A Year-long Pathway to Complete MATH 1101: Introduction to Mathematical Modeling

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1 A Year-long Pathway to Complete MATH 1101: Introduction to Mathematical Modeling A year-long path to complete MATH 1101 will consist of 2 Learning Support (LS) classes and MATH One of the LS courses is a prerequisite course, MATH 0988: Foundations for Mathematical Modeling, and the other is a co-requisite just in time support class, MATH 0998: Support for Mathematical Modeling. MATH 0998 will be taken during the same semester as MATH Students with lower placement scores will be placed in this year-long path. Semester 1: MATH 0988 Semester 2: MATH 1101 MATH 0998 MATH 0988 Foundations for Mathematical Modeling Credit: 4 hours Description: A study of the essential algebraic concepts and applications required for success in Math 1101: Introduction to Mathematical Modeling. Topics may include properties of real numbers, operations with algebraic expressions, linear equations and inequalities in a single variable, operations with polynomials, quadratic equations, graphs of linear and quadratic equations in two variables, and applications of linear and quadratic equations. Lecture/Lab Hours: Four hours per week MATH 0998 Support for Mathematical Modeling Credit: 2 hours Co-requisite: MATH 1101 Description: This course is designed to support a student taking MATH 1101 with just in time assistance. Topics will parallel topics being studied in MATH 1101 as well as the essential algebraic skills needed to be successful MATH Lecture/Lab Hours: 2 or 3

2 Recommended Models for MATH 0988 Foundations for Mathematical Modeling This 4 credit hour course could meet 4 times per week for 50 minutes each, 3 times per week for 70 minutes each, or 2 times per week for 100 minutes each. The later 2 accommodate assessments in an emporium model better. I. Emporium Models Emporium models require computer labs for scheduled class time and additionally, computers must be available to students for outside the scheduled class time. The class time lab can be small with about 20 computers for student use and an instructor, or large with the ratio of 1 instructor/tutor for every 20 students. These models also require instructional software. A. Emporium Model using ALEKS ALEKS is a uniquely designed software teaching product that is designed to assess each student and provide individualize, mastery-based instruction. The instruction can be associated with a McGraw-Hill text to provide an ebook and video instructional resources. Choose an ALEKS 360 book for these resources. Georgia Gwinnett College, Middle Georgia State College, and Georgia Perimeter College (and perhaps others) are currently using this web-based system. B. Emporium Model using MyMathLab MyMathLab is a rich online system that contains video instruction, and ebook and a robust homework/practice system. The system can be individualized by being set up in modules by creating pre and posttests. USG faculty experimenting with this instructional tool reported that it takes instructor time to customize the product, but that resulted in success for many students. College of Coastal Georgia and Georgia Perimeter College piloted emporium models using this product. II. Models without access to computer labs for scheduled class time A. Online homework systems Any of the current major book vender s beginning/intermediate algebra textbooks has topics appropriate to this course. Pearson s MyMathLab and McGraw-Hill s Connect Math provide robust online homework systems to help students working outside of class without an instructor. These systems explain how to work missed problems and offer similar problems. Students grades are recorded for the instructor and the instructor can tell at a glance whether a student is doing the assignments. Both systems mentioned also provide instructional videos. B. Flipped instruction Homework systems with videos can also be used for flipped instruction. Students can view instructional videos outside of class, and working problems in class with the help of the

3 instructor. An instructor could also record their own instructional videos to be used in this model. The entire class could be flipped or only flipped on certain days. III. Student support models An ancillary student support model is an important component of a successful pathway through gateway mathematics. Many institutions have an Academic Resource Center (ARC) with Mathematics tutors. Such tutors are useful for the timid student who fears instructors office hours. Middle Georgia State College on their Macon campus has created a Mathematics Academic Resource Center (MARC) is solely for mathematics tutoring and also has computers for students to use with ALEKS and online homework systems outside of class. The Director and Assistant Director have bachelor degrees in mathematics. Tutors are mathematics majors. Students are allowed to sign up for up to 2 hours of tutoring per week. This support model has been highly successful and an integral part in promoting increases in success rates in gateway mathematics courses. Recommended Course Content for MATH 0988 Foundations for Mathematical Modeling This course is designed to cover the topics in beginning and intermediate algebra that are essential to success in MATH Below is a list of recommended topics sorted by area. Basic Mathematics: Decimal place value Rounding decimals Distributive, commutative and associative properties Order of operations Placing integers and signed rational numbers on a number line Identifying numbers as integers, rational or irrational Identify prime numbers Prime factorizations of natural numbers GCD and LCM Basic operations on integers and signed rational (fraction and decimal) numbers Simplify expressions with Integer exponents Find absolute value of numeric expressions Applications Linear Equations and Inequalities in one variable: Solve simple linear equations symbolically Solve linear equations with several instances of the variable symbolically Translate an English sentence into an equation Solve problems involving area and perimeter Solve simple interest problems with a calculator Find an original price given a sale price

4 Solve percentage problems Solve average problems Solve simple linear inequalities and graph solution sets Find unions and intersections of intervals Solve basic compound inequalities and graph the solution sets Data analysis: Interpret a bar graph Interpret a pie chart Find the mean and median of a data set Interpret a line graph Linear equations in two-variables: Plot points in the Cartesian plane Find a solution to an equation in two variable Graph a linear equation by plotting points Find the slope given a graph, two points, or an equation Find the intercepts of a line given a graph or an equation Write the equation of a line given two points, or a point and the slope Recognize if a relation is function described by a mapping diagram or expressed as a set of ordered pairs Use the vertical line test to determine if a relation is a function Polynomial expressions: Roots: Add, subtract, multiply and divide simple polynomials Factor polynomials including difference of squares, trinomials, factor by grouping Solve quadratic equations by factoring or using the square root property Solve rational equations that simplify to linear equations Solve problems involving quadratic equations that solve by factoring Simplify a square root with a numeric radicand Simplify a higher roots with a numeric radicand Solve an equation with one radical Multiply two square roots with numeric radicands Rationalize a numeric denominator Use the Pythagorean Theorem to solve problems

5 Recommended Models for MATH 0998 Support for Mathematical Modeling This course is a support course/lab for MATH The goal is to give the emotional support and confidence, as well as the mathematical support to the student, in order to promote success in MATH The lab needs to be student centered, flexible, and provide success experiences in order to meet those needs. This should not be just a repeat of lectures from MATH The grade should be tied to the grade in MATH 1101 and tied to attendance and participation. Activities can include the following which we usually have limited time to do in the MATH 1101 classroom: Answering student MATH 1101 homework and content questions Demonstrating and practicing calculator skills Short lectures and/or work out more examples (up to 20 minutes) and topics which have been known to be difficult for many students Practice worksheets Group work solving problems Students explaining how to work a problem to another student Self-check quizzes Exam reviews The course can be scheduled in many different ways based on the scheduling needs and requirements of a particular campus. Here some models: MATH 1101 MWF, and MATH 0998 TR at the same hour with the same instructor MATH 1101 MW, and MATH 0998 TR at the same hour with the same instructor Courses at different hours with different instructors Appropriate content topics for the MATH 1101 example described next might be: First Quarter (MATH 1101 Chapter 1 support) Calculator for simplifying various varieties of expressions Calculator Scatterplots and line graphs Scientific Notation Calculator graphing functions in symbolic form Find Domain and Range of a Function represented graphically and symbolically Calculating average rate of change simplifying rational fractions Solve simple linear equations in preparation for Chapter 2 Second Quarter (MATH 1101 Chapter 2 support) Calculator Linear Regression and modeling Examine the different forms of a line Slope-intercept, point-slope, standard form Rewrite Standard form and point-slope form in slope-intercept form

6 Write the equation of a line given points, slope and other conditions including horizontal, vertical, parallel or perpendicular lines. Calculator Solve equations by Intersection of Graphs method Solve linear equations and inequalities Third Quarter (MATH 1101 Chapter 3 4 support) Graph Quadratic functions by hand Calculator Graph quadratic functions Calculator Quadratic Regression Review factoring Review simplifying numeric radicals Use factoring, the square root property, the quadratic formula and the calculator to solve quadratic equations Calculator Identify intervals of increasing, decrease for graphs Calculator Find Extrema Calculator Cubic Regression Fourth Quarter (MATH 1101 Chapter 5 support) Introduce inverses of operations (Add/subtract; Multiple/Divide) and then define logarithm as an inverse of exponentiation Compound Interest problems Calculator Exponential and Logarithmic regression Calculator Solve exponential and logarithmic equations by Intersection of graphs method Calculator Use the change of base formula Following is a description of a sample MATH 1101 course as implemented by Middle Georgia State College. This implementation has improved success rates significantly. This implementation used College Algebra with Modeling and Visualization, 5 th Edition by Gary Rockswold Pearson ISBN-10:

7 Middle Georgia State College s Philosophy on Math 1101 At Middle Georgia State College we offer two fundamentally different versions of College Algebra for our students. While the focus of Math 1111 is on preparing students for Calculus, Math 1101 is designed to explore algebraic concepts and problem solving using multiple representations: symbolic, graphical, numeric and verbal. It also makes heavy use the TI-84 calculator. Math 1101 is a departmental standardized course with a common final exam. Over the past few years, a committee on the Macon Campus, called Rescue Math 1101, established procedures which have been shown to improve student success. They are described below. HOMEWORK We all know that homework is an essential component to students learning and retaining material. We have provided suggested textbook exercises for each section of the textbook. We have also created a MyMathLab template for online homework that every instructor can copy. Faculty are encouraged to choose the online homework that is graded automatically for the instructor. It is recommended that the homework average, whether MyMathLab or pencil/paper, be part of the final grade (10 to 20%) so that students take this work seriously. IN-CLASS PRACTICE WORKSHEETS Current students are hesitant to ask questions about homework, so we recommend providing an opportunity for practice in class. Worksheets are available for each section covered in this course. These worksheets stimulate student/instructor engagement and provide immediate practice and feedback. The Rescue 1101 committee found these worksheets to be very popular among students. Our research showed that if the worksheets are used as an assessment tool, class attendance improved. CUMULATIVE EXAM GRADING In the sample syllabus you will see a cumulative grading scheme that uses 4 exams and a final exam. This grading scheme gives the student a chance to improve his/her grade by providing more time for mastery. It has been proven successful especially for those students with the steepest learning curves. A comprehensive final exam is required and should count at least 20% of the final grade.

8 COURSE CONTENT Introduction to Functions and Graphs (Chapter 1) Recognize common sets of numbers (1.1) Write and operate with numbers in scientific notation (1.1) Evaluate expressions by hand and with a TI83/84 calculator (1.1) Use problem-solving strategies in applications (1.1) Determine the mean and median of data (1.2) Graph (by hand) a relation in the xy-plane and from data using the TI83/84 calculator (1.2) Find the domain and range of a relation (1.2) Find the midpoint of a line segment (1.2) Find the domain and range of a function (1.3) Identify a function symbolically, numerically, graphically and verbally (1.3) Graph a function using the symbolic form and using the TI83/84 calculator (1.3) Evaluate functions given input both symbolically and on the TI83/84 calculator (1.3) Classify functions as constant, linear or nonlinear (1.4) Calculate slope and interpret it as a rate of change (1.4) Determine whether a function is increasing or decreasing (1.4) Use problem-solving strategies in applications and average rate of change (1.4) Linear Functions and Equations (Chapter 2) Use a linear function to model data by hand and using linear regression the TI83/84 calculator. Use the model to predict results. (2.1,2.4) Identify the characteristics of a linear function (slope, intercepts, zero) given a graph or equation (2.1) Write the equation of a line given points, slope and other conditions including horizontal, vertical, parallel or perpendicular lines. (2.1) Solve linear equations symbolically and graphically by the Intersection of Graphs Method.(2.2) Apply problem-solving strategies to applications involving linear functions (2.1, 2.2, 2.3, 2.4) Solve linear inequalities (including compound inequalities) symbolically, graphically, or numerically and show the solution set using interval notation (2.3) Quadratic Functions and Equations (Chapter 3) Identify quadratic functions and their basic characteristics: vertex, axis of symmetry, direction of opening (3.1) Apply the vertex formula (3.1) Graph a quadratic function by hand and using the TI-83/84 (3.1) Solve application problems including maximum/minimum problems (3.1) Use quadratic regression to model data and use the model to predict results (3.1) Use factoring, the square root property, the quadratic formula and the calculator to solve quadratic equations (3.2) Determine the number of real solutions using the discriminant or the graph (3.2) Solve application problems involving quadratic equations (3.1, 3.2)

9 Polynomial Functions and Models (Chapter 4) Describe the characteristics of a polynomial function: degree, leading coefficient, increasing/decreasing intervals (revisited from chapter 1) and extrema (4.1) Locate local and absolute extrema using TI-83/84 (4.1) Identify the correlation between the degree, x-intercepts and turning points (4.2) Determine the end behavior of a polynomial function (4.2) Use polynomial regression on the calculator to model data and predict input or output values (4.2) Exponential and Logarithmic Functions (Chapter 5) Distinguish between linear and exponential growth (5.3) Use exponential functions in applications problems including compound interest problems (5.3) Model data with exponential and logarithmic functions. Use the model to predict input and output values.(5.3, 5.4) Evaluate common and natural logarithmic functions (5.4) Solve exponential and logarithmic equations (5.4, 5.6) Evaluate logarithms with other bases using the change of base formula (5.4) Choose an appropriate function (linear, quadratic, polynomial, exponential, logarithmic) to model a given set of data (5.7) Other potential MATH 1101 textbooks: 1. College Algebra Graphs and Models, John W. Coburn, McGraw-Hill, 2012, ISBN Functions and Change: A Modeling Approach to College Algebra, 5 th Edition, Crauder, Evans,Nowell, Cengage, 2014, ISBN-10:

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