Mathematics in Action

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. Function Sense

Cluster 1. Modeling with Functions

Activity 1.1 Parking Problems

            Objectives:

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

            2. Identify a functional relationship between two variables.

            3. Identify the independent and dependent variables.

            4. Use a table to numerically represent a functional relationship between two variables.

            5. Write a function using function notation.

Activity 1.2 Fill ’er Up

            Objectives:

            1. Determine the equation (symbolic representation) that defines a function.

            2. Determine the domain and range of a function.

            3. Identify the independent and the dependent variables of a function.

Activity 1.3 Graphically Speaking

            Objectives:

            1. Represent a function verbally, symbolically, numerically and graphically.

            2. Distinguish between a discrete function and a continuous function.

            3. Graph a function using technology.

Activity 1.4 Stopping Short

            Objectives:

            1. Use a function as a mathematical model.

            2. Determine when a function is increasing, decreasing, or constant.

            3. Use the vertical line test to determine if a graph represents a function.

Project Activity 1.5 Graphs Tell Stories

            Objectives:

            1. Describe in words what a graph tells you about a given situation.

            2. Sketch a graph that best represents the situation described in words.

            3. Identify increasing, decreasing, and constant parts of a graph.

            4. Identify minimum and maximum points on a graph.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

 

Cluster 2. Linear Functions

Activity 1.6 Walking for Fitness

            Objective:

            1. Determine the average rate of change.

Activity 1.7 Depreciation

            Objectives:

            1. Interpret slope as an average rate of change.

            2. Use the formula to determine slope.

            3. Discover the practical meaning of vertical and horizontal intercepts.

            4. Develop the slope-intercept form of an equation of a line.

            5. Use the slope-intercept formula to determine vertical and horizontal intercepts.

            6. Determine the zeros of a function.

Activity 1.8 A New Computer

            Objectives:

            1. Write a linear equation in the slope-intercept form, given the initial value and the average rate of change.

            2. Write a linear equation given two points, one of which is the vertical intercept.

            3. Use the point-slope form to write a linear equation given two points, neither of which is the vertical intercept.

            4. Compare slopes of parallel lines.

Activity 1.9 Skateboard Heaven

            Objectives:

            1. Write an equation of a line in standard form Ax + By = C.

            2. Write the slope-intercept form of a linear equation given the standard form.

            3. Determine the equation of a horizontal line.

            4. Determine the equation of a vertical line.

Activity 1.10 College Tuition

            Objectives:

            1. Construct scatterplots from sets of data pairs.

            2. Recognize when patterns of points in a scatterplot have a linear form.

            3. Recognize when the pattern in the scatterplot shows that the two variables are positively related or negatively related.

            4. Estimate and draw a line of best fit through a set of points in a scatterplot.

            5. Use a graphing calculator to determine a line of best fit by the least-squares method.

            6. Measure the strength of the correlation (association) by a correlation coefficient.

            7. Recognize that a strong correlation does not necessarily imply a linear or a cause-and-effect relationship.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

 

Cluster 3. Systems of Linear Equations, Inequalities, and Absolute Value Functions

Activity 1.11 Moving Out

            Objectives:

            1. Solve a system of 2 x 2 linear equations numerically and graphically.

            2. Solve a system of 2 x 2 linear equations using the substitution method.

            3.Solve an equation of the form ax + b = cx + d for x.

Activity 1.12 Healthy Lifestyle

            Objectives:

            1. Solve a 2 x 2 linear system algebraically using the substitution method and the addition method.

            2. Solve equations containing parentheses.

Activity 1.13 Manufacturing Cell Phones

            Objectives:

            1. Solve a 3 x 3 linear system of equations.

Activity 1.14 Earth Week

            Objectives:

            1. Solve a linear system of equations using matrices.

Activity 1.15 How Long Can You Live?

            Objectives:

            1. Solve linear inequalities in one variable numerically and graphically.

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

            3. Solve compound inequalities algebraically.

            4. Use interval notation to represent a set of real numbers described by an inequality.

Activity 1.16 Sales Commission

            Objectives:

            1. Graph a piecewise linear function.

            2. Write a piecewise linear function to represent a given situation.

            3. Graph a function defined by y = |x - c|.

Cluster 3 What Have I Learned?

Cluster 3 How Can I Practice?

Chapter 1 Summary

Chapter 1 Gateway Review

 

Chapter 2. The Algebra of Functions

Cluster 1. Addition, Subtraction, and Multiplication of Polynomial Functions

Activity 2.1 Spending and Earning Money

            Objectives:

            1. Identify a polynomial expression.

            2. Identify a polynomial function.

            3. Add and subtract polynomial expressions.

            4. Add and subtract polynomial functions.

Activity 2.2 The Dormitory Parking Lot

            Objectives:

            1. Multiply two binomials using the FOIL method.

            2. Multiply two polynomial functions.

            3. Apply the property of exponents to multiply powers having the same base.

Activity 2.3 Stargazing

            Objectives:

            1. Convert scientific notation to decimal notation.

            2. Convert decimal notation to scientific notation.

            3. Apply the property of exponents to divide powers having the same base.

            4. Apply the definition of exponents a0 = 1 where a ≠ 0.

            5. Apply the definition of exponents a-n = 1/(an), where a ≠ 0 and n is any real number.

Activity 2.4 The Cube of a Square

            Objectives:

            1. Apply the property of exponents to simplify an expression involving a power to a power.

            2. Apply the property of exponents to expand the power of a product.

            3. Determine the nth root of a real number.

            4. Write a radical as a power having a rational exponent and write a base to a rational exponent as a radical.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

 

Cluster 2. Composition and Inverses of Functions

Activity 2.5 Inflated Balloons

            Objectives:

            1. Determine the composition of two functions.

            2. Explore the relationship between f (g(x)) and g (f(x)).

Activity 2.6 Finding a Bargain

            Objective:

            1. Solve problems using the composition of functions.

Activity 2.7 Study Time

            Objectives:

            1. Determine the inverse of a function represented by a table of values.

            2. Use the notation f -1 to represent an inverse function.

            3. Use the property f (f -1(x)) = f -1 (f (x)) = x to recognize inverse functions.

            4. Determine the domain and range of a function and its inverse.

Activity 2.8 Temperature Conversions

            Objectives:

            1. Determine the equation of the inverse of a function represented by an equation.

            2. Describe the relationship between graphs of inverse functions.

            3. Determine the graph of the inverse of a function represented by a graph.

            4. Use the graphing calculator to produce graphs of an inverse function.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

Chapter 2 Summary

Chapter 2 Gateway Review

 

Chapter 3. Exponential and Logarithmic Functions

Cluster 1. Exponential Functions

Activity 3.1 The Summer Job

            Objectives:

            1. Determine the growth factor of an exponential function.

            2. Identify the properties of the graph of an exponential function defined by y = bx, where b > 1.

            3. Graph an increasing exponential function.

Activity 3.2 Half-life of Medicine

            Objectives:

            1. Determine the decay factor of an exponential function.

            2. Graph a decreasing exponential function.

            3. Identify the properties an exponential function defined by y = bx, where b > 0 and b ≠ 1.

Activity 3.3 Cellular Phones

            Objectives:

            1. Determine the growth and decay factor for an exponential function represented by a table of values or an equation.

            2. Graph exponential functions defined by y = abx, where b > 0 and b ≠ 1, a ≠ 0.

            3. Determine the doubling and halving time.

Activity 3.4 Population Growth

            Objectives:

            1. Determine the annual growth or decay rate of an exponential function represented by a table of values or an equation.

            2. Graph an exponential function having equation y = a(1+r)x.

Activity 3.5 Time is Money

            Objective:

            1. Apply the compound interest and continuous compounding formulas to a given situation.

Activity 3.6 Continuous Growth and Decay

            Objectives:

            1. Discover the relationship between the equations of exponential functions defined by y = abt and the equations of continuous growth and decay exponential functions defined by y = aekt.

            2. Solve problems involving continuous growth and decay models.

            3. Graph base e exponential functions.

Activity 3.7 Bird Flu

            Objectives:

            1. Determine the regression equation of an exponential function that best fits the given data.

            2. Make predictions using an exponential regression equation.

            3. Determine whether a linear or exponential model best fits the data.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

 

Cluster 2. Logarithmic Functions

            Activity 3.8 The Diameter of Spheres

            Objectives:

            1. Define logarithm.

            2. Write an exponential statement in logarithmic form.

            3. Write a logarithmic statement in exponential form.

            4. Determine log and In values using a calculator.

Activity 3.9 Walking Speed of Pedestrians

            Objectives:

            1. Determine the inverse of the exponential function.

            2. Identify the properties of the graph of a logarithmic function.

            3. Graph the natural logarithmic functions.

Activity 3.10 Walking Speed of Pedestrians, continued

            Objectives:

            1. Compare the average rate of change of increasing logarithmic, linear, and exponential functions.

            2. Determine the regression equation of a natural logarithmic function having the equation y = a + bInx that best fits a set of data.

Activity 3.11 The Elastic Ball

            Objectives:

            1. Apply the log of a product property.

            2. Apply the log of a quotient property.

            3. Apply the log of a power property.

            4. Discover change of base formula.

Activity 3.12 Prison Growth

            Objective:

            1. Solve exponential equations both graphically and algebraically.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

Chapter 3 Summary

Chapter 3 Gateway Review

 

Chapter 4. Quadratic and Higher-Order Polynomial Functions

Cluster 1. Introduction to Quadratic Functions

Activity 4.1 Baseball and the Willis Tower

            Objectives:

            1. Identify functions of the form f(x) = ax² + bx + c, as quadratic functions.

            2. Explore the role of c as it relates to the graph of f(x) = ax² + bx + c.

            3. Explore the role of a as it relates to the graph of f(x) = ax² + bx + c.

            4. Explore the role of b as it relates to the graph of f(x) = ax² + bx + c.

            Note: a ≠ 0 in objectives 1-4.

Activity 4.2 The Shot Put

            Objectives:

            1. Determine the vertex or turning point of a parabola.

            2. Identify the vertex as the maximum or minimum.

            3. Determine the axis of symmetry of a parabola.

            4. Identify the domain and range.

            5. Determine the y-intercept of a parabola.

            6. Determine the x-intercept(s) of a parabola using technology.

            7. Interpret the practical meaning of the vertex and intercepts in a given problem.

Activity 4.3 Per Capita Personal Income

            Objectives:

            1. Solve quadratic equations numerically.

            2. Solve quadratic equations graphically.

            3. Solve quadratic inequalities graphically.

Activity 4.4 Sir Isaac Newton

            Objectives:

            1. Factor expressions by removing the greatest common factor.

            2. Factor trinomials using trial and error.

            3. Use the Zero-Product Property to solve equations.

            4. Solve quadratic equations by factoring.

Activity 4.5 Price of Gold

            Objectives:

            1. Solve quadratic equations by the quadratic formula.

Activity 4.6 Heat Index

            Objectives:

            1. Determine quadratic regression models using a graphing calculator.

            2. Solve problems using quadratic regression models.

Activity 4.7 Complex Numbers

            Objectives:

            1. Identify the imaginary unit i = √(-1).

            2. Identify a complex number.

            3. Determine the value of the discriminant b² - 4ac.

            4. Determine the types of solutions to a quadratic equation.

            5. Solve a quadratic equation in the complex number system.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

 

Cluster 2. Curve Fitting and Higher-Order Polynomial Functions

Activity 4.8 The Power of Power Functions

            Objectives:

            1. Identify a direct variation function.

            2. Determine the constant of variation.

            3. Identify the properties of graphs of power functions defined by y = kxn, where n is a positive integer, k ≠ 0.

Activity 4.9 Volume of a Storage Tank

            Objectives:

            1. Identify equations that define polynomial functions.

            2. Determine the degree of a polynomial function.

            3. Determine the intercepts of the graph of a polynomial function.

            4. Identify the properties of the graphs of polynomial functions.

Activity 4.10 Recycling

            Objective:

            1. Determine the regression equation of a polynomial function that best fits the data.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

Chapter 4 Summary

Chapter 4 Gateway Review

 

Chapter 5. Rational and Radical Functions

Cluster 1. Rational Functions

Activity 5.1 Speed Limits

            Objectives:

            1. Determine the domain and range of a function defined by y = k/x, where k is a nonzero real number.

            2. Determine the vertical and horizontal asymptotes of a graph of y = k/x.

            3. Sketch a graph of functions of the form y = k/x.

            4. Determine the properties of graphs having equation y = k/x.

Activity 5.2 Loudness of a Sound

            Objectives:

            1. Graph a function defined by an equation of the form y = k/x, where n is any positive integer and k is a nonzero real number, x ≠ 0.

            2. Describe the properties of graphs having equation y = k/x, x ≠ 0.

            3. Determine the constant of proportionality (also called the constant of variation).

Activity 5.3 Percent Markup

            Objectives:

            1. Determine the domain of a rational function defined by an equation of the form y = k/(g(x)), where k is a nonzero constant and g(x) is a first degree polynomial.

            2. Identify the vertical and horizontal asymptotes of y = k/(g(x)).

            3. Sketch graphs of rational functions defined by y = k/(g(x)).

Activity 5.4 Blood-Alcohol Levels

            Objectives:

            1. Solve an equation involving a rational expression using an algebraic approach.

            2. Solve an equation involving a rational expression using a graphing approach.

            3. Determine horizontal asymptotes of the graph y = f(x)/g(x), where f(x) and g(x) are first-degree polynomials.

Activity 5.5 Traffic Flow

            Objectives:

            1. Determine the least common denominator (LCD) of two or more rational expressions.

            2. Solve an equation involving rational expressions using an algebraic approach.

            3. Solve a formula for a specific variable.

Activity 5.6 Electrical Circuits

            Objectives:

            1. Multiply and divide rational expressions.

            2. Add and subtract rational expressions.

            3. Simplify a complex fraction.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

 

Cluster 2. Radical Functions

Activity 5.7 Sky Diving

            Objectives:

            1. Determine the domain of a radical function defined by y = √(g(x)), where g(x) is a polynomial.

            2. Graph functions having an equation y = √ (g(x)) and y = -√(g(x)).

            3. Identify properties of the graph of y = √(g(x)) and y = -√(g(x)).

Activity 5.8 Falling Objects

            Objective:

            1. Solve an equation involving a radical expression using a graphical and algebraic approach.

Activity 5.9 Propane Tank

            Objectives:

            1. Determine the domain of a function defined by an equation of the form y = n√(g(x)), where n is a positive integer and g(x) is a polynomial.

            2. Graph y = n√(g(x)).

            3. Identify properties of graphs of y = n√(g(x)).

            4. Solve radical equations that contain radical expressions with an index other than 2.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

Chapter 5 Summary

Chapter 5 Gateway Review

 

Chapter 6. Introduction to the Trigonometric Functions

Cluster 1. Introducing Sine, Cosine, and Tangent Functions

Activity 6.1 The Leaning Tower of Pisa

            Objectives:

            1. Identify the sides and corresponding angles of a right triangle.

            2. Determine the length of the sides of similar right triangles using proportions.

            3. Determine the sine, cosine, and tangent of an angle using a right triangle.

            4. Determine the sine, cosine, and tangent of an acute angle by using the graphing calculator.

Activity 6.2 A Gasoline Problem

            Objectives:

            1. Identify complementary angles.

            2. Demonstrate that the sine of one of the complementary angles equal the cosine of the other.

Activity 6.3 The Sidewalks of New York

            Objectives:

            1. Determine the inverse tangent of a number.

            2. Determine the inverse sine and cosine of a number using the graphing calculator.

            3. Identify the domain and range of the inverse sine, cosine, and tangent functions.

Activity 6.4 Solving a Murder

            Objective:

            1. Determine the measures of all sides and angles of a right triangle.

Project Activity 6.5 How Stable is that Tower?

            Objectives:

            1. Solve problems using right-triangle trigonometry.

            2. Solve optimization problems using right-triangle trigonometry with a graphing approach.

Cluster 1 What Have I Learned?

Cluster 1 How Can I Practice?

 

Cluster 2. Why Are the Trigonometric Functions Called Circular Functions?

Activity 6.6 Learn Trig or Crash!

            Objectives:

            1. Determine the coordinates of points on a unit circle using sine and cosine functions.

            2. Sketch a graph of y = sin x and y = cos x.

            3. Identify the properties of the graphs of the sine and cosine functions.

Activity 6.7 It Won’t Hertz

            Objectives:

            1. Convert between degrees and radian measure.

            2. Identify the period and frequency of a function defined by y = a sin (bx) or

            y = a cos (bx) using the graph.

Activity 6.8 Get in Shape

            Objectives:

            1. Determine the amplitude of the graph of y = a sin (bx) or y = a cos (bx).

            2. Determine the period of the graph of y = a sin (bx) or y = a cos (bx) using a formula.

Activity 6.9 The Carousel

            Objective:

            1. Determine the displacement of the y = a sin (bx + c) and y = a cos (bx + c) using a formula.

Activity 6.10 Texas Temperature

            Objectives:

            1. Determine the equation of a sine function that best fits the given data.

            2. Make predictions using a sine regression equation.

Cluster 2 What Have I Learned?

Cluster 2 How Can I Practice?

Chapter 6 Summary

Chapter 6 Gateway Review

 

Appendix A: Concept Review

Appendix B: Trigonometry

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

 

Glossary