomial division is analogous to that of the remainder in whole number division.
  • Different forms of rational expressions are useful to reveal important features of the expression.

Vocabulary

  • Rational expression
  • Degree of polynomial
  • Inspection
  • Long Division
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.19

Add, subtract, multiply, and divide rational expressions.

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Knowledge

Students know:
  • Techniques for performing operations on polynomials.

Skills

Students are able to:
  • Accurately perform addition, subtraction, multiplication, and division of rational expressions.

Understanding

Students understand that:
  • They can communicate a mathematical justification for all four operations on rational expressions being closed.
  • The structure of mathematics present in the system of rational numbers is also present in the system of rational expressions.

Vocabulary

  • Rational expression
  • Closed under an operation
  • Nonzero rational expression
  • Analogous
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.19a

Explain why rational expressions form a system analogous to the rational numbers, which is closed under addition, subtraction, multiplication, and division by a non-zero rational expression.

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.20

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.

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Knowledge

Students know:
  • How to solve equations using a reasoning process centered around inverse operations and order of operations

Skills

Students are able to:
  • Solve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric equations (including their inverses) using multiple solution strategies and explain each step in the solution path.
  • Construct a viable argument to justify a chosen solution path used to solve a linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric equation (including their inverses).
  • Compare the steps in each and determine which solution path is most efficient, given an equation with multiple solution paths.
  • Explain when an equation has no solution or infinitely many solutions.

Understanding

Students understand that:
  • The process of solving equations is a reasoning process to determine a solution that satisfies the equation rather than a procedural list of steps.
  • An equation has no solution because there is no value that can maintain equivalency and an equation has infinitely many solutions because all values used for the variable create a true equivalency statement

Vocabulary

  • equivalence
  • viable

Aligned Learning Resources

Recognizing Quadratic Functions
Finding Components of Quadratic Functions - Overview Video
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.21

Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

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Knowledge

Students know:
  • Algebraic rules for manipulating rational equations.
  • Conditions under which a solution is considered extraneous.

Skills

Students are able to:
  • Accurately rearrange rational equations to produce a set of values to test against the conditions of the original situation and equation, and determine whether or not the value is a solution.
  • Explain with mathematical reasoning from the context (when appropriate) why a particular solution is or is not extraneous.

Understanding

Students understand that:
  • Values which arise from solving equations may not satisfy the original equation.
  • Values which arise from solving the equations may not exist due to considerations in the context.

Vocabulary

  • Rational equations
  • Extraneous solutions
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.22

Represent a system of linear equations as a single matrix equation in a vector variable.

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Knowledge

Students know:
  • The structure of a matrix equation is the product of the coefficient matrix on the left and the vector variable on the right is equal to the constant vector.

Skills

Students are able to:
  • Transform a system of linear equations into a matrix equation.

Understanding

Students understand that:
  • Multiple representations of linear systems are needed to facilitate work with more complex systems.
  • Matrix representation allows for solutions with systems containing any number of variables.

Vocabulary

  • Vector variable
  • Matrix equation
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.23

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).

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Knowledge

Students know:
  • Procedures for converting systems of equations to matrix equations.
  • Conditions that determine the inverse of a matrix exists.
  • Techniques for determining the inverse of a matrix (including using technology).
  • Process for using the inverse of a matrix to find the solution to a matrix equation (2 x 2).
  • Matrix multiplication is not commutative.

Skills

Students are able to:
  • Perform row operations on a matrix to find the inverse of the matrix.
  • Efficiently and accurately find the inverse of a matrix using technology.

Understanding

Students understand that:
  • The solutions to a matrix equation are the solutions to the system of equations that produced the matrix equation.
  • Solving a matrix equation is analogous to solving a linear equation.
  • Technology is a useful tool that facilitates investigation and, once the initial process is understood, helps find solutions of more complex problems.

Vocabulary

  • Inverse of a matrix
  • Dimension of a matrix

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