Variables and Patterns Introducing Algebraic Reasoning 4 CONNECTED MATHEMATICS® Student Edition SAMPLE
xiii Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Investigation 1. Organizing a Bike Tour: Variables, Tables, and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem 1.1 Organizing a Bike Tour Experiment: Variables and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem 1.2 Organizing a Bike Tour: Variables, Tables, and Graphs . . . 6 Problem 1.3 Atlantic City to Lewes to Chincoteague: Time, Rate, and Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Problem 1.4 Chincoteague Island to Norfolk: Stories, Tables, and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 17 Investigation 2. Determining Tour Needs: Analyzing Relationships Among Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 Problem 2.1 Renting Bicycles: Independent and Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 Problem 2.2 Finding Customers: More Variables . . . . . . . . . . . . . . . . . . . 32 Problem 2.3 What’s the Story?: Interpreting Graphs . . . . . . . . . . . . . . . .34 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 37 Investigation 3. Returning Home: Relating Variables, Expressions, and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Problem 3.1 Returning Home: Equations with One Operation . . . . . . .49 Problem 3.2 Planning the Next Tour: More Equations with One Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 CONTENTS SAMPLE
xiv Contents Problem 3.3 Planning Ahead: Connecting Equations with Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 61 English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 SAMPLE
xv Relationships Between Variables. Begin informal understanding of variables and how they are related. • Explore problem situations that involve variables and the relationships among the variables including situations that change over time • Recognize variables in a real-world situation, identify the dependent and independent variables, and describe how they are related in a situation • Make sense of the “stories” told by patterns in tables and coordinate graphs of numeric (x, y) data and equations and make connections across the representations • Represent relationships between two variables with one operation (y = ax and y = b + x) using words and stories, data tables, graphs, and equations • Describe advantages and disadvantages of using words, tables, graphs, and equations to represent a relationship between two variables and to answer questions about the relationship • Solve problems that involve variables to answer questions about one variable given the value of the associated variable when represented using tables, graphs, equations, or words (a story context) Expressions and Equations. Begin informal understanding of expressions and equations. • Recognize that equations describe a relationship between two variables • Represent real-world relationships, stories, involving two variables and one operation (y = ax and y = b + x) with an equation and describe in words a relationship given in the form of an equation such as y = ax and y = b + x • Recognize that expressions like ax or b + x represent a relationship of a quantity or mathematical pattern • Identify parts of an equation using mathematical terms (sum, term, product, factor, quotient, coefficient) • Use an equation such as y = ax and y = b + x to determine the value of one variable given the value of the other using numeric guess and check, tables of (x, y) values, and graphs MATHEMATICAL GOALS SAMPLE
Number Connections Expressing Factors and Multiples Algebraically 4 CONNECTED MATHEMATICS® Student Edition SAMPLE
xv Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Investigation 1. Generalizing Factor and Multiple Patterns: Algebraic Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem 1.1 The Factor Game: Generalizing Factors with Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problem 1.2 The Product Game: Generalizing Multiples with Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 1.3 Generalizing Number Patterns with Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 15 Investigation 2. Common Multiple and Common Factor: Least or Greatest? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 Problem 2.1 Riding Ferris Wheels: Least Common Multiple or Greatest Common Factor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Problem 2.2 Looking at Cicada Cycles: Least Common Multiple or Greatest Common Factor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 Problem 2.3 Bagging Snacks: Least Common Multiple or Greatest Common Factor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . .34 CONTENTS SAMPLE
xvi Contents Investigation 3. Using Exponents to Express the Multiplicative Structure of Numbers . . . . . . . . . . . . . . . . . . . . . . . . .42 Problem 3.1 The Product Puzzle: Factor Strings and Exponential Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 Problem 3.2 Using Prime Factorizations with Exponents . . . . . . . . . . . .46 Problem 3.3 The Locker Problem: Putting It All Together . . . . . . . . . . . .50 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 53 Investigation 4. Using the Distributive Property to Write and Evaluate Equivalent Expressions . . . . . . . . . . . . . . . . . 60 Problem 4.1 Connecting Addition and Multiplication . . . . . . . . . . . . . . . 61 Problem 4.2 Ordering Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 Problem 4.3 Choosing an Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 73 English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 SAMPLE
xvii Generalizing Factors and Multiples. Generalize properties of the multiplicative and additive structure of number. • Apply and extend understanding of the multiplicative structure of number to multiples and factors to recognize situations that call for least common multiple and greatest common factor • Develop strategies for finding least common multiples and greatest common factors • Develop understanding of exponential notation to represent the multiplicative structure of number • Use exponential notation to write and evaluate expressions • Recognize the role of prime numbers in answering questions about factors and multiples, including least common multiples and greatest common factors • Recognize that the Distributive Property relates the multiplicative and additive structures of whole numbers • Relate the area of a rectangle as a representation and generalization of the Distributive Property • Solve problems involving greatest common factors and least common multiples Algebraic Reasoning. Express mathematical patterns and relationships between quantities described with expressions and equations. • Write, read, and evaluate expressions in which letters stand for numbers; evaluate expressions for specific values of the variables • Identify when two expressions are equivalent • Write and evaluate expressions involving whole-number exponents • Identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity • Use the properties of operations (including the Distributive Property and Order of Operations convention) to write and evaluate equivalent algebraic expressions • Use algebraic expressions and equations to represent the multiplicative and additive structure of numbers • Solve problems involving the structure of number and its operations MATHEMATICAL GOALS SAMPLE
Comparing Quantities Ratios, Rates, and Equivalence 4 CONNECTED MATHEMATICS® Student Edition SAMPLE
xv Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Investigation 1. Making Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem 1.1 Fundraising: Making Comparisons. . . . . . . . . . . . . . . . . . . . . . 2 Problem 1.2 Fundraising Update: Ratios as Comparisons . . . . . . . . . . . . 5 Problem 1.3 Making Cupcakes: Part-to-Part or Part-to-Whole Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Applications—Connections—Extension (ACE) . . . . . . . . . . . . . . . . . . . . . . . 12 Investigation 2. Using Ratios to Solve Problems . . . . . . . . . . . . . . . . . 17 Problem 2.1 Packaging Cupcakes: Using Ratios . . . . . . . . . . . . . . . . . . . . 18 Problem 2.2 Selling Cupcakes: Representing Ratios with Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Problem 2.3 Sharing Smoothie Bars: More Ratios . . . . . . . . . . . . . . . . . . 23 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 26 Investigation 3. Using Unit Rates and Rate Tables . . . . . . . . . . . . . . 31 Problem 3.1 Selling Smoothie Bars: Unit Rates . . . . . . . . . . . . . . . . . . . . . 32 Problem 3.2 Making Popcorn: More Unit Rates and Tables . . . . . . . . .34 Problem 3.3 Experimenting with Slime: Solving Problems with Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Problem 3.4 Walking Backward: More Ratio Problems . . . . . . . . . . . . .40 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . .44 CONTENTS SAMPLE
xvi Contents Investigation 4. For Every 100: Introducing Percent . . . . . . . . . . . . . 51 Problem 4.1 Percent: Out of 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Problem 4.2 Surveys, Percents, and Tape Diagrams . . . . . . . . . . . . . . . . 56 Problem 4.3 Genetic Traits: Finding Any Percent . . . . . . . . . . . . . . . . . . .60 Problem 4.4 Making Sense of Percent Situations . . . . . . . . . . . . . . . . . . . 63 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 67 English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 SAMPLE
xvii Reasoning with Ratios. Understand ratios. • Understand ratios as a multiplicative comparison of two quantities and distinguish them from difference (additive) relationships • Understand that ratios can be either part-to-part or part-to-whole comparison • Understand unit rate as a ratio where one of the two quantities being compared has a value of 1 • Understand that a percent is a part-to-whole ratio where the whole is 100 • Recognize and represent equivalent ratios, including percents and unit rates, with ratio tables, tape diagrams, double number lines, and equations • Identify ratio and rate situations and choose appropriate representations and/or strategies to reason with ratios when solving problems • Use concepts associated with recognizing, representing, and using relationships between two variables, begun in the Variables and Patterns and Number Connections units, to work with ratio relationships MATHEMATICAL GOALS SAMPLE
Bits of Rational Extending Fraction Operations and Solving Equations 4 CONNECTED MATHEMATICS® Student Edition SAMPLE
xv Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Investigation 1. Extending Addition and Subtraction of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 1.1 Getting Close Game: Estimating Sums and Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 1.2 Tupelo Township: Equations with Addition and Subtraction of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 1.3 Spice It Up: Solving Equations with Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx Investigation 2. Extending Multiplication of Fractions . . . . . . . . . . xx Problem 2.1 Selling Brownies: Finding Parts of Parts . . . . . . . . . . . . . . . . xx Problem 2.2 Modeling Multiplication Situations: Equations . . . . . . . . xx Problem 2.3 Buying the Biggest Lot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx Investigation 3. Making Sense of Fraction Division. . . . . . . . . . . . . . xx Problem 3.1 Preparing Food: Dividing a Fraction by a Fraction . . . . . . xx Problem 3.2 Cheese Pizzas: Dividing Whole Numbers or Mixed Numbers by a Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 3.3 Sharing Trail Mix: Dividing a Fraction by a Whole Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx CONTENTS SAMPLE
xvi Contents Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx Investigation 4. Wrapping Up Operations: Equivalent Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 4.1 Back to Tupelo Township: More Than One Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 4.2 Just the Facts: Using Fact Families to Solve Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 4.3 Becoming an Operations Sleuth . . . . . . . . . . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx SAMPLE
xvii Operating with Rational Numbers (Fractions). Make sense of and deepen understandings of the four operations. • Use prior knowledge of fractions, including equivalence of fractions, to model (area, fraction strips, and/or number lines) and develop algorithms for adding, subtracting, multiplying, and dividing fractions. • Use benchmarks and other strategies to estimate results of operations with fractions. Give reasons to estimate and identify when a situation calls for an overestimate or an underestimate. • Identify which operation, addition, subtraction, multiplication, or division, is needed to solve a problem. • Use estimates and exact solutions using operations on fractions to make decisions including real-world problems. Algebraic Reasoning (Solving Equations). Use variables to represent unknown values and equations to represent relationships. • Represent and interpret algebraic and numeric expressions that represent real-world and rational number situations. • Write equations (or number sentences) to represent relationships among real-world situations and properties of rational number. • Use understanding of the operations of rational numbers and their inverse relationship to solve equations. MATHEMATICAL GOALS SAMPLE
4 CONNECTED MATHEMATICS® Student Edition Covering and Surrounding Two- and Three-Dimensional Measurement SAMPLE
xv Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Investigation 1. Designing Parks: Building on Area and Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 1.1 Measuring Lakes: Choosing Units . . . . . . . . . . . . . . . . . . . . . xx Problem 1.2 Building Storm Shelters Experiment: Using Algebra toRepresent Constant Area, Changing Perimeter . . . . . . . . . . . . . . . . . . . xx Problem 1.3 Fencing In Spaces Experiment: Using Algebra to Represent Constant Perimeter, Changing Area . . . . . . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx Investigation 2. Measuring Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 2.1 Investigating Area of Polygons . . . . . . . . . . . . . . . . . . . . . . . xx Problem 2.2 Finding Area of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 2.3 Finding Area of Parallelograms . . . . . . . . . . . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx Investigation 3. Designing Under Constraints . . . . . . . . . . . . . . . . . . . xx Problem 3.1 Designing Polygons Under Constraints . . . . . . . . . . . . . . . . xx Problem 3.2 Designing Polygons on Coordinate Grids . . . . . . . . . . . . . . xx Problem 3.3 Designing Circular Fountains Experiment: Using Ratios to Find Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx CONTENTS SAMPLE
xvi Contents Investigation 4. Measuring Three-Dimensional Shapes . . . . . . . . . xx Problem 4.1 Filling the Box: Finding Volume . . . . . . . . . . . . . . . . . . . . . . . . xx Problem 4.2 Making a Box: Finding Surface Area . . . . . . . . . . . . . . . . . . xx Problem 4.3 Designing Gift Boxes: Using Surface Area. . . . . . . . . . . . . . xx Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . xx English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx SAMPLE
xvii Two- and Three-Dimensional Measurement. Deepen and use understanding of area and perimeter of polygonal shapes to explore surface area and volume of polygonal prisms and pyramids. • Relate area to covering, perimeter to surrounding, and filling to volume and explore methods for estimating and calculating these measures • Investigate the relationships among triangles, rectangles, and special quadrilaterals (composing or decomposing shapes into triangles and rectangles) to make sense of area and perimeter formulas and state these in words and symbols • Investigate and describe relationships between side lengths, perimeters, and areas of polygons under given constraints (e.g., fixed area, or fixed perimeter) using tables, graphs, and some equations • Use coordinates to represent polygons and, when possible, use the coordinates of vertices to find side lengths of polygons • Apply knowledge of ratios to converting length measurements in metric and English systems as needed to solve problems involving area and perimeter • Extend understanding of ratio and measurement by representing relationships between two quantities (tables, graphs, and equations) to make sense of perimeter in non-routine contexts • Connect the concepts and skills associated with area of twodimensional shapes to surface area of three-dimensional objects (pyramids and prisms) when represented as a two-dimensional net or as a two-dimensional drawing of the three-dimensional figure • Extend understanding of the strategies and formulas for finding the volume of rectangular prisms to accommodate fractional side lengths and other prisms • Solve problems that involve finding the area and/or perimeter of triangles, rectangles, parallelograms, trapezoids, and other polygonal shapes • Solve problems involving surface area (pyramids and prisms) and volume of prisms MATHEMATICAL GOALS SAMPLE
4 CONNECTED MATHEMATICS® Student Edition Points of Rational A Focus on Decimals and Algebraic Reasoning SAMPLE
Contents Mathematical Goals Looking Ahead Investigation 1. Representing and Ordering Decimals Fluently Problem 1.1 Drawing the Line: Representing Rational Numbers on a Number Line Problem 1.2 Dialing for Digits Game: Ordering Rational Numbers Problem 1.3 Bungee Jumps, Bandanas, Crackers, and Walnuts: Solving Inequalities Mathematical Reflection Applications—Connections—Extensions (ACE) Investigation 2. Operating Fluently with Decimals Problem 2.1 Buying Snacks: Decimal Addition and Subtraction Problem 2.2 Making Sense of Decimal Multiplication Problem 2.3 Dividing the Long Way: Long Division of Whole Numbers Problem 2.4 Swimming to Win: Relating Division of Decimals, Whole Numbers, and Fractions Mathematical Reflection Applications—Connections—Extensions (ACE) Investigation 3. Going Negative: Extending the Number Line and Coordinate Plane 3.1 Extending the Number Line: Negative Rational Numbers 3.2 Fishing, Bank Accounts, and Hiking: Absolute Values 3.3 Mapping Locations in the Town of Euclid: Using Four-Quadrant Coordinate Graphs 3.4 Pulling It All Together: Representing and Working with Rational Numbers Mathematical Reflection Applications—Connections—Extensions (ACE) English/Spanish Glossary Index SAMPLE
xv Rational Numbers. Extend the rational number system and deepen understandings of the four rational number operations. • Extend the rational number system to include negative numbers on the number line and coordinate plane, including identifying absolute values, distance, and opposite numbers • Identify which operation—addition, subtraction, multiplication, or division—is needed to solve problems that include multidigit decimals • Use estimates and exact solutions, with operations on decimals, to make decisions and solve real-world problems • Use absolute value to express distance on a number line • Order and compare rational numbers on a horizontal or vertical number line diagram • Extend graphing to include four quadrants in a coordinate plane • Solve real-world and mathematical problems with graphs, number lines, and rational numbers Algebraic Reasoning. Use variables to represent relationships with rational numbers. • Represent and interpret algebraic and numeric expressions that represent real-world and rational number situations • Write equations and inequalities to represent relationships in realworld situations; a + x = b, ax = b, and x = c, x > c, or x < c • Use understanding of the operations of rational numbers and their inverse relationship to solve equations and inequalities of the form a + x = b, ax = b, and x = c, x > c, or x < c MATHEMATICAL GOALS SAMPLE
4 CONNECTED MATHEMATICS® Student Edition Data About Us Statistics and Data Analysis SAMPLE
Contents Mathematical Goals Looking Ahead Investigation 1. What’s in a Name? Asking, Organizing, Representing, and Describing Data Problem 1.1 Beginning with the Question: What Is a Statistical Question? Problem 1.2 How Many Letters Are in a Name? Organizing Data Problem 1.3 Describing Name Lengths: What Are the Shape, Mode, and Range? Problem 1.4 Describing Name Lengths: What Is the Median? Mathematical Reflection Applications—Connections—Extensions (ACE) Investigation 2. Who’s in a Household? Making Sense of Measures of Center Problem 2.1 What’s a Mean Household Size? Problem 2.2 Comparing Distributions with the Same Mean Problem 2.3 The Number of Hours of Homework: Mean, Median, or Mode? Mathematical Reflection Applications—Connections—Extensions (ACE) Investigation 3. How Do They Differ? Measuring Variability Problem 3.1 Variability of Gaming Platforms: Keeping Our Sights on Range Problem 3.2 Cereal Shelf Location and Sugar Content: Describing Variability Using the IQR Problem 3.3 Is It Worth the Wait? Using the MAD Mathematical Reflection Applications—Connections—Extensions (ACE) Investigation 4. Using Data and Statistical Graphs to Describe Attributes Problem 4.1 Traveling to School: Histograms Problem 4.2 Hours of Sleep: Using Box Plots Problem 4.3 Pulling It All Together: Back to Jumping Jacks to Look at Variability Mathematical Reflection Applications—Connections—Extensions (ACE) English/Spanish Glossary Index SAMPLE
xv Reasoning About Data Distributions. Understanding the statistical problem-solving process. • Use the process of statistical problem-solving, including asking statistical questions, collecting, and analyzing data, and interpreting data to answer questions • Distinguish data and data types, including recognizing that data consist of counts or measurements of a variable, or an attribute, and distinguishing between categorical and numerical data • Use the analysis of the data to predict, compare, or identify relationships among the data and use the information to make decisions about the original question • Use measures of central tendency (mode, median, mean) as single numbers to describe what’s typical for a set of data and to provide an overall picture of what is going on in the entire data set and whose choice depends upon the type of data being analyzed • Use measures of spread or variability (e.g., range, IQR, MAD) as single numbers to characterize the degree of variability or spread in a set of data and whose choice depends upon the type of data being analyzed • Display data using multiple representations, recognizing there are dozens of charts and graphs to make to display data whose choice depends upon the type of data being displayed MATHEMATICAL GOALS SAMPLE