scope and sequence - core-plusmath.org · scope and sequence algebra and functions geometry and...

Scope and Sequence Algebra and Functions Geometry and Trigonometry Statistics and Probability Discrete Mathematics

ISBN: 978-0-07-892135-3MHID: 0-07-892135-X

www.glencoe.com

Visit us at www.glencoe.com

Christian R. Hirsch • James T. Fey • Eric W. Hart Harold L. Schoen • Ann E. Watkins

with

Beth E. Ritsema • Rebecca K. Walker • Sabrina Keller Robin Marcus • Arthur F. Coxford • Gail Burrill

Scope and Sequence

ii CPMP • Scope and Sequence

Copyright © 2009 by the McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher.

This material is based upon work supported, in part, by the National Science Foundation under grant no. ESI 0137718. Opinions expressed are those of the authors and not necessarily those of the Foundation.

Send all inquires to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 978-0-07-892135-3 Core-Plus MathematicsMHID: 0-07-892135-X Contemporary Mathematics in Context

Scope and Sequence

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CPMP • Scope & Sequence iii

About the Core-Plus Mathematics Project

The Core-Plus Mathematics Project (CPMP) was funded by the National Science Foundation to

develop student and teacher materials for a comprehensive Standards-based high school mathematics

curriculum. Courses 1–3 comprise a core program appropriate for all students. Course 4 continues the

preparation of students for college mathematics and statistics.

Senior Curriculum Developers

Christian R. Hirsch (Director)Western Michigan University

James T. FeyUniversity of Maryland

Eric W. HartMaharishi University of Management

Harold L. SchoenUniversity of Iowa

Ann E. WatkinsCalifornia State University,

Northridge

Contributing Curriculum

Developers

Beth E. RitsemaWestern Michigan University

Rebecca K. WalkerGrand Valley State University

Sabrina KellerMichigan State University

Robin MarcusUniversity of Maryland

Arthur F. Coxford (deceased)University of Michigan

Gail BurrillMichigan State University

(First edition only)

Principal Evaluator

Steven W. ZiebarthWestern Michigan University

Advisory Board

Diane BriarsPittsburgh Public Schools

Jeremy KilpatrickUniversity of Georgia

Robert E. MegginsonUniversity of Michigan

Kenneth RuthvenUniversity of Cambridge

David A. SmithDuke University

Mathematical Consultants

Deborah Hughes-HallettUniversity of Arizona /

Harvard University

Stephen B. MaurerSwarthmore College

William McCallumUniversity of Arizona

Doris SchattschneiderMoravian College

Richard ScheafferUniversity of Florida

Evaluation Consultant

Norman L. WebbUniversity of Wisconsin-Madison

Collaborating Teachers

Mary Jo MessengerHoward Country Public Schools,

Maryland

Jacqueline StewartOkemos, Michigan

Technical Coordinator

James LaserWestern Michigan University

Production and Support Staff

Angie Reiter

Matt Tuley

Teri ZiebarthWestern Michigan University

Graduate Assistants

Allison BrckaLorenz

Christopher HlasUniversity of Iowa

Madeline Ahearn

Geoffrey Birky

Kyle Cochran

Michael Conklin

Brandon Cunningham

Tim Fukawa-Connelly

Ming TomaykoUniversity of Maryland

Karen Fonkert

Dana Grosser

Anna Kruizenga

Nicole Lanie

Diane MooreWestern Michigan University

Undergraduate Assistants

Cassie DurginUniversity of Maryland

Rachael Kaluzny

Megan O’Connor

Jessica Tucker

Ashley WiersmaWestern Michigan University

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iv CPMP • Scope and Sequence

Core-Plus Mathematics 2Field-Test Sites

Core-Plus Mathematics 2 builds on the strengths of the 1st edition which was shaped by multi-year fi eld

tests in 36 high schools in Alaska, California, Colorado, Georgia, Idaho, Iowa, Kentucky, Michigan,

Ohio, South Carolina, and Texas. Each revised text is the product of a three-year cycle of research and

development, pilot testing and refi nement, and fi eld testing and further refi nement. Special thanks are

extended to the following teachers and their students who participated in the testing and evaluation of

the 2nd Edition materials.

Hickman High SchoolColumbia, Missouri

Peter Doll

Melissa Hundley

Stephanie Krawczyk

Cheryl Lightner

Amy McCann

Tiffany McCracken

Ryan Pingrey

Michael Westcott

Holland Christian High SchoolHolland, Michigan

Jeff Goorhouse

Tim Laverell

Brian Lemmen

Betsi Roelofs

John Timmer

Mike Verkaik

Jefferson Junior High SchoolColumbia, Missouri

Marla Clowe

Lori Kilfoil

Martha McCabe

Paul Rahmoeller

Evan Schilling

Malcolm Price Lab SchoolCedar Falls, Iowa

Megan Balong

Dennis Kettner

James Maltas

North Shore Middle SchoolHolland, Michigan

Sheila Schippers

Brenda Katerberg

Oakland Junior High SchoolColumbia, Missouri

Teresa Barry

Erin Little

Christine Sedgwick

Dana Sleeth

Riverside University High SchoolMilwaukee, Wisconsin

Cheryl Brenner

Dave Cusma

Alice Lanphier

Ela Kiblawi

Ulices Sepulveda

Rock Bridge High SchoolColumbia, Missouri

Nancy Hanson

Emily Hawn

Lisa Holt

Betsy Launder

Linda Shumate

Sauk Prairie High SchoolPrairie du Sac, Wisconsin

Joel Amidon

Shane Been

Kent Jensen

Joan Quenan

Scott Schutt

Dan Tess

Mary Walz

Sauk Prairie Middle SchoolSauk City, Wisconsin

Julie Dahlman

Janine Jorgensen

South Shore Middle SchoolHolland, Michigan

Lynn Schipper

Washington High SchoolMilwaukee, Wisconsin

Anthony Amoroso

Debbie French

West Junior High SchoolColumbia, Missouri

Katie Bihr

Josephus Johnson

Rachel Lowery

Mike Rowson

Amanda Schoenfeld

Patrick Troup

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CPMP • Scope and Sequence 1

A Balanced and Unifi ed Curriculum

Core-Plus Mathematics is a four-year unifi ed curriculum that replaces the Algebra-Geometry-Advanced

Algebra/Trigonometry-Precalculus sequence. Each course features interwoven strands of algebra

and functions, geometry and trigonometry, statistics and probability, and discrete mathematics.

Each of these strands is developed within coherent focused units connected by fundamental ideas

such as symmetry, functions, matrices, and data analysis and curve-fi tting. By actively investigating

mathematics and its applications every year from an increasingly more mathematically sophisticated

point of view, students’ understanding of the mathematics in each strand deepens across the four-

year curriculum. Mathematical connections between strands and ways of thinking mathematically

that are common across strands are emphasized. These mathematical habits of mind include visual

thinking, recursive thinking, searching for and explaining patterns, making and checking conjectures,

reasoning with multiple representations, and providing convincing arguments and proofs.

Algebra and Functions

The Algebra and Functions strand develops student ability to recognize,

represent, and solve problems involving relations among quantitative

variables. Central to the development is the use of functions as

mathematical models. The key algebraic models in the curriculum

are linear, exponential, power, polynomial, logarithmic, rational, and

trigonometric functions. Modeling with systems of equations, both

linear and nonlinear, is developed. Attention is also given to symbolic

reasoning and manipulation.

Geometry and Trigonometry

The primary goal of the Geometry and Trigonometry strand is to

develop visual thinking and ability to construct, reason with, interpret,

and apply mathematical models of patterns in visual and physical

contexts. The focus is on describing patterns in shape, size, and

location; representing patterns with drawings, coordinates, or vectors;

predicting changes and invariants in shapes under transformations;

and organizing geometric facts and relationships through deductive

reasoning.

Statistics and Probability

The primary role of the Statistics and Probability strand is to develop

student ability to analyze data intelligently, to recognize and measure

variation, and to understand the patterns that underlie probabilistic

situations. The ultimate goal is for students to understand how

inferences can be made about a population by looking at a random

sample from that population. Graphical methods of data analysis,

simulations, sampling, and experience with the collection and

interpretation of real data are featured.

Discrete Mathematics

The Discrete Mathematics strand develops student ability to solve problems

using vertex-edge graphs, recursion, matrices, systematic counting methods

(combinatorics), and mathematical methods for democratic decision

making and information processing. Key themes are discrete mathematical

modeling, optimization, and algorithmic problem solving.

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2 CPMP • Scope and Sequence

Organization of the Curriculum

The fi rst three courses in the Core-Plus Mathematics series provide a signifi cant core of broadly

useful mathematics for all students. They were developed to prepare students for success in

college, in careers, and in daily life in contemporary society. Course 4: Preparation for Calculus

formalizes and extends the core program, with a focus on the mathematics needed to be successful

in undergraduate programs requiring calculus. Unit titles for the four courses are given in the

following table. Focus and content of these units are described on pages 3–7.

Course 1

1 Patterns of Change

2 Patterns in Data

3 Linear Functions

4 Vertex-Edge Graphs

5 Exponential Functions

6 Patterns in Shape

7 Quadratic Functions

8 Patterns in Chance

Course 2

1 Functions, Equations, and Systems

2 Matrix Methods

3 Coordinate Methods

4 Regression and Correlation

5 Nonlinear Functions and Equations

6 Network Optimization

7 Trigonometric Methods

8 Probability Distributions

Course 3

1 Reasoning and Proof

2 Inequalities and Linear Programming

3 Similarity and Congruence

4 Samples and Variation

5 Polynomial and Rational Functions

6 Circles and Circular Functions

7 Recursion and Iteration

8 Inverse Functions

Course 4: Preparation for Calculus

1 Families of Functions

2 Vectors and Motion

3 Algebraic Functions and Equations

4 Trigonometric Functions and Equations

5 Exponential Functions, Logarithms, and

Data Modeling

6 Surfaces and Cross Sections

7 Concepts of Calculus

8 Counting Methods and Induction

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Core-Plus Mathematics 2nd Edition

Course 1 Units

Unit 1 Patterns of Change develops student ability to recognize and describe important patterns that relate

quantitative variables, to use data tables, graphs, words, and symbols to represent the relationships, and

to use reasoning and calculating tools to answer questions and solve problems.

Topics include variables and functions, algebraic expressions and recurrence relations, coordinate graphs,

data tables and spreadsheets, and equations and inequalities.

Unit 2 Patterns in Data develops student ability to make sense of real-world data through use of graphical

displays, measures of center, and measures of variability.

Topics include distributions of data and their shapes, as displayed in dot plots, histograms, and box plots;

measures of center including mean and median, and their properties; measures of variability including

interquartile range and standard deviation, and their properties; and percentiles and outliers.

Unit 3 Linear Functions develops student ability to recognize and represent linear relationships between variables

and to use tables, graphs, and algebraic expressions for linear functions to solve problems in situations that

involve constant rate of change or slope.

Topics include linear functions, slope of a line, rate of change, modeling linear data patterns, solving linear

equations and inequalities, equivalent linear expressions.

Unit 4 Vertex-Edge Graphs develops student understanding of vertex-edge graphs and ability to use these graphs

to represent and solve problems involving paths, networks, and relationships among a fi nite number of

elements, including fi nding effi cient routes and avoiding confl icts.

Topics include vertex-edge graphs, mathematical modeling, optimization, algorithmic problem solving,

Euler circuits and paths, matrix representation of graphs, vertex coloring and chromatic number.

Unit 5 Exponential Functions develops student ability to recognize and represent exponential growth and decay

patterns, to express those patterns in symbolic forms, to solve problems that involve exponential change,

and to use properties of exponents to write expressions in equivalent forms.

Topics include exponential growth and decay functions, data modeling, growth and decay rates, half-life

and doubling time, compound interest, and properties of exponents.

Unit 6 Patterns in Shape develops student ability to visualize and describe two- and three-dimensional shapes,

to represent them with drawings, to examine shape properties through both experimentation and careful

reasoning, and to use those properties to solve problems.

Topics include Triangle Inequality, congruence conditions for triangles, special quadrilaterals and

quadrilateral linkages, Pythagorean Theorem, properties of polygons, tilings of the plane, properties of

polyhedra, and the Platonic solids.

Unit 7 Quadratic Functions develops student ability to recognize and represent quadratic relations between

variables using data tables, graphs, and symbolic formulas, to solve problems involving quadratic functions,

and to express quadratic polynomials in equivalent factored and expanded forms.

Topics include quadratic functions and their graphs, applications to projectile motion and economic

problems, expanding and factoring quadratic expressions, and solving quadratic equations by the quadratic

formula and calculator approximation.

Unit 8 Patterns in Chance develops student ability to solve problems involving chance by constructing sample

spaces of equally-likely outcomes or geometric models and to approximate solutions to more complex

probability problems by using simulation.

Topics include sample spaces, equally-likely outcomes, probability distributions, mutually exclusive

(disjoint) events, Addition Rule, simulation, random digits, discrete and continuous random variables,

Law of Large Numbers, and geometric probability.



Course 2 Units

Unit 1 Functions, Equations, and Systems reviews and extends student ability to recognize, describe, and use

functional relationships among quantitative variables, with special emphasis on relationships that involve

two or more independent variables.

Topics include direct and inverse variation and joint variation; power functions; linear equations in

standard form; and systems of two linear equations with two variables, including solution by graphing,

substitution, and elimination.

Unit 2 Matrix Methods develops student understanding of matrices and ability to use matrices to represent and

solve problems in a variety of real-world and mathematical settings.

Topics include constructing and interpreting matrices, row and column sums, matrix addition, scalar

multiplication, matrix multiplication, powers of matrices, inverse matrices, properties of matrices, and

using matrices to solve systems of linear equations.

Unit 3 Coordinate Methods develops student understanding of coordinate methods for representing and analyzing

properties of geometric shapes, for describing geometric change, and for producing animations.

Topics include representing two-dimensional fi gures and modeling situations with coordinates, including

computer-generated graphics; distance in the coordinate plane, midpoint of a segment, and slope;

coordinate and matrix models of rigid transformations (translations, rotations, and line refl ections), of size

transformations, and of similarity transformations; animation effects.

Unit 4 Regression and Correlation develops student understanding of the characteristics and interpretation of

the least squares regression equation and of the use of correlation to measure the strength of the linear

association between two variables.

Topics include interpreting scatterplots; least squares regression, residuals and errors in prediction, sum

of squared errors, infl uential points; Pearson’s correlation coeffi cient and its properties, lurking variables,

and cause and effect.

Unit 5 Nonlinear Functions and Equations introduces function notation, reviews and extends student ability

to construct and reason with functions that model parabolic shapes and other quadratic relationships in

science and economics, with special emphasis on formal symbolic reasoning methods, and introduces

common logarithms and algebraic methods for solving exponential equations.

Topics include formalization of function concept, notation, domain and range; factoring and expanding

quadratic expressions, solving quadratic equations by factoring and the quadratic formula, applications

to supply and demand, break-even analysis; common logarithms and solving exponential equations using

base 10 logarithms.

Unit 6 Network Optimization develops student understanding of vertex-edge graphs and ability to use these

graphs to solve network optimization problems.

Topics include optimization, mathematical modeling, algorithmic problem solving, digraphs, trees,

minimum spanning trees, distance matrices, Hamilton circuits and paths, the Traveling Salesperson

Problem, critical paths, and the PERT technique.

Unit 7 Trigonometric Methods develops student understanding of trigonometric functions and the ability to use

trigonometric methods to solve triangulation and indirect measurement problems.

Topics include sine, cosine, and tangent functions of measures of angles in standard position in a coordinate

plane and in a right triangle; indirect measurement; analysis of variable-sided triangle mechanisms; Law

of Sines and Law of Cosines.

Unit 8 Probability Distributions further develops student ability to understand and visualize situations involving

chance by using simulation and mathematical analysis to construct probability distributions.

Topics include Multiplication Rule, independent and dependent events, conditional probability, probability

distributions and their graphs, waiting-time (or geometric) distributions, expected value, and rare events.



Course 3 Units

Unit 1 Reasoning and Proof develops student understanding of formal reasoning in geometric, algebraic, and

statistical contexts and of basic principles that underlie those reasoning strategies.

Topics include inductive and deductive reasoning strategies; principles of logical reasoning—Affi rming

the Hypothesis and Chaining Implications; relation among angles formed by two intersecting lines or by

two parallel lines and a transversal; rules for transforming algebraic expressions and equations; design

of experiments including the role of randomization, control groups, and blinding; sampling distribution,

randomization test, and statistical signifi cance.

Unit 2 Inequalities and Linear Programming develops student ability to reason both algebraically and graphically

to solve inequalities in one and two variables, introduces systems of inequalities in two variables, and

develops a strategy for optimizing a linear function in two variables within a system of linear constraints

on those variables.

Topics include inequalities in one and two variables (including absolute value and quadratic inequalities),

number line graphs, interval notation, systems of linear inequalities, and linear programming.

Unit 3 Similarity and Congruence extends student understanding of similarity and congruence and their ability

to use those relations to solve problems and to prove geometric assertions with and without the use of

coordinates.

Topics include connections between Law of Cosines, Law of Sines, and suffi cient conditions for similarity

and congruence of triangles, centers of triangles, applications of similarity and congruence in real-world

contexts, necessary and suffi cient conditions for parallelograms, suffi cient conditions for congruence of

parallelograms, and midpoint connector theorems.

Unit 4 Samples and Variation extends student understanding of the measurement of variability, develops

student ability to use the normal distribution as a model of variation, introduces students to the binomial

distribution and its use in decision making, and introduces students to the probability and statistical

inference involved in control charts used in industry for statistical process control.

Topics include normal distribution, standardized scores, binomial distributions (shape, expected value,

standard deviation), normal approximation to a binomial distribution, odds, statistical process control,

control charts, and the Central Limit Theorem.

Unit 5 Polynomial and Rational Functions extends student ability to represent and draw inferences about

polynomial and rational functions using symbolic expressions and manipulations.

Topics include defi nition and properties of polynomials, operations on polynomials; completing the

square, proof of the quadratic formula, solving quadratic equations (including complex number solutions),

vertex form of quadratic functions; defi nition and properties of rational functions, operations on rational

expressions.

Unit 6 Circles and Circular Functions develops student understanding of relationships among special lines,

segments, and angles in circles and the ability to use properties of circles to solve problems; develops

student understanding of circular functions and the ability to use these functions to model periodic change;

and extends student ability to reason deductively in geometric settings.

Topics include properties of chords, tangent lines, and central and inscribed angles of circles; linear and

angular velocity; radian measure of angles; and circular functions as models of periodic change.

Unit 7 Recursion and Iteration extends student ability to represent, analyze, and solve problems in situations

involving sequential and recursive change.

Topics include iteration and recursion as tools to model and analyze sequential change in real-world

contexts, including compound interest and population growth; arithmetic, geometric, and other sequences;

arithmetic and geometric series; fi nite differences; linear and nonlinear recurrence relations; and function

iteration, including graphical iteration and fi xed points.

Unit 8 Inverse Functions develops student understanding of inverses of functions with a focus on logarithmic

functions and their use in modeling and analyzing problem situations and data patterns.

Topics include inverses of functions; logarithmic functions and their relation to exponential functions,

properties of logarithms, equation solving with logarithms; and inverse trigonometric functions and their

applications to solving trigonometric equations.



Course 4 Units: Preparation for Calculus

Unit 1 Families of Functions extends student understanding of linear, exponential, quadratic, power, and

trigonometric functions to model data patterns whose graphs are transformations of basic patterns; and

develops understanding of operations on functions useful in representing and reasoning about quantitative

relationships.

Topics include linear, exponential, quadratic, power, and trigonometric functions; data modeling;

translation, refl ection, and stretching of graphs; and addition, subtraction, multiplication, division, and

composition of functions.

Unit 2 Vectors and Motion develops student understanding of two-dimensional vectors and their use in modeling

linear, circular, and other nonlinear motion.

Topics include concept of vector as a mathematical object used to model situations defi ned by magnitude

and direction; equality of vectors, scalar multiples, opposite vectors, sum and difference vectors, dot

product of two vectors, position vectors and coordinates; and parametric equations for motion along a line

and for motion of projectiles and objects in circular and elliptical orbits.

Unit 3 Algebraic Functions and Equations reviews and extends student understanding of properties of polynomial

and rational functions and skills in manipulating algebraic expressions and solving polynomial and rational

equations, and develops student understanding of complex number representations and operations.

Topics include polynomials, polynomial division, factor and remainder theorems, operations on complex

numbers, representation of complex numbers as vectors, solution of polynomial equations, rational function

graphs and asymptotes, and solution of rational equations and equations involving radical expressions.

Unit 4 Trigonometric Functions and Equations extends student understanding of, and ability to reason with,

trigonometric functions to prove or disprove two trigonometric expressions are identical and to solve

trigonometric equations; to geometrically represent complex numbers and complex number operations

and to fi nd roots of complex numbers.

Topics include the tangent, cotangent, secant, and cosecant functions; fundamental trigonometric identities,

sum and difference identities, double-angle identities; solving trigonometric equations and expression of

periodic solutions; rectangular and polar representations of complex numbers, absolute value, DeMoivre’s

Theorem, and the roots of complex numbers.

Unit 5 Exponential Functions, Logarithms, and Data Modeling extends student understanding of exponential

and logarithmic functions to the case of natural exponential and logarithmic functions, solution of exponential

growth and decay problems, and use of logarithms for linearization and modeling of data patterns.

Topics include exponential functions with rules in the form f(x) = Aekx, natural logarithm function,

linearizing bivariate data and fi tting models using log and log-log transformations.

Unit 6 Surfaces and Cross Sections extends student ability to visualize and represent three-dimensional shapes

using contours, cross sections, and reliefs, and to visualize and represent surfaces and conic sections

defi ned by algebraic equations.

Topics include using contours to represent three-dimensional surfaces and developing contour maps from

data; sketching surfaces from sets of cross sections; conics as planar sections of right circular cones and

as locus of points in a plane; three-dimensional rectangular coordinate system; sketching surfaces using

traces, intercepts and cross sections derived from algebraically-defi ned surfaces; and surfaces of revolution

and cylindrical surfaces.

Unit 7 Concepts of Calculus develops student understanding of fundamental calculus ideas through explorations

in a variety of applied problem contexts and their representations in function tables and graphs.

Topics include instantaneous rates of change, linear approximation, area under a curve, and applications

to problems in physics, business, and other disciplines.



Unit 8 Counting Methods and Induction extends student ability to count systematically and solve enumeration

problems, and develops understanding of, and ability to do, proof by mathematical induction.

Topics include systematic listing and counting, counting trees, the Multiplication Principle of Counting,

Addition Principle of Counting, combinations, permutations, selections with repetition; the binomial

theorem, Pascal’s triangle, combinatorial reasoning; the general multiplication rule for probability; the

Principle of Mathematical Induction; and the Least Number Principle.



Mathematics Content

Elaborated

The mathematical content of the four courses of Core-Plus Mathematics

are elaborated in the remainder of this booklet. Pages 9–21 provide strand

charts and pages 22–26 provide an alignment with the Common Core State

Standards Mathematical Practices.

Strand Charts The following charts provide an overview of the mathematical content and

fl ow of Courses 1–4 in the Core-Plus Mathematics curriculum. The charts

are organized by mathematical strand: algebra and functions, geometry

and trigonometry, statistics and probability, and discrete mathematics. Each

of the four strands has been divided into major content categories, and

under each of these categories you will fi nd the key mathematical topics

developed in the curriculum.

Many cells in the grid have either a “F” or a “C” to indicate the units in

which each topic is treated. The “F” indicates focus; this means that the

topic is initially developed or is extended beyond its initial development

or use. The “C” indicates connections, which means that a conceptual

basis for the topic is developed, the topic is informally introduced, or the

topic is revisited and used without further development.

To help build and maintain profi ciency with key topics in the charts,

Review tasks in each lesson of each unit provide students distributed

practice with related concepts and skills. These practice opportunities

are not referenced in the following strand charts.



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Rat

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Mult

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Alg

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Two-d

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GEO

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nsCours

e 1

Cours

e 2

Cours

e 3

Cours

e 4

12

34

56

78

12

34

56

78

12

34

56

78

12

34

56

78

Pro

bab

ility

Nor

mal

dis

trib

utio

nF

FSi

mul

atio

nF

CC

CRan

dom

dig

it ta

bles

and

ran

dom

num

ber

gene

rato

rsF

CLa

w o

f La

rge

Num

bers

FEm

pirica

l (ex

perim

enta

l) an

d th

eore

tical

pro

babi

litie

sF

FC

Inde

pend

ent

even

tsF

CB

inom

ial d

istr

ibut

ion

CF

CW

aitin

g-tim

e (g

eom

etric)

dis

trib

utio

nC

FC

Mul

tiplic

atio

n Rul

e fo

r in

depe

nden

t ev

ents

FC

CG

eom

etric

(are

a) m

odel

sF

CCon

ditio

nal p

roba

bilit

yF

CEx

pect

ed v

alue

FC

CRar

e ev

ents

CF

Sam

plin

g di

stribu

tions

FM

utua

lly e

xclu

sive

eve

nts

FC

Add

ition

Rul

e fo

r m

utua

lly e

xclu

sive

eve

nts

FC

Ven

n D

iagr

amF

CC

CCen

tral

Lim

it Th

eore

mF

Prob

abili

ty d

istr

ibut

ion

FF

FG

ener

al m

ultip

licat

ion

rule

FF

Bin

omia

l pro

babi

lity

form

ula

CN

orm

al a

ppro

xim

atio

n to

a b

inom

ial

F

Infe

rential

Sta

tist

ics

Popu

latio

n vs

. sam

ple

FC

FCon

trol

(ru

n) c

hart

sF

Test

s of

sig

nifi c

ance

incl

udin

g Ty

pe I

and

Type

II e

rror

sF

CSt

atis

tical

sig

nifi c

ance

FC

Gen

eral

izab

ility

of re

sults

FF

F



STA

TISTI

CS A

ND

PRO

BA

BIL

ITY

F : Fo

cus

C : C

onne

ctio

nsCours

e 1

Cours

e 2

Cours

e 3

Cours

e 4

12

34

56

78

12

34

56

78

12

34

56

78

12

34

56

78

Surv

eys

Sta

tist

ical

Stu

die

sSa

mpl

e su

rvey

FEx

perim

ent

FSa

mpl

e si

zeC

Sim

ple

rand

om s

ampl

eC

Obs

erva

tiona

l stu

dyF

Exp

erim

ents

Cha

ract

eris

tics

of a

wel

l-de

sign

ed e

xper

imen

ts:

trea

tmen

ts, c

ontr

ol g

roup

s, r

ando

m a

ssig

nmen

t, an

d re

plic

atio

nC

F

Sour

ce o

f bi

as a

nd c

onfo

undi

ng, i

nclu

ding

pla

cebo

ef

fect

and

blin

ding

F

Ran

dom

izat

ion

test

F



DIS

CRETE

MATH

EM

ATI

CS

F : Fo

cus

C : C

onne

ctio

nsCours

e 1

Cours

e 2

Cours

e 3

Cours

e 4

12

34

56

78

12

34

56

78

12

34

56

78

12

34

56

78

Ver

tex-

Edge

Gra

phs

Ver

tex-

edge

gra

ph m

odel

sF

CC

CF

CD

igra

phs

CC

FA

djac

ency

mat

rice

s fo

r ve

rtex

-edg

e gr

aphs

FF

Eule

r pa

ths

and

circ

uits

FC

Gra

ph c

olor

ing

FC

Critic

al p

ath

anal

ysis

and

PER

T ch

arts

FC

Tree

s an

d m

inim

um s

pann

ing

tree

sF

CC

Ham

ilton

pat

hs a

nd c

ircu

its (in

clud

ing

TSP)

F

Rec

urs

ion a

nd Ite

rati

on

Info

rmal

rep

rese

ntat

ion

with

wor

ds li

ke N

OW

and

NEX

TF

CF

CF

CC

Rec

ursi

ve r

epre

sent

atio

n of

line

ar fun

ctio

nsF

CF

Rec

ursi

ve r

epre

sent

atio

n of

exp

onen

tial f

unct

ions

FF

FRec

ursi

ve r

epre

sent

atio

n of

pol

ynom

ial f

unct

ions

FRec

urre

nce

rela

tions

FC

Sequ

ence

s an

d se

ries

CF

CFi

nite

diff

eren

ces

CF

Func

tion

itera

tion

FFi

xed

poin

tsF

Gra

phic

al it

erat

ion

FC

Frac

tals

CC

CC

Mat

rice

sM

atrix

mod

els

CF

CC

CRow

and

col

umn

sum

sC

FM

atrix

addi

tion

FSc

alar

mul

tiplic

atio

nF

CM

atrix

mul

tiplic

atio

nF

CC

CId

entit

y m

atrice

sF

Inve

rse

mat

rice

sF

CPr

oper

ties

of m

atrice

sF

Mat

rix

solu

tions

of lin

ear

syst

ems

FC

CA

djac

ency

mat

rice

s fo

r ve

rtex

-edg

e gr

aphs

FF

Tran

sfor

mat

ion

mat

rice

sF

CC

CSc

atte

rplo

t m

atrice

sF



DIS

CRETE

MATH

EM

ATI

CS

F : Fo

cus

C : C

onne

ctio

nsCours

e 1

Cours

e 2

Cours

e 3

Cours

e 4

12

34

56

78

12

34

56

78

12

34

56

78

12

34

56

78

Alg

ori

thm

sA

lgor

ithm

ic p

robl

em s

olvi

ngF

FC

Des

igni

ng a

lgor

ithm

sF

FC

Ana

lyzi

ng a

nd c

ompa

ring

alg

orith

ms

CF

Prog

ram

min

g al

gorith

ms

CC

Sys

tem

atic

Counti

ng M

ethods

Cou

ntin

g tr

ees

FF

Mul

tiplic

atio

n Pr

inci

ple

of c

ount

ing

CC

FF

Add

ition

Princ

iple

of co

untin

gF

FPi

geon

hole

princ

iple

CPe

rmut

atio

nsC

FCom

bina

tions

FSe

lect

ions

with

rep

etiti

onC

FPa

scal

’s t

rian

gle

FB

inom

ial t

heor

emF

Com

bina

torial

rea

soni

ngF

FC

F



Com

mon C

ore

Sta

te S

tandar

ds

for

Mat

hem

atic

s Corr

elat

ed t

o

Core

-Plu

s M

athem

atic

s:

Cours

e 1, Cours

e 2

, Cours

e 3, an

d C

ours

e 4: Pre

par

atio

n f

or

Cal

culu

s

The C

ore

-Plu

s M

ath

emati

cs c

urr

iculu

m,

by d

esi

gn,

inco

rpora

tes

the C

om

mon C

ore

Sta

te S

tandard

s (C

CSS)

Math

em

ati

cal Pra

ctic

es

into

each

less

on.

Desc

ripti

ons

of

the M

ath

em

ati

cal Pra

ctic

es

alo

ng w

ith s

ele

cted e

xam

ple

s of

these

pra

ctic

es

in e

ach

Core

-Plu

s M

ath

em

ati

cs t

ext

are

pro

vid

ed in t

he f

ollow

ing t

able

. A

com

ple

te c

orr

ela

tion t

hat

incl

udes

the c

onte

nt

standard

s is

available

fro

m y

our

Gle

nco

e s

ale

s re

pre

senta

tive o

r

at

ww

w.w

mic

h.e

du/c

pm

p.

Sta

ndar

ds

for

Mat

hem

atic

al P

ract

ice

Stu

den

t Editio

n L

esso

ns

Cours

e 1

Cours

e 2

Cours

e 3

Cours

e 4:

Pre

par

atio

n

for

Cal

culu

s

The

Stan

dard

s fo

r M

athe

mat

ical

Pra

ctic

e de

scribe

var

ietie

s of

exp

ertis

e th

at m

athe

mat

ics

educ

ator

s at

all

leve

ls s

houl

d se

ek t

o de

velo

p in

the

ir s

tude

nts.

The

se p

ract

ices

res

t on

impo

rtan

t “p

roce

sses

and

pro

fi cie

ncie

s” w

ith lo

ngst

andi

ng im

port

ance

in m

athe

mat

ics

educ

atio

n. T

he fi

rst

of t

hese

are

the

NCTM

pro

cess

sta

ndar

ds o

f pr

oble

m s

olvi

ng,

reas

onin

g an

d pr

oof,

com

mun

icat

ion,

rep

rese

ntat

ion,

and

con

nect

ions

. The

sec

ond

are

the

stra

nds

of m

athe

mat

ical

pro

fi cie

ncy

spec

ifi ed

in t

he N

atio

nal R

esea

rch

Cou

ncil’

s re

port

A

ddin

g It

Up:

ada

ptiv

e re

ason

ing,

str

ateg

ic c

ompe

tenc

e, c

once

ptua

l und

erst

andi

ng (co

mpr

ehen

sion

of m

athe

mat

ical

con

cept

s, o

pera

tions

and

rel

atio

ns),

proc

edur

al fl

uenc

y (s

kill

in c

arry

ing

out

proc

edur

es fl

exib

ly, a

ccur

atel

y, e

ffi ci

ently

and

app

ropr

iate

ly),

and

prod

uctiv

e di

spos

ition

(ha

bitu

al in

clin

atio

n to

see

mat

hem

atic

s as

sen

sibl

e, u

sefu

l, an

d w

orth

whi

le, c

oupl

ed w

ith a

bel

ief in

dili

genc

e an

d on

e’s

own

effi c

acy)

.

1.

Mak

e se

nse

of

pro

ble

ms

and p

erse

vere

in s

olv

ing t

hem

.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

star

t by

exp

lain

ing

to t

hem

selv

es t

he m

eani

ng o

f a

prob

lem

and

look

ing

for

entr

y po

ints

to

its s

olut

ion.

The

y an

alyz

e gi

vens

, con

stra

ints

, re

latio

nshi

ps, a

nd g

oals

. The

y m

ake

conj

ectu

res

abou

t th

e fo

rm a

nd m

eani

ng o

f th

e so

lutio

n an

d pl

an a

sol

utio

n pa

thw

ay r

athe

r th

an s

impl

y ju

mpi

ng in

to a

sol

utio

n at

tem

pt. T

hey

cons

ider

ana

logo

us p

robl

ems,

and

try

spe

cial

cas

es a

nd s

impl

er

form

s of

the

origi

nal p

robl

em in

ord

er t

o ga

in in

sigh

t in

to it

s so

lutio

n. T

hey

mon

itor

and

eval

uate

the

ir p

rogr

ess

and

chan

ge c

ours

e if

nece

ssar

y. O

lder

stu

dent

s m

ight

, de

pend

ing

on t

he c

onte

xt o

f th

e pr

oble

m, t

rans

form

alg

ebra

ic e

xpre

ssio

ns o

r ch

ange

th

e vi

ewin

g w

indo

w o

n th

eir

grap

hing

cal

cula

tor

to g

et t

he in

form

atio

n th

ey n

eed.

M

athe

mat

ical

ly p

rofi c

ient

stu

dent

s ca

n ex

plai

n co

rres

pond

ence

s be

twee

n eq

uatio

ns,

verb

al d

escr

iptio

ns, t

able

s, a

nd g

raph

s or

dra

w d

iagr

ams

of im

port

ant

feat

ures

and

re

latio

nshi

ps, g

raph

dat

a, a

nd s

earc

h fo

r re

gula

rity

or

tren

ds. Y

oung

er s

tude

nts

mig

ht r

ely

on u

sing

con

cret

e ob

ject

s or

pic

ture

s to

hel

p co

ncep

tual

ize

and

solv

e a

prob

lem

. Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

chec

k th

eir

answ

ers

to p

robl

ems

usin

g a

diffe

rent

met

hod,

and

the

y co

ntin

ually

ask

the

mse

lves

, “D

oes

this

mak

e se

nse?

” Th

ey

can

unde

rsta

nd t

he a

ppro

ache

s of

oth

ers

to s

olvi

ng c

ompl

ex p

robl

ems

and

iden

tify

corr

espo

nden

ces

betw

een

diffe

rent

app

roac

hes.

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 8–1

0,

56

–58

, 61

#1

1,

93

#8

, 11

1, 1

31,

15

9, 2

39

–24

2,

29

7 S

TM, 4

13

#

3, 4

20

–42

1

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 61–6

4, 1

21

#7, 1

24 #

11,

141 C

heck

You

r Und

erst

andi

ng

(CYU

) 280–2

85,

314 #

13,

364–3

67,

372 #

17,

474 #

2, #

3

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 32–3

3, 1

49 #

14,

233 #

7, 3

60

#15, 4

32 C

YU,

435 #

7, 4

40

#8, 4

50 #

33,

459–4

61, 4

70

#4, 5

54 #

22

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 71 #

22,

178–1

80, 2

31

CYU

, 288 #

7,

300–3

01,

318–3

21,

394–3

98,

470–4

73,

496–5

01,

591–5

95,

611–6

14



Com

mon C

ore

Sta

te S

tandar

ds

for

Mat

hem

atic

s Corr

elat

ed t

o

Core

-Plu

s M

athem

atic

s:

Cours

e 1, Cours

e 2

, Cours

e 3, an

d C

ours

e 4: Pre

par

atio

n f

or

Cal

culu

s

Sta

ndar

ds

for

Mat

hem

atic

al P

ract

ice

Stu

den

t Editio

n L

esso

ns

Cours

e 1

Cours

e 2

Cours

e 3

Cours

e 4:

Pre

par

atio

n

for

Cal

culu

s

2.

Rea

son a

bst

ract

ly a

nd q

uan

tita

tive

ly.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

mak

e se

nse

of q

uant

ities

and

the

ir r

elat

ions

hips

in

pro

blem

situ

atio

ns. T

hey

brin

g tw

o co

mpl

emen

tary

abi

litie

s to

bea

r on

pro

blem

s in

volv

ing

quan

titat

ive

rela

tions

hips

: the

abi

lity

to d

econ

text

ualiz

e—to

abs

trac

t a

give

n si

tuat

ion

and

repr

esen

t it

sym

bolic

ally

and

man

ipul

ate

the

repr

esen

ting

sym

bols

as

if th

ey h

ave

a lif

e of

the

ir o

wn,

with

out

nece

ssar

ily a

tten

ding

to

thei

r re

fere

nts—

and

the

abili

ty t

o co

ntex

tual

ize,

to

paus

e as

nee

ded

during

the

m

anip

ulat

ion

proc

ess

in o

rder

to

prob

e in

to t

he r

efer

ents

for

the

sym

bols

invo

lved

. Q

uant

itativ

e re

ason

ing

enta

ils h

abits

of cr

eatin

g a

cohe

rent

rep

rese

ntat

ion

of t

he

prob

lem

at

hand

; con

side

ring

the

uni

ts in

volv

ed; a

tten

ding

to

the

mea

ning

of

quan

titie

s, n

ot ju

st h

ow t

o co

mpu

te t

hem

; and

kno

win

g an

d fl e

xibl

y us

ing

diffe

rent

pr

oper

ties

of o

pera

tions

and

obj

ects

.

Thro

ugho

ut

Uni

ts 1

–3, 5

, 8

Exam

ples

: 2

7–3

1,

11

6–1

23

, 1

94

–19

7,

25

0–2

54

, 3

07

–31

1, 3

69

CYU

, 46

9–4

72

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 70–7

1,

104–1

17,

145 #

2, 2

23

#15, 3

31,

349–3

50 #

17,

472 #

4,

489–4

97,

552–5

53

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 118 #

1, #

2,

145–1

50, 2

20

#18, 3

38–3

39,

353 #

1, 5

62

CYU

, 468–4

71

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 68 #

10,

146 #

3,1

57–1

61,

239–2

42,

305–3

06,

364–3

67,

399–4

04,

439–4

42,

490–4

95

3.

Const

ruct

via

ble

arg

um

ents

and c

ritique

the

reas

onin

g o

f oth

ers.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

unde

rsta

nd a

nd u

se s

tate

d as

sum

ptio

ns,

defi n

ition

s, a

nd p

revi

ousl

y es

tabl

ishe

d re

sults

in c

onst

ruct

ing

argu

men

ts. T

hey

mak

e co

njec

ture

s an

d bu

ild a

logi

cal p

rogr

essi

on o

f st

atem

ents

to

expl

ore

the

trut

h of

the

ir

conj

ectu

res.

The

y ar

e ab

le t

o an

alyz

e si

tuat

ions

by

brea

king

the

m in

to c

ases

, and

ca

n re

cogn

ize

and

use

coun

tere

xam

ples

. The

y ju

stify

the

ir c

oncl

usio

ns, c

omm

unic

ate

them

to

othe

rs, a

nd r

espo

nd t

o th

e ar

gum

ents

of ot

hers

. The

y re

ason

indu

ctiv

ely

abou

t da

ta, m

akin

g pl

ausi

ble

argu

men

ts t

hat

take

into

acc

ount

the

con

text

fro

m

whi

ch t

he d

ata

aros

e. M

athe

mat

ical

ly p

rofi c

ient

stu

dent

s ar

e al

so a

ble

to c

ompa

re

the

effe

ctiv

enes

s of

tw

o pl

ausi

ble

argu

men

ts, d

istin

guis

h co

rrec

t lo

gic

or r

easo

ning

fr

om t

hat

whi

ch is

fl aw

ed, a

nd—

if th

ere

is a

fl aw

in a

n ar

gum

ent—

expl

ain

wha

t it

is. E

lem

enta

ry s

tude

nts

can

cons

truc

t ar

gum

ents

usi

ng c

oncr

ete

refe

rent

s su

ch a

s ob

ject

s, d

raw

ings

, dia

gram

s, a

nd a

ctio

ns. S

uch

argu

men

ts c

an m

ake

sens

e an

d be

co

rrec

t, ev

en t

houg

h th

ey a

re n

ot g

ener

aliz

ed o

r m

ade

form

al u

ntil

late

r gr

ades

. La

ter, s

tude

nts

lear

n to

det

erm

ine

dom

ains

to

whi

ch a

n ar

gum

ent

appl

ies.

Stu

dent

s at

all

grad

es c

an li

sten

or

read

the

arg

umen

ts o

f ot

hers

, dec

ide

whe

ther

the

y m

ake

sens

e, a

nd a

sk u

sefu

l que

stio

ns t

o cl

arify

or

impr

ove

the

argu

men

ts.

Thro

ugho

ut

Uni

ts 1

–7

Exam

ples

: 18

0

#3

0, 1

94

#4

, 3

35

–33

7, 3

48

#

29

, 37

4–3

77

, 3

87

, 49

3 #

2,

50

4 #

13

, 51

7

CYU

b

Thro

ugho

ut

Uni

ts 1

–7

Exam

ples

: 22

#14, 5

4–5

5,

111#

7, 1

23

#10, 1

68 #

9,

170–1

80, 1

79

#8, 2

07 #

3, #

4,

490–4

91

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 2–1

5,

166 #

5, 1

71

#5, 2

04–2

08,

233 #

7, 3

52 #

9,

405–4

06, 4

87

#11, 5

66–5

67

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 124

#19, 2

87–2

94,

315 #

5, 3

16 #

10,

326 #

22, 4

34 #

6,

472 #

8, 5

94–5

95,

598 #

8, 6

04–6

14



Com

mon C

ore

Sta

te S

tandar

ds

for

Mat

hem

atic

s Corr

elat

ed t

o

Core

-Plu

s M

athem

atic

s:

Cours

e 1, Cours

e 2

, Cours

e 3, an

d C

ours

e 4: Pre

par

atio

n f

or

Cal

culu

s

Sta

ndar

ds

for

Mat

hem

atic

al P

ract

ice

Stu

den

t Editio

n L

esso

ns

Cours

e 1

Cours

e 2

Cours

e 3

Cours

e 4:

Pre

par

atio

n

for

Cal

culu

s

4.

Model

with m

athem

atic

s.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

can

appl

y th

e m

athe

mat

ics

they

kno

w t

o so

lve

prob

lem

s ar

isin

g in

eve

ryda

y lif

e, s

ocie

ty, a

nd t

he w

orkp

lace

. In

early

grad

es, t

his

mig

ht b

e as

sim

ple

as w

ritin

g an

add

ition

equ

atio

n to

des

crib

e a

situ

atio

n. In

mid

dle

grad

es, a

stu

dent

mig

ht a

pply

pro

port

iona

l rea

soni

ng t

o pl

an a

sch

ool e

vent

or

anal

yze

a pr

oble

m in

the

com

mun

ity. B

y hi

gh s

choo

l, a

stud

ent

mig

ht u

se g

eom

etry

to

sol

ve a

des

ign

prob

lem

or

use

a fu

nctio

n to

des

crib

e ho

w o

ne q

uant

ity o

f in

tere

st

depe

nds

on a

noth

er. M

athe

mat

ical

ly p

rofi c

ient

stu

dent

s w

ho c

an a

pply

wha

t th

ey k

now

ar

e co

mfo

rtab

le m

akin

g as

sum

ptio

ns a

nd a

ppro

xim

atio

ns t

o si

mpl

ify a

com

plic

ated

si

tuat

ion,

rea

lizin

g th

at t

hese

may

nee

d re

visi

on la

ter.

They

are

abl

e to

iden

tify

impo

rtan

t qu

antit

ies

in a

pra

ctic

al s

ituat

ion

and

map

the

ir r

elat

ions

hips

usi

ng s

uch

tool

s as

dia

gram

s, t

wo-

way

tab

les,

gra

phs,

fl ow

char

ts a

nd for

mul

as. T

hey

can

anal

yze

thos

e re

latio

nshi

ps m

athe

mat

ical

ly t

o dr

aw c

oncl

usio

ns. T

hey

rout

inel

y in

terp

ret

thei

r m

athe

mat

ical

res

ults

in t

he c

onte

xt o

f th

e si

tuat

ion

and

refl e

ct o

n w

heth

er t

he r

esul

ts

mak

e se

nse,

pos

sibl

y im

prov

ing

the

mod

el if

it h

as n

ot s

erve

d its

pur

pose

.

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 1

61

–16

7, 2

80

#

10

, 32

3–3

32

, 3

63

–36

9, 3

83

, 4

63

–46

8,

55

1–5

61

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 158–1

60,

232–2

42,

260–2

68,

439–4

42,

518–5

19,

522–5

31,

569 C

YU,

573–5

76

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 81–8

8,

132–1

43, 2

18

#12, 2

31 #

5,

242–2

47, 3

60

#14, 4

35–4

37,

495–5

06

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 126 #

23,

157–1

61,

182–1

87,

425–4

29, 4

36

#9, 4

64–4

70,

482 #

26, 5

01,

512–5

16

5.

Use

appro

pri

ate

tools

str

ateg

ical

ly.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

cons

ider

the

ava

ilabl

e to

ols

whe

n so

lvin

g a

mat

hem

atic

al p

robl

em. T

hese

too

ls m

ight

incl

ude

penc

il an

d pa

per, c

oncr

ete

mod

els,

a r

uler

, a p

rotr

acto

r, a

cal

cula

tor, a

spr

eads

heet

, a c

ompu

ter

alge

bra

syst

em, a

sta

tistic

al p

acka

ge, o

r dy

nam

ic g

eom

etry

sof

twar

e. P

rofi c

ient

stu

dent

s ar

e su

ffi ci

ently

fam

iliar

with

too

ls a

ppro

pria

te for

the

ir g

rade

or

cour

se t

o m

ake

soun

d de

cisi

ons

abou

t w

hen

each

of th

ese

tool

s m

ight

be

help

ful,

reco

gniz

ing

both

the

in

sigh

t to

be

gain

ed a

nd t

heir li

mita

tions

. For

exa

mpl

e, m

athe

mat

ical

ly p

rofi c

ient

hi

gh s

choo

l stu

dent

s an

alyz

e gr

aphs

of fu

nctio

ns a

nd s

olut

ions

gen

erat

ed u

sing

a

grap

hing

cal

cula

tor. T

hey

dete

ct p

ossi

ble

erro

rs b

y st

rate

gica

lly u

sing

est

imat

ion

and

othe

r m

athe

mat

ical

kno

wle

dge.

Whe

n m

akin

g m

athe

mat

ical

mod

els,

the

y kn

ow

that

tec

hnol

ogy

can

enab

le t

hem

to

visu

aliz

e th

e re

sults

of va

ryin

g as

sum

ptio

ns,

expl

ore

cons

eque

nces

, and

com

pare

pre

dict

ions

with

dat

a. M

athe

mat

ical

ly p

rofi c

ient

st

uden

ts a

t va

riou

s gr

ade

leve

ls a

re a

ble

to id

entif

y re

leva

nt e

xter

nal m

athe

mat

ical

re

sour

ces,

suc

h as

dig

ital c

onte

nt lo

cate

d on

a w

ebsi

te, a

nd u

se t

hem

to

pose

or

solv

e pr

oble

ms.

The

y ar

e ab

le t

o us

e te

chno

logi

cal t

ools

to

expl

ore

and

deep

en t

heir

unde

rsta

ndin

g of

con

cept

s.

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 5

2–5

6, 1

37

#

16

, 16

1–1

67

, 2

81

#1

3, 2

97

, 4

80

–48

2,

56

8–5

70

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 49–5

7, 9

7

#18, 2

80–2

98,

385, 4

05 #

12,

498–5

01, 5

07

#11, 5

81

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 93–9

5,

220 #

19, 2

54

#13, 2

60–2

65,

462–4

67, 4

75

#11, 5

19–5

22

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 138–1

44, 4

51

#31, 4

68–4

70,

483 #

28,

505–5

07, 5

18

#20, 5

36–5

37,

591–5

95



Com

mon C

ore

Sta

te S

tandar

ds

for

Mat

hem

atic

s Corr

elat

ed t

o

Core

-Plu

s M

athem

atic

s:

Cours

e 1, Cours

e 2

, Cours

e 3, an

d C

ours

e 4: Pre

par

atio

n f

or

Cal

culu

s

Sta

ndar

ds

for

Mat

hem

atic

al P

ract

ice

Stu

den

t Editio

n L

esso

ns

Cours

e 1

Cours

e 2

Cours

e 3

Cours

e 4:

Pre

par

atio

n

for

Cal

culu

s

6.

Att

end t

o p

reci

sion.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

try

to c

omm

unic

ate

prec

isel

y to

oth

ers.

The

y tr

y to

us

e cl

ear

defi n

ition

s in

dis

cuss

ion

with

oth

ers

and

in t

heir o

wn

reas

onin

g. T

hey

stat

e th

e m

eani

ng o

f th

e sy

mbo

ls t

hey

choo

se, i

nclu

ding

usi

ng t

he e

qual

sig

n co

nsis

tent

ly

and

appr

opriat

ely.

The

y ar

e ca

refu

l abo

ut s

peci

fyin

g un

its o

f m

easu

re, a

nd la

belin

g ax

es t

o cl

arify

the

cor

resp

onde

nce

with

qua

ntiti

es in

a p

robl

em. T

hey

calc

ulat

e ac

cura

tely

and

effi

cien

tly, e

xpre

ss n

umer

ical

ans

wer

s w

ith a

deg

ree

of p

reci

sion

ap

prop

riat

e fo

r th

e pr

oble

m c

onte

xt. I

n th

e el

emen

tary

gra

des,

stu

dent

s gi

ve

care

fully

for

mul

ated

exp

lana

tions

to

each

oth

er. B

y th

e tim

e th

ey r

each

hig

h sc

hool

th

ey h

ave

lear

ned

to e

xam

ine

clai

ms

and

mak

e ex

plic

it us

e of

defi

niti

ons.

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 20

–21

#

15

, 17

5 #

17

, 3

74

–37

7, 3

87

#

10

, 39

0 #

17

, 3

91

#1

9, 4

66

#

5, 4

82

#9

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 32–3

3,

92 #

7, 1

72 #

6,

261 #

3, 3

82–

383, 4

67–4

77,

531

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 123

#16, 1

65–1

68,

245–2

47,

283–2

96,

321–3

23,

581–5

83

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 66 #

7,

110 #

4, 2

39 #

10,

366, 4

93–4

95,

517 #

15, 5

35

#3c,

576 #

11

7.

Look

for

and m

ake

use

of

stru

cture

.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

look

clo

sely

to

disc

ern

a pa

tter

n or

str

uctu

re.

Youn

g st

uden

ts, f

or e

xam

ple,

mig

ht n

otic

e th

at t

hree

and

sev

en m

ore

is t

he s

ame

amou

nt a

s se

ven

and

thre

e m

ore,

or

they

may

sor

t a

colle

ctio

n of

sha

pes

acco

rdin

g to

how

man

y si

des

the

shap

es h

ave.

Lat

er, s

tude

nts

will

see

7 ×

8 e

qual

s th

e w

ell

rem

embe

red

7 ×

5 +

7 ×

3, i

n pr

epar

atio

n fo

r le

arni

ng a

bout

the

dis

trib

utiv

e pr

oper

ty. I

n th

e ex

pres

sion

x2 +

9x

+ 1

4, o

lder

stu

dent

s ca

n se

e th

e 1

4 a

s 2 ×

7 a

nd t

he 9

as

2 +

7. T

hey

reco

gniz

e th

e si

gnifi

canc

e of

an

exis

ting

line

in a

ge

omet

ric

fi gur

e an

d ca

n us

e th

e st

rate

gy o

f dr

awin

g an

aux

iliar

y lin

e fo

r so

lvin

g pr

oble

ms.

The

y al

so c

an s

tep

back

for

an

over

view

and

shi

ft p

ersp

ectiv

e. T

hey

can

see

com

plic

ated

thi

ngs,

suc

h as

som

e al

gebr

aic

expr

essi

ons,

as

sing

le o

bjec

ts o

r as

bei

ng c

ompo

sed

of s

ever

al o

bjec

ts. F

or e

xam

ple,

the

y ca

n se

e 5

– 3

(x –

y)2

as

5

min

us a

pos

itive

num

ber

times

a s

quar

e an

d us

e th

at t

o re

aliz

e th

at it

s va

lue

cann

ot

be m

ore

than

5 for

any

rea

l num

bers

x a

nd y

.

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 17

8

#2

4, 1

95

, 2

39

–24

7,

29

1–2

97

, 30

3

STM

, 32

8 S

TM,

46

8, 4

78

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 26–2

9,

42 #

21, 6

0

STM

d, 9

6 C

YU,

178 #

6,

332–3

40, 3

51

#22, 4

81 #

20

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 60 #

5,

62–7

1, 1

12–1

17,

334, 3

48–3

52,

387 #

28,

489–4

95, 5

06

#19, 5

97 #

29

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 59–6

4,

141–1

44,

188–1

93, 2

02

#14, 2

50–2

56,

260 #

14, 2

62

#24, 3

69–3

72,

391–3

93,

464–4

70



Com

mon C

ore

Sta

te S

tandar

ds

for

Mat

hem

atic

s Corr

elat

ed t

o

Core

-Plu

s M

athem

atic

s:

Cours

e 1, Cours

e 2

, Cours

e 3, an

d C

ours

e 4: Pre

par

atio

n f

or

Cal

culu

s

Sta

ndar

ds

for

Mat

hem

atic

al P

ract

ice

Stu

den

t Editio

n L

esso

ns

Cours

e 1

Cours

e 2

Cours

e 3

Cours

e 4:

Pre

par

atio

n

for

Cal

culu

s

8. L

ook

for

and e

xpre

ss r

egula

rity

in r

epea

ted r

easo

nin

g.

Mat

hem

atic

ally

pro

fi cie

nt s

tude

nts

notic

e if

calc

ulat

ions

are

rep

eate

d, a

nd lo

ok

both

for

gen

eral

met

hods

and

for

sho

rtcu

ts. U

pper

ele

men

tary

stu

dent

s m

ight

no

tice

whe

n di

vidi

ng 2

5 b

y 1

1 t

hat

they

are

rep

eatin

g th

e sa

me

calc

ulat

ions

ove

r an

d ov

er a

gain

, and

con

clud

e th

ey h

ave

a re

peat

ing

deci

mal

. By

payi

ng a

tten

tion

to t

he c

alcu

latio

n of

slo

pe a

s th

ey r

epea

tedl

y ch

eck

whe

ther

poi

nts

are

on t

he

line

thro

ugh

(1, 2

) w

ith s

lope

3, m

iddl

e sc

hool

stu

dent

s m

ight

abs

trac

t th

e eq

uatio

n (y

– 2

)/(x

– 1

) =

3. N

otic

ing

the

regu

larity

in t

he w

ay t

erm

s ca

ncel

whe

n ex

pand

ing

(x –

1)(x

+ 1

), (x

– 1

)(x2 +

x +

1),

and

(x –

1)(x

3 +

x2 +

x +

1) m

ight

lead

the

m t

o th

e ge

nera

l for

mul

a fo

r th

e su

m o

f a

geom

etric

series

. As

they

wor

k to

sol

ve a

pr

oble

m, m

athe

mat

ical

ly p

rofi c

ient

stu

dent

s m

aint

ain

over

sigh

t of

the

pro

cess

, w

hile

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ng t

o th

e de

tails

. The

y co

ntin

ually

eva

luat

e th

e re

ason

able

ness

of th

eir

inte

rmed

iate

res

ults

.

Thro

ugho

ut

Uni

ts 1

–7

Exam

ples

: 2

7–3

5, 1

52

#

1e,

33

2–3

34

, 4

04

–40

6,

43

7–4

38

#

6, 4

73

–47

8,

49

7 #

5

Thro

ugho

ut

Uni

ts 1

–7

Exam

ples

: 23

#20, 4

0 #

15,

210–2

16, 3

71

#13, 5

70–5

72,

578 #

14

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 62

#4, #

5, 1

03 #

3,

329–3

31, 3

41

#12, 4

82–4

89,

507, 5

39–5

42

Thro

ugho

ut

Uni

ts 1

–8

Exam

ples

: 107 #

7,

165–1

68, 2

05

#28, 2

45 #

26,

318–3

21, 3

61 #

7,

499 1

0, 5

01–5

04,

535 #

3, 5

91–5

95


Scope and Sequence Algebra and Functions Geometry and Trigonometry Statistics and Probability Discrete Mathematics

ISBN: 978-0-07-892135-3MHID: 0-07-892135-X

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