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RCSS Adaptive Version-Draft 1- August 2016 (Includes Success Criteria/ I Can Statements adapted from Johnston County)

North Carolina Math 3

Collaborative Pacing Guide

This pacing guide is the collaborative work of math teachers, coaches, and curriculum leaders from 38 NC public school districts. The teams worked through two

face-to-face meetings and digitally to compile the information presented. NC Math 1, 2, and 3 standards were used to draft possible units of study for these

courses. This is a first draft living document. Teams plan to meet throughout the year to continually tweak, update and refine these guides. Updates will be posted

as available to this google document.

Please reference the NC Math 1, 2, or 3 standards for any questions or discrepancies. This document should be used only after reading the NC Math 1, 2, and 3

standards and instructional guides provided by NC DPI.

If you have suggestions or comments that you would like the collaborative writing team to consider for revisions, please email [email protected] or

[email protected].

Units for NC Math 3 Number of Days (Block) Number of Days (Traditional)

Unit 1: Functions and Their Inverses 10 20

Unit 2: Exponential and Logarithmic Functions 9 18

Unit 3: Polynomial Functions 9 18

Unit 4: Modeling with Geometry 7 14

Unit 5: Rational Functions 10 20

Unit 6: Reasoning with Geometry 15 30

Unit 7: Trigonometric Functions 10 20

Unit 8: Statistics 5 10

Total (allowing for flex days) 75 150

http://maccss.ncdpi.wikispaces.net/HS+Standards

mailto:[email protected]

mailto:[email protected]

Unit 1: Functions and Their Inverses

Estimated Days: 10 Semester or 20 Year Long

Rationale: This first unit builds upon students’ previous work with modeling functions in Math 1 and Math 2. This unit helps students transition

from modeling in the real world to more abstract mathematical concepts like polynomial and rational functions. It develops the notion of the

inverse function of quadratic, exponential, and linear functions and introduces piecewise-defined and absolute value functions through multiple

representations, i.e. graphing, equations, tables, verbal descriptions, etc. Since students in Math 1 and Math 2 have already worked with linear,

quadratic, and exponential functions, this allows teachers a chance to begin with content that is familiar to students. It also assists teachers in

identifying misconceptions, obstacles, and gaps in prior learning.

Overarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1b

Standards Learning Intentions

NC.M3.F-IF.4 NC.M3.F-IF.9 NC.M3.F-IF.2 NC.M3.F-IF.7 NC.M3.F-BF.1b NC.M3.F-BF.3

Compare functions using multiple representations of and understand key features to interpret, analyze, and find solutions.

NC.M3.F-BF.4 NC.M3.F-BF.4a NC.M3.F-BF.4b NC.M3.F-BF.4c

Understand inverse relationships, describe them algebraically, and use these relationships to solve, analyze and interpret.

NC.M3.A-CED.1 NC.M3.A-CED.2 NC.M3.A-CED.3 NC.M3.A-SSE.1 NC.M3.A-REI.11

Understand and recognize piecewise defined relationships. Understand and solve absolute value equations and inequalities. Understand and solve systems of equations that include piecewise defined relationships (including absolute value) and

can use a graphing tool when necessary. Identify key parts in expressions and equations.

Learning Intentions: These are big ideas, understandings, important math that needs to be developed. They are not necessarily measurable

statements. Ideally a unit will have a handful of learning intentions.

Success Criteria: These are directly associated with a learning intention and articulate to students measurable, tangible, observable

demonstrations of the learning intention. Typically one learning intention has around 3 to 5 success criteria.

Unit 1: Functions and Their Inverses


Suggested Order: 1 of 8

Suggested Time: 10 days semester block (90-minute classes)

Rationale: The first unit builds upon students’ previous work with modeling functions in Math 1 and Math 2. This unit helps students

transition from modeling in the real world to more abstract mathematical concepts like polynomial and rational functions. It develops

the notion of the inverse function of quadratic, exponential, and linear functions and introduces piecewise-defined and absolute value

functions through multiple representations, i.e. graphing, equations, tables, verbal descriptions, etc. Since students in Math 1 and

Math 2 have already worked with linear, quadratic, and exponential functions, this allows teachers a chance to begin with content that

is familiar to students. It also assists teachers in identifying misconceptions, obstacles, and gaps in prior learning.

Overarching Standards - NC.M3.A-SSE.1, NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-IF.4, NC.M3.F-BF.1

Major Work

Standards Learning Intentions Success Criteria

Priority Standard

NC.M3.F-IF.9 Compare key features of two

functions using different

representations by comparing

properties of two different

functions, each with a different

representation (symbolically,

graphically, numerically in tables,

or by verbal descriptions

Supporting Standard

NC.M3.F-IF.4 Interpret key features of graphs,

tables, and verbal descriptions in

context to describe functions that

arise in applications relating two

quantities to include periodicity

and discontinuities.

A. Compare and identify

functions’ characteristics of

interest including intercepts,

intervals of increasing,

decreasing, positive and

negative, relative maximums and

minimums, symmetries and end

behavior using a variety of

representations. Combine

standard functions using

arithmetic operations.

A1. I can identify the key features of linear, quadratic and exponential

functions using tables, graphs, and verbal descriptions.

Priority Standard

NC.M3.F-BF.4c If an inverse function exists for a

linear, quadratic and/or

exponential function, f, represent

the inverse function, f-1 with a

table, graph, or equation and use

it to solve problems in terms of a

context.

Supporting Standards

NC.M3.F-BF.4 Find an inverse function.

NC.M3.F-BF.4a Understand the inverse

relationship between exponential

and logarithmic, quadratic and

square root, and linear to linear

functions and use this

relationship to solve problems

using tables, graphs, and

equations.

NC.M3.F-BF.4b Determine if an inverse function exists by analyzing tables, graphs, and equations.

B. Find an inverse of a function.

Understand inverse relationships

exist and determine if the

inverse is a function. Use the

inverse to model real world

phenomena.

B1. I can find an inverse of a linear, quadratic function algebraically,

graphically and tabularly.

B2. I can find an inverse of an exponential graphically and tabularly.

B3. I can analyze tables, graphs, and equations to determine if an

inverse function exists.

B4. I can use the inverse of a function to solve problems in context.

Priority Standard

NC.M3.A-CED.2 Create and graph equations in

two variables to represent

absolute value, polynomial,

exponential and rational

relationships between quantities.

Supporting Standard

NC.M3.A-CED.1 Create equations and inequalities

in one variable that represent


exponential, and rational

relationships and use them to

solve problems algebraically and

graphically.

C. Graph and write equations for

absolute value functions. Use

absolute value functions to

model and solve problems. Find

the intersection of two functions

graphically.

C1. I can identify and describe the effects of transformations of absolute

value equations in relation to their parent functions.

C2. I can graph and describe key features of absolute value functions

with and without technology.

C3. I can use different representations to compare key features of

absolute value functions.

C4. I can build absolute value functions to represent relevant

relationships/quantities given a real-world situation or mathematical

process or graph.

C5. I can approximate the intersection of a system with absolute value

functions graphically.

Priority Standard

NC.M3.A-REI.11 Extend an understanding that the

�-coordinates of the points where

the graphs of two equations � =

(�) and � = (�) intersect are the

solutions of the equation (�) =

(�) and approximate solutions

using graphing technology or

successive approximations with a

table of values.

Supporting Standard

NC.M3.A-CED.3 Create systems of equations

and/or inequalities to model

situations in context.

Priority Standard

NC.M3.F-BF.3 Extend an understanding of the

effects on the graphical and

tabular representations of a

function when replacing �(�) with

�∙�(�), �(�) + �, �(� + �) to

include �(�∙�) for specific values

of ɑ (both positive and negative).

Priority Standard

NC.M3.F-IF.2 Use function notation to evaluate

piecewise defined functions for

inputs in their domains, and

interpret statements that use

function notation in terms of a

context.

D. Write equations for piecewise

functions. Analyze the key

features of piecewise functions.

Use piecewise to model and

solve problems. Find the

intersection of two functions

graphically. Evaluate a

piecewise function at select

values of x.

D1. I can evaluate piecewise functions both graphically and

algebraically.

D2. I can describe key features of a piecewise function with and without

technology.

D3. I can use different representations to compare key features of

piecewise functions.

D4. I can build piecewise functions to represent relevant


process or graph.

D5. I can identify discontinuities in a piecewise function.

Priority Standard

NC.M3.F-IF.7 Analyze piecewise, absolute

value, polynomials, exponential,

rational, and trigonometric

functions (sine and cosine) using

different representations to show

key features of the graph, by

hand in simple cases and using

technology for more complicated

cases, including: domain and

range; intercepts; intervals where

the function is increasing,

decreasing, positive, or negative;

rate of change; relative

maximums and minimums;

symmetries; end behavior; period;


Supporting Standard

NC.M3.A-SSE.1a Identify and interpret parts of a

piecewise, absolute value,

polynomial, exponential and

rational expressions including

terms, factors, coefficients, and

exponents.

Vocabulary: Inverse, relation, function, one-to-one, logarithm, horizontal line test, vertical line test, Absolute Value, Piecewise

function, system of equations, domain, range, x-intercept, y-intercept, increasing/decreasing intervals), evaluate, end behavior

Possible Honors Topics: Greatest Integer Function

Unit 2: Exponential and Logarithmic Functions


Rationale: Following the functions unit, this unit continues to build upon familiarity with exponents and exponential functions and introduces

logarithmic functions. Additionally, solving exponential and logarithmic equations involves using algebraic operations students have practiced in

Math 1 and Math 2, thus this unit seeks to build continued opportunities for students to be successful at the beginning of Math 3. Furthermore,

flexibility with exponential and logarithmic models is essential for competence in Precalculus and Calculus; therefore, teachers should stress a

modeling approach to this unit.



NC.M3.A-CED.1 NC.M3.A-CED.2 MC.M3.A-SSE.1 NC.M3.A-SSE.3c

Understand how to create exponential equations and graphs with one or two variables, and be able to identify the different

parts of an exponential equation and relate them to the real world.

NC.M3.F-BF.3 NC.M3.F-IF.4 NC.M3. F-IF.7 NC.M3. F-IF.9 NC.M3.F-BF.1a NC.M3.F-BF.4 NC.M3.F-LE.4

Recognize the relationship between exponential and logarithmic equations as inverses using multiple representations,

interpret the key features of the graph, and use them to solve equations and model real world phenomena.

Unit 2: Exponential and Logarithmic Functions



Suggested Time: 9 days on a semester schedule (90-minute classes)

Rationale: Following the functions unit, this unit continues to build upon familiarity with exponents and exponential functions and

introduces logarithmic functions. Additionally, solving exponential and logarithmic equations involves using algebraic operations

students have practiced in Math 1 and Math 2, thus this unit seeks to build continued opportunities for students to be successful at the

beginning of Math 3. Furthermore, flexibility with exponential and logarithmic models is essential for competence in a fourth math,

Precalculus and/or Calculus; therefore, teachers should stress a modeling approach to this unit.


Major Work





exponential and rational

relationships between quantities.

NC.M3.A-CED.1 Create equations and inequalities

in one variable that represent


exponential, and rational

relationships and use them to

solve problems algebraically and

graphically.

A. Understand how to create

exponential equations and graphs

with one or two variables, and be

able to identify the different parts

of an exponential equation as

well as relating them to the real

world.

A1. I can create and graph exponential equations with one or two

variables and use them to solve problems algebraically.

A2. I can identify and interpret parts of an exponential expression and

relate them to a real-world situation.

A3. I can use the properties of exponents to write equivalent forms of

exponential expressions and state the domain.

NC.M3.A-SSE.3c Write an equivalent form of an

exponential expression by using

the property of exponents to

transform expressions to reveal

rates based on different intervals

of the domain.




function when replacing �(�) with

�∙�(�), �(�) + �, �(� + �) to


of ɑ (both positive and negative).

NC.M3. F-IF.9 Compare key features of two


representations by comparing properties of two

different functions, each with a

different representation (symbolically,

graphically, numerically in tables,

or by verbal descriptions).

NC.M3. F-IF.7 Analyze piecewise, absolute



functions (sine and cosine) using

different representations to show

key features of the graph, by

hand in simple cases and using

technology for more complicated

cases, including: domain and

B. Identify and implement the

relationship among exponential

and logarithmic equations in

addition to graphing and

describing the key features of

both functions with and without

technology. Furthermore, using

properties of both functions to

find and implement the inverse

algebraically and relating to real

world phenomena.

B1. I can identify and describe the effects of transformations of

exponential and logarithmic equations in relation to their parent

functions.

B2. I can graph and describe key features of exponential and

logarithmic functions with and without technology.

B3. I can use different representations to compare key features of

logarithmic and exponential functions.

B4. I can build exponential functions to represent relevant


process.

B5: I can find and implement the inverse relationship between

exponential and logarithmic equations through different

representations.

B6: I can determine if the inverse function exists through different

representations and use it to solve real-world problems.

B7: I can use and identify the key features of logarithmic functions.

B8: I can use logarithms to solve exponential models.

range; intercepts; intervals where

the function is increasing,

decreasing, positive, or negative;

rate of change; relative

maximums and minimums;

symmetries; end behavior; period;


MC.M3.A-SSE.1a Identify and interpret parts of a piecewise, absolute value, polynomial, exponential and rational expressions including


exponents.

NC.M3.F-BF.1a Build polynomial and exponential

functions with real solution(s)

given a graph, a description of a

relationship, or ordered pairs

(include reading these from a

table).

NC.M3.F-BF.4 Find an inverse function.

(Exponential to Log and vice-

versa)

NC.M3.F-LE.4 Use logarithms to express the

solution to abct = d where a, c,

and d are numbers and evaluate the

logarithm using technology.

Resources for Exponential/Logarithmic Unit: Logarithmic & Exponential Problem Solving http://www.doe.virginia.gov/testing/solsearch/sol/math/AFDA/m_ess_afda-1~-2~-3~-4_loga.pdf http://www.carlisleschools.org/webpages/wolfer/files/lab%20modeling%20with%20mms.pdf

Vocabulary: Domain, range, Logarithm, natural logarithm, exponential function, compound interest, base, growth/decay factor, transformation(s), rate, principal, increasing/decreasing intervals, x-intercept, y-intercept, end behavior

Possible Honors Topics: Properties of Logarithms, Use exponents to solve logarithmic equations

https://drive.google.com/open?id=0B-i_U7CppmLhSnRXY3FBUUxIUVE

http://www.doe.virginia.gov/testing/solsearch/sol/math/AFDA/m_ess_afda-1~-2~-3~-4_loga.pdf

http://www.carlisleschools.org/webpages/wolfer/files/lab%20modeling%20with%20mms.pdf

Unit 3: Polynomial Functions


Rationale: Students will begin by continuing their modeling work (connected to the first unit), with expressions or functions that represent familiar

topics such as perimeter and area, and volume. Students have worked with quadratics in Math 1 and 2, so the model they create for area will be

familiar to them. The modeling of volume would introduce a cubic polynomial and present the opportunity to begin exploring polynomials of

higher degree more in depth. The placement of the this unit before Modeling with Geometry is strongly suggested.



NC.M3.G-GMD.3 NC.M3.G-MG.1

Students understand surface area and volume of geometric figures can be modeled by polynomial functions.

NC.M3.N-CN.9 Recognize parts of a polynomial, and apply the Fundamental Theorem of Algebra to determine the types and number

of solutions. NC.M3.A-SSE.1 NC.M3.A-APR.2 NC.M3.A-APR.3 NC.M3.A-CED.1 NC.M3.A-CED.2 NC.M3.F-BF.1a

Understand and apply the Remainder Theorem, the Factor Theorem, and the Division Algorithm. Create polynomial equations in one or two variables and use them to solve problems algebraically and graphically.

NC.M3.F-IF.4 NC.M3.F-IF.7 NC.M3.F-IF.9 NC.M3.F-BF.1a NC.M3.F-BF.1b NC.M3.F-BF.3 NC.M3.F-LE.3

Recognize key features, zeros, and transformations of polynomial functions. Analyze a polynomial function and

compare two or more functions by using their key features. Analyze and compare the relative rates of growth of exponential and polynomial functions.

Unit 3: Polynomial Functions




Rationale: Students will begin this unit by developing an understanding of the relevance for a new function, polynomial, through

modeling. Students will begin by continuing their modeling work with expressions or functions that represent familiar topics such as

perimeter and area, and then with expressions or functions that model volume. Students have worked with quadratics in Math 1 and

2. The modeling of volume would introduce a cubic polynomial and present the opportunity to begin exploring polynomials of higher

degree more in depth. The placement of the this unit after modeling functions and before Modeling with Geometry is strongly

suggested because of these connections.


Major Work


NC.M3.N-CN.9 Use the Fundamental Theorem

of Algebra to determine the

number and potential types of solutions for

polynomial functions.

NC.M3.A-SSE.1a Identify and interpret parts of a

piecewise, absolute value,

polynomial, exponential and

rational expressions including


exponents.

A. Identify parts of a polynomial,

and apply the Fundamental

Theorem of Algebra to determine

the types and number of solutions.

A1. I can identify the constant, linear, quadratic and lead terms, lead

coefficients and degree of a polynomial.

A2. I can identify the number of solutions to a polynomial as exactly

the degree of the polynomial, including multiplicities, Real, and

Complex solutions.

NC.M3.A-APR.3 Understand the relationship

among factors of a polynomial

expression, the solutions of a polynomial

equation and the zeros of a

polynomial function.

NC.M3.A-APR.2 Understand and apply the

Remainder Theorem.

NC.M3.A-APR.6

Rewrite simple rational

expressions in different forms;

write

in the form

,

where and are polynomials with the degree

of less than the degree of

.



absolute value, polynomial, exponential and

rational relationships between

quantities.

NC.M3.A-CED.1 Create equations and

inequalities in one variable that

represent absolute value, polynomial, exponential,

and rational relationships and

use them to solve problems algebraically

and graphically.

B. Understand and apply the

Remainder Theorem, the Factor

Theorem, and the Division

Algorithm. Create polynomial

equations in one or two variables

and use them to solve problems

algebraically and graphically.

B1. I can divide polynomials by using long division, or synthetic

division.

B2. I can understand and apply the remainder theorem.

B3. I can understand and apply the factor theorem.

B4. I can create and solve one and two variable polynomial equations

algebraically and graphically..

NC.M3.F-IF.9 Compare key features of two


representations by comparing

properties of two different

functions, each with a different

representation (symbolically,

graphically, numerically in

tables, or by verbal

descriptions).

NC.M3.F-IF.7 Analyze piecewise, absolute



functions (sine and cosine)

using different representations

to show key features of the

graph, by hand in simple cases

and using technology for more

complicated cases, including:

domain and range; intercepts;

intervals where the function is

increasing, decreasing, positive,

or negative; rate of change;

relative maximums and

minimums; symmetries; end

behavior; period; and

discontinuities.

NC.M3.F-LE.3 Compare the end behavior of

functions using their rates of

change over intervals of the

same length to show that a

quantity increasing exponentially

eventually exceeds a quantity

increasing as a polynomial

function.




function when replacing �(�)

with �∙�(�), �(�) + �, �(� + �) to


of ɑ (both positive and

negative).

C. Analyze a polynomial function

and compare two or more

functions by using their key

features. Given solutions or a

graph write the equation of

polynomial function. Graph

transformations. Compare the

relative rates of growth of

exponential and polynomial

functions.

C1. I can identify the key features of a polynomial function.

C2. I can compare key features of two given functions in a variety of

representations.

C3. I can create a polynomial function given the zeros or a graph.

C4. I can determine whether an exponential or polynomial function

increases more rapidly.

C5. I can use rules of transformations on polynomial functions.

NC.M3.F-BF.1a Build polynomial and exponential functions with real solution(s) given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

NC. M3. F-BF.1b.

Build a new function, in terms of a context, by combining standard function types using arithmetic operations.

Vocabulary: Polynomial, leading coefficient, degree, term, factor, Fundamental Theorem of Algebra, domain, range, x-intercept, y-intercept, increasing/decreasing intervals, relative vs. absolute maximum(s) & minimum(s), end behavior, zero, root, synthetic division, long division, Remainder Theorem, Factor Theorem, complex number, imaginary number, multiplicity

Task: Popcorn task

Unit 4: Modeling with Geometry


Rationale: This unit transitions from polynomial work to geometric concepts that require the use of algebra. It is intentionally placed after the

polynomials unit because the polynomials unit is suggested to begin with geometric modeling that results in a polynomial. Teaching this unit right

after the conclusion of polynomials, allows you to circle back to the geometric modeling concept and study it to its full depth. The placement of

this unit also gives students a break from the heavy algebra work of polynomials prior to beginning rational functions.



NC.M3.G-GPE.1 Derive the equation of a circle as well as distinguishing the center and radius of a circle from an equation.

NC.M3.G-GMD.3 NC.M3.G-GMD.4 NC.M3.G-MG.1 NC.M3.G-CO.14

Implement surface area and volume of geometric figures and model using polynomial functions. Furthermore, relating

cross sections with two-dimensional and three-dimensional figures.

https://drive.google.com/a/jcsnc.org/file/d/0B_rn0GkwtIKeeXFkbG5zVTNQNzA/view?usp=sharing

Unit 4: Modeling with Geometry




Rationale: This unit transitions from polynomial work to geometric concepts that require the use of algebra. It is intentionally placed

after the polynomials unit because the polynomials unit is suggested to begin with geometric modeling that results in a polynomial.

Teaching this unit right after the conclusion of polynomials, allows you to circle back to the geometric modeling concept and study it to

its full depth. The placement of this unit also gives students a break from the heavy algebra work of polynomials prior to beginning

rational functions.

Overarching Standards - NC.M3.A-SSE.1, NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-IF.4, NC.M3.F-BF.1b

Major Work


NC.M3.G-GMD.3 Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems. NC.M3.G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects NC.M3.G-MG.1 Apply geometric concepts in modeling situations:

● Use geometric and algebraic

Implement surface area and

volume of geometric figures and

model using polynomial functions.

Furthermore, relating cross

sections with two-dimensional and

three-dimensional figures

A1: I can use volume formulas to solve problems.

A2: I can identify the two-dimensional shape formed by the cross-

section of a three dimensional figure.

A3. I can identify the three-dimensional shape formed by the rotation of

a two-dimensional figure.

A4: I can use geometric concepts to model and solve real-world

situations.

concepts to solve problems in modeling situations:

● Use geometric shapes, their measures, and their properties, to model real-life objects.

● Use geometric formulas and algebraic functions to model relationships.

● Apply concepts of density based on area and volume.

● Apply geometric concepts to solve design and optimization problems.

NC.M3.G-CO.14 Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems.

Vocabulary: Parallelogram, diagonal, median, altitude, angle bisector, perpendicular bisector, prism, cylinder, pyramid, cone, sphere,

cross section, rotation

● Math Resources:

○ Shodor

● Cross Section Resources:

○ Cross Section Flyer - Shodor

● 3D Rotation Resources:

○ 3D Transmographer - Shodor

http://www.shodor.org/

http://www.shodor.org/interactivate/activities/CrossSectionFlyer/

http://www.shodor.org/interactivate/activities/3DTransmographer/

● Pixar Video for CCSS:G-MG.1 (Really cool! A must watch)

○ “Pixar the Math Behind the Movies”

● Volume Density Video (Really cool!)

○ “Why does ice float in water?”

● Area Density Activity

○ Reroofing Your Uncle’s House

● Task 1: 3D Rotations: This activity has students use Shodor’s 3D Transmographer to graph a 2D figure and rotate it.

Students are then required to find heights, diameters, radius, surface area, and volumes of each created figure.

● Task 2: Pool Party Project: In this project students are required to:

○ Label a 3D Sketch of a pool

○ Find the volume of water in the pool if the water is 6 inches below the top

○ Find the surface area of the pool and use the surface are to figure out how much paint is needed to paint the

interior of the pool

○ Find how much material is needed for the pool cover that extends 2 ft on either side of the top of the pool

and then use the area of the cover to find the density of flies resting on the cover.

○ Find the density of a truck that drove into the pool

● Illustrative Mathematics

● Inside Mathematics

● Khan Academy

● OER Commons

● Mathematics Vision Project - Utah

● JMAP

http://ed.ted.com/lessons/pixar-the-math-behind-the-movies-tony-derose

http://ed.ted.com/lessons/why-does-ice-float-in-water-george-zaidan-and-charles-morton

http://www.pbslearningmedia.org/resource/ate10.sci.engin.systems.reroof/reroofing-your-uncles-house/

https://docs.google.com/document/d/1oQmNHlO7h6xEgjA-k3iEXIH4E18dYlcBqldnzHpxk2U/edit#bookmark=id.z7kl4xko6zhz

https://docs.google.com/document/d/1oQmNHlO7h6xEgjA-k3iEXIH4E18dYlcBqldnzHpxk2U/edit#bookmark=kix.bl6pog8r9k00

https://www.illustrativemathematics.org/content-standards/6

http://www.insidemathematics.org/performance-assessment-tasks

https://www.khanacademy.org/

https://www.oercommons.org/courseware/2550

https://www.mathematicsvisionproject.org/curriculum.html

http://www.jmap.org/

Unit 5: Rational Functions


Rationale: This unit is intended to develop students’ understanding of rational functions. It is suggested to be taught in close proximity to the

polynomials unit because of the connection of rational expressions to the division of polynomials. This unit should begin with reviewing both

simplification of fractions and all arithmetic operations to help students understand the similarities and differences between rational numbers and

expressions.



NC.M3.A-SSE.1 NC.M3.A-APR.6 NC.M3.A-APR.7 NC.M3.A-CED.1 NC.M3.A-CED.2 NC.M3.A-REI.2

Recognize rational expressions as the division of two polynomials and use properties of simple fractions to analyze,

perform arithmetic operations, create and solve equations that model real world phenomena.

NC.M3.F-IF.4 NC.M3.F-IF.7 NC.M3.F-IF.9

Understand and interpret the key features, uses and limitations of multiple representations of a rational function.

Unit 5: Rational Functions




Rationale: This unit is intended to develop students’ understanding of rational functions. It is suggested to be taught in close proximity

to the polynomials unit because of the connection of rational expressions to the division of polynomials. This unit should begin with

reviewing both simplification of fractions and all arithmetic operations to help students understand the similarities and differences

between rational numbers and expressions.


Major Work


NC.M3.A-APR.7 Understand the similarities between arithmetic with rational expressions and arithmetic with rational numbers.

a. Add and subtract two rational expressions, (�) and (�), where the denominators of both (�) and (�) are linear expressions.

b. Multiply and divide two rational expressions.

NC.M3.A-SSE.1a Identify and interpret parts of a piecewise, absolute value, polynomial, exponential and rational expressions including terms, factors, coefficients, and exponents. NC.M3.A-CED.2 Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities.

NC.M3.A-CED.1 Create equations and inequalities in one variable that represent absolute value, polynomial, exponential, and rational relationships and use them to solve problems algebraically and graphically.

A. Recognize rational expressions

as the division of two polynomials

and use properties of simple

fractions to analyze, perform

arithmetic operations, create and

solve equations that model real

world phenomena.

A1. I can rewrite a rational expression in 2 ways by factoring and

simplifying or write as a quotient and remainder.

A2. I can explain how operations on rational expressions are the same as

simple fractions.

A3. I can multiply and divide rational expressions.

A4. I can find LCD in order to add and subtract rational expression when

the denominators are linear.

A5. I can recognize the difference between adding rational expressions

and solving rational expressions.

A6. I can solve a one variable rational equation algebraically or using a

graph.

A7. I can give examples showing how extraneous solutions may arise

when solving rational equation.

A8. I can create and solve a rational equation to solve an application

problem.

A9. I can interpret the terms, factors, and coefficients of rational

expressions.

NC.M3.A-REI.2 Solve and interpret one variable rational equations arising from a context, and explain how extraneous solutions may be produced.

NC.M3.F-IF.9 Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

NC.M3.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities. NC.M3.F-IF.7 Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.

B. Understand and interpret the

key features, uses and limitations

of multiple representations of a

rational function.

B1. I can determine the domain and discontinuities of a rational

expression by finding vertical asymptotes and holes of the function.

B2. I can determine the range of a rational expression by finding the

horizontal asymptotes and holes of a function.

B3. I can find the end behaviors of a rational function by looking at a

graph or table.

B4. I can find x and y intercepts from a graph, while recognizing that they

may not exist.

Vocabulary: Rational function, common denominator, complex fraction, proportion, extraneous solution, domain restriction, domain, range, x-intercept, y-intercept, increasing/decreasing intervals, end behavior, asymptote, point of discontinuity (hole)

Unit 6: Reasoning with Geometry


Rationale: This unit transitions into geometric concepts with an emphasis on reasoning, justification, and formalizing proof. Students will extend

upon their work with proof in Math 2 (NC.M2.G.CO.9 and NC.M2.G.CO.10) focusing on both paragraph and flow proofs. Students are familiar

with the properties of parallelograms from middle school and have categorized parallelograms and informally verified parallelogram properties

through coordinate geometry in Math 1. Students will prove more theorems about triangles including the centers of triangles. This concept can be

used as a transition into reasoning with circles. The Reasoning with Geometry Unit purposefully concludes with circles. In students’ work with

circles, they will develop their understanding of radian measure through proportions in circles. This sets up a connection of circular motion to

trigonometric functions in the next unit.



NC.M3.G-CO.11

NC.M3.G-CO.14 NC.M3.G-CO.10

Construct logical arguments and explain reasoning with two-dimensional figures to prove geometric theorems about

parallelograms and solve problems. Demonstrate an understanding of the properties of three of a triangle’s points of

concurrency.

NC.M3.G-GPE.1 NC.M3.G-C.2 NC.M3.G-C.5 NC.M3.G-CO.14 NC.M3.G-MG.1

Understand properties of circles and how to apply them algebraically and geometrically. Demonstrate understanding that

within circles, segments, lines, and angles create special relationships and use these to solve geometric problems.



Unit 6A: REASONING WITH GEOMETRY


Suggested Time : 9 days on a semester schedule (90-minute classes)

Rationale: This unit transitions into more geometric concepts with an emphasis on reasoning, justification, and formalizing proof. In

Math III, students formalize proof, focusing on both paragraph and flow proofs. Emphasis should be placed on reasoning and

justification. Students are familiar with the properties of parallelograms from middle school and have categorized parallelograms and

informally verified parallelogram properties through coordinate geometry in Math I. We strongly suggest ending the unit with centers of

triangles to transition into the next unit - Circles. Centers of Triangles allows the concept of circles to emerge and transitions into

further work with circles in the next unit.


Major Work


NC.M3.G-CO.14 Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems.

NC.M3.G-CO.11 Prove theorems about parallelograms.

● Opposite sides of a parallelogram are congruent.

● Opposite angles of a parallelogram are congruent.

● Diagonals of a parallelogram bisect each other.

● If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

A. Prove theorems (paragraph

and flow) and solve problems

using parallelograms

A1. I can prove a figure is a parallelogram iff opposite sides are congruent.

A2. I can prove a figure is a parallelogram iff opposite angles are

congruent.

A3. I can prove a figure is a parallelogram iff diagonals bisect each other.

A4. I can prove a figure is a rectangle iff the diagonals of a parallelogram

are congruent

A5. I can use the theorems of parallelograms to solve problems.

NC.M3.G-CO.10 Verify experimentally properties of the centers of triangles (centroid, incenter, and circumcenter).

B. Use the concept of centroid,

incenter and circumcenter of a

triangle as it pertains to centers of

appropriate circles.

B1. I can determine and draw the centroid, incenter and circumcenter of a

triangle using appropriate centers of circles.

Vocabulary: Parallelogram, diagonal, bisector, radius, diameter, center (of a circle), centroid, incenter, circumcenter, central angle, inscribed angle, circumscribed angle.



Unit 6B: CIRCLES



Rationale: This unit, suggested to follow the reasoning with geometry because of the connection between circles and centers of

triangles, should precede the trigonometry unit. This is suggested because the concept of radian measure should first be developed

through proportions in circles and trigonometric functions can be connected to circular motion. In Math III, students formalize proof,

focusing on both paragraph and flow proofs. Relationships among line segments, angles, and circles should be developed

conceptually through constructions, reasoning, and justification.


Major Work


NC.M3.G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

A. Write the equation of a circle

given radius, and center or a

graph of a circle in standard and

general form. Determine radius

and center of a circle in general

form by completing the square.

A1. I can write the equation of a circle given center and radius in standard

and general form.

A2. I can determine the radius and center of a circle given an equation in

standard or general form , or the graph itself.

NC.M3.G-CO.14 Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems. NC.M3.G-C.2 Understand and apply theorems about circles.

● Understand and apply theorems about relationships with angles and circles, including central, inscribed and circumscribed angles.

● Understand and apply theorems about relationships with line segments and circles including, radii, diameters, secants, tangents and chords

NC.M3.G-C.5 Using similarity, demonstrate that the length of an arc, , for a given central angle is proportional to the radius, , of the circle. Define radian measure of the central angle as the ratio of the length of the arc to the radius of the

circle,

. Find arc lengths

and areas of sectors of circles.

B. Quantify the relationships

between angles and arcs of a

circle, whether those angles are

inscribed, circumscribed, or

central angles. Use theorems on

right angles inscribed in circles,

tangent lines, secants and chords

in circles to solve problems and

prove theorems. Use similarity to

verify arc-length formula. Use arc-

length and area formulas to solve

problems generally and in context.

B1. I can use theorems on angles and arcs of circles to solve problems

and prove theorems.

B2. I can use theorems on inscribed right angles, tangent lines, chords

and secant lines in circles to solve problems and prove theorems.

B3. I can determine the arc-length and area of a sector.

B4. I can use the similarity of circles to measure an angle by the ratio of

the radius and the arc length.

NC.M3.G-MG.1 Apply geometric concepts in modeling situations:

● Use geometric and algebraic concepts to solve problems in modeling situations:

● Use geometric shapes, their measures, and their properties, to model real-life objects.

● Use geometric formulas and algebraic functions to model relationships

. ● Apply

concepts of density based on area and volume.

Apply geometric concepts to

solve design and

optimization problems.

Vocabulary: tangent line, chord, secant line, radian (angle measure), inscribed, circumscribed, arc length, sector.

Unit 7: Trigonometric Functions


Rationale: This unit should immediately follow the Reasoning with Geometry unit. Students’ understanding of radians and the idea of circular

motion are connections that can help students better understand trigonometric functions.



NC.M3.F-IF.1 NC.M3.F-IF.4 NC.M3.F-IF.7 NC.M3.F-IF.9

Understand that triangular trigonometric functions are related to circular trigonometric functions in the coordinate

plane.

NC.M3.F-BF.3 NC.M3.F-TF.1 NC.M3.F-TF.2 NC.M3.F-TF.5

Understand and interpret the key features, uses and limitations of multiple representations of trigonometric functions

that model real world periodic behavior.

Unit 7: Trigonometric Functions




Rationale: This unit should immediately follow the unit Circles. Students’ understanding of radians and the idea of circular motion are

connections that can help students better understand trigonometric functions.


Major Work


NC.M3.F-TF.1 Understand radian measure of an angle as:

● The ratio of the length of an arc on a circle subtended by the angle to its radius.

● A dimensionless measure of length defined by the quotient of arc length and radius that is a real number.

● The domain for trigonometric functions.

NC.M3.F-IF.1 Extend the concept of a function by recognizing that trigonometric ratios are functions of angle measure. NC.M3.F-TF.2 Build an understanding of trigonometric functions by using tables, graphs and technology to represent the cosine and sine functions.

a. Interpret the sine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate.

b. Interpret the cosine function as the relationship

A. Understand that (right)

triangular trigonometric

functions are related to

circular trigonometric

functions in the coordinate

plane. Develop the sine

graph using the unit circle.

A1. I can define a radian measure of an angle as the length of the arc on the unit

circle subtended by the angle.

A2. I can explain how a ratio represents a value of a trig function for an angle.

A3. I can work with angles in standard position to find coterminal and reference

angles.

A4. I can sketch a sine graph from the values of the unit circle.

between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its x coordinate.

NC.M3.F-IF.9 Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

NC.M3.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities. NC.M3.F-IF.7 Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.

B. Understand and interpret

the key features, uses and

limitations of multiple

representations of

trigonometric functions that

model real world periodic

behavior.

B1. I can use key features to construct the graph of sine function.

B2. I can state the amplitude, period, and midline of the sine function.

B3. I can use technology, graphs, and tables to compare sine graphs.

B4. I can describe the effect of a transformation on the graph of the sine function.

B6. I can use technology to interpret the key features of the sine graph in a real

world situation.

NC.M3.F-TF.5 Use technology to investigate the parameters, �, �, and ℎ of a sine function, (�) = �∙ sin (�∙�) + ℎ , to represent periodic phenomena and interpret key features in terms of a context. NC.M3.F-BF.3 Extend an understanding of

the effects on the graphical

and tabular representations

of a function when

replacing �(�) with �∙�(�),

�(�) + �, �(� + �) to include

�(�∙�) for specific values of

ɑ (both positive and

negative).

Vocabulary: Initial side of an angle, terminal side of an angle, ray, coterminal angles, sine, cosine, tangent, radian (angle measure), unit circle, domain, range, period, midline, amplitude, frequency, cycle, phase shift.

Unit 8: Statistics


Rationale: This unit, Statistics, is more flexible in the pacing, than the other suggested units. Statistics can be taught as a stand-alone unit, since

there is less integration and connection between standards. However, it is suggested that you do not break up the coherency of units that have

intentionally been suggested to be taught in a certain order. (i.e - do not teach this unit between Reasoning with Geometry and Trigonometric

Functions)


NC.M3.S-IC.1 NC.M3.S-IC.3

Understand statistics as a process of making inferences about a population (parameter) based on results from a

random sample (statistic). Acknowledge the role of randomization in using sample surveys, experiments, and observational studies to collect

data and understand the limitations of generalizing results to populations (related to randomization).

Unit 8: Statistics




Rationale: This unit, Statistics, is more flexible in the pacing, than the other suggested units. Statistics can be taught as a stand-alone

unit, since there is less integration and connection between standards. However, it is suggested that you do not break up the

coherency of units that have intentionally been suggested to be taught in a certain order. (i.e - do not teach this unit between Circles

and Trigonometric Functions)


Major Work


Priority Standard

NC.M3.S-IC.1 Understand the process of making inferences about a population based on a random sample from that population.

Supporting Standard

NC.M3.S-IC.3 Recognize the

A. Understand statistics as a

process of making inferences

about a population (parameter)

based on results from a random

sample (statistic).

Acknowledge the role of

randomization in using sample

A1. I can distinguish between a sample (statistic) and a population

(parameter).

A2. I can describe how to select a random sample from a given

population.

A3. I can explain the purposes and the differences of sample surveys,

observational studies, and experiments, including how randomization

applies to each.

A4. I can distinguish between sample surveys, observational studies, and

experiments.

NC.M3.S-IC.4 NC.M3.S-IC.5

Understand simulation is useful for using data to make decisions. Understand that samples can differ by chance.

NC.M3.S-IC.6 Understand not all data that is reported is valid. Reports should be evaluated based on source, design of the study, and

data displays.

purposes of and differences between sample surveys, experiments, and observational studies and understand how randomization should be used in each.

surveys, experiments, and

observational studies to collect

data and understand the

limitations of generalizing results

to populations (related to

randomization).

A5. I can determine how results of a statistical study can be generalized to

make conclusions about a population based on the sample.

Priority Standard

NC.M3.S-IC.4 Use simulation to understand how samples can be used to estimate a population mean or proportion and how to determine a margin of error for the estimate. Supporting Standard

NC.M3.S-IC.5 Use simulation to determine whether observed differences between samples from two distinct populations indicate that the two populations are actually different in terms of a parameter of interest.

B. Understand simulation is useful

for using data to make decisions.

Know how to carry out a

simulation with data for the

purposes of: estimating

population means or proportions,

determining the margin of error for

those estimates, and determining

statistical significance.

Understand that samples can

differ by chance.

B1. I can use data from a sample survey to estimate a population mean or

proportion with a margin of error.

B2. I can determine and justify if results from an experiment are

statistically significant.

● I can identify the parameter of interest in an experiment.

● I can select and calculate sample statistics.

● I can calculate the difference between the sample statistics.

● I can set up and complete a simulation re-randomizing the groups.

● I can compare the actual difference to the simulated differences to

determine statistical significance.

B3. I can state a conclusion about the effectiveness or accuracy of a claim

based on a sample.

Priority Standard

NC.MC.S-IC.6 Evaluate articles and websites that report data by identifying the source of the data, the design of the study, and the way the data are graphically displayed.

C. Understand not all data that is

reported is valid. Reports should

be evaluated based on source,

design of the study, and data

displays.

C1. I can evaluate and make sense of a statistical article or website.

Vocabulary: Sample, population, margin of error, standard deviation, simple random sample, systematic sample, stratified random sample, cluster

sample, self-selected sample, convenience sample, lurking variable, mean, standard deviation, survey, experiment, observation, simulation

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