NYS COMMON CORE MATHEMATICS CURRICULUM A Story of

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Exemplar Module Analysis Grade 9 – Module 1 Grade 11 – Module 1 Grade 12 – Module 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • G 9 -M 1 • Topic Exploration – a Sampling of Exercises and Key Concepts • Exercise: Assessment & Scoring Rubric • Overview of Other G 9 Modules • Summary of Key Shifts of Instruction • G 11 -M 1 • G 12 -M 1 2 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What’s in G 9 -M 1? • Topic A: Explore the main functions of the year (linear, exponential and quadratic) through graphing stories (making graphs of situations) • Topic B: Study the structure of expressions, define what it means for expressions to be equivalent • Topic C: Precisely explain each step in the process of solving an equation • Topic D: The modeling cycle – solving problems using equations and inequalities in one variable, systems of equations in two variables 3 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A Introduction to Functions Studied this Year: Graphing Stories Sample Exercise: • Describe the motion of the man in the video 4 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The Video: • http: //mrmeyer. com/graphingstories 1/graphingstories 2. mov (watch only the first 1: 08 minutes) 5 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A Introduction to Functions Studied this Year: Graphing Stories Sample Exercise: • Describe the motion of the man in the video 6 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Piecewise Defined Functions (cont. ) A Follow-On Activity: • Here is an elevation vs. . time graph of a person’s motion, can we describe what the person might have been doing? 7 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A Introduction to Functions Studied this Year: Piecewise Defined Functions Define Real piece-wise defined linear function: • Given a finite number of non-overlapping intervals on the real number line, a real piecewise-linear function is a function from the union of the intervals, to the real number line, such that the function is defined by (possibly different) linear functions on each interval. 8 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B The Structure of Expressions Sample Exercises from Lesson 7: • Use this diagram to build an expression using these symbols and operators. 9 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B The Structure of Expressions DEFINE VARIABLE: A variable symbol is a symbol that is a placeholder for a number from a specified set of numbers. The set of numbers is called the domain of variability. If a variable symbol represents a number from a set of numbers with more than one element, it is typically just called a variable. If the domain of variability is exactly one element, then the variable symbol is called a constant. 10 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B The Structure of Expressions DEFINE ALGEBRAIC EXPRESSION: An algebraic expression is either a numerical symbol or a variable symbol, or the result of placing previously generated algebraic expressions into the two blanks of one of the four operators (__+__, __-__, __×__, __÷__) or into the base blank of an exponentiation with an exponent that is a rational number. A numerical expression is a algebraic expression that does not have any variables in it and that evaluates to a real number value. 11 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C Solving Equations and Inequalities ALGEBRAIC EQUATION: An algebraic equation is a statement of equality between two algebraic expressions. NUMBER SENTENCE. A number sentence is a statement of equality between two numerical expressions. TRUTH VALUES OF A NUMBER SENTENCE. A number sentence is said to be true if both numerical expressions evaluate to the same number; it is said to be false otherwise. 12 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C Solving Equations and Inequalities The vocabulary of Solution Sets: • An equation with variables is often viewed as a question asking for which values of the variables is the equation true. • The equation serves as a filter that sifts through all the numbers in the domain of the variability and sorts those into two disjoint sets: The Solution Set and the set of values for which the equation is false. 13 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C Solving Equations and Inequalities Key concepts and definitions: • “Solve”– identifying all the values of the solution set for a given system of one or more equations. • Solving an equation: … starts with the assumption that the original equation has a solution, example: x + 5 = x - 3 … and then strategically uses the associative, commutative and distributive properties and “If-then” moves that apply the properties of equalities (or inequalities). 14 © 2012 Common Core, Inc. All rights reserved. commoncore. org

A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Topic D Creating Equations to Solve Problems Sample exercise: • Create a piecewise defined function to represent the total effective Federal Income tax due for a married couple filing jointly. Schedule Y-1 — Married filing Jointly or Qualifying Widow(er) If taxable income is over-- But not over-- The tax is: of the amount over-- $0 $17, 000 10% $0 $17, 000 $69, 000 $1, 700. 00 + 15% $17, 000 $69, 000 $139, 350 $9, 500. 00 + 25% $69, 000 $139, 350 $212, 300 $27, 087. 50 + 28% $139, 350 $212, 300 $379, 150 $47, 513. 50 + 33% $212, 300 $102, 574. 00 + 35% $379, 150 15 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Sample from the Assessments: 1. Jacob lives on a street that runs east and west. The grocery store is to the east and the post office is to the west of his house. Both are on the same street as his house. Answer the questions below about the following story: At 1: 00 p. m. , Jacob hops in his car and drives at a constant speed of 25 mph for 6 minutes to the post office. After 10 minutes at the post office, he realizes he is late, and drives at a constant 30 mph to the grocery store, arriving at 1: 28 p. m. He then spends 20 minutes buying groceries. a. Draw a graph that shows the distance Jacob’s car is from his house with respect to time. Remember to label your axes with the units you chose and any important points (home, post office, grocery store). 16 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Other Grade 9 Modules: G 9 -M 2: Descriptive Statistics • Distributions and their shapes • Measures of center and spread • Modeling relationships of numerical data on two variables 17 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Other Grade 9 Modules: G 9 -M 3: Linear and Exponential Relationships • Formal function notation – a study of arithmetic and geometric sequences • Rates of change – contrasting linear and exponential • Interpreting graphs of functions – domain, range, maxima, minima • Relating equation notation to function notation • Absolute value function – studying transformations – how graphs change when equations change • Applying functions to real world contexts – systems of equations 18 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Other Grade 9 Modules: G 9 -M 4: Expressions and Equations • Explaining properties of quantities represented by an expression based on contextual situation • Identify ways to rewrite quadratics and the usefulness of each • Operations with polynomials • Symmetry in quadratic graphs • Solving quadratic equations, deriving the quadratic formula POLYNOMIAL. A polynomial is any algebraic expression generated in the following way: (1) declare all variable symbols and all numerical expressions to be polynomials. (2) any algebraic expression created by substituting two polynomials into the blanks of an addition operator or multiplication operator is also a polynomial. 19 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Other Grade 9 Modules: G 9 -M 5: Quadratic Functions • Key features of quadratic and non-quadratic graphs • Exponential vs. . quadratic growth • Modeling with quadratic functions 20 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Shifts of G 9 Curriculum: • A focus on the solution set. • How does the solution set stay the same or change as we modify the equation. • Graphs of equations are pictorial representations of solution sets. • The graph of the function, f, is a pictorial representation of the solution set of y = f(x). • How does the graph of the function stay the same or change as we modify the function. • Students experience learning and modeling: Start with an intuitive notion -> play with examples and look for structure --> find rogue examples and figure out what to do with them ---> arrive at a nice definition. 21 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • G 9 -M 1 • G 11 -M 1 • Topic Exploration – a Sampling of Exercises and Key Concepts • Summary of Key Shifts of Instruction • G 12 -M 1 22 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What’s in G 11 -M 1? • Topic A: POLYNOMIALS: extending from, and analogous to, base 10 arithmetic; division with polynomials • Topic B: FACTORING: Its use and its obstacles, leading to modeling with polynomials • Topic C: Solving and applying polynomial and rational equations • Topic D: A surprise from geometry: complex numbers overcome all obstacles; radical equations 23 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A Polynomials from Base Ten to Base X Exercise: • Write the number 8943 in base 20 • Let’s be as general as possible – not identify which base we are in, just call it x. 1 x x 3 + 2 x x 2 + 7 x x + 3 x 1 24 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A Polynomials from Base Ten to Base X • Two models for division with polynomials: • Long division algorithm that is analogous to the division algorithm in base 10 arithmetic. • Reverse Galley method that is analogous to the area model • Extensive exploration with division serves as a precursor to factoring: What happens when we divide x 2 – a 2 by x – a? 25 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B Factoring: Its Use and Its Obstacles • Exercise: Write 501 as a product of prime numbers. • 1 st Obstacle: What if a factor is not given to us first… how can we achieve factored form when we don’t know what to divide by? • Motivated by graphing (A-APR. 3): Can we write an equation whose graph would look like this? 26 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C Solving and Applying Equations: Polynomial and Rational 27 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic D A Surprise from Geometry—Complex Numbers Overcome All Obstacles • Some quadratic equations do not have solutions, why? • Does every cubic have a solution? • A surprising boost from geometry – transformations on the number line: • What is the geometric effect of adding 2 to all the numbers on the number line? • … to multiplying all the numbers by -1? • What transformation would create a 90°rotation? 28 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Summary of Key Shifts in G 11 -M 1 • Polynomials are analogous to the integers (foundational from G 9 M 4’s A-APR. 1) • Rational expressions are analogous to rational numbers (precursor to G 12 -M 3’s A-APR. 7) • Graphs of equations are pictorial representations of solution sets. • The graph of the function, f, is a pictorial representation of the solution set of y = f(x). • Students experience learning and modeling: Start with an intuitive notion --> play with examples and look for structure --> find rogue examples and figure out what to do with them ---> arrive at a nice definition. 29 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • G 9 -M 1 • G 11 -M 1 • G 12 -M 1 • Topic Exploration – a Sampling of Exercises and Key Concepts • Supporting content from grades 8 -11 • Summary of Key Shifts of Instruction 30 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions By the end of G 12 -M 1 students: • Use matrix notation to define and interpret transformations of the coordinate plane 31 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions How do they get there? Topic A: A Question of Linearity • Investigate the qualities of linear transformations (from reals to the reals) Topic B: Complex Number Operations as Transformations • Geometric representations of complex numbers and operations thereon (in the complex plane) • Extend linearity to transformations from coordinate plane to coordinate the plane (aka, functions that take an (x, y) pair as an input and produce another (x, y) pair as an output. ) • Connect to the geometric effect of dilation and/or rotation Topic C: The Power of the Right Notation 32 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A A Question of Linearity 33 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A A Question of Linearity 34 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B Complex Number Operations as Transformations What ‘happens’ in the geometric representation • When we add two complex numbers? • When we multiply a complex number by -1? • When we subtract one complex number from another? • When we multiply two complex numbers? • When we divide a complex number by another? 35 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B Complex Number Operations as Transformations 36 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B Complex Number Operations as Transformations 37 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C The Power of the Right Notation 38 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Supporting Content of Grade 9 -11: Grade 11 • Motivates a discovery of complex numbers as students graph polynomials and thus look for their zeros (G 11 -M 1) • Complex numbers are connected to the complex plane via, “If multiplying by -1 has the effect of rotating the number line 180 degrees, what could I multiply by to rotate 90 -degrees” (G 11 -M 1) • Provides a study of the trigonometric functions (G 11 -M 2) 39 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Supporting Content of Grade 9 -11: Grade 10 • In depth study of basic rigid motions: rotations, translations, reflections (G 10 -M 1) • In depth study of similarity transformations (dilations) (G 10 -M 2) Grade 9 • Introduces function notation; • Gives focused attention to interpreting the structure of expressions 40 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Shifts of G 12 -M 1: • Introduction motivation for use of matrices: • The need for notation of transformations in the plane (their primary mathematical use) – every type of 3 dimensional game uses this type of rotation and dilation matrices. • Introduces the idea of an inverse function. 41 © 2012 Common Core, Inc. All rights reserved. commoncore. org