Mathematics in Action

1. Introduction to Problem Solving



1.1 The Bookstore

1.1.1 Practice communication skills

1.1.2 Organize information

1.1.3 Write a solution in sentences

1.1.4 Develop problem-solving skills



1.2 The Classroom

1.2.1 Organize information

1.2.2 Develop problem-solving strategies: Draw a picture, Recognize a pattern, Do a simpler problem

1.2.3 Communicate problem-solving ideas



1.3 Properties of Arithmetic

1.3.1 Identify and use the commutative property in calculations.

1.3.2 Use the associative property to evaluate arithmetic expressions

1.3.3 Use the order of operations convention to evaluate arithmetic expressions

1.3.4 Identify and use the properties of exponents in calculations

1.3.5 Convert numbers to and from scientific notation



1.4 Top Chef

1.4.1 Add and Subtract Fractions

1.4.2 Multiply and Divide Fractions



1.5 Course Grades and Your GPA

1.5.1 Recognize and Calculate a Weighted Average

1.5.2 Express Fractions in Decimal Format



1.6 Everything is Relative

1.6.1 Distinguish between absolute and relative measure

1.6.2 Write ratios in fraction, decimal, and percent formats

1.6.3 Determine equivalence of ratios



1.7 The Devastation of AIDS in Africa

1.7.1 Use proportional reasoning to apply a known ratio to a given piece of information



1.8 Who Really Did Better?

1.8.1 Define actual and relative change

1.8.2 Distinguish between actual and relative change

1.8.3 Calculate relative change as a percent increase or percent decrease



1.9 Going Shopping

1.9.1 Define growth factor

1.9.2 Determine growth factors from percent increases

1.9.3 Apply growth factors to problems involving percent increases

1.9.4 Define decay factor

1.9.5 Determine decay factors from percent decreases

1.9.6 Apply decay factors to problems involving percent decreases



1.10 Take an Additional 20% Off

1.10.1 Define consecutive growth and decay factors

1.10.2 Determine a consecutive growth or decay factor from two or more consecutive percent changes

1.10.3 Apply consecutive growth and/or decay factors to solve problems involving percent changes



1.11 Fuel Economy

1.11.1 Apply rates directly to solve problems

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



1.12 Celsius Thermometers

1.12.1 Identify signed numbers

1.12.2 Use signed numbers to represent quantities in real-world situations

1.12.3 Compare signed numbers

1.12.4 Calculate the absolute value of numbers

1.12.5 Identify and use properties of addition and subtraction of signed numbers

1.12.6 Add and subtract signed numbers using absolute value



1.13 Shedding the Extra Pounds

1.13.1 Multiply and divide signed numbers



1.14 Order of Operations Revisited

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

1.14.2 Evaluate expressions that involve negative exponents

1.14.3 Distinguish between such expressions as -54 and (-5)4

1.14.4 Write very small numbers in scientific notation



2. Variable Sense



2.1 Symbolizing Arithmetic

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

2.1.2 Evaluate algebraic expressions



2.2 Blood Alcohol Levels

2.2.1 Identify input and output in situations involving two variable quantities

2.2.2 Determine the replacement values for a variable within a given situation

2.2.3 Represent and interpret an input/output relationship algebraically by formula, numerically as data pairs in a table and graphically as points on a graph



2.3 College Expenses

2.3.1 Write verbal rules that represent relationships between input and output variables

2.3.2 Translate verbal rules into symbolic rules

2.3.3 Determine input/output values from a graph



2.4 Are They the Same?

2.4.1 Identify equivalent algebraic expressions by examining their outputs

2.4.2 Identify equivalent algebraic expressions by comparing their graphs

2.4.3 Write algebraic expressions that involve grouping symbols



2.5 Let's Go Shopping

2.5.1 Solve an equation of the form ax = b, a 0, for x using an algebraic approach

2.5.2 Solve an equation of the form x + a = b for x using an algebraic approach

2.5.3 Use the four-step process to solve problems



2.6 Leasing a Copier

2.6.1 Model contextual situations with symbolic rules of the form y = ax + b, a 0

2.6.2 Solve equations of the form ax + b = c, a 0



2.7 The Algebra of Weather

2.7.1 Solve a formula for a specified variable



2.8 Four out of Five Dentists Prefer Crest

2.8.1 Recognize that equivalent fractions lead to proportions

2.8.2 Use proportions to solve problems involving ratios and rates



2.9 Do it Two Ways

2.9.1 Apply the distributive property

2.9.2 Use areas of rectangles to interpret the distributive property geometrically

2.9.3 Identify the greatest common factor in an expression

2.9.4 Factor out the greatest common factor in an expression

2.9.5 Recognize like terms

2.9.6 Simplify an expression by combining like terms



2.10 Decoding

2.10.1 Recognize an algebraic expression as a code of instruction

2.10.2 Simplify algebraic expressions



2.11 Comparing Energy Costs

2.11.1 Develop mathematical models to solve problems

2.11.2 Write and solve equations of the form ax + b = cx + d, where a 0 and c 0

2.11.3 Use the distributive property to solve equations involving grouping symbols

2.11.4 Solve formulas for a specified variable



2.12 Summer Job Opportunities

2.12.1 Use problem solving skills to make decisions based on solutions of mathematical models



3. Function Sense and Linear Functions



3.1 Gold, Silver, and Bronze

3.1.1 Use the input-output concept to define a function

3.1.2 Identify a functional relationship between two variables

3.1.3 Identify the independent and dependent variables

3.1.4 Use function notation to express the input-output correspondence of a function

3.1.5 Identify the domain and range of a function

3.1.6 Expand the rectangular coordinate system to include all four quadrants in the plane

3.1.7 Represent functions in tabular and graphical forms



3.2 How Fast Did You Lose?

3.2.1 Determine the average rate of change of an output variable with respect to the input variable



3.3 Symbolically Defined Functions and their Graphs

3.3.1 Define functions by symbolic rules

3.3.2 Recognize that an output value is determined by evaluating the symbolic rule

3.3.3 Recognize that an input value is determined by solving an associated equation

3.3.4 Understand the connection between the input-output pairs generated by the symbolic rule and the plotted points that form its graph

3.3.5 Understand the connection between the average rates of change of a function and the shape of its graph

3.3.6 Explore and compare the algebraic and graphical methods for evaluating a function

3.3.7 Explore and compare the algebraic and graphical methods for solving an associated equation



3.4 Course Grade

3.4.1 Determine the symbolic rule that defines a function

3.4.2 Identify the practical domain and range of a function



3.5.1 Identify linear functions by a constant average rate of change of the output variable with respect to the input variable

3.5.2 Interpret the slope as an average rate of change

3.5.3 Determine the slope of a nonvertical line drawn through two points

3.5.4 Identify increasing and decreasing linear functions using slope

3.5.5 Determine horizontal and vertical intercepts of a linear function from its graph

3.5.6 Interpret the meaning of horizontal and vertical intercepts in a contextual situations



3.6 Software Sales

3.6.1 Identify whether a situation can be represented by a linear function

3.6.2 Write an equation of a line in slope-intercept form

3.6.3 Use the y-intercept and the slope to graph a linear function

3.6.4 Determine horizontal intercepts of linear functions using an algebraic approach

3.6.5 Use intercepts to graph a linear function

3.6.6 Identify parallel lines



3.7 Predicting Populations

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

3.7.2 Determine the relative error in a measurement or prediction using a linear model



3.8 College Tuition

3.8.1 Determine the equation for a linear function when given two points, neither of which is the vertical intercept

3.8.2 Use the point-slope form, y - y1 = m(1x - x12), to write the equation of a nonvertical line

3.8.3 Identify lines having zero or undefined slopes

3.8.4 Determine the equation of horizontal and vertical lines



3.9 Education Pays

3.9.1 Recognize when patterns of points in a scatterplot are approximately linear

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

3.9.3 Use technology to determine a line of best fit by the least-squares method

3.9.4 Estimate the error of representing a set of data by a line of best fit



3.10 Body Parts

3.10.1 Collect and organize data in a table

3.10.2 Recognize linear patterns in paired data



3.11 Smartphone Plan Options

3.11.1 Solve a system of two linear equations numerically

3.11.2 Solve a system of two linear equations graphically

3.11.3 Solve a system of two linear equations using the substitution method

3.11.4 Recognize the connections between the three methods of solutions

3.11.5 Interpret the solution to a system of two linear equations in terms of the problem's content



3.12 Healthy Lifestyle

3.12.1 Solve a system of two linear equations algebraically using the substitution method

3.12.2 Solve a system of two linear equations algebraically using the addition (or elimination) method



3.13 Modeling a Business

3.13.1 Solve a system of two linear equations by any method

3.13.2 Define and interpret the break-even point as the point at which the cost and revenue functions are equal

3.13.3 Determine the break-even point of a linear cost and revenue system algebraically and graphically



3.14 How Long Can You Live?

3.14.1 Use properties of inequalities to solve linear inequalities in one variable algebraically



4. An Introduction to Nonlinear Problem Solving



4.1 Fatal Crashes

4.1.1 Identify polynomials and polynomial functions

4.1.2 Classify a polynomial as a monomial, binomial, or trinomial

4.1.3 Determine the degree of a polynomial

4.1.4 Simplify a polynomial by identifying and combining like terms

4.1.5 Add and subtract polynomials

4.1.6 Evaluate and interpret polynomials



4.2 Volume of a Storage Box

4.2.1 Use properties of exponents to simplify expressions and combine powers that have the same base

4.2.2 Use the distributive property and properties of exponents to write expressions in expanded form



4.3 Room for Work

4.3.1 Expand and simplify the product of two binomials

4.3.2 Expand and simplify the product of any two polynomials

4.3.3 Recognize and expand the product of conjugate binomials: difference of squares

4.3.4 Recognize and expand the product of identical binomials: perfect-square trinomials



4.4 The Amazing Property of Gravity

4.4.1 Evaluate quadratic functions of the form y = ax2

4.4.2 Graph quadratic functions of the form y = ax2

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

4.4.4 Solve a quadratic equation of the form ax2 = c graphically

4.4.5 Solve a quadratic equation of the form ax2 = c algebraically by taking square roots.

4.4.6 Solve a quadratic equation of the form (x+/-a)2 = c algebraically by taking square roots



4.5 What Goes Up, Comes Down

4.5.1 Evaluate quadratic functions of the form y = ax2 + bx, a 0

4.5.2 Graph quadratic functions of the form y = ax2 + bx, a 0

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

4.5.4 Interpret the x-intercepts of a quadratic function in context

4.5.5 Factor a binomial of the form ax2 + bx

4.5.6 Solve an equation of the form ax2 + bx = 0 using the zero-product property



4.6 How High Did it Go?

4.6.1 Recognize and write a quadratic equation in standard form, ax2 + bx + c = 0,a 0

4.6.2 Factor trinomials of the form x2 + bx + c

4.6.3 Solve a factorable quadratic equation of the form x2 + bx + c = 0 using the zero-product property

4.6.4 Identify a quadratic function from its algebraic form.



4.7 More Ups and Downs

4.7.1 Use the quadratic formula to solve quadratic equations

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



4.8 Inflation

4.8.1 Recognize an exponential function as a rule for applying a growth factor or a decay factor

4.8.2 Graph exponential functions from numerical data

4.8.3 Recognize exponential functions from equations

4.8.4 Graph exponential functions from equations



4.9 A Thunderstorm

4.9.1 Recognize the equivalent forms of the direct variation statement

4.9.2 Determine the constant of proportionality in a direct variation problem

4.9.3 Solve direct variation problems



4.10 Diving Under Pressure, or Don't Hold Your Breath

4.10.1 Recognize functions of the form y = k/x , x 0, as nonlinear

4.10.2 Recognize equations of the form xy = k as inverse variation

4.10.3 Graph an inverse variation relationship from symbolic rules

4.10.4 Solve equations of the form a/x = b, x 0

4.10.5 Identify functions of the form y = k / x2, x 0, as an inverse variation relationship



4.11 Hang Time

4.11.1 Recognize functions of the form y = a (x) as nonlinear

4.11.2 Evaluate and solve equations that involve square roots

4.11.3 Graph square root functions from numerical data

4.11.4 Graph square root functions from symbolic rules



Appendices

A. Fractions

B. Decimals

C. Skills Checks

D. Algebraic Extensions

E. Getting Started with the TI-84 Plus Family of Calculators



Selected Answers

Glossary

Index