STEM-Prep Pathway SLOs Background: The STEM-Prep subgroup of the MMPT adopts a variation of the student learning outcomes for STEM from the courses Reasoning with Functions I and Reasoning with Functions II by the New Math Pathway as a baseline. The New Math Pathway s Reasoning with Functions I and Reasoning with Functions II are designed for students who intend to pursue a degree in the fields of science, technology, engineering or mathematics, as well as other fields that require a high level of algebraic reasoning. Communication Goal: Students will be able to interpret and communicate quantitative information and mathematical and statistical concepts using language appropriate to the context and intended audience. Specifically, students will be able to: Communicate effectively about function processes using function notation. Describe the behavior of a function on entire intervals. Describe dynamic scenarios using graphs, tables, symbolically and in writing using appropriate mathematical language. Communicate conclusions both orally and in written form and support these conclusions by providing appropriate mathematical justification. Problem Solving Goal: Students will be able to make sense of problems, develop strategies to find solutions and persevere in solving problems. Specifically, students will be able to: Identify a variety of strategies to solve a problem, persist in applying a strategy, and reflect on the outcome of that strategy. Solve multistep problems in a variety of contexts related to science, technology, engineering and mathematics. Reasoning Goal: Students will be able to reason, model and draw conclusions or make decisions using mathematical, statistical and quantitative information. Specifically, students will be able to:
Examine and explore functions in various contexts and representations and draw appropriate conclusions. Apply mathematical reasoning to create appropriate functional models and use these models to make decisions. Create mathematical models in a variety of applications and use these models to make decisions. Evaluation Goal: Students will be able to critique and evaluate quantitative arguments that utilize mathematical, statistical and quantitative information. Specifically, students will be able to: Identify constraints and limitations for mathematical models in a variety of contexts and representations. Critically reflect on the reasonableness of their solutions. Technology Goal: Students will be able to use appropriate technology in such a manner as to not hinder their ability to perform algebraic manipulations by hand. Specifically, students will be able to: Use technology effectively and appropriately to analyze multiple representations of a function. Our subgroup is recommending that we refer to the courses Precalculus A and Precalculus B for the purposes of this document. Individual institutions may select their own course title. Content Learning Outcomes for Precalculus A: The content learning outcomes ensure students develop a firm foundation in functions and algebraic reasoning. The learning outcomes are fall under these three topics: Foundations of Functions Analysis of Functions Algebraic Reasoning Conic Sections (optional) Foundations of Functions:
Outcome: Students will use multiple representations of different function types to investigate quantities and describe relationships between quantities. Recognize quantities and define variables that are present in a given situation. For example: Clearly define variables relevant to a problem, use appropriate units of measure and use delta notation to denote changes in quantities. Use multiple representations of functions to interpret and describe how two quantities change together. For example: Justify the presence of a relationship, identify constraints on quantities and domains, distinguish between dependent and independent variables, identify domains and ranges, and draw diagrams of dynamic situations. Measure, compute, describe and interpret rates of change of quantities embedded in multiple representations. For example: Identify constant rates of change, determine average rates of change and be able to estimate instantaneous rates of change. Effectively communicate using function notation. For example: Recognize why function notation is used and be able to represent functions in multiple ways. Use appropriate tools and representations to investigate the patterns and relationships present in multiple function types. For example: Work effectively with linear, quadratic, exponential, logarithmic, rational, periodic, piecewise and absolute value functions. Analysis of Functions: Outcome: Students will describe characteristics of different function types and convert between different representations and algebraic forms to analyze and solve meaningful problems.
Create, use and interpret linear equations and convert between forms as appropriate. For example: Identify important values (e.g. slope & intercepts) from multiple representations, determine equations of lines given one point and the slope, two points, or statements about proportional relationships. Create, use and interpret exponential equations and convert between forms as appropriate. For example: Model constant percent change and be able to interpret half-life and doubling time to create a growth model. Recognize similarities and differences between linear functions, exponential functions, the role of e as a natural base, describe long-term behavior of growth models. Be able to invert the exponential function to create a logarithmic function. Use and interpret polynomial, power and rational functions. For example: Recognize how power functions are different from exponential functions. Be able to graph rational functions. Determine whether a graph has symmetry and whether a function is even or odd. Determine maximum, minimum and turning points of a graph. Find roots of a function and correctly graph them. Find vertical and horizontal asymptotes and describe an asymptote in terms of limits. Construct, use and describe transformations and operations of functions. For example: Describe how the graph of a function can be the result of vertical and horizontal shifts, stretches and compressions of the graph of a basic function. Construct, use and describe composition of functions. For example: Create new functions by composing basic functions, describe the domain of the new function, and decompose a composite function into basic functions. Construct, use and describe inverses of functions. For example: For a given function, determine if an inverse exists. Find the inverse function and describe its domain and range. Algebraic Reasoning
Outcome: Students will identify and apply algebraic reasoning to write equivalent expressions, solve equations and interpret inequalities. Use factoring techniques to simplify expressions and locate roots. For example: Use the distributive property, multiply polynomials, complete the square, factor and work with inequalities (as they arise from absolute value, distances and other similar geometric interpretations). Be familiar with complex roots and basic properties of complex numbers. Use algebraic reasoning to simplify a variety of expressions and find roots of equations involving multiple function types. For example: Correctly apply properties of exponents and logarithms. Be able to work with polynomial, radical and rational functions. Describe asymptotic behavior of functions near roots of the denominator and as x increases/decreases without bound. Use rational exponents to express and simplify a variety of expressions and solve equations. For example: Translate between radicals and rational exponents, factor out common rational powers, simplify fractional expressions involving rational exponents. Recognize, solve and apply systems of linear equations using matrices. For example: Set up and solve systems of linear equations using simple substitution and Gaussian elimination. Conic Sections (optional): Outcome: Students will recognize and work with the four conic sections: circles, ellipses, parabolas and hyperbolas. Identify the four conic sections and basic properties. For example: Given a graph or equation of a conic, identify the conic as a circle, ellipse, parabola or hyperbola. For circles, identify the center and radius. For ellipses, identify the major axis, minor axis, vertices and foci. For parabolas, identify the vertex, focus and axis of symmetry. For hyperbolas, identify the center and asymptotes. Graph basic conic sections. For example: Graph a conic given its equation or a list of properties.
Write equations of conics given a list of properties. For example: Given the center and a point on a circle, write its equation. Given the focus and vertex of a parabola, write its equation. Given enough information about an ellipse, write its equation.
Content Learning Outcomes for Precalculus B: The learning outcomes for Precalculus B fall under these three topics: Geometric reasoning Trigonometry Conic Sections (optional) Geometric Reasoning: Outcome: Students will apply geometric reasoning to model and solve problems involving length, area and volume. Use geometric formulas for length, area and volume of common shapes For example: Apply the Pythagorean Theorem and determine the distance between points in the plane. Compute circumference and area of circles. Compute perimeter and area of triangles and rectangles. Calculate the volume of spheres, rectangular solids, cones and cylinders. Use proportional reasoning to describe and identify geometric quantities. For example: Determine the arc length of a sector. Find missing lengths or angles in similar triangles. Conic Sections (optional): Outcome: Students will recognize and work with the four conic sections: circles, ellipses, parabolas and hyperbolas. Identify the four conic sections and basic properties. For example: Given a graph or equation of a conic, identify the conic as a circle, ellipse, parabola or hyperbola. For circles, identify the center and radius. For ellipses, identify the major axis, minor axis, vertices and foci. For parabolas, identify the vertex, focus and axis of symmetry. For hyperbolas, identify the center and asymptotes. Graph basic conic sections.
For example: Graph a conic given its equation or a list of properties. Write equations of conics given a list of properties. For example: Given the center and a point on a circle, write its equation. Given the focus and vertex of a parabola, write its equation. Given enough information about an ellipse, write its equation. Trigonometry: Outcome: Students will model and solve meaningful problems using trigonometric functions and their properties. Demonstrate an understanding of the properties of angles and of the basic trigonometric functions. For example: Convert between degrees and radians. Interpret sine and cosine as coordinates on a unit circle. Understand definitions of tangent, cotangent, secant and cosecant. Apply right triangle trigonometry and recognize when a trigonometric function appropriately represents the relationship between two variables. Prove and use trigonometric identities For Example: Use the Pythagorean identity (and its variations), double and half angle identities, and angle addition and subtraction formulas to convert and simplify trigonometric expressions. Identify important properties of the graphs of trigonometric functions. For example: Identify amplitude, period, frequency, phase shift (domain shift), and vertical and horizontal shifts and stretches. Solve equations involving trigonometric functions. For example: Use identities, properties and factoring to simplify a trigonometric equation. Find general solutions to a trigonometric equation, as well as solutions within a given interval. Solve for missing lengths or angles of oblique triangles. For example: Apply the Law of Sines and Cosines Use and describe inverse trigonometric functions. For example: Use a calculator and reference angle to evaluate inverse trigonometric functions. Solve equations using properties of inverse trigonometric functions.