Discovering Algebra : An Investigative Approach

by Murdock, Jerald

• ISBN13: 9781559533409
• ISBN10: 1559533404
• Format: Hardcover
• Copyright: 2001-08-01
• Publisher: Springer Verlag

Summary

To meet the challenges of success in school or today's technology-rich work world, students must develop mathematical skills in a meaningful and retrievable way. With Discovering Algebra: An Investigative Approach, the sequence of topics, content emphasis, mathematics integration, hands-on activities, and use of technology combine to form a program that gives students a solid, lasting foundation in algebra and truly reflects the NCTM Principles and Standards for School Mathematics. Discovering Algebra integrates the traditional algebra curriculum with statistics, data analysis, functions, discrete mathematics, geometry, probability, and trigonometry. Students work with data-rich, real-world situations in a curriculum that places algebra in an applications-based context and where investigations precede introduction of formulas and expressions. From topics like fractals, iteration, and self-similarity to technology-accessible techniques like use of the graphing calculator to do statistical analysis, the curriculum encompasses cutting-edge mathematics as well as time-honored topics and concepts. Investigations are the heart of Discovering Algebra. Field tested and revised with extensive teacher feedback, the investigations promote understanding of the material as students move from the concrete to the abstract. Ongoing practice and review ensure that students acquire lasting skills, along with conceptual understanding. The most current technologies prepare students for today's world. The text utilizes graphing calculators (necessary for success with this curriculum), The Geometer's Sketchpad and Fathom Dynamic Statistics software, and sensor technology like the CBR and CBL. The comprehensive Teaching Resources package provides you with a host of resources to enrich learning and to make the teaching experience more rewarding.

Table of Contents

A Note to Students from the Authors	xii
Fractions and Fractals	1	(36)
The Same yet Smaller	2	(7)
Investigation: Connect the Dots	2	(7)
More and More	9	(5)
Investigation: How Many?	9	(5)
Shorter yet Longer	14	(8)
Investigation: How Long Is This Fractal?	14	(5)
Improving Your Reasoning Skills	19	(2)
Project: Invent a Fractal!	21	(1)
Going Somewhere?	22	(6)
Investigation: A Strange Attraction	22	(6)
Out of Chaos	28	(5)
Investigation: A Chaotic Pattern?	28	(5)
Review	33	(4)
Take Another Look	35	(1)
Assessing What You've Learned	36	(1)
Data Exploration	37	(55)
Bar Graphs and Dot Plots	38	(6)
Investigation: Picturing Pulse Rates	39	(4)
Improving Your Reasoning Skills	43	(1)
Summarizing Data with Measures of Center	44	(6)
Investigation: Making ``Cents'' of the Center	44	(6)
Five-Number Summaries and Box Plots	50	(7)
Investigation: Pennies in a Box	51	(6)
Histograms and Stem-and-Leaf Plots	57	(8)
Investigation: Hand Spans	58	(6)
Project: Compare Communities	64	(1)
Activity Day: Exploring a Conjecture	65	(2)
Activity: The Conjecture	65	(2)
Two-Variable Data	67	(7)
Investigation: Backed into a Corner	67	(6)
Improving Your Reasoning Skills	73	(1)
Estimating	74	(6)
Investigation: Guesstimating	74	(5)
Project: Baseball Relationships	79	(1)
Using Matrices to Organize and Combine Data	80	(7)
Investigation: Row-by-Column Matrix Multiplication	82	(5)
Review	87	(5)
Take Another Look	90	(1)
Assessing What You've Learned	91	(1)
Proportional Reasoning and Probability	92	(46)
Proportions	93	(7)
Investigation: Multiply and Conquer	94	(5)
Project: The Golden Ratio	99	(1)
Capture-Recapture	100	(5)
Investigation: Fish in the Lake	100	(4)
Improving Your Reasoning Skills	104	(1)
Proportions and Measurement Systems	105	(5)
Investigation: Converting Centimeters to Inches	105	(4)
Improving Your Visual Thinking Skills	109	(1)
Increasing and Decreasing	110	(6)
Investigation: Enlarging and Reducing	111	(4)
Improving Your Reasoning Skills	115	(1)
Circle Graphs and Relative Frequency Graphs	116	(6)
Investigation: Circle Graphs and Bar Graphs	116	(6)
Probability Outcomes and Trials	122	(7)
Investigation: Candy Colors	123	(5)
Project: Probability, Genes, and Chromosomes	128	(1)
Random Outcomes	129	(5)
Investigation: Calculator Coin Toss	123	(11)
Review	134	(4)
Take Another Look	136	(1)
Assessing What You've Learned	137	(1)
Variation and Graphs	138	(43)
Using Rates	139	(7)
Investigation: Off to Work We Go	139	(6)
Improving Your Reasoning Skills	145	(1)
Direct Variation	146	(8)
Investigation: Ship Canals	146	(7)
Improving Your Reasoning Skills	153	(1)
Scale Drawings and Similar Figures	154	(9)
Investigation: Floor Plans	154	(8)
Project: Make Your Own Map	162	(1)
Inverse Variation	163	(9)
Investigation: Seesaw Nickels	163	(8)
Project: Families of Rectangles	171	(1)
Activity Day: Variation with a Bicycle	172	(3)
Activity: The Wheels Go Round and Round	172	(3)
Review	175	(6)
Mixed Review	177	(2)
Take Another Look	179	(1)
Assessing What You've Learned	180	(1)
Linear Equations	181	(69)
Order of Operations and the Distributive Property	182	(8)
Investigation: Cross-number Puzzle	183	(7)
Writing Expressions and Undoing Operations	190	(9)
Investigation: The Math Behind the Number Trick	190	(9)
Recursive Sequences	199	(7)
Investigation: Recursive Toothpick Patterns	200	(6)
Linear Plots	206	(7)
Investigation: On the Road Again	207	(6)
Activity Day: Time-Distance Relationships	213	(3)
Activity: Walk the Line	213	(2)
Project: Pascal's Triangle	215	(1)
Linear Equations and the Intercept Form	216	(9)
Investigation: Working Out with Equations	216	(9)
Linear Equations and Rate of Change	225	(8)
Investigation: Wind Chill	226	(6)
Project: Legal Limits	232	(1)
Solving Equations Using the Balancing Method	233	(9)
Investigation: Balancing Pennies	233	(9)
Activity Day: Modeling Data	242	(2)
Activity: Tying Knots	242	(1)
Improving Your Reasoning Skills	243	(1)
Review	244	(6)
Take Another Look	248	(1)
Improving Your Reasoning Skills	248	(1)
Assessing What You've Learned	249	(1)
Fitting a Line to Data	250	(57)
A Formula for Slope	251	(10)
Investigation: Points and Slope	251	(9)
Project: Step Right Up	260	(1)
Writing a Linear Equation to Fit Data	261	(9)
Investigation: Beam Strength	262	(7)
Improving Your Visual Thinking Skills	269	(1)
Point-Slope Form of a Linear Equation	270	(6)
Investigation: The Point-Slope Form for Linear Equations	271	(5)
Equivalent Algebraic Equations	276	(8)
Investigation: Equivalent Equations	276	(7)
Improving Your Geometry Skills	283	(1)
Writing Point-Slope Equations to Fit Data	284	(4)
Investigation: Life Expectancy	284	(4)
More on Modeling	288	(8)
Investigation: Fire!!!!	288	(7)
Project: State of the States	295	(1)
Applications of Modeling	296	(5)
Investigation: What's My Line?	296	(4)
Improving Your Reasoning Skills	300	(1)
Activity Day: Data Collection and Modeling	301	(2)
Activity: The Toyland Bungee Jump	301	(2)
Review	303	(4)
Take Another Look	305	(1)
Assessing What You've Learned	306	(1)
Systems of Equations and Inequalities	307	(59)
Solving Systems of Equations	308	(8)
Investigation: Where Will They Meet?	310	(6)
Solving Systems of Equations Using Substitution	316	(8)
Investigation: All Tied Up	317	(7)
Solving Systems of Equations Using Elimination	324	(7)
Investigation: Paper Clips and Pennies	325	(6)
Solving Systems of Equations Using Matrices	331	(8)
Investigation: Diagonalization	333	(6)
Inequalities in One Variable	339	(8)
Investigation: Toe the Line	340	(5)
Improving Your Visual Thinking Skills	345	(1)
Project: Temperatures	346	(1)
Graphing Inequalities in Two Variables	347	(7)
Investigation: Graphing Inequalities	347	(7)
Systems of Inequalities	354	(8)
Investigation: A ``Typical'' Envelope	357	(4)
Improving Your Reasoning Skills	361	(1)
Review	362	(4)
Take Another Look	364	(1)
Assessing What You've Learned	365	(1)
Exponents and Exponential Models	366	(57)
Recursive Routines	367	(7)
Investigation: Bugs, Bugs, Everywhere Bugs	367	(6)
Improving Your Geometry Skills	373	(1)
Exponential Equations	374	(8)
Investigation: Growth of the Koch Curve	374	(7)
Project: Automobile Depreciation	381	(1)
Multiplication and Exponents	382	(6)
Investigation: Moving Ahead	383	(5)
Scientific Notation for Large Numbers	388	(5)
Investigation: A Scientific Quandary	388	(4)
Improving Your Reasoning Skills	392	(1)
Looking Back with Exponents	393	(6)
Investigation: The Division Property of Exponents	393	(6)
Zero and Negative Exponents	399	(7)
Investigation: More Exponents	399	(6)
Improving Your Reasoning Skills	405	(1)
Fitting Exponential Models to Data	406	(8)
Investigation: Radioactive Decay	406	(7)
Project: Moore's Law	413	(1)
Activity Day: Decreasing Exponential Models and Half-Life	414	(2)
Activity: Bouncing and Swinging	414	(2)
Review	416	(7)
Mixed Review	418	(3)
Take Another Look	421	(1)
Assessing What You've Learned	422	(1)
Functions	423	(52)
Secret Codes	424	(8)
Investigation: TFDSFU DPEFT	424	(7)
Project: Computer Number Systems	431	(1)
Functions and Graphs	432	(8)
Investigation: Testing for Functions	433	(7)
Graphs of Real-World Situations	440	(6)
Investigation: Identify the Variables	441	(5)
Function Notation	446	(6)
Investigation: A Graphic Message	447	(5)
Interpreting Graphs	452	(8)
Investigation: Matching Up	453	(7)
Defining the Absolute Value Function	460	(6)
Investigation: Deviations from the Mean	461	(4)
Improving Your Reasoning Skills	465	(1)
Squares, Squaring, and Parabolas	466	(5)
Investigation: Graphing a Parabola	466	(4)
Improving Your Visual Thinking Skills	470	(1)
Review	471	(4)
Take Another Look	473	(1)
Assessing What You've Learned	474	(1)
Transformations	475	(56)
Translating Points	476	(7)
Investigation: Figures in Motion	476	(6)
Project: Animating with Transformations	482	(1)
Translating Graphs	483	(9)
Investigation: Translations of Functions	483	(8)
Improving Your Visual Thinking Skills	491	(1)
Reflecting Points and Graphs	492	(9)
Investigation: Flipping Graphs	492	(8)
Improving Your Reasoning Skills	500	(1)
Stretching and Shrinking Graphs	501	(9)
Investigation: Changing the Shape of a Graph	502	(7)
Improving Your Reasoning Skills	509	(1)
Activity Day: Using Transformations to Model Data	510	(3)
Activity: Roll, Walk, or Sum	510	(3)
Introduction to Rational Functions	513	(7)
Investigation: I'm Trying to Be Rational	513	(6)
Improving Your Visual Thinking Skills	519	(1)
Transformations with Matrices	520	(6)
Investigation: Matrix Transformations	520	(5)
Project: Tiles	525	(1)
Review	526	(5)
Take Another Look	529	(1)
Assessing What You've Learned	530	(1)
Quadratic Models	531	(51)
Solving Quadratic Equations	532	(6)
Investigation: Rocket Science	534	(4)
Finding the Roots and the Vertex	538	(6)
Investigation: Making the Most of It	538	(5)
Improving Your Visual Thinking Skills	543	(1)
From Vertex to General Form	544	(7)
Investigation: Sneaky Squares	545	(6)
Factored Form	551	(6)
Investigation: Getting to the Root of the Matter	551	(6)
Activity Day: Projectile Motion	557	(3)
Activity: Jump or Roll	557	(2)
Project: Parabola by Definition	559	(1)
Completing the Square	560	(6)
Investigation: Searching for Solutions	560	(6)
The Quadratic Formula	566	(6)
Investigation: Deriving the Quadratic Formula	566	(5)
Improving Your Reasoning Skills	571	(1)
Cubic Functions	572	(6)
Investigation: Rooting for Factors	573	(5)
Review	578	(4)
Take Another Look	581	(1)
Assessing What You've Learned	581	(1)
Introduction to Geometry	582	(57)
Parallel and Perpendicular	583	(5)
Investigation: Slopes	583	(5)
Finding the Midpoint	588	(5)
Investigation: In the Middle	588	(4)
Improving Your Geometry Skills	592	(1)
Squares, Right Triangles, and Areas	593	(5)
Investigation: What's My Area?	594	(3)
Improving Your Visual Thinking Skills	597	(1)
The Pythagorean Theorem	598	(6)
Investigation: The Sides of a Right Triangle	599	(4)
Project: Pythagoras Revisited	603	(1)
Operations with Roots	604	(7)
Investigation: Radical Expressions	604	(6)
Project: Show Me Proof	610	(1)
A Distance Formula	611	(6)
Investigation: Amusement Park	612	(5)
Similar Triangles and Trigonometric Functions	617	(9)
Investigation: Ratio, Ratio, Ratio	619	(7)
Trigonometry	626	(6)
Investigation: Reading Topographic Maps	627	(5)
Review	632	(7)
Mixed Review	634	(4)
Take Another Look	638	(1)
Assessing What You've Learned	638	(1)
Selected Answers	639	(51)
Glossary	690	(13)
Index	703	(10)
Photo Credits	713