Connected Mathematics 2, 8th Grade Units 2009 Correlated to: Connecticut Mathematics Curriculum Framework Companion, 2005 (Grade 8)

Transcription

1 Grade 8 ALGEBRAIC REASONING: PATTERNS AND FUNCTIONS Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies. 1.1 Understand and describe patterns and functional relationships. a. Analyze physical phenomena, functions and patterns to identify relationships and make generalizations. (1) Write recursive and explicit functions to generalize patterns. Investigation 1: Exploring Data Patterns (10 11, 20), Investigation 2: Linear Models and Equations (27 29, 32, 43), Investigation 3: Inverse Variation (48 52); Growing, Growing, Growing Investigation 1: Exponential Growth (6 10), Investigation 2: Examining Growth Patterns (20 23), Investigation 3: Growth Factors and Growth Rates (34 37), Investigation 4: Exponential Decay (48 52); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (19 20), Investigation 3: Quadratic Patterns of Change (40 43, 51); Say It With Symbols Investigation 4: Looking Back at Functions (57 58), Investigation 5: Reasoning With Symbols (74 75); Shapes of Algebra Investigation 3: Equations With Two or More Variables (37 38, 40 41) Investigation 1: Exploring Data Patterns (25 30, 35), Investigation 2: Linear Models and Equations (43 46, 51 54, 59), Investigation 3: Inverse Variation (62 74); Growing, Growing, Growing Investigation 1: Exponential Growth (25 38), Investigation 2: Examining Growth Patterns (45 56), Investigation 3: Growth Factors and Growth Rates (62 74), Investigation 4: Exponential Decay (81 92); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (20 24), Investigation 3: Quadratic Patterns of Change (89 105, 113); Say It With Symbols Investigation 4: Looking Back at Functions (97 102), Investigation 5: Reasoning With Symbols ( ); Shapes of Algebra Investigation 3: Equations With Two or More Variables (56 60, 65 68) 1

2 (2) Identify relationships that are linear and nonlinear and compare and contrast their properties using tables, graphs, equations and verbal descriptions. (3) Recognize and solve problems of direct variation. Investigation 1: Exploring Data Patterns (8 9, 16), Investigation 2: Linear Models and Equations (40), Investigation 3: Inverse Variation (49 51), Looking Back Looking Ahead (62 64); Growing, Growing, Growing Investigation 1: Exponential Growth (10); Frogs, Fleas, and Painted Cubes Investigation 4: What Is a Quadratic Function? (60 61); Say It With Symbols Investigation 2: Combining Expressions (23 25), Investigation 4: Looking Back at Functions (57 59) Investigation 1: Exploring Data Patterns (21 24, 33 34), Investigation 2: Linear Models and Equations (58), Investigation 3: Inverse Variation (67 70), Looking Back Looking Ahead (80 81); Growing, Growing, Growing Investigation 1: Exponential Growth (35 38), Investigation 2: Examining Growth Patterns; Frogs, Fleas, and Painted Cubes Investigation 4: What Is a Quadratic Function? ( ); Say It With Symbols Investigation 2: Combining Expressions (45 52) Investigation 1: Exploring Data Patterns (10 11), Investigation 2: Linear Models and Equations (24 32), Investigation 3: Inverse Variation (58); Say It With Symbols Investigation 2: Combining Expressions (23 25), Investigation 3: Solving Equations (39), Investigation 4: Looking Back at Functions (56 57); Shapes of Algebra Investigation 2: Linear Equations and Inequalities (24 25), Investigation 3: Equations With Two or More Variables (37 38, 40 41), Investigation 5: Linear Inequalities (70 72) (4 6, Investigation 1: Exploring Data Patterns (25 30), Investigation 2: Linear Models and Equations (38 54), Investigation 3: Inverse Variation (78); Say It With Symbols Investigation 2: Combining Expressions (45 52), Investigation 3: Solving Equations (69 72), Investigation 4: Looking Back at Functions (94 96); Shapes of Algebra Investigation 2: Linear Equations and Inequalities (43 46), Investigation 3: Equations With Two or More Variables (56 60, 65 68), Investigation 5: Linear Inequalities ( ) 2

3 1.2 Represent and analyze quantitative relationships in a variety of ways. a. Describe the effects of characteristics of linear relationships on the way the relationships are represented verbally and in tables, graphs and equations. (1) Determine the constant rate of change in a linear relationship and recognize this as the slope of a line. (2) Compare and contrast the graphs of lines with the same slope versus those with different slopes. (3) Interpret slope and y-intercepts from contextual situations, graphs and linear equations. (4) Given two linear relationships in context, recognize that they may have a common solution. (27 29); Say It With Symbols Investigation 4: Looking Back at Functions (56 57); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (8 11) (43 46); Say It With Symbols Investigation 4: Looking Back at Functions (94 96); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (23 30) SE: Shapes of Algebra Investigation 1: Equations for Circles and Polygons (8 11) TG: Shapes of Algebra Investigation 1: Equations for Circles and Polygons (23 30) (27 29); Say It With Symbols Investigation 2: Combining Expressions (23 25), Investigation 4: Looking Back at Functions (56 57) (43 46); Say It With Symbols Investigation 2: Combining Expressions (45 52), Investigation 4: Looking Back at Functions (94 96) (32); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (35); Say It With Symbols Investigation 1: Equivalent Expressions (5 7), Investigation 3: Solving Equations (39); Shapes of Algebra Investigation 2: Linear Equations and Inequalities (24 27), Investigation 3: Equations With Two or More Variables (37 38, 40 41), Investigation 5: Linear Inequalities (76 77) 3

4 (51 54); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (84); Say It With Symbols Investigation 1: Equivalent Expressions (20 28), Investigation 3: Solving Equations (69 72); Shapes of Algebra Investigation 2: Linear Equations and Inequalities (43 50), Investigation 3: Equations With Two or More Variables (56 60, 65 68), Investigation 5: Linear Inequalities ( ) 1.3 Use operations, properties and algebraic symbols to determine equivalence and solve problems. a. Solve problems using various algebraic methods and properties. (1) Solve multistep equations using algebraic properties. Investigation 1: Exploring Data Patterns (19), (30 31); Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (49); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (21 25), Investigation 4: What Is a Quadratic Function? (56); Say It With Symbols Investigation 1: Equivalent Expressions (7), Investigation 2: Combining Expressions (24 27), Investigation 3: Solving Equations (37 39, 42 43); Shapes of Algebra Investigation 2: Linear Equations and Inequalities (24 25), Investigation 3: Equations With Two or More Variables (37 40), Investigation 4: Solving Systems of Linear Equations Symbolically (52 58) Investigation 1: Exploring Data Patterns (35), (47 50); Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (79 82); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (21 25), Investigation 4: What Is a Quadratic Function? (85 88); Say It With Symbols Investigation 1: Equivalent Expressions (25 28), Investigation 2: Combining Expressions (49 56), Investigation 3: Solving Equations (65 72, 81 84); Shapes of Algebra Investigation 2: Linear Equations and Inequalities (47 50), Investigation 3: Equations With Two or More Variables (56 64), Investigation 4: Solving Systems of Linear Equations Symbolically (76 94) 4

5 (2) Use tables, graphs and equations to represent mathematical relationships and solve real-world problems. Investigation 1: Exploring Data Patterns (5 11), (24 32), Investigation 3: Inverse Variation (47 52); Growing, Growing, Growing Investigation 1: Exponential Growth (6 10), Investigation 2: Examining Growth Patterns (20 23), Investigation 3: Growth Factors and Growth Rates (34 37), Investigation 4: Exponential Decay (48 52); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (19 20), Investigation 3: Quadratic Patterns of Change (40 43), Investigation 4: What Is a Quadratic Function? (56 58, 62 63); Shapes of Algebra Investigation 3: Equations With Two or More Variables (37 38, 40 41), Investigation 5: Linear Inequalities (69 72, 76 77) Investigation 1: Exploring Data Patterns (16 30), Investigation 2: Linear Models and Equations (38 54), Investigation 3: Inverse Variation (62 74); Growing, Growing, Growing Investigation 1: Exponential Growth (16 30), Investigation 2: Examining Growth Patterns (45 56), Investigation 3: Growth Factors and Growth Rates (67 74), Investigation 4: Exponential Decay (81 92); Frogs, Fleas, and Painted Cubes Investigation 2: Quadratic Expressions (20 24), Investigation 3: Quadratic Patterns of Change (89 105), Investigation 4: What Is a Quadratic Function? ( , ); Shapes of Algebra Investigation 3: Equations With Two or More Variables (56 60, 65 68), Investigation 5: Linear Inequalities ( , ) NUMERICAL AND PROPORTIONAL REASONING: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies. 2.1 Understand that a variety of numerical representations can be used to describe quantitative relationships. a. Compare and order integers, powers and roots using number lines and grids. (1) Compare, locate, label and order rational numbers on number lines, scales, coordinate grids and measurement tools. (40), Investigation 3: Inverse Variation (57); Looking for Pythagoras Investigation 2: Squaring Off (26) (59), Investigation 3: Inverse Variation (78); Looking for Pythagoras Investigation 2: Squaring Off (48) 5

6 (2) Identify another rational number between any two rational numbers. (3) Solve a variety of problems involving integers, powers, roots and scientific notation. b. Extend the understanding of scientific notation to very small numbers. (1) Use powers of ten and negative exponents to write decimal fractions. (2) Use powers of ten and positive and negative exponents to express and compare magnitude of very large and very small numbers and connect to scientific notation. (3) Find the results of multiplication and division with powers of ten using patterns in operating with exponents. (4) Develop, describe and use a variety of methods to operate with very large and very small numbers. 2: Examining Growth Patterns (30) 2: Examining Growth Patterns (59) SE: Looking for Pythagoras Investigation 2: Squaring Off (29), Investigation 3: The Pythagorean Theorem (44), Investigation 4: Using the Pythagorean Theorem (57); Growing, Growing, Growing Investigation 1: Exponential Growth (12, 17), Investigation 2: Examining Growth Patterns (28), Investigation 4: Exponential Decay (55) TG: Looking for Pythagoras (3 9), Investigation 2: Squaring Off (50), Investigation 3: The Pythagorean Theorem (71), Investigation 4: Using the Pythagorean Theorem (92 93); Growing, Growing, Growing Investigation 1: Exponential Growth (42), Investigation 2: Examining Growth Patterns (58), Investigation 4: Exponential Decay (96) 5: Patterns with Exponents (61 62) 5: Patterns with Exponents ( ) 1: Exponential Growth (6 10, 17), Investigation 2: Examining Growth Patterns (28), Investigation 4: Exponential Decay (55), Investigation 5: Patterns with Exponents (70 71) 1: Exponential Growth (25 38, 42), Investigation 2: Examining Growth Patterns (58), Investigation 4: Exponential Decay (95), Investigation 5: Patterns with Exponents (115) 1: Exponential Growth (17) 1: Exponential Growth (42) 1: Exponential Growth (6 10, 17, 18) 1: Exponential Growth (25 38, 42, 43) 6

7 2.2 Use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities. a. Solve problems involving fractions, decimals, ratios and percents. (1) Estimate and solve problems involving percent of increase and decrease. 1: Exponential Growth (8 10), Investigation 2: Examining Growth Patterns (20 23), Investigation 3: Growth Factors and Growth Rates (34-38, 43), Investigation 4: Exponential Decay (48 52), Investigation 5: Patterns with Exponents (60 61) 1: Exponential Growth (29 38), Investigation 2: Examining Growth Patterns (45 56), Investigation 3: Growth Factors and Growth Rates (62 70, 77), Investigation 4: Exponential Decay (81 92), Investigation 5: Patterns with Exponents (99 104) b. Make generalizations about operations with very large and very small numbers. (1) Use the rules for exponents to multiply and divide with powers of ten, including negative exponents. (2) Develop, describe and use a variety of methods to estimate and calculate mentally with very large and very small numbers. 5: Patterns with Exponents (61 62) 5: Patterns with Exponents ( ) SE: Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (57) TG: Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (92) c. Connect the exponential growth and decay models to repeated multiplication by the same factor. (1) Solve problems that involve repetitive patterns and iterations, such as compound interest, using tables, spreadsheets and calculators. SE: Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (59); Growing, Growing, Growing Investigation 1: Exponential Growth (6 7), Investigation 3: Growth Factors and Growth Rates (43) TG: Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (94); Growing, Growing, Growing Investigation 1: Exponential Growth (25 30), Investigation 3: Growth Factors and Growth Rates (77) 7

8 Grade 8 GEOMETRY AND MEASUREMENT Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies. 3.1 Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems. a. Explore the relationships among sides, angles, perimeters, areas, surface areas and volumes of congruent and similar polygons and solids. (1) Explore the effect of scale factors on the length (2) Make and test conjectures about the relationships among angles (43), Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (58 59); Growing, Growing, Growing Investigation 2: Examining Growth Patterns (29), Investigation 3: Growth Factors and Growth Rates (43) (59), Looking for Pythagoras Investigation 4: Using the Pythagorean Theorem (93 94); Growing, Growing, Growing Investigation 2: Examining Growth Patterns (59), Investigation 3: Growth Factors and Growth Rates (77) SE: Kaleidoscopes, Hubcaps, and Mirrors Investigation 2: Symmetry Transformations (27 33), Investigation 3: Exploring Congruence (50 53); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (5 11) TG: Kaleidoscopes, Hubcaps, and Mirrors Investigation 2: Symmetry Transformations (45 62), Investigation 3: Exploring Congruence (79 86); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (18 30) 3.2 Use spatial reasoning, location and geometric relationships to solve problems. a. Model geometric relationships in a variety of ways. (1) Use coordinate geometry to explore and test geometric relationships of parallel and perpendicular lines and polygons and their transformations. SE: Looking for Pythagoras Investigation 1: Coordinate Grids (10); Kaleidoscopes, Hubcaps, and Mirrors Investigation 5:Transforming Coordinates (79 87); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (8 11) TG: Looking for Pythagoras Investigation 1: Coordinate Grids (25 28); Kaleidoscopes, Hubcaps, and Mirrors Investigation 5:Transforming Coordinates ( ); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (23 30) 8

9 3.3 Develop and apply units, systems, formulas and appropriate tools to estimate and measure. a. Use a variety of concrete methods, including displacement, to find volumes of solids. (1) Develop measurement strategies to find the surface area and volume of pyramids, cones, spheres and irregular solids. (2) Use estimation and measurement strategies to solve problems involving the volumes of solids. Investigation 1: Exploring Data Patterns (20); Looking for Pythagoras Investigation 3: The Pythagorean Theorem (40 41); Say It With Symbols Investigation 2: Combining Expressions (34) Investigation 1: Exploring Data Patterns (35); Looking for Pythagoras Investigation 3: The Pythagorean Theorem (70); Say It With Symbols Investigation 2: Combining Expressions (62) Investigation 1: Exploring Data Patterns (20); Kaleidoscopes, Hubcaps, and Mirrors Investigation 1: Three Types of Symmetry (21) Investigation 1: Exploring Data Patterns (35); Kaleidoscopes, Hubcaps, and Mirrors Investigation 1: Three Types of Symmetry (40) b. Solve problems involving measurement through the use of appropriate tools, techniques and strategies. (1) Use the Pythagorean Theorem to solve indirect measurement problems. (2) Describe the accuracy of estimates and measures and the precision of measurement tools. SE: Looking for Pythagoras Investigation 3: The Pythagorean Theorem (35, 42, 44), Investigation 4: Using the Pythagorean Theorem (49 52, 60); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (5 11) TG: Looking for Pythagoras Investigation 3: The Pythagorean Theorem (61 64, 70 72), Investigation 4: Using the Pythagorean Theorem (79 86, 94); Shapes of Algebra Investigation 1: Equations for Circles and Polygons (18 30) SE: Looking for Pythagoras Investigation 3: The Pythagorean Theorem (43 44) TG: Looking for Pythagoras Investigation 3: The Pythagorean Theorem (71) (3) Solve dimensional analysis problems. Online Activity: ME-d Dimensional Analysis 9

10 Grade 8 WORKING WITH DATA: PROBABILITY AND STATISTICS Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies. 4.1 Collect, organize and display data using appropriate statistical and graphical methods. a. Construct appropriate representations of data based on the size and kind of data set and the purpose for its use. (1) Collect, organize, display, compare and analyze large data sets. (2) Construct a variety of data displays, including box-and-whisker plots, and identify where measures of central tendency and dispersion are found in graphical displays. SE: Samples and Populations Investigation 1: Comparing Data Sets (5 16), Investigation 2: Choosing a Sample From a Population (31 35), Investigation 3: Solving Real-World Problems (47 53), Investigation 4: Relating Two Variables (62 68) TG: Samples and Populations Investigation 1: Comparing Data Sets (20 46), Investigation 2: Choosing a Sample From a Population (31 35), Investigation 3: Solving Real-World Problems (63 70), Investigation 4: Relating Two Variables (96 106) Investigation 1: Exploring Data Patterns (21); Investigation 1: Exploring Data Patterns (35 36); Online Activity: DA-b Analyzing Data Using Pictographs and Box-and-Whisker Plots 4.2 Analyze data sets to form hypotheses and make predictions. a. Make and evaluate statistical claims and justify conclusions with evidence. (1) Make predictions from scatter plots using or estimating a line-of-best-fit. (2) Make inferences and evaluate reasonable hypotheses based on experimental data. (3) Analyze and interpret data using descriptive statistics, including range, mode, median, quartiles, outliers and mean. SE: Samples and Populations Investigation 4: Relating Two Variables (62 68) TG: Samples and Populations Investigation 4: Relating Two Variables (96 106) SE: Samples and Populations Investigation 3: Solving Real-World Problems (51 53) TG: Samples and Populations Investigation 3: Solving Real-World Problems (83 89) Investigation 3: Inverse Variation (57); Samples and Populations Investigation 1: Comparing Data Sets (12 16), Investigation 2: Choosing a Sample From a Population (31 33), Investigation 3: Solving Real-World Problems (47 53) 10

11 Investigation 3: Inverse Variation (78); Samples and Populations Investigation 1: Comparing Data Sets (31 46), Investigation 2: Choosing a Sample From a Population (63 66), Investigation 3: Solving Real-World Problems (77 89) (4) Determine the accuracy of statistical claims. SE: Samples and Populations Investigation 3: Solving Real-World Problems (59), Investigation 4: Relating Two Variables (66 68, 69, 75) (5) Describe the role of random sampling, random number generation and the effects of sample size in statistical claims. 4.3 Under-stand and apply basic concepts of probability. a. Determine possible outcomes using a variety of counting techniques. (1) Distinguish between combinations and permutations as ways to predict possible outcomes in certain situations. (2) Use combinations and permutations, trees and networks (counting strategies) in a variety of contexts, and identify when order is irrelevant in determining a solution. TG: Samples and Populations Investigation 3: Solving Real-World Problems (93), Investigation 4: Relating Two Variables ( , 107, 108) SE: Samples and Populations Investigation 2: Choosing a Sample From a Population (26 35) TG: Samples and Populations Investigation 2: Choosing a Sample From a Population (54 70) 11