A Correlation of Connected Mathematics Project 3 2014 To the Florida MAFS Mathematics Standards (2014)
A Correlation of Connected Mathematics Project 3, 2014 to the (2014) Introduction This document demonstrates how Connected Mathematics Project 3 (CMP3), 2014 meets the (2014),. Correlation references are to the units of the Student and Teacher s Editions. The goal of Connected Mathematics Project 3 is to help students develop mathematical knowledge, conceptual understanding, and procedural skills, along with an awareness of the rich connections between math topics across grades and across Common Core content areas. Through the Launch-Explore-Summarize model, students investigate and solve problems that develop rigorous higher-order thinking skills and problem-solving strategies. Curriculum development for CMP3 has been guided by an important mathematical idea: All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representation, materials, tools, techniques, and intellectual methods of mathematics. This includes the ability to define and solve problems with reason, insight, inventiveness, and technical proficiency. CMP3 uses technology to help teachers implement with fidelity, thus raising student achievement. Easy-to-use mobile tools help with classroom management and capture student work on the go. ExamView delivers a full suite of assessment tools, and MathXL provides individualized skills practice. 21st century social networking technology connects CMP3 teachers, while students benefit from interactive digital student pages that allow for instantaneous sharing and effective group work. Copyright 2014 Pearson Education, Inc. or its affiliate(s). All rights reserved
(2014) MAFS.912.N-RN.1.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define to be the cube root of 5 because we want = to hold, so must equal 5. Growing, Growing, Growing: 5.1: Looking for Patterns Among Exponents; 5.2: Rules of Exponents; 5.3: Extending the Rules of Exponents MAFS.912.N-RN.1.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. Growing, Growing, Growing: 5.3: Extending the Rules of Exponents MAFS.912.N-RN.2.3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Students learn that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers can be approximated using decimals. They also learn about the density properties of irrational and rational numbers. Looking for Pythagoras: 4.3: Representing Decimals as Fractions; 4.4: Getting Real 1
(2014) MAFS.912.N-Q.1.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Students use units appropriately throughout the series. They use scientific notation to write very large or very small measurements, they measure accurately with a ruler or protractor, they conduct multi-step experiments which involve measurement, and they graph real measurement data on a coordinate plane. See, for example: 1.1: Bridge Thickness and Strength; 1.2: Bridge Length and Strength; 3.2: Distance, Speed, and Time; 4.1: Vitruvian Man: Relating Body Measurements; 4.2: Older and Faster Growing, Growing, Growing: 2.1: Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1; 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 3.2: Investing for the Future: Growth Rates; 4.3: Cooling Water: Modeling Exponential Decay; 5.4: Operations With Scientific Notation Frogs, Fleas, and Painted Cubes: 2.1: Trading Land: Representing Areas of Rectangles; 2.2: Changing Dimensions: The Distributive Property 2
(2014) MAFS.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling. Students practice mathematical modeling and define appropriate quantities for descriptive modeling throughout the series; in fact, mathematical modeling of mathematical systems and applied concepts is a primary focus of the program. Additionally, Thinking With Mathematical Models is a unit of the 8 th grade course that is entirely devoted to modeling problem situations and real-world data with linear and variation functions. Similarly, Growing, Growing, Growing is a unit devoted to modeling with exponential (growth and decay) functions. See, for example: 2.1: Modeling Linear Data Patterns; 3.4: Modeling Data Patterns Looking for Pythagoras: 2.2: Square Roots; 4.1: Analyzing the Wheel of Theodorus: Square Roots on a Number Line; 5.2: Analyzing Triangles Growing, Growing, Growing: 3.1: Reproducing Rabbits: Fractional Growth Patterns; 4.1: Making Smaller Ballots: Introducing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay 4.4: Using Similar Triangles It's in the System: 2.1: Shirts and Caps Again: Solving Systems With y = mx + b 3
(2014) MAFS.912.N-Q.1.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Precision is an underlying focus of the Connected Mathematics Project 3 series, and a fundamental mathematical practice. Students are expected to be precise in their calculations, and to use exact or approximate answers as appropriate to the problem situation. See, for example: 2.4: Boat Rental Business: Solving Linear Equations; 4.3: Correlation Coefficients and Outliers Looking for Pythagoras: 2.2: Square Roots; 4.2: Representing Fractions as Decimals; 4.3: Representing Decimals as Fractions; 4.4: Getting Real: Irrational Numbers Growing, Growing, Growing: 1.2: Requesting a Reward: Representing Exponential Functions; 5.4: Operations With Scientific Notation 2.2: Supporting the World: Congruent Triangles I It's In the System: 3.3: Operating at a Profit: Systems of Lines and Curves 4
(2014) MAFS.912.A-SSE.1.1: Interpret expressions that represent a quantity in terms of its context. Students continuously write, evaluate, simplify, and interpret expressions in Connected Mathematics Project 3. The unit, Say It With Symbols, focuses entirely on algebraic expressions. Students use properties to write equivalent expressions, students combine and compose expressions, students factor expressions and solve equations, and students write function definitions that include linear, exponential, and quadratic expressions. Frogs, Fleas, and Painted Cubes is an algebra unit devoted to quadratic functions. Say It With Symbols: 1.1: Tiling Pools: Writing Equivalent Expressions; 1.2: Thinking in Different Ways: Determining Equivalence; 1.3: The Community Pool Problem: Interpreting Expressions; 1.4: Diving In: Revisiting the Distributive Property; 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions; 3.3: Factoring Quadratic Equations; 4.3: Generating Patterns: Linear, Exponential, Quadratic; 5.1: Using Algebra to Solve a Puzzle Frogs, Fleas, and Painted Cubes: 2.1: Trading Land: Representing Areas of Rectangles; 2.2: Changing Dimensions: The Distributive Property; 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs 5
(2014) a. Interpret parts of an expression, such as terms, factors, and coefficients. Say It With Symbols: 1.1: Tiling Pools: Writing Equivalent Expressions; 1.2: Thinking in Different Ways: Determining Equivalence; 1.3: The Community Pool Problem: Interpreting Expressions; 1.4: Diving In: Revisiting the Distributive Property; 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions; 3.3: Factoring Quadratic Equations; 4.3: Generating Patterns: Linear, Exponential, Quadratic; 5.1: Using Algebra to Solve a Puzzle Frogs, Fleas, and Painted Cubes: 2.1: Trading Land: Representing Areas of Rectangles; 2.2: Changing Dimensions: The Distributive Property; 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret as the product of P and a factor not depending on P. Say It With Symbols: 1.1: Tiling Pools: Writing Equivalent Expressions; 1.2: Thinking in Different Ways: Determining Equivalence; 1.3: The Community Pool Problem: Interpreting Expressions; 1.4: Diving In: Revisiting the Distributive Property; 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions; 3.3: Factoring Quadratic Equations; 4.3: Generating Patterns: Linear, Exponential, Quadratic; 5.1: Using Algebra to Solve a Puzzle Growing, Growing, Growing: 3.2: Investing for the Future: Growth Rates; 3.3: Making a Difference: Connecting Growth Rate and Growth Factor; 4.2: Fighting Fleas: Representing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay Frogs, Fleas, and Painted Cubes: 2.1: Trading Land: Representing Areas of Rectangles; 2.2: Changing Dimensions: The Distributive Property 6
(2014) MAFS.912.A-SSE.1.2: Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). Say It With Symbols: 1.1: Tiling Pools: Writing Equivalent Expressions; 1.2: Thinking in Different Ways: Determining Equivalence; 1.4: Diving In: Revisiting the Distributive Property; 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions; 3.3: Factoring Quadratic Equations Frogs, Fleas, and Painted Cubes: 2.1: Trading Land: Representing Areas of Rectangles; 2.2: Changing Dimensions: The Distributive Property; 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs MAFS.912.A-SSE.2.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Say It With Symbols: 1.1: Tiling Pools: Writing Equivalent Expressions; 1.2: Thinking in Different Ways: Determining Equivalence; 1.4: Diving In: Revisiting the Distributive Property; 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions; 3.3: Factoring Quadratic Equations Frogs, Fleas, and Painted Cubes: 2.1: Trading Land: Representing Areas of Rectangles; 2.2: Changing Dimensions: The Distributive Property; 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs a. Factor a quadratic expression to reveal the zeros of the function it defines. Say It With Symbols: 3.3: Factoring Quadratic Equations Frogs, Fleas, and Painted Cubes: 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Frogs, Fleas, and Painted Cubes: 2.4: Quadratic Functions and Their Graphs Function Junction: 4.2: Completing the Square 7
(2014) c. Use the properties of exponents to transform expressions for exponential functions. For example the expression can be rewritten as to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Growing, Growing, Growing: 1.1: Making Ballots: Introducing Exponential Functions; 1.2: Requesting a Reward: Representing Exponential Functions; 1.3: Making a New Offer: Growth Factors; 2.1: Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1; 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 3.1: Reproducing Rabbits: Fractional Growth Patterns; 3.2: Investing for the Future: Growth Rates; 3.3: Making a Difference: Connecting Growth Rate and Growth Factor; 4.1: Making Smaller Ballots: Introducing Exponential Decay; 4.2: Fighting Fleas: Representing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay; 5.2: Rules of Exponents; 5.3: Extending the Rules of Exponents MAFS.912.A-APR.1.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Say It With Symbols: 1.4: Diving In: Revisiting the Distributive Property; 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions Frogs, Fleas, and Painted Cubes: 2.2: Changing Dimensions: The Distributive Property; 2.3: Factoring Quadratic Expressions Function Junction: 5.2: Combining Profit Functions: Operating With Polynomials I; 5.3: Product Time: Operating With Polynomials II MAFS.912.A-APR.2.3: Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. Frogs, Fleas, and Painted Cubes: 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs Function Junction: 5.3: Product Time: Operating With Polynomials II 8
(2014) MAFS.912.A-CED.1.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Looking for Pythagoras: 2.2: Square Roots; 2.3: Using Squares; 2.4: Cube Roots; 5.1: Stopping Sneaky Sally Growing, Growing, Growing: 1.3: Making a New Offer; 2.1: Killer Plant Strikes Lake Victoria; 2.3: Studying Snake Populations; 3.3: Making a Difference; 4.2: Fighting Fleas; 5.5: Revisiting Exponential Functions Say It With Symbols: 3.1: Selling Greeting Cards; 3.2: Comparing Costs; 3.4: Solving Quadratic Equations MAFS.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase; 2.3: Tree Top Fun; 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Growing, Growing, Growing: 1.3: Making a New Offer; 2.1: Killer Plant Strikes Lake Victoria; 2.3: Studying Snake Populations; 3.3: Making a Difference; 4.2: Fighting Fleas; 5.5: Revisiting Exponential Functions It s In The System: 1.1: Shirts and Caps; 1.2: Connecting Ax + By = C and y = mx + b; 1.3: Booster Club Members Frogs, Fleas, and Painted Cubes: 2.4: Quadratic Functions and Their Graphs 9
(2014) MAFS.912.A-CED.1.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase; 2.3: Tree Top Fun; 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Growing, Growing, Growing: 1.3: Making a New Offer; 2.1: Killer Plant Strikes Lake Victoria; 2.3: Studying Snake Populations; 3.3: Making a Difference; 4.2: Fighting Fleas; 5.5: Revisiting Exponential Functions It s In The System: 1.1: Shirts and Caps; 3.3: Operating at a Profit: Systems of Lines and Curves; 4.1: Limiting Driving Miles: Inequalities With Two Variables Frogs, Fleas, and Painted Cubes: 4.1: Tracking a Ball: Interpreting a Table and an Equation MAFS.912.A-CED.1.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. 3.1: Rectangles With Fixed Area; 3.2: Distance, Speed, and Time Say It With Symbols: 3.1: Solving Equations It's In the System: 1.2 Connecting Ax + By = C and y = mx + b MAFS.912.A-REI.1.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Students write verbal explanations of their equation solving practices, rather than stepby-step property justifications. 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Say It With Symbols: 3.1: Selling Greeting Cards; 3.2: Comparing Costs; 5.1: Using Algebra to Solve a Puzzle 10
(2014) MAFS.912.A-REI.2.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Say It With Symbols: 3.1: Selling Greeting Cards; 3.2: Comparing Costs; 5.1: Using Algebra to Solve a Puzzle It's In the System: 3.1: Comparing Security Services; 3.2: Solving Linear Inequalities Symbolically; 3.3: Operation at a Profit MAFS.912.A-REI.2.4: Solve quadratic equations in one variable. Say It With Symbols: 3.4: Solving Quadratic Equations Function Junction: 4.1: Applying Square Roots; 4.2: Completing the Square; 4.3: The Quadratic Formula; 4.4: Complex Numbers a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)² = q that has the same solutions. Derive the quadratic formula from this form. Function Junction: 4.2: Completing the Square b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Say It With Symbols: 3.4: Solving Quadratic Equations Function Junction: 4.1: Applying Square Roots; 4.2: Completing the Square; 4.3: The Quadratic Formula; 4.4: Complex Numbers MAFS.912.A-REI.3.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. It's In the System: 2.2: Taco Truck Lunch: Solving Systems by Combining Equations I; 2.3: Solving Systems by Combining Equations II 11
(2014) MAFS.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. It's In the System: 1.1: Shirts and Caps: Solving Equations With Two Variables; 1.2: Connecting Ax + By = C and y = mx + b; 1.3: Booster Club Members: Intersecting Lines; 2.1: Shirts and Caps Again: Solving Systems With y = mx + b; 2.2: Taco Truck Lunch: Solving Systems by Combining Equations I; 2.3: Solving Systems by Combining Equations II MAFS.912.A-REI.4.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase; 2.3: Tree Top Fun; 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Growing, Growing, Growing: 1.3: Making a New Offer; 2.1: Killer Plant Strikes Lake Victoria; 2.3: Studying Snake Populations; 3.3: Making a Difference; 4.2: Fighting Fleas; 5.5: Revisiting Exponential Functions It s In The System: 1.1: Shirts and Caps; 1.2: Connecting Ax + By = C and y = mx + b; 1.3: Booster Club Members Frogs, Fleas, and Painted Cubes: 2.4: Quadratic Functions and Their Graphs MAFS.912.A-REI.4.11: Explain why the x- coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. It s In the System: 1.1: Shirts and Caps: Solving Equations With Two Variables; 1.2: Connecting Ax + By = C and y = mx + b; 1.3: Booster Club Members: Intersecting Lines 12
(2014) MAFS.912.A-REI.4.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. MAFS.912.F-IF.1.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). It's In the System: 4.1: Limiting Driving Miles: Inequalities With Two Variables; 4.2: What Makes a Car Green?: Solving Inequalities by Graphing I; 4.3: Feasible Points: Solving Inequalities by Graphing II; 4.4: Miles of Emissions: Systems of Linear Inequalities 2.1: Modeling Linear Data Patterns; 3.2: Distance, Speed and Time Growing, Growing, Growing: 2.1: Killer Plant Strikes Lake Victoria; 2.2: Growing Mold; 5.5: Revisiting Exponential Functions Say It With Symbols: 3.1: Selling Greeting Cards; 4.4: What s the Function? Frogs, Fleas, and Painted Cubes: 4.1: Tracking a Ball: Interpreting a Table and an Equation; 4.2: Measuring Jumps: Comparing Quadratic Relationships; 4.3: Putting It All Together: Functions and Patterns of Change; 4.4: Painted Cubes: Looking at Several Functions Function Junction: 1.1: Filling Functions; 1.2: Domain, Range, and Function Notation; 1.3: Taxi Fares, Time Payments, and Step Functions; 5.1: Properties of Polynomial Expressions and Functions MAFS.912.F-IF.1.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Frogs, Fleas, and Painted Cubes: 4.1: Tracking a Ball: Interpreting a Table and an Equation; 4.2: Measuring Jumps: Comparing Quadratic Relationships; 4.3: Putting It All Together: Functions and Patterns of Change; 4.4: Painted Cubes: Looking at Several Functions Function Junction: 1.1: Filling Functions; 1.2: Domain, Range, and Function Notation; 1.3: Taxi Fares, Time Payments, and Step Functions; 1.4: Piecewise-Defined Functions; 5.1: Properties of Polynomial Expressions and Functions; 5.2: Combining Profit Functions: Operating With Polynomials II 13
(2014) MAFS.912.F-IF.1.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. MAFS.912.F-IF.2.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. MAFS.912.F-IF.2.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. MAFS.912.F-IF.2.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Function Junction: 2.1: Arithmetic Sequences; 2.2: Geometric Sequences Frogs, Fleas, and Painted Cubes: 4.1: Tracking a Ball: Interpreting a Table and an Equation; 4.2: Measuring Jumps: Comparing Quadratic Relationships; 4.3: Putting It All Together: Functions and Patterns of Change; 4.4: Painted Cubes: Looking at Several Functions Function Junction: 1.1: Filling Functions; 1.2: Domain, Range, and Function Notation; 1.3: Taxi Fares, Time Payments, and Step Functions; 1.4: Piecewise-Defined Functions; 5.1: Properties of Polynomial Expressions and Functions; 5.2: Combining Profit Functions: Operating With Polynomials II Frogs, Fleas, and Painted Cubes: 4.1: Tracking a Ball: Interpreting a Table and an Equation; 4.2: Measuring Jumps: Comparing Quadratic Relationships; 4.3: Putting It All Together: Functions and Patterns of Change; 4.4: Painted Cubes: Looking at Several Functions Function Junction: 1.1: Filling Functions; 1.2: Domain, Range, and Function Notation; 1.3: Taxi Fares, Time Payments, and Step Functions; 1.4: Piecewise-Defined Functions; 5.1: Properties of Polynomial Expressions and Functions; 5.2: Combining Profit Functions: Operating With Polynomials II 1.3: Custom Construction Parts; 2.2: Up and Down the Staircase; 2.3: Tree Top Fun; 2.4: Boat Rental Business; 2.5: Amusement Park or Movies Say It With Symbols: 3.1: Selling Greeting Cards; 3.2: Comparing Costs 14
(2014) MAFS.912.F-IF.3.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y =, y =, y =, y =, and classify them as representing exponential growth or decay. Say It With Symbols: 1.2: Thinking in Different Ways: Determining Equivalence; 1.3: The Community Pool Problem: Interpreting Expressions; 2.2: Predicting Profit: Substituting Expressions; 3.3: Factoring Quadratic Equations Say It With Symbols: 3.3: Factoring Quadratic Equations Frogs, Fleas, and Painted Cubes: 2.3: Factoring Quadratic Expressions; 2.4: Quadratic Functions and Their Graphs Growing, Growing, Growing: 1.1: Making Ballots: Introducing Exponential Functions; 1.2: Requesting a Reward: Representing Exponential Functions; 1.3: Making a New Offer: Growth Factors; 2.1: Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1; 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 3.1: Reproducing Rabbits: Fractional Growth Patterns; 3.2: Investing for the Future: Growth Rates; 3.3: Making a Difference: Connecting Growth Rate and Growth Factor; 4.1: Making Smaller Ballots: Introducing Exponential Decay; 4.2: Fighting Fleas: Representing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay; 5.2: Rules of Exponents; 5.3: Extending the Rules of Exponents MAFS.912.F-IF.3.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Students are given two equations for two quadratic functions, one in vertex form and one in standard form, and asked to graph and identify various properties of the functions and their graphs. Frogs, Fleas, and Painted Cubes: 5.1: Properties of Polynomial Expressions and Functions 15
(2014) MAFS.912.F-BF.1.1: Write a function that describes a relationship between two quantities. Say It With Symbols: 3.1: Selling Greeting Cards: Solving Linear Equations; 3.2: Comparing Costs: Solving More Linear Equations; 4.2: Area and Profit What s the Connection?: Using Equations; 4.4: What's the Function?: Modeling With Functions Growing, Growing, Growing: 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 3.1: Reproducing Rabbits: Fractional Growth Patterns; 4.3: Cooling Water: Modeling Exponential Decay Frogs, Fleas, and Painted Cubes: 3.2: Counting Handshakes: Another Quadratic Function; 4.1: Tracking a Ball: Interpreting a Table and an Equation; 4.2: Measuring Jumps: Comparing Quadratic Functions; 4.3: Painted Cubes: Looking at Several Functions Function Junction: 1.3: Taxi Fairs, Time Payments, and Step Functions; 2.1: Arithmetic Sequences; 4.2: Completing the Square; 5.2: Combining Profit Functions a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Function Junction: 2.1: Arithmetic Sequences; 2.2: Geometric Sequences b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Say It With Symbols: 2.1: Walking Together: Adding Expressions; 2.2: Predicting Profit: Substituting Expressions Function Junction: 5.2: Combining Profit Functions: Operating With Polynomials I; 5.3: Product Time: Operating With Polynomials II 16
(2014) c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. MAFS.912.F-BF.2.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. MAFS.912.F-LE.1.1: Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Say It With Symbols: 2.2: Predicting Profit: Substituting Expressions Function Junction: 3.1: Sliding Up and Down: Vertical Translations of Functions; 3.2: Stretching and Flipping Up and Down: Multiplicative Transformations of Functions; 3.3: Sliding Left and Right: Horizontal Translations of Functions; 3.4: Getting From Here to There: Transforming y = x 2 Say It With Symbols: 3.1: Selling Greeting Cards: Solving Linear Equations; 3.2: Comparing Costs: Solving More Linear Equations; 4.3: Generating Patterns: Linear, Exponential, Quadratic; 4.4: What's the Function?": Modeling With Functions Say It With Symbols: 4.3: Generating Patterns: Linear, Exponential, Quadratic 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase: Exploring Slope; 2.3: Tree Top Fun: Equations for Linear Functions; 2.4: Boat Rental Service: Solving Linear Equations; 2.5: Amusement Park or Movies: Intersecting Linear Functions Function Junction: 2.1: Arithmetic Sequences 17
(2014) c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Growing, Growing, Growing: 2.1: Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1; 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 2.3: Studying Snake Populations: Interpreting Graphs of Exponential Functions; 3.1: Reproducing Rabbits: Fractional Growth Patterns; 3.2: Investing for the Future: Growth Rates; 3.3: Making a Difference: Connecting Growth Rate and Growth Factor; 4.1: Making Smaller Ballots: Introducing Exponential Decay; 4.2: Fighting Fleas: Representing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay Function Junction: 2.2: Geometric Sequences MAFS.912.F-LE.1.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase: Exploring Slope; 2.3: Tree Top Fun: Equations for Linear Functions; 2.4: Boat Rental Service: Solving Linear Equations; 2.5: Amusement Park or Movies: Intersecting Linear Functions Growing, Growing, Growing: 2.1: Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1; 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 3.1: Reproducing Rabbits: Fractional Growth Patterns; 3.2: Investing for the Future: Growth Rates; 3.3: Making a Difference: Connecting Growth Rate and Growth Factor; 4.1: Making Smaller Ballots: Introducing Exponential Decay; 4.2: Fighting Fleas: Representing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay Function Junction: 2.1: Arithmetic Sequences; 2.2: Geometric Sequences 18
(2014) MAFS.912.F-LE.1.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. MAFS.912.F-LE.2.5: Interpret the parameters in a linear or exponential function in terms of a context. 2.1: Modeling Linear Data Patterns; 3.4: Modeling Data Patterns Say It With Symbols: 4.1: Pumping Water: Looking at Patterns of Change; 4.2: Area and Profit What's the Connection?: Using Equations; 4.3: Generating Patterns: Linear, Exponential, Quadratic; 4.4: What's the Function?: Modeling With Functions 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase: Exploring Slope; 2.3: Tree Top Fun: Equations for Linear Functions; 2.4: Boat Rental Service: Solving Linear Equations; 2.5: Amusement Park or Movies: Intersecting Linear Functions Growing, Growing, Growing: 2.1: Killer Plant Strikes Lake Victoria: y-intercepts Other Than 1; 2.2: Growing Mold: Interpreting Equations for Exponential Functions; 3.1: Reproducing Rabbits: Fractional Growth Patterns; 3.2: Investing for the Future: Growth Rates; 3.3: Making a Difference: Connecting Growth Rate and Growth Factor; 4.1: Making Smaller Ballots: Introducing Exponential Decay; 4.2: Fighting Fleas: Representing Exponential Decay; 4.3: Cooling Water: Modeling Exponential Decay MAFS.912.S-ID.1.1: Represent data with plots on the real number line (dot plots, histograms, and box plots). MAFS.912.S-ID.1.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 4.4: Measuring Variability: Standard Deviation; 4: Variability and Associations in Data (ACE 10, 15, 23); 5: Variability and Associations in Categorical Data (ACE 32-33) 4.4: Measuring Variability: Standard Deviation; 4: Variability and Associations in Data (ACE 15-20, 23-24); 5: Variability and Associations in Categorical Data (ACE 26-29, 33) 19
(2014) MAFS.912.S-ID.1.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). MAFS.912.S-ID.2.5: Summarize categorical data for two categories in twoway frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. MAFS.912.S-ID.2.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. MAFS.912.S-ID.3.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 4.4: Measuring Variability: Standard Deviation; 4: Variability and Associations in Data (ACE 15-20, 23-24); 5: Variability and Associations in Categorical Data (ACE 26-29, 33) 5.1: Wood or Steel? That's the Question: Relationships in Categorical Data; 5.2: Politics of Girls and Boys: Analyzing Data in Two-Way Tables; 5.3: After-School Jobs and Homework: Working Backward: Setting Up a Two-Way Table 4.1: Vitruvian Man: Relating Body Measurements; 4.2: Older and Faster: Negative Correlation; 4.3: Correlation Coefficients and Outliers 4.1: Vitruvian Man: Relating Body Measurements; 4.2: Older and Faster: Negative Correlation; 4.3: Correlation Coefficients and Outliers 4.1: Vitruvian Man: Relating Body Measurements 4.1: Vitruvian Man: Relating Body Measurements; 4.2: Older and Faster: Negative Correlation 2.1: Modeling Linear Data Patterns; 2.2: Up and Down the Staircase: Exploring Slope; 2.3: Tree Top Fun: Equations for Linear Functions; 2.4: Boat Rental Business: Solving Linear Functions; 4.1: Vitruvian Man: Relating Body Measurements; 4.2: Older and Faster: Negative Correlation 20
(2014) MAFS.912.S-ID.3.8: Compute (using technology) and interpret the correlation coefficient of a linear fit. MAFS.912.S-ID.3.9: Distinguish between correlation and causation. LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9 10 topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively. 4.3: Correlation Coefficients and Outliers Students discuss what the correlation coefficient tells them about their linear models, and they discuss what variables are strongly related to the top speed of a roller coaster; however, they do not specifically refer to the fact that correlation and association are not the same as cause and effect. 4.3: Correlation Coefficients and Outliers Mathematical Reflections is a feature that appears at the conclusion of every investigation. Students are asked to reflect, participate in discussions with their classmates and with their teacher, and write summaries of their responses to open-ended questions related to the completed investigation. See, for example: 1: Exploring Data Patterns (Mathematical Reflections); 4: Variability and Association in Numeric Data (Mathematical Reflections); 5: Variability and Association in Categorical Data (Mathematical Reflections) Looking for Pythagoras: 3: The Pythagorean Theorem (Mathematical Reflections) Growing, Growing, Growing: 1: Exponential Growth (Mathematical Reflections); 2: Examining Growth Patterns (Mathematical Reflections) 1: Symmetry and Transformations (Mathematical Reflections); 2: Transformations and Congruence (Mathematical Reflections) Say It With Symbols: 2: Combining Expressions (Mathematical Reflections); 3: Solving Equations (Mathematical Reflections) 21
(2014) a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas. Mathematical Reflections is a feature that appears at the conclusion of every investigation. Students are asked to reflect, participate in discussions, and write summaries of their responses to open-ended questions related to the completed investigation. Students have prepared for the discussions by having completed the investigation and by reflecting on the posed questions. See, for example: 1: Exploring Data Patterns (Mathematical Reflections); 4: Variability and Association in Numeric Data (Mathematical Reflections); 5: Variability and Association in Categorical Data (Mathematical Reflections) Looking for Pythagoras: 3: The Pythagorean Theorem (Mathematical Reflections) Growing, Growing, Growing: 1: Exponential Growth (Mathematical Reflections); 2: Examining Growth Patterns (Mathematical Reflections) 1: Symmetry and Transformations (Mathematical Reflections); 2: Transformations and Congruence (Mathematical Reflections) Say It With Symbols: 2: Combining Expressions (Mathematical Reflections); 3: Solving Equations (Mathematical Reflections) 22
(2014) b. Work with peers to set rules for collegial discussions and decision-making (e.g., informal consensus, taking votes on key issues, presentation of alternate views), clear goals and deadlines, and individual roles as needed. Mathematical Reflections is a feature that appears at the conclusion of every investigation. Students are asked to reflect, participate in discussions, and write summaries of their responses to open-ended questions related to the completed investigation. This is an activity which should be conducted at the end of every investigation, so students learn and practice appropriate guidelines for discussion and exchange of ideas. See, for example: 1: Exploring Data Patterns (Mathematical Reflections); 4: Variability and Association in Numeric Data (Mathematical Reflections); 5: Variability and Association in Categorical Data (Mathematical Reflections) Looking for Pythagoras: 3: The Pythagorean Theorem (Mathematical Reflections) Growing, Growing, Growing: 1: Exponential Growth (Mathematical Reflections); 2: Examining Growth Patterns (Mathematical Reflections) 1: Symmetry and Transformations (Mathematical Reflections); 2: Transformations and Congruence (Mathematical Reflections) Say It With Symbols: 2: Combining Expressions (Mathematical Reflections); 3: Solving Equations (Mathematical Reflections) 23
(2014) c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. Mathematical Reflections is a feature that appears at the conclusion of every investigation. Students are asked to reflect, participate in discussions, and write summaries of their responses to specific open-ended questions related to the completed investigation. See, for example: 1: Exploring Data Patterns (Mathematical Reflections); 4: Variability and Association in Numeric Data (Mathematical Reflections); 5: Variability and Association in Categorical Data (Mathematical Reflections) Looking for Pythagoras: 3: The Pythagorean Theorem (Mathematical Reflections) Growing, Growing, Growing: 1: Exponential Growth (Mathematical Reflections); 2: Examining Growth Patterns (Mathematical Reflections) 1: Symmetry and Transformations (Mathematical Reflections); 2: Transformations and Congruence (Mathematical Reflections) Say It With Symbols: 2: Combining Expressions (Mathematical Reflections); 3: Solving Equations (Mathematical Reflections) 24
(2014) d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. Mathematical Reflections is a feature that appears at the conclusion of every investigation. Students are asked to reflect, participate in discussions, and write summaries of their responses to open-ended questions related to the completed investigation. Part of this discourse might include the expression of new information by others which can influence the student's own conclusions. See, for example: 1: Exploring Data Patterns (Mathematical Reflections); 4: Variability and Association in Numeric Data (Mathematical Reflections); 5: Variability and Association in Categorical Data (Mathematical Reflections) Looking for Pythagoras: 3: The Pythagorean Theorem (Mathematical Reflections) Growing, Growing, Growing: 1: Exponential Growth (Mathematical Reflections); 2: Examining Growth Patterns (Mathematical Reflections) 1: Symmetry and Transformations (Mathematical Reflections); 2: Transformations and Congruence (Mathematical Reflections) Say It With Symbols: 2: Combining Expressions (Mathematical Reflections); 3: Solving Equations (Mathematical Reflections) 25
(2014) LAFS.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats (e.g., visually, quantitatively, orally) evaluating the credibility and accuracy of each source. Throughout the course, either in the instructional pages or in the ACE problem sets, students are exposed to diverse media formats (including diagrams, text, technological displays, maps, cartoons, and advertising) to communicate mathematical information. See, for example: 1.1: Bridge Thickness and Strength; 5.3: After-School Jobs and Homework: Working Backward: Setting Up a Two-Way Table Looking for Pythagoras: 1: Coordinate Grids (Opener); 4.1: Analyzing the Wheel of Theodorus: Square Roots on a Number Line Growing, Growing, Growing: 1.2: Requesting a Reward: Representing Exponential Functions; 3: Growth Factors and Growth Rates (ACE 9, 19); 5.3: Extending the Rules of Exponents; 5: Patterns With Exponents (ACE 41) Say It With Symbols: 3.1: Selling Greeting Cards: Solving Linear Equations; 3.4: Solving Quadratic Equations 26
(2014) LAFS.910.SL.1.3: Evaluate a speaker s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. The instructional pages and the ACE problem sets frequently refer to a mathematical argument, claim, or reasoning by an individual; students are asked to evaluate this reasoning and conclusion with supporting evidence or counterexamples. See, for example: 2.3: Tree Top Fun: Equations for Linear Functions; 5.3: After-School Jobs and Homework: Working Backward: Setting Up a Two-Way Table Looking for Pythagoras: 5.2: Analyzing Triangles (Did You Know?) Growing, Growing, Growing: 1: Exponential Growth (ACE 2); 2.3: Studying Snake Populations: Interpreting Graphs of Exponential Functions; 3: Growth Factors and Growth Rates (ACE 33-34); 4: Exponential Decay (ACE 2, 16, 18); 5.3: Extending the Rules of Exponents; 5.4: Operations With Scientific Notation 2.3: Minimum Measurement: Congruent Triangles II 27
(2014) LAFS.910.SL.2.4: Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task. Mathematical Reflections is a feature that appears at the conclusion of every investigation. Students are asked to participate in discussions in response to open-ended questions related to the completed investigation. See, for example: 1.1: Bridge Thickness and Strength; 1.2: Bridge Length and Strength; 4.3: Correlation Coefficients and Outliers Looking for Pythagoras: 3.1: Discovering the Pythagorean Theorem; 3.4: Measuring the Egyptian Way: Lengths That Form a Right Triangle Growing, Growing, Growing: 2.3: Studying Snake Populations: Interpreting Graphs of Exponential Functions; 5.3: Extending the Rules of Exponents; 5.4: Operations With Scientific Notation 3.1: Flipping on a Grid: Coordinate Rules for Reflections; 3.5: Parallel Lines, Transversals, and Angle Sums 28
(2014) LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. Throughout the investigative curriculum of the Connected Mathematics Project 3 series, students are asked to perform multistep procedures in the course of playing a game or conducting an experiment. Then, students are expected to draw conclusions, make mathematical connections, and report their findings. See, for example: 1.1: Bridge Thickness and Strength; 1.2: Bridge Length and Strength; 3.2: Distance, Speed, and Time; 3: Inverse Variation (ACE 42) Looking for Pythagoras: 3.1: Discovering the Pythagorean Theorem Growing, Growing, Growing: 1.1: Making Ballots: Introducing Exponential Functions; 2.1: Killer Plant Strikes Lake Victoria: y- Intercepts Other Than 1; 4.3: Cooling Water: Modeling Exponential Decay; 4: Exponential Decay (ACE 25) Unit Project: Making a Wreath and a Pinwheel 29
(2014) LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9 10 texts and topics. Students are introduced to domain-specific mathematical symbols and vocabulary throughout the course, both in the instructional pages and also in the ACE problem sets. Students are then expected to immediately use the vocabulary or symbols to communicate their knowledge of mathematical concepts and properties. See, for example: 4.3: Correlation Coefficients and Outliers; 4: Variability and Association in Numeric Data (Mathematical Reflections) 1.1: Butterfly Symmetry: Line Reflections Say It With Symbols: 1.1: Tiling Pools: Writing Equivalent Expressions; 1.2: Thinking in Different Ways: Determining Equivalence; 1.3: The Community Pool Problem: Interpreting Expressions; 2.2: Predicting Profit: Substituting Expressions; 4.2: Area and Profit What's the Connection?: Using Equations; 5: Reasoning With Symbols (Mathematical Reflections) It's In the System: 4.2: What Makes a Car Green?: Solving Inequalities by Graphing I 30
(2014) LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. Students model information using visual models, including Venn diagrams, tables, graphs, and physical models throughout the course. They interpret the coefficients and constants in an equation to refer to rate of change or initial value, for example, in a linear function equation. See, for example: 2.1: Modeling Linear Data Patterns; 3.4: Modeling Data Patterns; 5.2: Politics of Girls and Boys: Analyzing Data in Two-Way Tables Looking for Pythagoras: 2.2: Square Roots; 4.1: Analyzing the Wheel of Theodorus: Square Roots on a Number Line; 5.2: Analyzing Triangles Growing, Growing, Growing: 3.1: Reproducing Rabbits: Fractional Growth Patterns 4.4: Using Similar Triangles It's In the System: 4.2: What Makes a Car Green?: Solving Inequalities by Graphing I; 4.4: Miles of Emissions: Systems of Linear Inequalities 31