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

(Note: Each cluster ends with a “What Have I Learned?” and a “How Can I Practice?” Each chapter ends with a Concepts and Skills Summary and a Chapter Gateway.)

1. Number Sense

Cluster 1: Introduction to Problem Solving with Whole Numbers.

Activity 1.1. The Bookstore.

Objectives:

Practice communication skills.

Organize information.

Write a solution in sentences.

Develop problem-solving skills.

Activity 1.2. The Classroom.

Objectives:

Organize information.

Develop problem-solving strategies: Draw a picture, Recognize patterns, Do a simpler problem.

Communicate Problem-solving ideas.

Activity 1.3. Problem Solving and Your Calculator.

Objectives:

Identify and use commutative property in calculations.

Use the distributive property to evaluate arithmetic expressions.

Use order of operations to evaluate arithmetic expressions.

Use order of operations in formulas involving whole numbers.

Identify and use the properties of exponents in calculation.

Convert numbers to and from scientific notation.

Identify, understand and use formulas.

Activity 1.4. The Class Schedule.

Objective: Use algorithms to problem solve.

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

Activity 1.5. Delicious Recipes.

Objectives:

Add and subtract fractions.

Multiply and divide fractions.

Activity 1.6. Course Grades and Your GPA.

Objectives:

Problem solving usingfractions and decimals.

Understand and calculate a weighted average.

Activity 1.7. Income and Expenses.

Objective: Problem solve with fractions and decimals.

Cluster 3: Comparisons and Proportional Reasoning.

Activity 1.8. Everything is Relative.

Objectives:

Distinguish between absolute and relative measure.

Write ratios in fraction, decimal, and percent formats.

Determine equivalence of ratios.

Activity 1.9. The Devastation of AIDS in Africa.

Objectives:

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

Write a proportion and then solve the resulting proportion.

Activity 1.10. Who Really Did Better?

Objectives:

Define actual and relative change, conceptually and computationally.

Distinguish between actual and relative change, conceptually and computationally.

Calculate relative change as a percent increase or percent decrease.

Activity 1.11. Going Shopping?

Objectives:

Define growth factor.

Determine growth factors from percent increases.

Apply growth factors to problems involving percent increases.

Define decay factor.

Determine decay factors from percent decreases.

Apply decay factors to problems involving percent decreases.

Activity 1.12. Take an Additional 20% Off.

Objectives:

Define consecutive growth and decay factors.

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

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

Project 1.13. Take it to Heart.

Objective: Solve problems involving percents and proportional reasoning.

Activity 1.14. Fuel Economy?

Objectives:

Apply rates directly to solve problems.

Use proportion equations to solve problems involving rates.

Use Unit (Dimensional) Analysis to solve problems that involve consecutive rates.

Project 1.15. Time Management.

Objectives:

Solve problems involving percents.

Use a protractor to represent percentages in a pie chart.

Lab 1.16. Uncle Sam's Place.

Objective: Solve problems involving unit analysis.

Activity 1.17. Home Improvements.

Objective: Use geometry formulas (perimeter and area) in context.

Cluster 4: Problem Solving with Signed Numbers.

Activity 1.18. Celsius Thermometers.

Objectives:

Identify signed numbers.

Use signed numbers to represent quantities in real world situations.

Compare signed numbers.

Calculate absolute value of numbers.

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

Activity 1.19. Shedding Extra Pounds.

Objective: Multiply and divide signed numbers.

Activity 1.20. Order of Operations Revisited.

Objectives:

Use order of operations to evaluate expressions involving signed numbers.

Evaluate expressions that involve negative exponents.

Distinguish between -5 4 and (-5)4.

Write numbers in scientific notation.

2. Variable Sense.

Cluster 1: Interpreting and Constructing Tables and Graphs.

Activity 2.1. Blood Alcohol Levels.

Objectives:

Identify input and output in situations involving two variable quantities.

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

Represent a relationship between two variables graphically.

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

Activity 2.2. Earth's Temperature.

Objectives:

Construct a graph of data pairs using an appropriately scaled and labeled rectangular coordinate system.

Determine the coordinates of a point on a graph.

Construct a table of data pairs using the data points on a graph.

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

Activity 2.3. College Expenses.

Objectives:

Identify input variables and output variables.

Determine possible replacement values for the input.

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

Construct tables of input/output values.

Activity 2.4. The Write Way to Learn Algebra.

Objectives:

Translate English phrases into symbolic (algebraic) expressions.

Distinguish between a factor and a term.

Evaluate an algebraic expression for a given value(s).

Lab 2.5. How Many Cups Are in That Stack?

Objectives:

Collect input/output data.

Represent input/output data numerically in tables.

Construct tables of data pairs for graphs.

Project 2.6. Body Parts.

Objectives:

Collect input/output data.

Represent input/output data numerically in tables.

Construct tables of data pairs for graphs.

Cluster 2: Solving Equations Numerically, Graphically, and Algebraically.

Activity 2.7. Fund-Raiser.

Objectives:

Translate verbal rules (statements using words) to symbolic rules (equations).

Distinguish between an algebraic expression and an equation.

Evaluate an expression for a given input value.

Solve an equation numerically and graphically.

Determine whether a graph is continuous or a collection of distinct points.

Activity 2.8. Let's Go Shopping.

Objectives:

Translate verbal rules to symbolic rules (equations).

Solve an equation of the form ax = b for x using an algebraic approach.

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

Activity 2.9. Are They the Same?

Objectives:

Translate English phrases into algebraic expressions.

Use symbols to translate English statements into equations.

Write algebraic expressions that involve grouping symbols.

Lab 2.10. Sherlock Holmes.

Objective: Recognize conceptual basis for solving equations algebraically.

Activity 2.11. Leasing a Copier.

Objectives:

Translate English statements into equations.

Solve equations of the form ax + b = c.

Solve problems involving equations of the form ax + b = c.

Activity 2.12. How Long Can You Live?

Objectives:

Represent sets of data by a regression equation of the form y = ax + b.

Solve equations of the form ax + b = c.

Solve problems involving equations of the form ax + b = c.

Activity 2.13. The Algebra of Weather.

Objectives:

Evaluate formulas for specified input values.

Solve a formula for a specified variable.

Cluster 3: Mathematical Modeling Involving Algebraic Expressions.

Activity 2.14. Do It Two Ways.

Objectives:

Apply the distributive property.

Use areas of rectangles to interpret the distributive property geometrically.

Identify equivalent expressions.

Activity 2.15. Ring it Up!

Objectives:

Identify the greatest common factor in an expression.

Factor out the greatest common factor in an expression.

Recognize like terms.

Simplify an expression by combining like terms.

Lab 2.16. Math Magic.

Objectives:

Recognize and algebraic expression as a code of instruction.

Simplify algebraic expressions.

Activity 2.17. Comparing Energy Costs.

Objectives:

Translate English statements into equations.

Solve equations of the form ax + b = cx + d.

Use the distributive property to solve equations involving grouping symbols.

Develop mathematical models to solve problems.

Solve formulas for a specified letter.

Activity 2.18. Summer Job Opportunities.

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

3. Function Sense and Linear Functions.

Cluster 1: Function Sense.

Activity 3.1. Graphs Tell Stories.

Objectives:

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

Sketch a graph that best represents a given situation that is described in words.

Identify increasing, decreasing and constant parts of a graph.

Identify minimum and maximum points on a graph.

Define a function.

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

Activity 3.2. Course Grade.

Objectives:

Represent functions numerically, graphically, and symbolically.

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

Use function notation to represent functions symbolically.

Identify the domain and range of a function.

Identify the practical domain and range of a function.

Activity 3.3. How Fast Did You Lose?

Objective: Determine the average rate of change of an output variable with respect to the input variable.

Cluster 2: Functions and Their Properties.

Activity 3.4. The Snowy Tree Cricket.

Objectives:

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

Determine the slope of the line drawn through two points.

Identify increasing linear functions using slopes.

Activity 3.5. Descending in an Airplane.

Objectives:

Identify lines having negative, zero, or undefined slope.

Identify a decreasing linear function from its graph or slope.

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

Interpret the meaning of horizontal and vertical intercepts of a line.

Activity 3.6. Charity Event.

Objectives:

Determine a symbolic rule for a linear function from contextual information.

Identify the practical meanings of the slope and intercepts of a linear function.

Determine the slope-intercept form of a linear function.

Identify functions as linear by numerical, graphical, and algebraic characteristics.

Activity 3.7. Software Sales.

Objectives:

Identify the slope and vertical intercept from the equation of a line written in slope-intercept form.

Write an equation of a line in the slope-intercept form.

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

Determine horizontal intercepts of linear functions using an algebraic approach.

Use intercepts to graph a linear function.

Cluster 3: Problem Solving with Linear Functions.

Activity 3.8. Predicting Population.

Objectives:

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

Write linear functions in slope-intercept form.

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

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

Activity 3.9. Housing Prices.

Objectives:

Write an equation for a linear function given two points.

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

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

Activity 3.10. Oxygen for Fish.

Objectives:

Construct scatterplots from sets of data.

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

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

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

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

Cluster 4: Systems of Two Linear Equations.

Activity 3.11. Business Checking Account.

Objectives:

Solve a system of two linear equations numerically.

Solve a system of two linear equations graphically.

Solve a system of two linear equations symbolically by the substitution method.

Recognize the connections between the three methods of solution.

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

Activity 3.12. Healthy Life-Style.

Objectives:

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

Solve equations containing parenthesis.

Activity 3.13. Modeling a Business.

Objectives:

Solve a system of two linear equations by any method.

Solve a system of linear inequalities graphically.

Determine the break-even point of a linear system algebraically and graphically.

Interpret break-even points in contextual situations.

Activity 3.14. Supply and Demand.

Objectives:

Solve problems that involve systems of two linear equations.

Determine the equilibrium point in a system of linear equations.

Interpret equilibrium points in contextual situations.

Activity 3.15. How Long Can You Live.

Objectives:

Use properties of inequalities to solve linear inequalities algebraically.

Solve compound inequalities algebraically and graphically.

Solve linear inequalities numerically and graphically.

4. An Introduction to Non-Linear Problem Solving.

Cluster 1: Mathematical Modeling Involving Polynomials.

Activity 4.1. Fatal Crashes.

Objectives:

Identify Polynomials.

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

Simplify a polynomial by identifying and combining like terms.

Add and subtract polynomials.

Model a situation using a polynomial expression.

Evaluate a polynomial.

Activity 4.2. Volume of a Storage Tank.

Objectives:

Use property of exponents to multiply powers having the same base.

Use the distributive property and property of exponents to write an expression as an equivalent expression in expanded form.

Identify the greatest common factor (GCF) in an algebraic expression.

Factor algebraic expressions completely.

Check factored algebraic expressions using the distributive property.

Use property of exponents to divide powers having the same base.

Use property of exponents to raise a power to a power.

Activity 4.3. Room for Work.

Objectives:

Expand and simplify the product of two binomials.

Expand and simplify the product of any two polynomials.

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

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

Cluster 2: Problem Solving with Quadratic Equations.

Activity 4.4. The Amazing Property of Gravity.

Objectives:

Evaluate functions of the form y = ax².

Graph functions of the form y = ax² on a graphing calculator.

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

Solve an equation of the form ax² = c graphically.

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

Solve an equation of the form (x + a)² = c algebraically by taking square roots.

Activity 4.5. What Goes Up, Comes Down.

Objectives:

Evaluate functions of the form y = ax² + bx.

Graph functions of the form y = ax² + bx on a graphing calculator.

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

Interpret the x-intercepts of a function in context.

Factor a binomial of the form ax² + bx.

Solve an equation of the form ax² + bx = 0 using the Zero Product Rule.

Activity 4.6. How High Did it Go?

Objectives:

Graph functions of the form y = ax² + bx on a graphing calculator.

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

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

Factor a trinomial of the form x² + bx = c.

Solve a factorable quadratic equation using the Zero Product Rule.

Identify a quadratic function from its algebraic form.

Activity 4.7. More Ups and Downs.

Objectives:

Use the quadratic formula to solve a quadratic equation.

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

Cluster 3: Other Non-Linear Functions.

Activity 4.8. Inflation.

Objectives:

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

Graph exponential functions from numerical data.

Recognize exponential functions from symbolic rules.

Graph exponential functions from symbolic rules.

Activity 4.9. Diving Under Pressure or Don't Hold Your Breath.

Objectives:

Recognize functions of the form y = k/x as non-linear.

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

Graph rational functions from symbolic rule.

Activity 4.10. Hang Time.

Objectives:

Recognize functions of the form y = a x as non-linear.

Evaluate and solve equations that involve square roots.

Graph square root functions from numerical data.

Graph square root functions from symbolic rules.

Appendix A: Fractions.

Appendix B: Decimals.

Appendix C: Algebra Extenstions.

Appendix D: The TI-83 Graphing Calculator.