Unit 1

Algebra II

Unit Description:

Building on their work with linear, quadratic, and exponential functions in Algebra I, students extend their repertoire of functions to include polynomial, rational, and radical functions. In this Algebra II course, rational functions are limited to those whose numerators are of degree at most 1 and denominators of degree at most 2.  Radical functions are limited to square roots or cube roots of at most quadratic polynomials.  Certain standards in this course require students to revisit the topics of linear, quadratic and/or exponential functions to build conceptual understanding.

Unit One develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers.  Students revisit the topic of systems of linear and quadratic equations to continue to build conceptual understanding. (Mathematics Appendix A, p.36-38, with adjustments)

Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

Louisiana Student Standards for Mathematics (LSSM)

Enduring Understandings:

·         Odd functions begin and end in opposite directions.

·         Even functions begin and end in the same direction.

·         Zeros are the x intercepts of an equation and can be used to find factors and write a polynomial equation.

·         Odd and even functions have graphs that are symmetric with respect to the origin or y-axis.

·         Zeros of a polynomial can be found by factoring or by graphing the polynomial.

·         Rational functions can be represented as fractional exponents and follow the same rules as regular exponents.

·         Solve rational expressions.

·         Computational skills applicable to numerical fractions also apply to rational expressions involving variables.

·         Radical expressions can be written and simplified using rational exponents.

·         Graphs of radical functions look like curves, are not symmetric and can be transformed using predictive indicators.

·         Radicals are the opposite of exponents.

·         Radical equations can be solved by graphing or inverses.

Essential Questions:

·         How do exponent value, zeros, and factors affect the appearance of a graph?

·         How can a polynomial inequality be solved?

·         How can real world data be used to generate a polynomial model?

·         What does it mean to be an odd or even function?

·         How do the elements of a polynomial equation determine its general shape?

·         How can the solutions or zeros of a polynomial be found?

·         How do the zeros of a polynomial relate to its graph?

·         What is a rational function and what does its graph look like?

·         How are operations extended to rational functions?

·         How are rational expressions simplified?

·         How are rational equations solved?

·         How are rational equations graphed?

·         What are radicals and how can they be simplified?

·         What do graphs of radical functions look like?

·         How can a radical function be identified from its graph?

·         How can a radical expression be simplified and combined?

·         How can a radical equation be solved?

·         How can operations be extended to radical expressions and equations?