Fostering Conceptual Understanding in a Developmental Algebra Classroom AMATYC 2009 Las Vegas, NV Nov 13 th 2009 Gowribalan A. Vamadeva University of Cincinnati

BACKGROUND o ASU – Developmental Algebra & Intermediate Algebra – Traditional Curriculum – Text Used Pat – Mckeague o BCC – Pre Algebra, Two Part Pre Algebra, Introductory Algebra, Algebraic Skills, Elementary Algebra Mixture of reform curriculum & traditional curriculum – Text Used Aufman, Martin Gay, & Alice Kaseburg More Reformists than Traditionalists o UC – Present Elementary Algebra I, Elementary Algebra II, & Elementary Algebra III Traditional Curriculum – Text Used Martin Gay & Lial Hornsby

ELEMENTARY ALGEBRA I –MATH 091 Topics include Fractions, Decimals, Integers, Rational Numbers, Variable Expressions, Ratios, Rates, Proportions, Percents, Geometry, Probability & Statistics

ELEMENTARY ALGEBRA II –MATH 092 Topics include First degree equations & inequalities, Functions & linear functions, Linear inequalities, Systems of linear equations & inequalities, Rules of Exponents, and Polynomials.

E LEMENTARY A LGEBRA III –MATH 101 Topics include Factoring & solving polynomial equations, Rational expressions & equations, Rational exponents & Radical expressions & equations, Quadratic equations, & quadratic functions

G RAPHING E QUATIONS & I NEQUALITIES The Rectangular Coordinate System Graphing Linear Equations Intercepts Slope Equations of Lines Introduction to Functions Graphing Linear Inequalities in two variables

T RADITIONAL D EFINITIONS OF THE S LOPE OF A LINE & F UNCTIONS Slope o The slope of a line is a measure of the slant or steepness of a line o The slope is the ratio of the vertical change in y to the horizontal change in x o The slope of a line is a ratio that compares the vertical change with the corresponding horizontal change as we move along the line from one point to another Functions o A function is a relation in which no two ordered pairs have the same first coordinates and different second coordinates o A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component

C ONCEPTUAL & R EFORM IDEAS IN M ATH 092 Distinguish between Ratios and Rates Average Rate of Change rate of change & slope slope slope & equations of lines Function basics lesson plan functions I.pdf functions II.pdf absolute value function.pdf Outside Class Project end of year project 092.pdf

C ONCEPTUAL & R EFORM I DEAS IN MATH 101 course outline rational function introduction radical functions Innovative Approach  introduction to rational equations – DLA, ILD, Student Practice, & Summary – handouthandout quadratic translations leading to vertex form basic functions & properties end of year project

S OME I DEAS FROM MATH 091 o Number Sense If a million seconds went by, how many days have gone by? If a billion seconds went by, how many days have gone by? If a trillion seconds went by, how many days have gone by? o Linear Equations in one variable - applications fulcrum idea college football bowl game

L EARNING O UTCOMES Identify and distinguish among different families of functions: linear, quadratic, rational, and radical by interpreting graphical, symbolic, numerical, and verbal representations. Demonstrate the mathematical skills appropriate for each course Use the appropriate function model to analyze and solve application problems Use a graphing calculator when appropriate to explore and solve problems Explain how a function model relates to an applied situation and interpret problem solutions in the context of the situation.

A CTIVE L EARNING M ETHODS Examples drawn from real life and skills learned in context Group learning activities, instructor led discussions Active participation and team work are essential Problems are approached using a variety of perspectives Summary Examples:1)slope and linear functions(DLA).pdfslope and linear functions(DLA).pdf 2) Rational Equations.pdfRational Equations.pdf 3)Dick Aufman – MaterialsDick Aufman – Materials

CONCLUSION DON’T LET GO YOUR PHILOSOPHIES THERE IS ALWAYS SOME TIME FOR REFORM METHODS PROJECTSPROJECTS ARE NOT JUST IN CALCULUS AND BEYOND PEER LED LEARNING APPRECIATED OVER TIME HOW MANY FACTORING, SIMPLIFYING, AND SOLVING PROBLEMS?FACTORING, SIMPLIFYING ONE STRUCTURE DOES NOT FIT ALL

THANK YOU MATHEMATICS DEPARTMENTS AT BROOKDALE CC AND THE UNIVERSITY OF CINCINNATI