PROPOSED COURSE OF STUDY HOLMDEL TOWNSHIP PUBLIC SCHOOLS

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1

PROPOSED COURSE OF STUDY

HOLMDEL TOWNSHIP PUBLIC SCHOOLS



Course Title
:

Grade 6 Mathematics



Curriculum Area
:

Mathematics


Credits:




Length of Course: Full Year

Half Year



New Course

Revision of Existing Course


Course Pre
-
Requisites:





1
.

Course Description


The sixth grade mathematics program develops and extends mathematical and algebraic skills in calculating, reasoning

and problem
solving.
This course offers a broad, rich curriculum which engages students in solving problems in such i
mportant branches of
mathematics as geometry, measurement, algebra, statistics, and probability.
The desire to continue learning is encouraged.


2.

Course Philosophy


The primary focus of the Grade 6 Mathematics curriculum is to continue the development o
f mathematical concepts in contexts that
promote problem solving, reasoning, communication, making connections, and designing and analyzing representations. Students
should be encouraged to be active learners and to work in both small and large groups as w
ell as individually to explore and develop
their understanding of important mathematical concepts. Concrete experiences with manipulative materials will be integrated i
nto the
curriculum because they provide the means by which students construct knowledge.

Students move from the concrete to the abstract
in the ongoing development of conceptual understanding. Consequently, assessment must also be an ongoing process.


X


X



2

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, su
btopics, outcome proficiencies and performance assessments. These outcome proficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standard
s (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for t
his unit.


NJCCCS:

4.1.6B, 4.3.6A, 4.3.6D

CECFLS:

9.1.6A


Unit
1: Properties of Whole Numbers

Duration: 2 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Compare and Order Whole Numbers


2.

Exponents

3.

Order of Operations

4.

Properties of Operations




5. Whole number patterns and sequences

1.

Demonstrate understanding of number line to compare
and order whole numbers


2.

Represent nu
mbers using exponents


3
.

Evaluate numerical expressions using order of operations

4
.

Demonstrate an understanding of and apply the
prope
rties of operations and numbers



Associative, commutative, identity



Distributive property to represent products

5
. Recognize, describe, and extend patterns and sequences

Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individ
ual at 80% or better; for the class: 80% of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included in the Evaluation S
ection


By the end of u
nit 1, 80% of the students will demonstrate a satisfactory p
erformance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.



3

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. Th
ese outcome proficiencie
s should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be sp
ecific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.1.6A, 4.1.6B, 4.3.6C, 4.3.6D

TL:

8.1.6D


Unit
2: Introduction
to Algebra

Duration: 3 we
eks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Algebraic Expressions




2.

Function Tables

3. Solving Equa
tions with whole numbers

1.

Identify, evaluate, and create algebraic expressions using
variables



Identify and evaluate algebraic expressions



Translate between variable and numeric expressions

2.

Write an algebraic expression to describe a function table

3.

So
lve e
q
uations



Determine whether a number is a solution of an
equation



Solve one
-
step equations using inverse
operations

Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cognitive skills
.


Student proficiency (for

a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 80% of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included in the Evaluation S
ection


By the end o
f u
nit
2
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.


4

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, subtopics, outcome proficie
ncies and performance assessments. These outcome proficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New J
ersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.1.6
A, 4.1.6B, 4.1.6C, 4.2.6D, 4.3.6D

CECFLS:

9.1.6A

TL:

8.1.6A


Unit 3:
Applications of Decimals

Duration: 3 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identif
y
mastery

level

skills
bold
).

1.

Concept of decimals

2.

Estimation


3.

Operations with decimals




4.

Scientific Notation


1.

Read, write, compare, order and round decimals


2.

Estimate decimal sums, diffe
rences, products, and
quotients

3.

Perform computations involving
dec
imals



A
dditi
on and subtraction of decimals



M
ultipli
cation and division of decimals



M
ultiply and divide decimals by powers of 10

4.

Write large numbers using scientific notation


Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of

content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0% of the
students attain the established minimum standard; an exemplar or rubric should be referen
ced and included in the Evaluation Section


By the end of u
nit
3
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.


5


3.

Scope and Sequence




This section of the curriculum
specifies the course’s units, subtopics, outcome proficiencies and performance assessments. These outcome proficiencies shoul
d be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer,
Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (
Section 7), TL (Section 8) for this unit.


NJCCCS:

4.1.6A

CECFLS:

9.1.6A

TL:

8.1.6C


Unit
4: Number Theory and Fractions

Dura
tion: 2 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
n
d cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Divisibility Rules



2.

Factors



3.

Concept of Fractions






4.

Meaning of Fraction Operations

1.

Understand and apply number theory concepts



Primes, composites
,



Divisibility rules for 2, 3, 4, 5, 6, 9, 10


2.

Write prime factorization of composite numbers



F
ind the greatest common factor of a set of
numbers

3.

Understand and apply the concept of fractions



Fraction
-
decimal conversion



Equivalent fraction
s



Compare and order fractions



Mixed numbers and improper frac
tions

4.

Develop meaning for addition and subtraction



Add and subtract fractions with like
denominators


6



Estimating sums and differences

Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cognitive skills
.


Student pro
ficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 80% of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included in the Evaluation S
ection


By the end of u
nit
4
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.




7

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, subtopics, o
utcome proficiencies and performance assessments. These outcome proficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS)
, and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.1.6A, 4.1.6B, 4.1.6C, 4.3.6A


Unit
5: Fraction Operations

Duration: 3 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Least Common Multiple

2.

Operations with fractions





3.

Equations with fractions


1.

Find the least common multiple

2.

Perform operations with fractions



Addition



Subtraction



Multiplication



Division

3.

Solve equations involving fractions



Addition and Subtraction



Mul
tiplication and Division

Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the cl
ass: 80% of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included in the Evaluation S
ection


By the end of u
nit
5
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.



8

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. Th
ese outcome proficiencies should be
indexed to the releva
nt New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform in
terpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.1.6A, 4.1.6B, 4.1.6C

CECFLS:

9.1.6A

TL:

8.1.A; 8.1.C


Unit
6: Ratio, Proportion and Percent

Du
ration: 3 Weeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Ratios and rates


2.

Proportions

3.

Similar figures


4.

Scale drawing


5.

Concept

of percents


6.

Percent problems

7.

Application of percent

1.

Write ratios and rates and find unit rates



Equivalent Ratios

2.

Write and

solve

proportions

3.

Use proportions to find missing measures in similar
figures

4.

Use proportions to solve problems involving scale
dr
awings

5.

Represent the same number using decimals, percents,
and fractions

6.

Find the missing value in a percent problem

7.

Solve percent problems involving discount, tip and sales
tax.


9


Instructional Benchmarks
:

A
ctivities to provide evidence for student maste
ry of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0% of the
students attain the established minimum standard; an exemplar or rubric should be re
ferenced and included in the Evaluation Section


By the end of u
nit
6
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.




10

3.

Scope and Sequence




This section of the curri
culum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. These outcome proficiencies

should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Cons
umer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CE
CFLS (Section 7), TL (Section 8) for this unit.

NJCCCS:

4.2.6A, 4.2.6B, 4.2.6E

CECFLS:

9.3.6A

TL:

8.1.6C

Unit
7: Geometry

Duration: 3 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Geometry Vocabulary




2.

Angles






3.

Triangles

4.

Quadrilaterals

5.

Other Polygons

6.

Geometric Patterns

7.

Congruence

8.

Transformations


9.

Line Symmetry

1.

Describe figures using geometry vocabulary and
notation



P
oints, lines, rays, segments
, parallel,
perpendicular, and
skew


2.

Name, measure, classify, estimate, and draw angles
using a protractor



Classify by angle measure



Supplementary, complementary
, adjacent,
vertical,
congruent

3.

Classify triangles by sides

and angle measure

4.

Identify, classify, and compare quadrilaterals

5.

Identify and find the angle measures of regular polygons

6.

Recognize, describe, and extend geometric patterns

7.

Identify congruent figures

8.

Use translations, reflections, and rotations to transfo
rm
geometric figures

9.

Identify line symmetry


11


Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80%
or better; for the class: 80% of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included in the Evaluation S
ection


By the end of u
nit
7
, 80% of the students will demonstrate a satisfactory performance
of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.

























12

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. Th
ese outcome pr
oficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and sh
ould be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.2.6A, 4.2.6E

CECFLS:

9.1.6A

TL:

8.1.6A, 8.1.6C


Uni
t
8: Measurement: Perimeter, Area, Volume

Duration: 4 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Perimeter

2.

Area and Circumf
erence of circles

3.

Area






4.

Three
-
dimensional figures

5.

Volume

6.

Surface Area

1.

Find the perimeter and missing side lengths of polygons

2.

Find the area and circumference of circles

3.

Determine the area of the following polygons:



R
ectangle



P
arallelogram



T
riang
le



T
rapezoid



C
omposite figures

4.

Identify and describe three
-
dimensional figures

5.

Find the volume of rectangular prisms and cylinders

6.

Determine the surface area of three
-
dimensional figures



Rectangular prisms



Cylinders


13

Instructional Benchmarks
:

A
ctivities t
o provide evidence for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0% of the
students attain the established minimum standard
; an exemplar or rubric should be referenced and included in the Evaluation Section


By the end of u
nit
8,

80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.




14

3.

Scope and S
equence




This section of the curriculum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. Th
ese outcome proficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the
New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.1.6A, 4.1.6B, 4.2.6C, 4.3.6C, 4.3.6D

TL:

8.1.6D


Unit
9: Integers

Duration: 3 w
eeks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Concept of Integers

2.

Compare and Order Integers

3.

Coordinate Plane

4.

Operations with integers





5.

Solving equations with integers

6.

Function Tables

1.

Iden
tify, model, an
d graph integers

2.

Compare and order integers

3.

Locate and graph points on a coordinate plane

4.

Demonstrate an understanding of integer operation



Addition



Subtraction



Multiplication



Division

5.

Solve equations using integers

6.

Write an algebraic equation to describe
a function table

Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0%

of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included in the Evaluation S
ection


By the end of u
nit
9
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oect
ives 潮 an en搠潦
cha灴er assessment.


15

3.

Scope and Sequence



This section of the curriculum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. Th
ese outcome proficiencies should be
indexed to the relevant New Jer
sey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretatio
n among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.4.6A

CECFLS:

9.1.6A

TL:

8.1.6A


Unit
10: Collect and Display Data

Duration: 3 w
eeks


Topical outline of
course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Frequency table

2.

Measures of Central Tendency






3.

Graphic Presentation of Data

1.

Create a frequency table

2.

Use measures of central tendency to analyze data



M
ean



M
edian



M
ode



R
ange



O
utliers

3.

Display and analyze data using:



Bar graphs



Line graphs



Stem and leaf


16


Instructional Benchmarks
:

A
ctivities to provide evidence for student mastery of content and cogn
itive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0% of the
students attain the established minimum standard; an exemplar or rubric should be referenced and included
in the Evaluation Section


By the end of u
nit 1
0
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.




17

3.

Scope and Sequence




This section of the curriculum specifies the
course’s units, subtopics, outcome proficiencies and performance assessments. These outcome proficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standards

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Li
fe Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use this curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL

(Section 8) for this unit.


NJCCCS:

4.4.6B, 4.4.6C

CECFLS:

9.1.6A

TL:

8.1.6A, 8.1.6C


Unit
11: Probability

Duration: 2 we
eks


Topical outline of course
content in the suggested order of
presentation


Outcome proficiencies
for
content a
nd cognitive sk
ills
(
Identify
mastery

level

skills
bold
).

1.

Probability of a simple event


2.

Experimental probability




3.

Determining the number of possible outcomes





4.

Theoretical probability





1.

Estimate the likelihood of an event and wr
ite and
compare probabilities

2.

Find the experimental probability of an event




Experiment



Outcome



Sample space


3.

Develop strategies to find all possible outcomes



Organized Lists



Tree Diagrams



Multiplication Principle of Counting



Permutations and Combination
s

4.

Find the theoretical probability of an event



Equally likely



Fair



Complement

of an event


18

5. Compound events


6. Predictions

5.

Find the theoretical probability of a compound event



Independent and Dependent Events

6.

Use probability to predict an event

Instructional Benchmarks
:

A
ctivities to pr
ovide evidence for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0% of the
students attain the established minimum standard; an

exemplar or rubric should be referenced and included in the Evaluation Section


By the end of u
nit
1
1, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.
























19

3.

Scope and Sequence




This section of the curriculum specifies the course’s units, subtopics, outcome proficiencies and performance assessments. Th
ese outcome proficiencies should be
indexed to the relevant New Jersey Core Curriculum Content Standar
ds

(NJCCCS),

the New Jersey
Career Education and Consumer, Family, and Life Skills
Standards (CECFLS), and the New Jersey Technological Literacy Standards (TL),

and should be specific enough to allow uniform interpretation among the
teachers who will use t
his curriculum.


Reference to NJCCCS (Section 6), CECFLS (Section 7), TL (Section 8) for this unit.


NJCCCS:

4.2.6D

CECFLS:

9.1.6A


Unit 12:
Systems of Measurement

Duration: 2 w
eeks


Topical outline of course
content in the suggested order of
presenta
tion


Outcome proficiencies
for
content a
nd cognitive skills
(
Identify
mastery

level

skills
bold
).

1.

Customary Units of Measure


2.

Metric Units of Measure

3.

Converting Customary Units

4.

Converting Metric Units

5.

Time and Temperature


1.

Understand and select appropria
te customary units of
measure

2.

Understand and select appropriate metric units of measure

3.

Convert customary units of measure

4.

Convert metric units of measure

5.

Find measures of time and temperature

Instructional Benchmarks
:

A
ctivities to provide evidence
for student mastery of content and cognitive skills
.


Student proficiency (for a specific unit or multiple units) is defined for the individual at 80% or better; for the class: 8
0% of the
students attain the established minimum standard; an exemplar or ru
bric should be referenced and included in the Evaluation Section


By the end of unit

1
2
, 80% of the students will demonstrate a satisfactory performance of a “B” on the state
d

潢oectives 潮 an en搠潦
cha灴er assessment.





20



4
.

Required Instructional Reso
urces



Bennett, Jennie M.
Holt Mathematics Course 1
.
Holt, Rinehart and Winston: Orlando. 2007


5
.

Evaluation and Grading



80% of the students will demonstrate on a quarterly basis a satisfactory performance of a “B” on the stated outcome
proficiencies
. The marking period grade will be determined by tests, quizzes, homework, projects, and/or reports.


6.

New Jersey Core Curriculum Content Standards


Standard 4.1 (Number And Numerical Operations) All Students Will Develop Number Sense And Will Perform
Standard Numerical
Operations And Estimations On All Types Of Numbers In A Variety Of Ways.


Descriptive Statement
: Numbers and arithmetic operations are what most of the general public think about when they think of
mathematics; and, even though other ar
eas like geometry, algebra, and data analysis have become increasingly important in recent
years, numbers and operations remain at the heart of mathematical teaching and learning. Facility with numbers, the ability t
o choose
the appropriate types of number
s and the appropriate operations for a given situation, and the ability to perform those operations as
well as to estimate their results, are all skills that are essential for modern day life.


Number Sense
. Number sense is an intuitive feel for numbers a
nd a common sense approach to using them. It is a comfort with what
numbers represent that comes from investigating their characteristics and using them in diverse situations. It involves an
understanding of how different types of numbers, such as fraction
s and decimals, are related to each other, and how each can best be
used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called nu
meration
and thus includes the important concepts of place valu
e, number base, magnitude, and approximation and estimation.


Numerical Operations
. Numerical operations are an essential part of the mathematics curriculum, especially in the elementary
grades. Students must be able to select and apply various computatio
nal methods, including mental math, pencil
-
and
-
paper techniques,
and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, dec
imals,
and other kinds of numbers. With the availability of calcul
ators that perform these operations quickly and accurately, the instructional

21

emphasis now is on understanding the meanings and uses of these operations, and on estimation and mental skills, rather than
solely
on the development of paper
-
and
-
pencil profici
ency.


Estimation
. Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily mastered by
children. It involves an educated guess about a quantity or an intelligent prediction of the outcome of a computati
on. The growing use
of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to mak
e that
determination is through the use of strong estimation skills. Equally important is an awareness of the m
any situations in which an
approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these judgments and use
mathematics more powerfully as a result.


Number and operation skills continue to be a critical piece of t
he school mathematics curriculum and, indeed, a very important part of
mathematics. But, there is perhaps a greater need for us to rethink our approach here than to do so for any other curriculum
component. An enlightened mathematics program for today’s ch
ildren will empower them to use all of today’s tools rather than
require them to meet yesterday’s expectations.


Standard 4.2 (Geometry And Measurement) All Students Will Develop Spatial Sense And The Ability To Use Geometric Properties,
Relationships, An
d Measurement To Model, Describe And Analyze Phenomena.


Descriptive Statement
: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve describing the
shapes we see all around us in art, nature, and the things we make
. Spatial sense, geometric modeling, and measurement can help us to
describe and interpret our physical environment and to solve problems.


Geometric Properties.

This includes identifying, describing and classifying standard geometric objects, describing
and comparing
properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjec
tures
and theorems. Also included here are such concepts as symmetry, congruence, and similarity.


Transforming Shap
es.

Analyzing how various transformations affect geometric objects allows students to enhance their spatial
sense. This includes combining shapes to form new ones and decomposing complex shapes into simpler ones. It includes the stan
dard
geometric transfor
mations of translation (slide), reflection (flip), rotation (turn), and dilation (scaling). It also includes using
tessellations and fractals to create geometric patterns.

Coordinate Geometry. Coordinate geometry provides an important connection between g
eometry and algebra. It facilitates the
visualization of algebraic relationships, as well as an analytical understanding of geometry.



22

Units of Measurement.

Measurement helps describe our world using numbers. An understanding of how we attach numbers to r
eal
-
world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowle
dge of
measurement tools and techniques are critical for students' understanding of the world around them.


Measuring Geometri
c Objects.

This area focuses on applying the knowledge and understandings of units of measurement in order to
actually perform measurement. While students will eventually apply formulas, it is important that they develop and apply stra
tegies
that derive fr
om their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement
skills, using, for example, similar triangles and trigonometry.


Students of all ages should realize that geometry and measurement are

all around them. Through study of these areas and their
applications, they should come to better understand and appreciate the role of mathematics in their lives.


Standard 4.3 (Patterns And Algebra) All Students Will Represent And Analyze Relationships A
mong Variable Quantities And Solve
Problems Involving Patterns, Functions, And Algebraic Concepts And Processes.


Descriptive Statement:

Algebra is a symbolic language used to express mathematical relationships. Students need to understand how
quantities
are related to one another, and how algebra can be used to concisely express and analyze those relationships. Modern
technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a
more
visual pe
rspective, with graphs of equations displayed on a screen. Students can then focus on understanding the relationship between
the equation and the graph, and on what the graph represents in a real
-
life situation.


Patterns.

Algebra provides the language th
rough which we communicate the patterns in mathematics. From the earliest age, students
should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to mak
e
mathematical discoveries. They should hav
e opportunities to analyze, extend, and create a variety of patterns and to use pattern
-
based
thinking to understand and represent mathematical and other real
-
world phenomena.


Functions and Relationships.

The function concept is one of the most fundament
al unifying ideas of modern mathematics. Students
begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability
to abstract
matures, students form rules, display information in a table or chart
, and write equations which express the relationships they have
observed. In high school, they use the more formal language of algebra to describe these relationships.


Modeling.

Algebra is used to model real situations and answer questions about them. Th
is use of algebra requires the ability to
represent data in tables, pictures, graphs, equations or inequalities, and rules. Modeling ranges from writing simple number
sentences

23

to help solve story problems in the primary grades to using functions to descri
be the relationship between two variables, such as the
height of a pitched ball over time. Modeling also includes some of the conceptual building blocks of calculus, such as how qu
antities
change over time and what happens in the long run (limits).


Proce
dures.

Techniques for manipulating algebraic expressions


procedures


remain important, especially for students who may
continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and app
lying
pro
perties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and
inequalities, and solving equations and inequalities.

Algebra is a gatekeeper for the future study of mathematics, science, the social s
ciences, business, and a host of other areas. In the
past, algebra has served as a filter, screening people out of these opportunities. For New Jersey to be part of the global so
ciety, it is
important that algebra play a major role in a mathematics program

that opens the gates for all students.


Standard 4.4 (Data Analysis, Probability, And Discrete Mathematics) All Students Will Develop An Understanding Of The Concept
s
And Techniques Of Data Analysis, Probability, And Discrete Mathematics, And Will Use Th
em To Model Situations, Solve
Problems, And Analyze And Draw Appropriate Inferences From Data.


Descriptive Statement:

Data analysis, probability, and discrete mathematics are important interrelated areas of applied mathematics.
Each provides students wit
h powerful mathematical perspectives on everyday phenomena and with important examples of how
mathematics is used in the modern world. Two important areas of discrete mathematics are addressed in this standard; a third
area,
iteration and recursion, is add
ressed in Standard 4.3 (Patterns and Algebra).


Data Analysis (or Statistics).

In today’s information
-
based world, students need to be able to read, understand, and interpret data in
order to make informed decisions. In the early grades, students should b
e involved in collecting and organizing data, and in presenting
it using tables, charts, and graphs. As they progress, they should gather data using sampling, and should increasingly be exp
ected to
analyze and make inferences from data, as well as to analy
ze data and inferences made by others.


Probability.

Students need to understand the fundamental concepts of probability so that they can interpret weather forecasts, avoid
unfair games of chance, and make informed decisions about medical treatments whose

success rate is provided in terms of
percentages. They should regularly be engaged in predicting and determining probabilities, often based on experiments (like f
lipping a
coin 100 times), but eventually based on theoretical discussions of probability tha
t make use of systematic counting strategies. High
school students should use probability models and solve problems involving compound events and sampling.



24

Discrete Mathematics

Systematic Listing and Counting
. Development of strategies for listing and co
unting can progress through
all grade levels, with middle and high school students using the strategies to solve problems in probability. Primary student
s, for
example, might find all outfits that can be worn using two coats and three hats; middle school s
tudents might systematically list and
count the number of routes from one site on a map to another; and high school students might determine the number of three
-
person
delegations that can be selected from their class to visit the mayor.


Discrete Mathema
tics

Vertex
-
Edge Graphs and Algorithms
. Vertex
-
edge graphs, consisting of dots (vertices) and lines joining
them (edges), can be used to represent and solve problems based on real
-
world situations. Students should learn to follow and devise
lists of instru
ctions, called “algorithms,” and use algorithmic thinking to find the best solution to problems like those involving vertex
-
edge graphs, but also to solve other problems.

These topics provide students with insight into how mathematics is used by decision
-
makers in our society, and with important tools
for modeling a variety of real
-
world situations. Students will better understand and interpret the vast amounts of quantitative data that
they are exposed to daily, and they will be able to judge the validity

of data
-
supported arguments.


Standard 4.5 (Mathematical Processes) All Students Will Use Mathematical Processes Of Problem Solving, Communication,
Connections, Reasoning, Representations, And Technology To Solve Problems And Communicate Mathematical Ide
as.


Descriptive Statement:

The mathematical processes described here highlight ways of acquiring and using the content knowledge and
skills delineated in the first four mathematics standards.


Problem Solving.

Problem posing and problem solving involve
examining situations that arise in mathematics and other disciplines
and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and us
ing a
variety of strategies to find solutions. Through problem

solving, students experience the power and usefulness of mathematics.
Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas.


Communication.

Communication of mathematical ideas involves stud
ents’ sharing their mathematical understandings in oral and
written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their under
standing
of mathematics and develop confidence in themselves as mathema
tics learners. It also enables teachers to better monitor student
progress.


Connections.

Making connections involves seeing relationships between different topics, and drawing on those relationships in future
study. This applies within mathematics, so th
at students can translate readily between fractions and decimals, or between algebra and

25

geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, a
nd the arts;
and to the everyday world, so

that students can connect school mathematics to daily life.


Reasoning.

Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the
development of mathematical reasoning, students recognize t
hat mathematics makes sense and can be understood. They learn how to
evaluate situations, select problem
-
solving strategies, draw logical conclusions, develop and describe solutions, and recognize how
those solutions can be applied.


Representations.

Repr
esentations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent
mathematical concepts and problem situations. By using various representations, students will be better able to communicate t
heir
thinking and solve probl
ems. Using multiple representations will enrich the problem solver with alternative perspectives on the
problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, b
ase ten
blocks, geoboards,
and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop
mathematics.


Technology. Calculators and computers need to be used along with other mathematical tools by students in both instructional a
nd
ass
essment activities. These tools should be used, not to replace mental math and paper
-
and
-
pencil computational skills, but to
enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts

with
calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance

in
solving real
-
world problems).


7.

New Jersey Career Education and Consumer, Family, and Life Skills Standards


Standard 9.1: (
Career And Technical Education) All Students Will Develop Career Awareness And Planning, Employability Skills,
And Foundational Knowledge Necessary For Success In The Workplace.


Descriptive Statement: All students will explore career opportunities an
d make informed choices based on aptitudes and interests.
Students will identify and pursue career goals, apply communications skills in work
-
relevant situations, demonstrate the ability to
combine ideas or information in new ways, make connections between

unrelated ideas, organize and present information, and allocate
financial and other resources efficiently and effectively. Students will identify and use various print and non
-
print resources in the
home, school, and community to seek and plan for employm
ent. They will be able to use the job application process, including
resumes, forms, and interviews.



26

Career and technical education, formerly called practical arts, is the application of life, academic, and occupational skills

demonstrated
by student
-
cen
tered experiences in courses related to the sixteen States. Career Clusters. The intent at the elementary and middle
school levels is to prepare all students for the option of further study in career and technical education at the high school

level. These
courses typically include business education, family and consumer sciences, and other courses related to careers and life ski
lls. Career
and technical education programs establish necessary pathways for secondary vocational
-
technical education programs, en
tering the
world of work, continuing education (such as college, post secondary vocational
-
technical education, specialized certification and/or
registered apprenticeships), and lifelong learning.


Those students electing courses in career and technical e
ducation should demonstrate both teamwork and problem
-
solving skills
through a structured learning experience. This could consist of an experiential, supervised educational activity designed to
provide
students with exposure to the requirements and respons
ibilities of specific job titles or job groupings, and to assist them in gaining
employment skills and making career and educational choices. The experience may be either paid or unpaid, depending on the ty
pe of
activities in which the student is involved.

Examples include, but are not limited to: apprenticeships, community service, cooperative
education, internships, job shadowing, school
-
based experiences, vocational student organizations, paid employment, and volunteer
activities. Structured learning exp
eriences must meet all state and federal child labor laws and regulations.


Standard 9.2 (Consumer, Family, And Life Skills) All Students Will Demonstrate Critical Life Skills In Order To Be Functiona
l
Members Of Society.


Descriptive Statement: All stu
dents need to develop consumer, family, and life skills necessary to be functioning members of society.
All students will develop original thoughts and ideas, think creatively, develop habits of inquiry, and take intellectual and

performance
risks. They wi
ll recognize problems, devise a variety of ways to solve these problems, analyze the potential advantages and
disadvantages of each alternative, and evaluate the effectiveness of the method ultimately selected. Students will understand

the
components of fi
nancial education and make economic choices. Students will demonstrate self
-
awareness and the ability to respond
constructively to criticism and potential conflict. In addition, students will work collaboratively with a variety of groups
and
demonstrate th
e essential components of character development and ethics, including trustworthiness, responsibility, respect, fairness,
caring, and citizenship. Students apply principles of resource management and skills that promote personal and professional w
ell
-
being
. Wellness, nutrition, child development, and human relationships are an important part of consumer, family, and life skills.

However, wellness, nutrition, and human relationship cumulative progress indicators are not listed here as it would duplicate

thos
e in
Comprehensive Health and Physical Education Standards.


27


8.

New Jersey Technological Literacy Standards


Standard 8.1 (Computer And Information Literacy) All Students Will Use Computer Applications To Gather And Organize
Information And To Solve Pro
blems.


Descriptive Statement: Using computer applications and technology tools students will conduct research, solve problems, impro
ve
learning, achieve goals, and produce products and presentations in conjunction with standards in all content areas, incl
uding career
education and consumer family, and life skills. They will also develop, locate, summarize, organize, synthesize, and evaluate

information for lifelong learning.



Standard 8.2 (Technology Education) All Students Will Develop An Understanding
Of The Nature And Impact Of Technology,
Engineering, Technological Design, And The Designed World As They Relate To The Individual, Society, And The Environment



Descriptive Statement: The following indicators are based on the Standards for Technological
Literacy (STL, 2000) and support the
National Academy of Engineering’s (2002) call for students to gain technological literacy. Students will be expected to under
stand the
various facets of technology and the design process. They will analyze and evaluate
design options and then apply the design process
to solve problems. A systems perspective is employed to emphasize the interconnectedness of all knowledge and the impact of
technology and technological change. Students will be expected to use technology as

it applies to physical systems, biological
systems, and information and communication systems. The intent at the elementary and middle school levels is that all student
s
develop technological literacy and are prepared for the option of further study in th
e field of technology education. At the elementary
level, the foundation for technology education is found in the science standards, particularly standards 5.2 and 5.4.





28

9
.

Scope and Sequence

Overview
:


1

2

3

4

5

6

7

8

9


Properties of Whole
Numbers


Algebra


Decimals


Number
Theory

10

11

12

13

14

15

16

17

18


Number
Theory



Fraction Operations


Ratio, Proportion, Percent



Geometry


19

20

21

22

23

24

25

26

27


Geometry



Measurement Area Perimeter Volume



TESTING


Integers

28

29

30

31

32

33

34

35

36


Collect & Display Data




Probability




Systems of Measurement



Problem
Solving


Discrete
Math
Activities



Submitted by
: Shawn Boehmcke and Jane Hannon


Date
: September 28, 2006



Board of Education Curriculum and Instruction
Committee:


Approved

Date
:

October 11, 2006



Boar
d of Education:


Approved

Date
: October 18, 2006