exemplar module analysis

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Functions

Exemplar Module AnalysisGrade 9 – Module 1

Grade 11 – Module 1

Grade 12 – Module 1


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Functions

AGENDA

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• G9-M1• Topic Exploration – a Sampling of Exercises and Key Concepts• Exercise: Assessment & Scoring Rubric• Overview of Other G9 Modules• Summary of Key Shifts of Instruction

• G11-M1• G12-M1



What’s in G9-M1?

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• 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



Topic A Introduction to Functions Studied this Year: Graphing Stories

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Sample Exercise:• Describe the motion of the man in the video



The Video:

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• http://mrmeyer.com/graphingstories1/graphingstories2.mov(watch only the first 1:08 minutes)

http://mrmeyer.com/graphingstories1/graphingstories2.mov

http://mrmeyer.com/graphingstories1/graphingstories2.mov



Topic A Introduction to Functions Studied this Year: Graphing Stories

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Sample Exercise:• Describe the motion of the man in the video



Piecewise Defined Functions (cont.)

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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?



Topic A Introduction to Functions Studied this Year: Piecewise Defined Functions

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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.



Topic B The Structure of Expressions

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Sample Exercises from Lesson 7:• Use this diagram to build an expression using these symbols and operators.




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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.




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DEFINE ALGEBRAIC EXPRESSION: An algebraic expression is eithera 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.



Topic CSolving Equations and Inequalities

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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.



Topic C Solving Equations and Inequalities

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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.



Topic C Solving Equations and Inequalities

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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).



Topic D Creating Equations to Solve Problems

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Sample exercise:• Create a piecewise defined function to represent the total effective Federal

Income tax due for a married couple filing jointly.

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

$379,150 $102,574.00 + 35% $379,150

Schedule Y-1 — Married filing Jointly or Qualifying Widow(er)



Sample from the Assessments:

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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).



Other Grade 9 Modules:

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G9-M2: Descriptive Statistics• Distributions and their shapes• Measures of center and spread• Modeling relationships of numerical data on two variables




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G9-M3: 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




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G9-M4: 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.




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G9-M5: Quadratic Functions• Key features of quadratic and non-quadratic graphs• Exponential vs.. quadratic growth• Modeling with quadratic functions



Key Shifts of G9 Curriculum:

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• 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.



AGENDA

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• G9-M1• G11-M1

• Topic Exploration – a Sampling of Exercises and Key Concepts• Summary of Key Shifts of Instruction

• G12-M1



What’s in G11-M1?

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• 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



Topic APolynomials from Base Ten to Base X

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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 x3 + 2 x x2 + 7 x x + 3 x 1



Topic APolynomials from Base Ten to Base X

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• 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 x2 – a2 by x – a?



Topic BFactoring: Its Use and Its Obstacles

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• Exercise: Write 501 as a product of prime numbers.• 1st 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?



Topic CSolving and Applying Equations: Polynomial and Rational

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• Rational Expressions are analogous to rational numbers• Solving rational equations in one variable – IF there is an

answer, what must the answer be?

• Solving systems of equations in real-world contexts• Deriving the equation of a parabola



Topic DA Surprise from Geometry—Complex Numbers Overcome All Obstacles

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• 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?



Summary of Key Shifts in G11-M1

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• Polynomials are analogous to the integers (foundational from G9-M4’s A-APR.1)

• Rational expressions are analogous to rational numbers (precursor to G12-M3’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.



AGENDA

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• G9-M1• G11-M1• G12-M1

• Topic Exploration – a Sampling of Exercises and Key Concepts• Supporting content from grades 8-11• Summary of Key Shifts of Instruction



By the end of G12-M1 students:

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• Use matrix notation to define and interpret transformations of the coordinate plane



How do they get there?

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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 rotationTopic C: The Power of the Right Notation



Topic A A Question of Linearity

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Wouldn’t it be lovely if functions were “nice” and just did what we expected them to do? Exercise: Here are some common student mistakes:

a. Substitute in some values to show these statements are not in general true.

b. Are there any values for which these statements, by coincidence, happen to work? Find all such values a and b for which these statements are true.



Topic AA Question of Linearity

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• If only all functions behaved in this way:

• Functions that behave in this way are called:

LINEAR TRANFORMATIONS

• Are all linear functions also linear transformations?

• No, but functions of the form are linear transformations.



Topic B Complex Number Operations as Transformations

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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?



Topic B Complex Number Operations as Transformations

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Can we come up with a function that takes any point in the complex plane and transforms it to another point in the complex plane based on the rule of the function?

YES!

We have seen that translates in the plane. We could write this as

And that rotates and dilates in the plane. We could write this as



Topic BComplex Number Operations as Transformations

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Can we agree that complex numbers, can be matched with or thought of as points?Consider our function that rotates and dilates about the origin:

What are the coordinates of the resulting point in the complex plane in terms of a, b, x and y?



Topic CThe Power of the Right Notation

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Doesn’t this look ugly?

After 70 years of struggle, they came up with this:

We know that: has to end up as: Can we decipher what would end up as?



Supporting Content of Grade 9-11:

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Grade 11• Motivates a discovery of complex numbers as students graph

polynomials and thus look for their zeros (G11-M1)• 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” (G11-M1)

• Provides a study of the trigonometric functions (G11-M2)



Supporting Content of Grade 9-11:

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Grade 10• In depth study of basic rigid motions: rotations, translations,

reflections (G10-M1)• In depth study of similarity transformations (dilations) (G10-M2)Grade 9• Introduces function notation; • Gives focused attention to interpreting the structure of expressions



Key Shifts of G12-M1:

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• 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.

exemplar module analysis

Documents

year linear

key conceptsexercise

sampling of exercises

structure of expressions

graphing stories

discussion of key concept

key shift of instruction

graphs of situationstopic