The second book of a three-part series, An Introduction to Algebraic, Graphical, and Numerical Problem Solving, Fourth Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and the accompanying practice exercises. Along with the activities and the exercises within the text, MathXL®and MyMathLab®have been enhanced to create a better overall learning experience for the reader. Technology integrated throughout the text helps readers interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops readers' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines.

The Consortium for Foundation Mathematics is a team of fourteen co-authors, primarily from the State University of New York and the City University of New York systems. Using the AMATYC *Crossroads* standards, the team developed an activity-based approach to mathematics in an effort to reach the large population of college students who, for whatever reason, have not yet succeeded in learning mathematics.

**Chapter 1. Number Sense**

**Cluster 1. Introduction to Problem Solving**

**Activity 1.1 The Bookstore**

Objectives:

1. Practice communication skills.

2. Organize information.

3. Write a solution in sentences.

4. Develop problem-solving skills.

**Activity 1.2 The Classroom**

Objectives:

1. Organize information.

2. Develop problem-solving strategies.

• Draw a picture.

• Recognize a pattern.

• So a simpler problem.

3. Communicate problem-solving ideas.

**Activity 1.3 Properties of Arithmetic**

Objectives:

1. Identify and use the commutative property in calculations.

2. Use the distributive property to evaluate arithmetic expressions.

3. Use the order of operations convention to evaluate arithmetic expressions.

4. Identify and use the properties of exponents in calculations

5. Covert numbers to and from scientific notation.

6. Identify, understand, and use formulas.

7. Use the order of operations convention in formulas involving whole numbers.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

**Cluster 2. Problem Solving with Fractions and Decimals (Rational Numbers)**

**Activity 1.4 Delicious Recipes**

Objectives:

1. Add and subtract fractions.

2. Multiply and divide fractions.

**Activity 1.5 Course Grades and Your GPA**

Objectives:

1. Recognize and calculate a weighted average.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

**Cluster 3. Comparisons and Proportional Reasoning **

**Activity 1.6 Everything is relative**

Objectives:

1. Distinguish between absolute and relative measure.

2. Write ratios in fraction, decimal, and percent formats.

3. Determine equivalence of rations.

**Activity 1.7 The Devastation of AIDS in Africa**

Objectives:

1. Use proportional reasoning to apply a known ration to a given piece of information.

**Activity 1.8 Who Really Did Better?**

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 1.9 Going Shopping**

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 1.10 Take an Additional 20% Off**

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 1.11 Fuel Economy**

Objectives:

1. Apply rates directly to solve problems.

2. Use unit or dimensional analysis to solve problems that involve consecutive rates.

Cluster 3 What Have I Learned?

Cluster 3 How Can I Practice?

Skills Check 1

**Cluster 4. Problem Solving with Signed Numbers**

Activity 1.12 Celsius Thermometers

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 1.13 Shedding the Extra Pounds

Objectives:

1. Multiply and divide signed numbers.

Activity 1.14 Order of Operations Revisited

Objectives:

1. Use the order of operations convention to evaluate expressions involving signed numbers.

2. Evaluate expressions that involve negative exponents.

3. Distinguish between such expressions asand

4. Write very small numbers in scientific notation.

Cluster 4 What Have I Learned?

Cluster 4 How Can I Practice?

Skills Check 2

Chapter 1 Summary

Chapter 1 Gateway Review

**Chapter 2. Variable Sense**

**Cluster 1. Interpreting and Constructing Tables and Graphs**

**Activity 2.1 Blood-Alcohol Levels**

Objectives:

1. Identify input and output in situations involving two variable quantities.

2. Determine the replacement values for a variable within a given situation.

3. Use a table to numerically represent a relationship between two variables.

4. Represent a relationship between two variables graphically.

5. Identify trends in data pairs that are represented numerically and graphically.

**Activity 2.2 Earth’s Temperature**

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. Identify points that lie in a given quadrant or on a given axis.

**Activity 2.3 College Expenses**

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 Symbolizing Arithmetic**

Objectives:

1. Generalize from an arithmetic calculation to a symbolic representation by utilizing variables.

2. Evaluate algebraic expressions.

**Lab Activity 2.5 How Many Cups Are in that Stack?**

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.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

**Cluster 2. Solving Equations**

**Activity 2.6 Let’s Go Shopping**

Objectives:

1. Translate verbal rules into symbolic rules.

2. Solve an equation of the form *ax* = *b*, *a* ≠ 0, for *x* using an algebraic approach.

3. Solve an equation of the form *x* + *a* + *b* for *x* using an algebraic approach.

**Activity 2.7 Leasing a Copier**

Objectives:

1. Model contextual situations with symbolic rules of the form *y* = *ax* + *b*, *a* ≠ 0.

2. Solve equations of the form *ax* + *b* + *c*, *a* ≠ 0.

**Activity 2.8 The Algebra of Weather**

Objectives:

1. Evaluate formulas for specified input values.

2. Solve a formula for a specified variable.

**Activity 2.9 Four out of Five Dentists Prefer Crest**

Objectives:

1. Recognize that equivalent fractions lead to proportions.

2. Use proportions to solve problems involving ratios and rates.

3. Solve proportion equations.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

**Cluster 3. Mathematical Modeling and Problem Solving**

**Activity 2.10 Are They the Same?**

Objectives:

1. Translate verbal rules into symbolic (algebraic) rules.

2. Write algebraic expressions that involve grouping symbols.

3. Evaluate algebraic expressions containing two of more operations.

4. Identify equivalent algebraic expressions by examining their outputs.

**Activity 2.11 Do It Two Ways**

Objectives:

1. Apply the distributive property.

2. Use areas of rectangles to interpret the distributive property geometrically.

3. Identify equivalent expressions.

4. Identify the greatest common factor in an expression.

5. Factor out the greatest common factor in an expression.

6. Recognize like terms.

7. Simplify an expression by combining like terms.

**Lab Activity 2.12 Math Magic**

Objectives:

1. Recognize an algebraic expression as a code of instruction.

2. Simplify algebraic expressions.

**Activity 2.13 Comparing Energy Costs**

Objectives:

1. Translate verbal rules into symbolic statements.

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 2.14 Summer Job Opportunities**

Objectives:

1. Use critical-thinking skills to make decisions based on solutions of systems of two linear equations.

Cluster 3 What Have I Learned?

Cluster 3 How Can I Practice?

Chapter 2 Summary

Chapter 2 Gateway Review

**Chapter 3. Function Sense and Linear Functions**

**Cluster 1. Function Sense**

**Activity 3.1 Graphs Tell Stories**

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**

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 domain and range of a function.

5. Identify the practical domain and range of a function.

**Activity 3.3 How Fast Did You Lose?**

Objectives:

1. Determine the average rate of change of an output variable with respect to the input receipt.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

**Cluster 2. Introduction to Linear Functions**

**Activity 3.4 The Snowy Tree Cricket**

Objectives:

1. Identify linear functions by a constant average 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 slope.

**Activity 3.5 Descending in an Airplane**

Objectives:

1. Identify lines as 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**

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**

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.

**Activity 3.8 Predicting Population**

Objectives:

1. Write an equation for a linear function given its slope and y-intercept.

2. Write linear functions in slope-intercept form, y = mx + b.

3. Interpret slope and y-intercept of linear functions in contextual situations.

4. Use the slope-intercept form of linear equations to solve problems.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

**Cluster 3. Problem Solving with Linear Functions**

**Activity 3.9 Housing Prices**

Objectives:

1. Determine the slope and y-intercept of a line algebraically and graphically.

2. Determine the equation for a linear function when given two points.

3.Interpret the slope and y-intercept of a linear function in contextual situations.

**Project Activity 3.10 Oxygen for Fish**

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.

**Lab Activity 3.11 Body Parts**

Objectives:

1. Collect and organize data in a table.

2. Plot data in a scatterplot.

3. Recognize linear patterns in paired data.

Cluster 3 What Have I Learned?

Cluster 3 How Can I Practice?

**Cluster 4. Systems of Two Linear Equations**

**Activity 3.12 Business Checking Account**

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 among the three methods of solution.

5. Interpret the solution to a system of two linear equations in terms of the problem’s content.

**Activity 3.13 Healthy Lifestyle**

Objectives:

1. Solve a system of two linear equations algebraically using the substitution method and the addition method.

2. Solve equations containing parentheses.

**Project Activity 3.14 Modeling a Business**

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 3.15 How Long Can You Live?**

Objectives:

1. Use properties of inequalities to solve linear inequalities in one variable. algebraically.

2. Solve linear inequalities graphically.

Cluster 4 What Have I Learned?

Cluster 4 How Can I Practice?

Chapter 3 Summary

Chapter 3 Gateway Review

**Chapter 4. An Introduction to Nonlinear Problem Solving**

**Cluster 1. Mathematical Modeling Involving Polynomials**

**Activity 4.1 Fatal Crashes**

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 Box**

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 expressions in expanded form.

**Activity 4.3 Room for Work**

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.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

**Cluster 2. Problem Solving with Quadratic Equations and Functions**

**Activity 4.4 The Amazing Property of Gravity**

Objectives:

1. Evaluate functions of the form y = ax².

2. Graph functions of the form y = ax².

3. Interpret the coordinates of points on the graph of y = ax² in context.

4. Solve an equation of the form ax² = c graphically.

5. Solve an equation of the form ax² = c algebraically by taking the square roots.

6. Solve an equation of the formalgebraically by taking square roots.

**Activity 4.5 What Goes Up, Comes Down**

Objectives:

1. Evaluate quadratic functions of the form y = ax² + bx, a ≠ 0.

2. Graph functions of the form y = ax² + bx, a ≠ 0.

3. Identify the x-intercepts of the graph of y = ax² + bx graphically and algebraically.

4. Interpret the x-intercepts of a quadratic function in context.

5. Factor a binomial of the form ax² + bx..

6. Solve an equation of the form ax² + bx = 0 using the zero-product rule.

**Activity 4.6 How High Did it Go?**

Objectives:

1. Recognize and write a quadratic equation in standard form, ax² + bx = c, a ≠ 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 property.

4. Identify a quadratic function from its algebraic form.

**Activity 4.7 More Ups and Downs**

Objectives:

1. Use the quadratic formula to solve quadratic equations.

2. Identify the solutions of a quadratic equation with points on the corresponding graph.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

**Cluster 3. Other Nonlinear Functions**

**Activity 4.8 Exponential Growth**

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 A Thunderstorm**

Objectives:

1. Recognize the equivalent forms of the direct variation statement.

2. Determine the constant of proportionality in a direct variation problem.

3. Solve direct variation problems.

**Activity 4.10 Diving Under Pressure, or Don’t Hold Your Breath**

Objectives:

1. Recognize functions of the formas nonlinear.

2. Recognize equations of the form xy = k as inverse variation.

3. Graph an inverse variation relationship from symbolic rules.

4. Solve equations of the form .

**Activity 4.11 Hang Time**

Objectives:

1. Recognize functions of the formas 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.

Cluster 3 What Have I Learned?

Cluster 3 How Can I Practice?

Chapter 4 Summary

Chapter 4 Gateway Review

Appendix A: Fractions

Appendix B: Decimals

Appendix C: Algebraic Extensions

Appendix D: Getting Started with the TI-83/TI-84 Plus Family of Calculators

Appendix E: Learning Math Opens Doors: Twelve Keys to Success