Mathematics in Action : An Introduction to Algebraic, Graphical, and Numerical Problem Solving

by Consortium for Foundation Mathematics

• ISBN13: 9780321149183
• ISBN10: 0321149181
• Edition: 2nd
• Format: Paperback
• Copyright: 2004-01-01
• Publisher: Addison Wesley

Summary

This introductory algebra text, based on standards in the AMATYC Crossroads document, motivates college math students to develop mathematical literacy and a solid foundation for future study in mathematics and other disciplines. This second book of a three-book series presents mathematical concepts and skills through relevant activities derived from real-life situations; these activities are meaningful to students because they illustrate how mathematics arises naturally from real-world situations and problems. The Mathematics in Action series is based on the assumption that students learn mathematics best by doing mathematics in a meaningful context. Students take an active role in their own learning by working in groups, thereby developing communication skills, a sense of independence, and a can-do attitude about mathematics. Technology is integrated throughout the book so that students learn to interpret real-life data numerically, symbolically, and graphically. Regardless of their level of preparation for the course, students can use this text to increase their knowledge of mathematics, their problem-solving skills, and their overall confidence in their ability to learn.

Table of Contents

PREFACE	xiii
CHAPTER 1 NUMBER SENSE	1	(160)
Cluster 1 Introduction to Problem Solving with Whole Numbers
Activity 1.1 The Bookstore	1	(2)
Objectives:
1. Practice communication skills.
2. Organize information.
3. Write a solution in sentences.
4. Develop problem-solving skills.
Activity 1.2 The Classroom	3	(6)
Objectives:
1. Organize information.
2. Develop problem-solving strategies.
Draw a picture.
Recognize patterns.
Do a simpler problem.
3. Communicate problem-solving ideas.
Activity 1.3 Problem Solving and Your Calculator	9	(13)
Objectives:
1. Identify and use the commutative property in calculations.
2. Use the distributive property to evaluate arithmetic expressions.
3. Use order of operations to evaluate arithmetic expressions.
4. Identify and use the properties of exponents in calculation.
5. Convert numbers to and from scientific notation.
6. Identify, understand, and use formulas.
7. Use the order of operations in formulas involving whole numbers.
Project Activity 1.4 The Class Schedule	22	(3)
Objective:
1. Use algorithms to solve problems.
What Have I Learned?	25	(2)
How Can I Practice?	27	(2)
Cluster 2 Problem solving with Fractions and Decimals (Rational Numbers)	29	(21)
Activity 1.5 Delicious Recipes	29	(5)
Objectives:
1. Add and subtract fractions.
2. Multiply and divide fractions.
Activity 1.6 Course Grades and Your GPA	34	(5)
Objectives:
1. Solve problems using fractions and decimals.
2. Recognize and calculate a weighted average.
Project Activity 1.7 Income and Expenses	39	(7)
Objective:
1. Solve problems with fractions and decimals.
What Have I Learned?	46	(1)
How Can I Practice?	47	(3)
Cluster 3 Comparisons and Proportional Reasoning	50	(57)
Activity 1.8 Everything Is Relative	50	(9)
Objectives:
1. Distinguish between absolute and relative measure.
2. Write ratios in fraction, decimal, and percent formats.
3. Determine equivalence of ratios.
Activity 1.9 The Devastation of AIDS in Africa	59	(7)
Objectives:
1. Use proportional reasoning to apply a known ratio to a given piece of information.
2. Write a proportion and then solve the resulting proportion.
Activity 1.10 Who Really Did Better?	66	(5)
Objectives:
1. Define actual and relative change.
2. Distinguish between actual and relative change.
3. Calculate relative change as a percent increase or percent decrease.
Activity 1.11 Going Shopping	71	(11)
Objectives:
1. Define growth factor.
2. Determine growth factors from percent increases.
3. Apply growth factors to problems involving percent increases.
4. Define decay factor.
5. Determine decay factors from percent decreases.
6. Apply decay factors to problems involving percent decreases.
Activity 1.12 Take an Additional 20% Off	82	(5)
Objectives:
1. Define consecutive growth and decay factors.
2. Determine a consecutive growth or decay factor from two or more consecutive percent changes.
3. Apply consecutive growth and/or decay factors to solve problems involving percent changes.
Activity 1.13 Fuel Economy	87	(9)
Objectives:
1. Apply rates directly to solve problems.
2. Use proportion equations to solve problems involving rates.
3. Use unit or dimensional analysis to solve problems that involve consecutive rates.
Project Activity 1.14 Take It To Heart	96	(3)
Objective:
1. Solve problems involving percents, formulas, and proportional reasoning.
What Have I Learned?	99	(2)
How Can I Practice?	101	(3)
Skills Check 1	104	(3)
Cluster 4 Problem Solving with Signed Numbers	107	(54)
Activity 1.15 Celsius Thermometers	107	(13)
Objectives:
1. Identify signed numbers.
2. Use signed numbers to represent quantities in real-world situations.
3. Compare signed numbers.
4. Calculate the absolute value of numbers.
5. Identify and use properties of addition and subtraction of signed numbers.
6. Add and subtract signed numbers using absolute value.
Activity 1.16 Shedding the Extra Pounds	120	(9)
Objective:
1. Multiply and divide signed numbers.
Activity 1.17 Order of Operations Revisited	129	(9)
Objectives:
1. Use the order of operations to evaluate expressions involving signed numbers.
2. Evaluate expressions that involve negative exponents.
3. Distinguish between -54 and -(54).
4. Write very small numbers in scientific notation.
What Have I Learned?	138	(3)
How Can I Practice?	141	(3)
Skills Check 2	144	(5)
Chapter 1 Summary	149	(6)
Chapter 1 Gateway Review	155	(6)
CHAPTER 2 VARIABLE SENSE	161	(164)
Cluster 1: Interpreting and Constructing Tables and Graphs	161	(40)
Activity 2.1 Blood-Alcohol Levels	161	(8)
Objectives:
1. Identify input and output in situations involving two variable quantities.
2. Use a table to numerically represent a relationship between two variables.
3. Represent a relationship between two variables graphically.
4. Identify trends in data pairs that are represented numerically and graphically.
Activity 2.2 Earth's Temperature	169	(9)
Objectives:
1. Construct a graph of data pairs using an appropriately scaled and labeled rectangular coordinate system.
2. Determine the coordinates of a point on a graph.
3. Construct a table of data pairs using the data points on a graph.
4. Identify points that lie in a given quadrant or on a given axis.
Activity 2.3 College Expenses	178	(10)
Objectives:
1. Identify input variables and output variables.
2. Determine possible replacement values for the input.
3. Write verbal rules that represent relationships between input and output variables.
4. Construct tables of input/output values.
5. Construct graphs from input/output tables.
Activity 2.4 The Write Way to Learn Algebra	188	(6)
Objectives:
1. Translate verbal phrases into symbolic (algebraic) expressions.
2. Distinguish between a factor and a term in an algebraic expression.
3. Evaluate algebraic expressions for specified input values.
Lab Activity 2.5 How Many Cups Are in That Stack?	194	(1)
Objectives:
1. Collect input/output data.
2. Represent input/output data numerically in tables.
3. Construct tables of data pairs for graphing.
4. Graph input/output data pairs.
Project Activity 2.6 Body Parts	195	(1)
Objectives:
1. Collect input/output data.
2. Represent input/output data numerically in a table.
3. Construct graphs from tables of data pairs.
What Have I Learned?	196	(1)
How Can I Practice?	197	(4)
Cluster 2: Solving Equations Numerically, Graphically, and Algebraically	201	(60)
Activity 2.7 Fund-Raiser	201	(11)
Objectives:
1. Translate verbal rules (statements using words) to symbolic rules (equations).
2. Distinguish an algebraic expression from an equation and from a symbolic rule.
3. Solve an equation numerically and graphically.
4. Distinguish between a graph of distinct points and a graph of a continuous line.
Activity 2.8 Let's Go Shopping	212	(9)
Objectives:
1. Translate verbal rules into symbolic rules.
2. Solve an equation of the form ax = b for x using an algebraic approach.
3. Solve an equation of the form x + a = b for x using an algebraic approach.
Activity 2.9 Are They the Same?	221	(8)
Objectives:
1. Translate verbal rules into symbolic (algebraic) rules.
2. Write algebraic expressions that involve grouping symbols.
3. Evaluate algebraic expressions containing two or more operations.
4. Identify equivalent algebraic expressions by examining their outputs.
Activity 2.10 Sherlock Holmes	229	(2)
Objective:
1. Recognize the conceptual basis for solving equations algebraically.
Activity 2.11 Leasing a Copier	231	(9)
Objectives:
1. Model contextual situations with symbolic rules of the form y = ax + b,a not equal to 0.
2. Solve equations of the form ax + b = c, a not equal to 0.
Activity 2.12 How Long Can You Live?	240	(5)
Objectives:
1. Represent sets of data by a regression equation of the form y = ax + b,a not equal to 0.
2. Solve equations of the form ax + b = c, a not equal to 0.
3. Solve problems involving equations of the form ax + b = c, a not equal to 0.
Activity 2.13 The Algebra of Weather	245	(9)
Objectives:
1. Evaluate formulas for specified input values.
2. Solve a formula for a specified variable.
What Have I Learned?	254	(2)
How Can I Practice?	256	(5)
Cluster 3: Mathematical Modeling and Problem Solving	261	(64)
Activity 2.14 Do It Two Ways	261	(7)
Objectives:
1. Apply the distributive property.
2. Use areas of rectangles to interpret the distributive property geometrically.
3. Identify equivalent expressions.
Activity 2.15 Ring It Up!	268	(8)
Objectives:
1. Identify the greatest common factor in an expression.
2. Factor out the greatest common factor in an expression.
3. Recognize like terms.
4. Simplify an expression by combining like terms.
Lab Activity 2.16 Math Magic	276	(7)
Objectives:
1. Recognize an algebraic expression as a code of instruction.
2. Simplify algebraic expressions.
Activity 2.17 Comparing Energy Costs	283	(13)
Objectives:
1. Translate verbal rules into symbolic rules.
2. Write and solve equations of the form ax + b = cx + d.
3. Use the distributive property to solve equations involving grouping symbols.
4. Develop mathematical models to solve problems.
5. Solve formulas for a specified variable.
Project Activity 2.18 Summer Job Opportunities	296	(4)
Objective:
1. Use critical thinking skills to make decisions based on solutions of systems of two linear equations.
What Have I Learned?	300	(3)
How Can I Practice?	303	(6)
Chapter 2 Summary	309	(6)
Chapter 2 Gateway Review	315	(10)
CHAPTER 3 FUNCTION SENSE AND LINEAR FUNCTIONS	325	(184)
Cluster 1 Function Sense	325	(38)
Activity 3.1 Graphs Tell Stories	325	(10)
Objectives:
1. Describe in words what a graph tells you about a given situation.
2. Sketch a graph that best represents a situation that is described in words.
3. Identify increasing, decreasing, and constant parts of a graph.
4. Identify minimum and maximum points on a graph.
5. Define a function.
6. Use the vertical line test to determine whether a graph represents a function.
Activity 3.2 Course Grade	335	(10)
Objectives:
1. Represent functions numerically, graphically, and symbolically.
2. Determine the symbolic rule that defines a function.
3. Use function notation to represent functions symbolically.
4. Identify the domain and range of a function.
5. Identify the practical domain and range of a function.
Activity 3.3 How Fast Did You Lose?	345	(8)
Objective:
1. Determine the average rate of change of an output variable with respect to the input variable.
What Have I Learned?	353	(3)
How Can I Practice?	356	(7)
Cluster 2 Introduction to Linear Functions	363	(53)
Activity 3.4 The Snowy Tree Cricket	363	(11)
Objectives:
1. Identify linear functions by a constant rate of change of the output variable with respect to the input variable.
2. Determine the slope of the line drawn through two points.
3. Identify increasing linear functions using slopes.
Activity 3.5 Descending in an Airplane	374	(13)
Objectives:
1. Identify lines having negative, zero, or undefined slopes.
2. Identify a decreasing linear function from its graph or slope.
3. Determine horizontal and vertical intercepts of a linear function from its graph.
4. Interpret the meaning of horizontal and vertical intercepts of a line.
Activity 3.6 Charity Event	387	(9)
Objectives:
1. Determine a symbolic rule for a linear function from contextual information.
2. Identify the practical meanings of the slope and intercepts of a linear function.
3. Determine the slope-intercept form of a linear function.
4. Identify functions as linear by numerical, graphical, and algebraic characteristics.
Activity 3.7 Software Sales	396	(12)
Objectives:
1. Identify the slope and vertical intercept from the equation of a line written in slope-intercept form.
2. Write an equation of a line in the slope-intercept form.
3. Use the y-intercept and the slope to graph a linear function.
4. Determine horizontal intercepts of linear functions using an algebraic approach.
5. Use intercepts to graph a linear function.
What Have I Learned?	408	(1)
How Can I Practice?	409	(7)
Cluster 3 Problem Solving with Linear Functions	416	(35)
Activity 3.8 Predicting Population	416	(7)
Objectives:
1. Write an equation for a linear function given its slope and y-intercept.
2. Write linear functions in slope-intercept form.
3. Determine the relative error in a measurement or prediction using a linear model.
4. Interpret the slope and y-intercept of linear functions in contextual situations.
5. Use the slope-intercept form of linear equations to solve problems.
Activity 3.9 Housing Prices	423	(9)
Objectives:
1. Determine the slope and y-intercept of a line algebraically and graphically.
2. Determine the equation for a linear function that includes any two given points.
3. Interpret the slope and y-intercept of a linear function in contextual situations.
Activity 3.10 Oxygen for Fish	432	(10)
Objectives:
1. Construct scatterplots from sets of data.
2. Recognize when patterns of points in a scatterplot are approximately linear.
3. Estimate and draw a line of best fit through a set of points in a scatterplot.
4. Use a graphing calculator to determine a line of best fit by the least-squares method.
5. Estimate the error of representing a set of data by a line of best fit.
What Have I Learned?	442	(1)
How Can I Practice?	443	(8)
Cluster 4 Systems of Two linear Equations	451	(58)
Activity 3.11 Business Checking Account	451	(9)
Objectives:
1. Solve a system of two linear equations numerically.
2. Solve a system of two linear equations graphically.
3. Solve a system of two linear equations symbolically by the substitution method.
4. Recognize the connections between the three methods of solution.
5. Interpret the solution to a system of two linear equations in terms of the problem's content.
Activity 3.12 Healthy Lifestyle	460	(9)
Objectives:
1. Solve a system of two linear equations algebraically using the substitution method and the addition method.
2. Solve equations containing parentheses.
Activity 3.13 Modeling a Business	469	(7)
Objectives:
1. Solve a system of two linear equations by any method.
2. Determine the break-even point of a linear system algebraically and graphically.
3. Interpret break-even points in contextual situations.
Activity 3.14 How Long Can You Live?	476	(9)
Objectives:
1. Use properties of inequalities to solve linear inequalities algebraically.
2. Solve compound inequalities algebraically and graphically.
3. Solve linear inequalities graphically.
What Have I Learned?	485	(1)
How Can I Practice?	486	(5)
Chapter 3 Summary	491	(3)
Chapter 3 Gateway Review	494	(15)
CHAPTER 4 AN INTRODUCTION TO NONLINEAR PROBLEM SOLVING	509
Cluster 1 Mathematical Modeling Involving Polynomials	509	(41)
Activity 4.1 Fatal Crashes	509	(12)
Objectives:
1. Identify polynomials and polynomial functions.
2. Classify a polynomial as a monomial, binomial, or trinomial.
3. Determine the degree of a polynomial.
4. Simplify a polynomial by identifying and combining like terms.
5. Add and subtract polynomials.
6. Evaluate and interpret polynomials.
Activity 4.2 Volume of a Storage Tank	521	(13)
Objectives:
1. Use properties of exponents to simplify expressions and combine powers that have the same base.
2. Use the distributive property and properties of exponents to write an expression as an equivalent expression in expanded form.
3. Identify the greatest common factor (GCF) in an algebraic expression.
4. Factor algebraic expressions completely.
5. Use the distributive property to check factored algebraic expressions.
Activity 4.3 Room for Work	534	(10)
Objectives:
1. Expand and simplify the product of two binomials.
2. Expand and simplify the product of any two polynomials.
3. Recognize and expand the product of conjugate binomials: difference of squares.
4. Recognize and expand the product of identical binomials: perfect-square trinomials.
What Have I Learned?	544	(2)
How Can I Practice?	546	(4)
Cluster 2: Problem Solving with Quadratic Equations	550	(46)
Activity 4.4 The Amazing Property of Gravity	550	(12)
Objectives:
1. Evaluate functions of the form y = axe.
2. Graph functions of the form y = ax2.
3. Interpret the coordinates of points on the graph of y = axe in context.
4. Solve an equation of the form axe = c graphically.
5. Solve an equation of the form axe = c algebraically by taking square roots.
6. Solve an equation of the form (x + a)2 = c algebraically by taking square roots.
Activity 4.5 What Goes Up, Comes Down	562	(8)
Objectives:
1. Evaluate quadratic functions of the form y = axe + bx.
2. Graph functions of the form y = axe + bx.
3. Identify the x-intercepts of y = axe + bx graphically and algebraically.
4. Interpret the x-intercepts of a function in context.
5. Factor a binomial of the form axe + bx.
6. Solve an equation of the form axe + bx = 0 using the zero-product rule.
Activity 4.6 How High Did It Go?	570	(11)
Objectives:
1. Recognize and write a quadratic equation in standard form, ax2 + bx + c = 0.
2. Factor trinomials of the form x2 + bx + c.
3. Solve a factorable quadratic equation of the form x2 + bx + c = 0 using the zero-product rule.
4. Identify a quadratic function from its algebraic form.
Activity 4.7 More Ups and Downs	581	(7)
Objectives:
1. Use the quadratic formula to solve quadratic equations.
2. Identify the solutions of a quadratic equation with points on the corresponding graph.
What Have I Learned?	588	(2)
How Can I Practice?	590	(6)
Cluster 3 Other Nonlinear Functions	596
Activity 4.8 Inflation	596	(9)
Objectives:
1. Recognize an exponential function as a rule for applying a growth factor or a decay factor.
2. Graph exponential functions from numerical data.
3. Recognize exponential functions from symbolic rules.
4. Graph exponential functions from symbolic rules.
Activity 4.9 Diving Under Pressure, or Don't Hold Your Breath	605	(8)
Objectives:
1. Recognize functions of the form y = k/x as nonlinear.
2. Recognize equations of the form xy = k as inverse variation.
3. Graph an inverse relationship from symbolic rules.
4. Solve equations of the form a/k = b.
Activity 4.10 Hang Time	613	(8)
Objectives:
1. Recognize functions of the form y = a square root of x as nonlinear.
2. Evaluate and solve equations that involve square roots.
3. Graph square root functions from numerical data.
4. Graph square root functions from symbolic rules.
What Have I Learned?	621	(4)
How Can I Practice?	625	(4)
Chapter 4 Summary	629	(4)
Gateway Review	633
APPENDIXES
Appendix A: Fractions	A-1
Appendix B: Decimals	A-9
Appendix C: Algebraic Extensions	A-17
Appendix D: Getting Started with the TI-83 Plus	A-25
Appendix E: Learning Math Opens Doors: Twelve Keys to Success	A-39
Selected Answers	A-53
Glossary	A-65
Index	A-71