N.J. CCC Standards Mathematics (Adopted 2002)
| 4.4.
Data Analysis, Probability, and Discrete Mathematics: ALL
STUDENTS WILL
DEVELOP AN
UNDERSTANDING OF
THE CONCEPTS
AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS,
AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW
APPROPRIATE INFERENCES FROM DAT |
4.5.
Mathematical Processes: |
| Grade 2 (Strands A,B,C,D) | Grade 2 (Strands A,B,C,D,E,F) |
| Grade 3 (Strands A,B,C,D) | Grade 3 (Strands A,B,C,D,E,F) |
| Grade 4 (Strands A,B,C,D) | Grade 4 (Strands A,B,C,D,E,F) |
| Grade 5 (Strands A,B,C,D) | Grade 5 (Strands A,B,C,D,E,F) |
| Grade 6 (Strands A,B,C,D) | Grade 6 (Strands A,B,C,D,E,F) |
| Grade 7 (Strands A,B,C,D) | Grade 7 (Strands A,B,C,D,E,F) |
| Grade 8 (Strands A,B,C,D) | Grade 8 (Strands A,B,C,D,E,F) |
| Grade 12 (Strands A,B,C,D) | Grade 12 (Strands A,B,C,D,E,F) |
Cumulative
Progress Indicators
4.1
By the end of
Grade
2, students will:
Strand A.
Number Sense
1.
Use real-life experiences,
physical materials, and technology to construct meanings for numbers (unless
otherwise noted, all indicators for grade 2 pertain to these sets of numbers as
well).
·
Whole numbers through
hundreds
·
Ordinals
·
Proper fractions
(denominators of 2, 3, 4, 8, 10)
2.
Demonstrate an understanding of
whole number place value concepts.
3.
Understand that numbers have a
variety of uses.
4.
Count and perform simple
computations with coins.
·
Amounts up to $1.00 (using
cents notation)
5.
Compare and order whole numbers.
Strand B.
Numerical
Operations
1.
Develop the meanings of addition
and subtraction by concretely modeling and
discussing a large
variety of problems.
·
Joining, separating, and
comparing
2.
Explore the meanings of
multiplication and division by modeling and discussing problems.
3.
Develop proficiency with basic
addition and subtraction number facts using a variety of fact strategies (such
as .counting on. and .near doubles.) and then
commit them
to memory.
4.
Construct,
use, and
explain procedures
for performing addition and subtraction calculations with:
·
Pencil-and-paper
·
Mental math
·
Calculator
5.
Use efficient and accurate
pencil-and-paper procedures for
computation with whole numbers.
·
Addition of 2-digit
numbers
·
Subtraction of 2-digit
numbers
6.
Select pencil-and-paper, mental
math, or a calculator as the appropriate computational
method in a given situation depending on the context and numbers.
7.
Check the reasonableness of
results of computations.
8.
Understand and use the inverse
relationship between addition and subtraction.
C.
Estimation
1.
Judge without counting whether a
set of objects has less than, more than, or the same number of objects as a
reference set.
2.
Determine
the reasonableness
of an
answer by
estimating the
result of
computations (e.g., 15 + 16 is not 211).
3.
Explore
a variety
of strategies for estimating
both quantities (e.g., the
number of marbles in a jar)
and results of computation.
BuilB
4.1
Building
upon
knowledge and skills gained in preceding grades, by the end of
Grade 3, students will:
Strand A.
Number Sense
1.
Use real-life experiences,
physical materials, and technology to construct meanings for numbers (unless
otherwise noted, all indicators for grade 3 pertain to these sets of numbers as
well).
·
Whole numbers through
hundred thousands
·
Commonly used fractions
(denominators of 2, 3, 4, 5, 6, 8, 10) as part of a whole, as a subset of a set,
and as a location on a number line
2.
Demonstrate an understanding of
whole number place value concepts.
3.
Identify whether any whole number
is odd or even.
4.
Explore the extension of the
place value system to decimals through hundredths.
5.
Understand the various uses of
numbers.
·
Counting, measuring,
labeling (e.g., numbers on baseball uniforms)
6.
Compare and order numbers.
Strand B.
Numerical
Operations
1.
Develop the meanings of the four
basic arithmetic operations by modeling and discussing a large variety of
problems.
·
Addition and subtraction:
joining, separating, comparing
·
Multiplication:
repeated addition, area/array
·
Division:
repeated subtraction, sharing
2.
Develop proficiency with basic
multiplication and division number facts using a variety of fact strategies
(such as .skip counting. and .repeated subtraction.).
3.
Construct, use, and explain
procedures for performing whole number calculations with:
·
Pencil-and-paper
·
Mental math
·
Calculator
4.
Use efficient and accurate
pencil-and-paper procedures for
computation with whole numbers.
·
Addition of 3-digit
numbers
·
Subtraction of 3-digit
numbers
·
Multiplication of 2-digit
numbers by 1-digit numbers
5.
Count and perform simple
computations with money.
·
Cents notation (¢)
6.
Select pencil-and-paper, mental
math, or a calculator as the appropriate computational method in a given
situation depending on the context and numbers.
7.
Check the reasonableness of
results of computations.
C.
Estimation
1.
Judge without counting whether a
set of objects has less than, more than, or the same number of objects as a
reference set.
2.
Construct and use a variety of
estimation strategies (e.g., rounding and mental math) for estimating both
quantities and the result of computations.
3.
Recognize
when an
estimate is
appropriate, and
understand the
usefulness of
an estimate as distinct from an exact answer.
4.
Use estimation to determine
whether the result of a computation (either by calculator or by hand) is
reasonable.
Build
4.1
Building upon
knowledge and skills gained in preceding grades, by the end of
Grade 4, students will:
Strand A.
Number Sense
1.
Use real-life experiences,
physical materials, and technology to construct meanings for numbers (unless
otherwise noted, all indicators for grade 4 pertain to these sets of numbers as
well).
·
Whole numbers through
millions
·
Commonly used fractions
(denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole, as a
subset of a set, and as a location on a number line
·
Decimals through
hundredths
2.
Demonstrate an understanding of
place value concepts.
3.
Demonstrate a sense of the
relative magnitudes of numbers.
4.
Understand the various uses of
numbers.
·
Counting, measuring,
labeling (e.g., numbers on baseball uniforms), locating (e.g., Room 235 is on
the second floor)
5.
Use concrete and pictorial models
to relate whole numbers, commonly used fractions, and decimals to each other,
and to represent equivalent forms of the same number.
6.
Compare and order numbers.
7.
Explore settings that give rise
to negative numbers.
·
Temperatures below 0o,
debts
·
Extension of the number
line
Strand B.
Numerical
Operations
1.
Develop the meanings of the four
basic arithmetic operations by modeling and discussing a large variety of
problems.
·
Addition and subtraction:
joining, separating, comparing
·
Multiplication:
repeated addition, area/array
·
Division: repeated
subtraction, sharing
2.
Develop proficiency with basic
multiplication and division number facts using a variety of fact strategies
(such as .skip counting. and .repeated subtraction.) and then commit them to
memory.
3.
Construct, use, and explain
procedures for performing whole number calculations and with:
·
Pencil-and-paper
·
Mental math
·
Calculator
4.
Use
efficient and
accurate pencil-and-paper
procedures for
computation with
whole numbers.
·
Addition of 3-digit
numbers
·
Subtraction of 3-digit
numbers
·
Multiplication of 2-digit
numbers
·
Division of 3-digit
numbers by 1-digit numbers
5.
Construct and use procedures for
performing decimal addition and subtraction.
6.
Count and perform simple
computations with money.
·
Standard dollars and cents
notation
7.
Select pencil-and-paper, mental
math, or a calculator as the appropriate computational method in a given
situation depending on the context and numbers.
8.
Check the reasonableness of
results of computations.
9.
Use concrete models to explore
addition and subtraction with fractions.
10.
Understand and use the inverse
relationships between addition and subtraction and
between multiplication and division.
C.
Estimation
1.
Judge without counting whether a
set of objects has less than, more than, or the same number of objects as a
reference set.
2.
Construct and use a variety of
estimation strategies (e.g., rounding and mental math) for estimating both
quantities and the results of computations.
3.
Recognize
when an
estimate is
appropriate, and
understand the
usefulness of
an estimate as distinct from an exact answer.
4.
Use estimation to determine
whether the result of a computation (either by calculator or by hand) is
reasonable.
4.1
Building
upon
knowledge and skills gained in preceding grades, by the end of
Grade 5, students
will:
Strand A.
Number Sense
1.
Use real-life experiences,
physical materials, and technology to construct meanings for numbers (unless
otherwise noted, all indicators for grade 5 pertain to these sets of numbers as
well).
·
All fractions as part of a
whole, as subset of a set, as a location on a number line, and as divisions of
whole numbers
·
All decimals
2.
Recognize the decimal nature of
United States currency and compute with money.
3.
Demonstrate a sense of the
relative magnitudes of numbers.
4.
Use whole numbers, fractions, and
decimals to represent equivalent forms of the same
number.
5.
Develop and apply number theory
concepts in problem solving situations.
·
Primes, factors, multiples
6.
Compare and order numbers.
Strand B.
Numerical
Operations
1.
Recognize the appropriate use of
each arithmetic operation in problem situations.
2.
Construct, use, and explain
procedures for performing
addition and
subtraction with
fractions and
decimals with:
·
Pencil-and-paper
·
Mental math
·
Calculator
3.
Use
an efficient
and accurate
pencil-and-paper procedure
for division
of a
3-digit number by a 2-digit number.
4.
Select pencil-and-paper, mental
math, or a calculator as the appropriate computational method in a given
situation depending on the context and numbers.
5.
Check the reasonableness of
results of computations.
6.
Understand
and use
the various
relationships among
operations and properties of operations.
C.
Estimation
1.
Use a variety of estimation
strategies for both number and computation.
2.
Recognize
when an estimate
is appropriate,
and understand
the usefulness
of an estimate as distinct
from an exact answer.
3.
Determine the reasonableness of
an answer by estimating the result of operations.
4.
Determine whether a given
estimate is an overestimate or an underestimate.
Bu
4.1
Building upon
knowledge and skills gained in preceding grades, by the end of Grade
6, students will:
Strand A.
Number Sense
1.
Use real-life experiences,
physical materials, and technology to construct meanings for numbers (unless
otherwise noted, all indicators for grade 6 pertain to these sets of numbers as
well).
·
All integers
·
All fractions as part of a
whole, as subset of a set, as a location on a number line, and as divisions of
whole numbers
·
All decimals
2.
Recognize the decimal nature of
United States currency and compute with money.
3.
Demonstrate a sense of the
relative magnitudes of numbers.
4.
Explore the use of ratios and
proportions in a variety of situations.
5.
Understand
and use
whole-number percents between 1 and 100
in a variety of situations.
6.
Use whole numbers, fractions, and
decimals to represent equivalent forms of the same number.
7.
Develop and apply number theory
concepts in problem solving situations.
·
Primes, factors, multiples
·
Common multiples, common
factors
8.
Compare and order numbers.
Strand B.
Numerical
Operations
1.
Recognize the appropriate use of
each arithmetic operation in problem situations.
2.
Construct, use, and explain
procedures for performing
calculations with fractions
and decimals with:
·
Pencil-and-paper
·
Mental math
·
Calculator
3.
Use
an efficient
and accurate
pencil-and-paper procedure
for division
of a
3-digit number by a 2-digit number.
4.
Select pencil-and-paper, mental
math, or a calculator as the appropriate computational method in a given
situation depending on the context and numbers.
5.
Find squares and cubes of whole
numbers.
6.
Check the reasonableness of
results of computations.
7.
Understand
and use
the various
relationships among
operations and properties of operations.
8.
Understand
and apply
the standard algebraic order
of operations for
the four
basic operations, including appropriate use of parentheses.
C.
Estimation
1.
Use a variety of strategies for
estimating both quantities and the results of computations.
2.
Recognize
when an
estimate is
appropriate, and
understand the
usefulness of
an estimate as distinct from an exact answer.
3.
Determine the reasonableness of
an answer by estimating the result of operations.
4.
Determine whether a given
estimate is an overestimate or an underestimate.
Buildi
4.1
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 7, students will:
Strand A.
Number Sense
1.
Extend understanding of the
number system by constructing meanings for the following (unless otherwise noted, all indicators for
grade 7 pertain to these sets of numbers as well):
·
Rational numbers
·
Percents
·
Whole numbers with
exponents
2.
Demonstrate a sense of the
relative magnitudes of numbers.
3.
Understand
and use
ratios, proportions,
and percents
(including percents greater
than 100 and less than 1) in a variety of situations.
4.
Compare and order numbers of all
named types.
5.
Use whole numbers, fractions,
decimals, and percents to represent equivalent forms of the same number.
6.
Understand that all fractions can
be represented as repeating or terminating decimals.
Strand B.
Numerical
Operations
1.
Use
and explain
procedures for
performing calculations with integers
and all
number types named above with:
·
Pencil-and-paper
·
Mental math
·
Calculator
2.
Use exponentiation to find whole
number powers of numbers.
3.
Understand and apply the standard
algebraic order of operations, including appropriate use of parentheses.
C.
Estimation
1.
Use equivalent representations of
numbers such as fractions, decimals, and percents to facilitate estimation.
Buildi
4.1
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 8, students will:
Strand A.
Number Sense
1.
Extend understanding of the
number system by constructing meanings for the following (unless otherwise noted, all indicators for
grade 8 pertain to these sets of numbers as well):
·
Rational numbers
·
Percents
·
Exponents
·
Roots
·
Absolute values
·
Numbers represented in
scientific notation
2.
Demonstrate a sense of the
relative magnitudes of numbers.
3.
Understand
and use
ratios, proportions,
and percents
(including percents greater
than 100 and less than 1) in a variety of situations.
4.
Compare and order numbers of all
named types.
5.
Use whole numbers, fractions,
decimals, and percents to represent equivalent forms of the same number.
6.
Recognize that repeating decimals
correspond to fractions and determine their fractional
equivalents.
·
5/7 = 0. 714285714285.
= 0. 714285
7.
Construct meanings for common
irrational numbers, such as ð
(pi) and the square root of 2.
Strand B.
Numerical
Operations
1.
Use and explain procedures for
performing calculations involving addition, subtraction, multiplication,
division, and exponentiation with integers and all number types named above
with:
·
Pencil-and-paper
·
Mental math
·
Calculator
2.
Use exponentiation to find whole
number powers of numbers.
3.
Find square and cube roots of
numbers and understand the inverse nature of powers and roots.
4.
Solve problems involving
proportions and percents.
5.
Understand and apply the standard
algebraic order of operations, including appropriate use of parentheses.
C.
Estimation
1.
Estimate square and cube roots of
numbers.
2.
Use equivalent representations of
numbers such as fractions, decimals, and percents to facilitate estimation.
3.
Recognize
the limitations
of estimation
and assess the amount of
error resulting from estimation.
Build
4.1
Building
upon knowledge
and skills
gained in
preceding grades,
by the
end of Grade
12, students will:
Strand A.
Number Sense
1.
Extend understanding of the
number system to all real numbers.
2.
Compare and order rational and
irrational numbers.
3.
Develop conjectures and informal
proofs of properties of number systems
and sets
of numbers.
Strand B.
Numerical
Operations
1.
Extend understanding and use of
operations to real numbers and algebraic procedures.
2.
Develop,
apply, and
explain methods
for solving
problems involving
rational and negative
exponents.
3.
Perform operations on matrices.
·
Addition and subtraction
·
Scalar multiplication
4.
Understand and apply the laws of
exponents to simplify expressions involving numbers raised to powers.
C.
Estimation
1.
Recognize
the limitations
of estimation,
assess the amount of error resulting from estimation, and determine
whether the error is within acceptable tolerance limits.
Cumulative
Progress Indicators
By
t 4.2 By
the end of Grade
2, students will:
Strand A.
Geometric
Properties
1.
Identify
and describe
spatial relationships
among objects
in space
and their
relative shapes and sizes.
·
Inside/outside,
left/right, above/below, between
·
Smaller/larger/same size,
wider/ narrower, longer/shorter
·
Congruence (i.e., same
size and shape)
2.
Use
concrete objects,
drawings, and computer
graphics to
identify, classify,
and describe standard three-dimensional and two-dimensional shapes.
·
Vertex, edge, face, side
·
3D figures . cube,
rectangular prism, sphere, cone, cylinder, and pyramid
·
2D figures . square,
rectangle, circle, triangle
·
Relationships
between three-
and two-dimensional shapes (i.e.,
the face of a 3D shape is a 2D shape)
3.
Describe, identify and create
instances of line symmetry.
4.
Recognize, describe, extend and
create designs and patterns with geometric objects of different shapes and
colors.
Strand B.
Transforming
Shapes
1.
Use simple shapes to make
designs, patterns, and pictures.
2.
Combine and subdivide simple
shapes to make other shapes.
C.
Coordinate
Geometry
1.
Give and follow directions for
getting from one point to another on a map or grid.
D.
Units of
Measurement
1.
Directly compare and order
objects according to measurable attributes.
·
Attributes . length,
weight, capacity, time, temperature
2.
Recognize the need for a uniform
unit of measure.
3.
Select
and use
appropriate standard
and non-standard units of
measure and
standard measurement tools to solve real-life problems.
·
Length . inch, foot, yard,
centimeter, meter
·
Weight . pound, gram,
kilogram
·
Capacity . pint, quart,
liter
·
Time . second, minute,
hour, day, week, month, year
·
Temperature . degrees
Celsius, degrees Fahrenheit
4.
Estimate measures.
E.
Measuring
Geometric Objects
1.
Directly measure the perimeter of
simple two-dimensional shapes.
2.
Directly
measure the
area of
simple two-dimensional shapes by
covering them
with squares.
Buildi 4.2 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 3, students will:
Strand A.
Geometric
Properties
1.
Identify and describe spatial
relationships of two or more objects in space.
·
Direction, orientation,
and perspectives (e.g., which object is on your left when you are standing
here?)
·
Relative shapes and sizes
2.
Use
properties of
standard three-dimensional
and two-dimensional shapes to
identify, classify, and describe them.
·
Vertex, edge, face, side,
angle
·
3D figures . cube,
rectangular prism, sphere, cone, cylinder, and pyramid
·
2D figures . square,
rectangle, circle, triangle, pentagon, hexagon, octagon
3.
Identify and describe
relationships among two-dimensional shapes.
·
Same size, same shape
·
Lines of symmetry
4.
Understand and apply concepts
involving lines, angles, and circles.
·
Line, line segment,
endpoint
5.
Recognize, describe, extend, and
create space-filling patterns.
Strand B.
Transforming
Shapes
1.
Describe and use geometric
transformations (slide, flip, turn).
2.
Investigate the occurrence of
geometry in nature and art.
C.
Coordinate
Geometry
1.
Locate and name points in the
first quadrant on a coordinate grid.
D.
Units of
Measurement
1.
Understand
that everyday
objects have a variety of attributes, each of which can be measured in
many ways.
2.
Select
and use
appropriate standard
units of
measure and
measurement tools
to solve real-life problems.
·
Length . fractions of an
inch (1/4, 1/2), mile, decimeter, kilometer
·
Area . square inch, square
centimeter
·
Weight . ounce
·
Capacity . fluid ounce,
cup, gallon, milliliter
3.
Incorporate estimation in
measurement activities (e.g., estimate before measuring).
E.
Measuring
Geometric Objects
1.
Determine the area of simple
two-dimensional shapes on a square grid.
2.
Determine the perimeter of simple
shapes by measuring all of the sides.
3.
Measure and compare the volume of
three.dimensional objects using materials such as rice or cubes.
Buildi
4.2 Building upon knowledge and skills gained in preceding grades, by the end of Grade
4, students will:
Strand A.
Geometric
Properties
1.
Identify and describe spatial
relationships of two or more objects in space.
·
Direction, orientation,
and perspectives (e.g., which object is on your left when you are standing
here?)
·
Relative shapes and sizes
·
Shadows (projections) of
everyday objects
2.
Use
properties of
standard three-dimensional
and two-dimensional shapes to
identify,
classify, and
describe them.
·
Vertex, edge, face, side,
angle
·
3D figures . cube,
rectangular prism, sphere, cone, cylinder, and pyramid
·
2D
figures .
square, rectangle, circle,
triangle, quadrilateral, pentagon, hexagon, octagon
·
Inclusive relationships .
squares are rectangles, cubes are rectangular prisms
3.
Identify and describe
relationships among two-dimensional shapes.
·
Congruence
·
Lines of symmetry
4.
Understand and apply concepts
involving lines, angles, and circles.
·
Point, line, line segment,
endpoint
·
Parallel, perpendicular
·
Angles . acute, right,
obtuse
·
Circles . diameter,
radius, center
5.
Recognize, describe, extend, and
create space-filling patterns.
B.
Transforming
Shapes
1.
Use simple shapes to cover an
area (tessellations).
2.
Describe and use geometric
transformations (slide, flip, turn).
3.
Investigate the occurrence of
geometry in nature and art.
C.
Coordinate
Geometry
1.
Locate and name points in the
first quadrant on a coordinate grid.
2.
Use coordinates to give or follow
directions from one point to another on a map or grid.
D.
Units of
Measurement
1.
Understand
that everyday
objects have
a variety of
attributes, each
of which
can be measured in many
ways.
2.
Select
and use
appropriate standard
units of
measure and
measurement tools
to solve real-life problems
·
Length . fractions of an
inch (1/8, 1/4, 1/2), mile, decimeter, kilometer
·
Area . square inch, square
centimeter
·
Volume . cubic inch, cubic
centimeter
·
Weight . ounce
·
Capacity . fluid ounce,
cup, gallon, milliliter
3.
Develop and use personal
referents to approximate standard
units of
measure (e.g.,
a common paper clip is about an inch long).
4.
Incorporate estimation in
measurement activities (e.g., estimate before measuring).
5.
Solve problems involving elapsed
time.
E.
Measuring
Geometric Objects
1.
Determine the area of simple
two-dimensional shapes on a square grid.
2.
Distinguish between perimeter and
area and use each appropriately in problem-solving situations.
3.
Measure and compare the volume of
three.dimensional objects using materials such as rice or cubes.
Buildi 4.2 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 5, students will:
Strand A.
Geometric
Properties
1.
Understand and apply concepts
involving lines and angles.
·
Notation for line, ray,
angle, line segment
·
Properties of parallel,
perpendicular, and intersecting lines
·
Sum of the measures of the
interior angles of a triangle is 180°
2.
Identify, describe, compare, and
classify polygons.
·
Triangles by angles and
sides
·
Quadrilaterals, including
squares, rectangles, parallelograms, trapezoids, rhombi
·
Polygons by number of
sides
·
Equilateral, equiangular,
regular
·
All points equidistant
from a given point form a circle
3.
Identify similar figures.
4.
Understand and apply the concepts
of congruence and symmetry (line and rotational).
Strand B.
Transforming
Shapes
1.
Use a translation, a reflection,
or a rotation to map one figure onto another congruent
figure.
2.
Recognize, identify, and describe
geometric relationships and properties as they exist in nature, art, and other
real-world settings.
C.
Coordinate
Geometry
1.
Create geometric shapes with
specified properties in the first quadrant on a coordinate grid.
D.
Units of
Measurement
1.
Select and use appropriate units
to measure angles and area.
2.
Convert measurement units within
a system (e.g., 3 feet = ___ inches).
3.
Know
approximate equivalents
between the standard and
metric systems (e.g., one kilometer is approximately 6/10 of a mile).
4.
Use measurements and estimates to
describe and compare phenomena.
E.
Measuring
Geometric Objects
1.
Use
a protractor to measure angles.
2.
Develop and apply strategies and
formulas for finding perimeter and area.
·
Square
·
Rectangle
3.
Recognize
that rectangles
with the
same perimeter
do not necessarily have
the same area and vice
versa.
4.
Develop informal ways of
approximating the measures of familiar objects (e.g., use a grid to approximate
the area of the bottom of one’s foot).
Buildi 4.2 Building upon knowledge and skills gained in preceding grades, by the end of Grade
6, students will:
Strand A.
Geometric
Properties
1.
Understand and apply concepts
involving lines and angles.
·
Notation for line, ray,
angle, line segment
·
Properties of parallel,
perpendicular, and intersecting lines
·
Sum of the measures of the
interior angles of a triangle is 180°
2.
Identify, describe, compare, and
classify polygons and circles.
·
Triangles by angles and
sides
·
Quadrilaterals, including
squares, rectangles, parallelograms, trapezoids, rhombi
·
Polygons by number of
sides.
·
Equilateral, equiangular,
regular
·
All points equidistant
from a given point form a circle
3.
Identify similar figures.
4.
Understand and apply the concepts
of congruence and symmetry (line and rotational).
5.
Compare properties of cylinders,
prisms, cones, pyramids, and spheres.
6.
Identify,
describe, and
draw the
faces or
shadows (projections) of
three-dimensional geometric objects from different perspectives.
7.
Identify a three-dimensional
shape with given projections (top, front and side views).
8.
Identify a three-dimensional
shape with a given net (i.e., a flat pattern that folds into a 3D shape).
Strand B.
Transforming
Shapes
1.
Use a translation, a reflection,
or a rotation to map one figure onto another congruent
figure.
2.
Recognize, identify, and describe
geometric relationships and properties as they exist in nature, art, and other
real-world settings.
C.
Coordinate
Geometry
1.
Create geometric shapes with
specified properties in the first quadrant on a coordinate grid.
D.
Units of
Measurement
1.
Select and use appropriate units
to measure angles, area, surface area, and volume.
2.
Use a scale to find a distance on
a map or a length on a scale drawing.
3.
Convert measurement units within
a system (e.g., 3 feet = ___ inches).
4.
Know
approximate equivalents
between the standard and
metric systems (e.g., one kilometer is approximately 6/10 of a mile).
5.
Use measurements and estimates to
describe and compare phenomena.
E.
Measuring
Geometric Objects
1.
Use a protractor to measure
angles.
2.
Develop and apply strategies and
formulas for finding perimeter and area.
·
Triangle, square,
rectangle, parallelogram, and trapezoid
·
Circumference and area of
a circle
3.
Develop and apply strategies and
formulas for finding the surface area and volume of rectangular prisms and
cylinders.
4.
Recognize that shapes with the
same perimeter do not necessarily have the same area and vice versa.
5.
Develop informal ways of
approximating the measures of familiar objects (e.g., use a grid to approximate
the area of the bottom of one.s foot).
Buildi
4.2 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 7, students will:
Strand A.
Geometric
Properties
1.
Understand and apply properties
of polygons.
·
Quadrilaterals, including
squares, rectangles, parallelograms, trapezoids, rhombi
·
Regular polygons
2.
Understand and apply the concept
of similarity.
·
Using proportions to find
missing measures
·
Scale drawings
·
Models of 3D objects
3.
Use logic and reasoning to make
and support conjectures about geometric objects.
Strand B.
Transforming
Shapes
1.
Understand and apply
transformations.
·
Finding the image, given
the pre-image, and vice-versa
·
Sequence of
transformations needed to map one figure onto another
·
Reflections, rotations,
and translations result in images congruent to the pre-image
·
Dilations
(stretching/shrinking) result in images similar to the pre-image
C.
Coordinate
Geometry
1.
Use coordinates in four quadrants
to represent geometric concepts.
2.
Use
a coordinate
grid to
model and
quantify transformations (e.g., translate right 4 units).
D.
Units of
Measurement
1.
Solve
problems requiring
calculations that involve
different units
of measurement within a
measurement system (e.g., 4.3. plus 7.10. equals 12.1.).
2.
Select and use appropriate
units and tools to measure quantities to the degree of precision needed
in a particular problem-solving situation.
3.
Recognize that all measurements
of continuous quantities are approximations.
E.
Measuring
Geometric Objects
1.
Develop and apply strategies for
finding perimeter and area.
·
Geometric
figures made by combining
triangles, rectangles
and circles
or parts
of circles
·
Estimation of area using
grids of various sizes
2.
Recognize that the volume of a
pyramid or cone is one-third of the volume of the prism or cylinder with the
same base and height (e.g., use rice to compare volumes of figures with same
base and height).
Buildi 4.2 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 8, students will:
Strand A.
Geometric
Properties
1.
Understand and apply concepts
involving lines, angles, and planes.
·
Complementary and
supplementary angles
·
Vertical angles
·
Bisectors and
perpendicular bisectors
·
Parallel, perpendicular,
and intersecting planes
·
Intersection of plane with
cube, cylinder, cone, and sphere
2.
Understand and apply the
Pythagorean theorem.
3.
Understand and apply properties
of polygons.
·
Quadrilaterals, including
squares, rectangles, parallelograms, trapezoids, rhombi
·
Regular polygons
·
Sum of measures of
interior angles of a polygon
·
Which polygons can be used
alone to generate a tessellation and why
4.
Understand and apply the concept
of similarity.
·
Using proportions to find
missing measures
·
Scale drawings
·
Models of 3D objects
5.
Use logic and reasoning to make
and support conjectures about geometric objects.
Strand B.
Transforming
Shapes
1.
Understand and apply
transformations.
·
Finding the image, given
the pre-image, and vice-versa
·
Sequence of
transformations needed to map one figure onto another
·
Reflections, rotations,
and translations result in images congruent to the pre-image
·
Dilations
(stretching/shrinking) result in images similar to the pre-image
2.
Use iterative procedures to
generate geometric patterns.
·
Fractals (e.g., the Koch
Snowflake)
·
Self-similarity
·
Construction of initial
stages
·
Patterns in successive
stages (e.g., number of triangles in each stage of Sierpinski.s Triangle)
C.
Coordinate
Geometry
1.
Use coordinates in four quadrants
to represent geometric concepts.
2.
Use
a coordinate
grid to
model and
quantify transformations (e.g., translate right 4 units).
D.
Units of
Measurement
1.
Solve
problems requiring
calculations that involve
different units
of measurement within a
measurement system (e.g., 4.3. plus 7.10. equals 12.1.).
2.
Use
approximate equivalents
between standard
and metric
systems to
estimate measurements (e.g., 5 kilometers is about 3 miles).
3.
Recognize
that the
degree of
precision needed in
calculations depends on how the results will be used and the instruments used to
generate the measurements.
4.
Select and use appropriate
units and tools to measure quantities to the degree of precision needed
in a particular problem-solving situation.
5.
Recognize that all measurements
of continuous quantities are approximations.
6.
Solve
problems that
involve compound
measurement units,
such as
speed (miles
per hour), air pressure (pounds per square inch), and population density
(persons per square mile).
E.
Measuring
Geometric Objects
1.
Develop and apply strategies for
finding perimeter and area.
·
Geometric
figures made by combining
triangles, rectangles
and circles
or parts
of circles
·
Estimation of area using
grids of various sizes
·
Impact of a dilation on
the perimeter and area of a 2-dimensional figure
2.
Recognize that the volume of a
pyramid or cone is one-third of the volume of the prism or cylinder with the
same base and height (e.g., use rice to compare volumes of figures with same
base and height).
3.
Develop and apply strategies and
formulas for finding the surface area and volume of a thre-dimensional figure.
·
Volume - prism, cone,
pyramid
·
Surface
area -
prism (triangular
or rectangular
base), pyramid
(triangular or rectangular
base)
·
Impact of a dilation on
the surface area and volume of a three-dimensional figure
4.
Use formulas to find the volume
and surface area of a sphere.
4.2 Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
Strand A.
Geometric
Properties
1.
Use
geometric models
to represent
real-world situations
and objects
and to
solve problems using
those models
(e.g., use
Pythagorean Theorem
to decide
whether an object can fit
through a doorway).
2.
Draw perspective views of 3D
objects on isometric dot paper, given 2D representations (e.g., nets or
projective views).
3.
Apply the properties of geometric
shapes.
·
Parallel lines .
transversal, alternate interior angles, corresponding angles
·
Triangles
a.
Conditions for congruence
b.
Segment joining midpoints of two
sides is parallel to and half the length of the third side
c.
Triangle Inequality
·
Minimal conditions for a
shape to be a special quadrilateral
·
Circles . arcs, central
and inscribed angles, chords, tangents
·
Self-similarity
4.
Use reasoning and some form of
proof to verify or refute conjectures and theorems.
·
Verification or refutation
of proposed proofs
·
Simple proofs involving
congruent triangles
·
Counterexamples to
incorrect conjectures
Strand B.
Transforming
Shapes
1.
Determine,
describe, and
draw the
effect of
a transformation,
or a
sequence of transformations,
on a geometric or algebraic object, and, conversely, determine whether and how
one object can be transformed to another by a transformation or a sequence of
transformations.
2.
Recognize
three-dimensional figures
obtained through
transformations of
two-dimensional figures (e.g., cone as rotating an isosceles triangle
about an altitude), using software as an aid to visualization.
3.
Determine whether two or more
given shapes can be used to generate a tessellation.
4.
Generate and analyze iterative
geometric patterns.
·
Fractals (e.g.,
Sierpinski.s Triangle)
·
Patterns in areas and
perimeters of self-similar figures
·
Outcome of extending
iterative process indefinitely
C.
Coordinate
Geometry
1.
Use coordinate geometry to
represent and verify properties of lines.
·
Distance between two
points
·
Midpoint and slope of a
line segment
·
Finding the intersection
of two lines
·
Lines with the same slope
are parallel
·
Lines that are
perpendicular have slopes whose product is .1
2.
Show position and represent
motion in the coordinate plane using vectors.
·
Addition and subtraction
of vectors
D.
Units of
Measurement
1.
Understand and use the concept of
significant digits.
2.
Choose
appropriate tools
and techniques
to achieve the specified
degree of precision and error needed in a situation.
·
Degree of accuracy of a
given measurement tool
·
Finding the interval in
which a computed measure (e.g., area or volume) lies, given the degree of
precision of linear measurements
E.
Measuring
Geometric Objects
1.
Use techniques of indirect
measurement to represent and solve problems.
·
Similar triangles
·
Pythagorean theorem
·
Right triangle
trigonometry (sine, cosine, tangent)
2.
Use a variety of strategies to
determine perimeter and area of plane figures and surface area and volume of 3D
figures.
·
Approximation of area
using grids of different sizes
·
Finding which shape has
minimal (or maximal) area, perimeter, volume, or surface area
under given
conditions using graphing calculators,
dynamic geometric software,
and/or spreadsheets
·
Estimation of area,
perimeter, volume, and surface area
STANDARD
4.3 PATTERNS AND ALGEBRA:
ALL STUDENTS
WILL
Cumulative
Progress Indicators
By
4.3 By
the end of Grade
2, students will:
Strand A.
Patterns
1.
Recognize, describe, extend, and
create patterns.
·
Using concrete materials (manipulatives),
pictures, rhythms, & whole numbers
·
Descriptions using words
and symbols (e.g., .add two. or .+ 2.)
·
Repeating patterns
·
Whole
number patterns
that grow
or shrink
as a
result of
repeatedly adding
or subtracting a fixed number (e.g., skip counting forward or backward)
Strand B.
Functions and
Relationships
1.
Use concrete and pictorial models
of function machines to explore the basic concept of a function.
C.
Modeling
1.
Recognize and describe changes
over time (e.g., temperature, height).
2.
Construct and solve simple open
sentences involving addition or subtraction.
·
Result unknown (e.g., 6 .
2 = __
or n = 3 + 5)
·
Part unknown (e.g., 3 + !
= 8)
D.
Procedures
1.
Understand and apply (but don.t
name) the following properties of addition:
·
Commutative (e.g., 5 + 3 =
3 + 5)
·
Zero as the identity
element (e.g., 7 + 0 = 7)
·
Associative (e.g., 7 + 3 +
2 can be found by first adding either 7 + 3 or 3 + 2)
Bui
4.3 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 3, students will:
Strand A.
Patterns
1.
Recognize, describe, extend, and
create patterns.
·
Descriptions using words
and number sentences/expressions
·
Whole
number patterns
that grow
or shrink as a result of
repeatedly adding, subtracting, multiplying by, or dividing by a fixed number
(e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)
Strand B.
Functions and
Relationships
1.
Use concrete and pictorial models
to explore the basic concept of a function.
·
Input/output tables,
T-charts
C.
Modeling
1.
Recognize and describe change in
quantities.
·
Graphs representing change
over time (e.g., temperature, height)
2.
Construct
and solve
simple open
sentences involving
addition or
subtraction (e.g., 3 + 6 =
__, n = 15 . 3,
3 + __ = 3, 16 . c = 7).
D.
Procedures
1.
Understand and apply the
properties of operations and numbers.
·
Commutative (e.g., 3 x 7 = 7 x
3)
·
Identity element for
multiplication is 1 (e.g., 1 x
8 = 8)
·
Any number multiplied by
zero is zero
2.
Understand and use the concepts
of equals, less than, and greater than to describe relations between numbers.
·
Symbols ( = , < , >
)
B
4.3 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 4, students will:
Strand A.
Patterns
1.
Recognize, describe, extend, and
create patterns.
·
Descriptions
using words,
number sentences/expressions,
graphs, tables, variables (e.g., shape, blank, or letter)
·
Sequences that stop or
that continue infinitely
·
Whole
number patterns
that grow
or shrink as a result of
repeatedly adding, subtracting, multiplying by, or dividing by a fixed number
(e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)
·
Sequences can often be
extended in more than one way (e.g., the next term after 1, 2, 4, . . . could be
8, or 7, or . )
Strand B.
Functions and
Relationships
1.
Use concrete and pictorial models
to explore the basic concept of a function.
·
Input/output tables,
T-charts
·
Combining two function
machines
·
Reversing a function
machine
C.
Modeling
1.
Recognize and describe change in
quantities.
·
Graphs representing change
over time (e.g., temperature, height)
·
How
change in
one physical quantity
can produce
a corresponding change in
another (e.g., pitch of a sound depends on the rate of vibration)
2.
Construct and solve simple open
sentences involving any one operation (e.g., 3 x
6 = __, n = 15 ÷
3, 3 x __ = 0,
16 . c = 7).
D.
Procedures
1.
Understand, name, and apply the
properties of operations and numbers.
·
Commutative (e.g., 3 x 7 = 7 x
3)
·
Identity element for
multiplication is 1 (e.g., 1 x
8 = 8)
·
Associative (e.g., 2 x 4 x
25 can be found by first multiplying either 2 x
4 or 4 x 25)
·
Division by zero is
undefined
·
Any number multiplied by
zero is zero.
2.
Understand and use the concepts
of equals, less than, and greater than in simple number sentences.
·
Symbols ( = , < , >
)
Buildi
4.3 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 5, students will:
Strand A.
Patterns
1.
Recognize, describe, extend, and
create patterns involving whole numbers.
·
Descriptions using tables,
verbal rules, simple equations, and graphs
Strand B.
Functions &
Relationships
1.
Describe
arithmetic operations as
functions, including combining
operations and reversing
them.
2.
Graph
points satisfying
a function
from T-charts,
from verbal
rules, and
from simple equations.
C.
Modeling
1.
Use number sentences to model
situations.
·
Using variables to
represent unknown quantities
·
Using
concrete materials,
tables, graphs, verbal
rules, algebraic
expressions/equations
2.
Draw freehand sketches of graphs
that model real phenomena and use such graphs to predict and interpret events.
·
Changes over time
·
Rates
of change
(e.g., when
is plant
growing slowly/rapidly,
when is
temperature dropping most rapidly/slowly)
D.
Procedures
1.
Solve simple linear equations
with manipulatives and informally
·
Whole-number coefficients
only, answers also whole numbers
·
Variables on one side of
equation
Buildi
4.3 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 6, students will:
Strand A.
Patterns
1.
Recognize, describe, extend, and
create patterns involving whole numbers and rational numbers.
·
Descriptions using tables,
verbal rules, simple equations, and graphs
·
Formal iterative formulas
(e.g., NEXT = NOW * 3)
·
Recursive patterns,
including Pascal.s Triangle (where each entry is the sum of the entries
above it) and
the Fibonacci
Sequence: 1,
1, 2, 3, 5,
8, . . . (where
NEXT = NOW + PREVIOUS)
Strand B.
Functions and
Relationships
1.
Describe the general behavior of
functions given by formulas or verbal rules (e.g., graph to determine whether
increasing or decreasing, linear or not).
C.
Modeling
1.
Use patterns, relations, and
linear functions to model situations.
·
Using variables to
represent unknown quantities
·
Using concrete materials,
tables, graphs, verbal rules, algebraic expressions/equations/inequalities
2.
Draw freehand sketches of graphs
that model real phenomena and use such graphs to predict and interpret events.
·
Changes over time
·
Relations between
quantities
·
Rates
of change
(e.g., when
is plant
growing slowly/rapidly,
when is
temperature dropping most rapidly/slowly)
D.
Procedures
1.
Solve simple linear equations
with manipulatives and informally.
·
Whole-number coefficients
only, answers also whole numbers
·
Variables on one or both
sides of equation
2.
Understand and apply the
properties of operations and numbers.
·
Distributive property
·
The product of a number
and its reciprocal is 1
3.
Evaluate numerical expressions.
4.
Extend understanding and use of
inequality.
·
Symbols ( =
, .
, =
)
Buildi
4.3 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 7, students will:
Strand A.
Patterns
1.
Recognize,
describe, extend,
and create
patterns involving
whole numbers,
rational numbers,
and integers.
·
Descriptions using tables,
verbal and symbolic rules, graphs, simple equations or expressions
·
Finite and infinite
sequences
·
Generating sequences by
using calculators to repeatedly apply a formula
Strand B.
Functions and
Relationships
1.
Graph functions, and understand
and describe their general behavior.
·
Equations involving two
variables
C.
Modeling
1.
Analyze functional relationships
to explain how a change in one quantity can result in a change in another, using
pictures, graphs, charts, and equations.
2.
Use patterns, relations, symbolic
algebra, and linear functions to model situations.
·
Using manipulatives,
tables, graphs, verbal rules, algebraic expressions/equations/inequalities
·
Growth
situations, such
as population
growth and
compound interest,
using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and
social studies standard 6.6)
D.
Procedures
1.
Use graphing techniques on a
number line.
·
Absolute value
·
Arithmetic
operations represented
by vectors
(arrows) (e.g.,
.-3 + 6. is .left 3, right
6.)
2.
Solve simple linear equations
informally and graphically.
·
Multi-step, integer
coefficients only (although answers may not be integers)
·
Using
paper-and-pencil, calculators, graphing calculators, spreadsheets, and other
technology
3.
Create, evaluate, and simplify
algebraic expressions involving variables.
·
Order of operations,
including appropriate use of parentheses
·
Substitution of a number
for a variable
4.
Understand and apply the
properties of operations, numbers, equations, and inequalities.
·
Additive inverse
·
Multiplicative inverse
Buildi
4.3 Building upon knowledge and skills gained in preceding grades, by the end of
Grade 8, students will:
Strand A.
Patterns
1.
Recognize, describe, extend, and
create patterns involving whole numbers, rational numbers,
and integers.
·
Descriptions using tables,
verbal and symbolic rules, graphs, simple equations or expressions
·
Finite and infinite
sequences
·
Arithmetic sequences
(i.e., sequences generated by repeated addition of a fixed number, positive or
negative)
·
Geometric sequences (i.e.,
sequences generated by repeated multiplication by a fixed positive ratio,
greater than 1 or less than 1)
·
Generating sequences by
using calculators to repeatedly apply a formula
Strand B.
Functions and
Relationships
1.
Graph functions, and understand
and describe their general behavior.
·
Equations involving two
variables
·
Rates of change (informal
notion of slope)
2.
Recognize
and describe
the difference
between linear and exponential growth, using tables, graphs, and
equations.
C.
Modeling
1.
Analyze functional relationships
to explain how a change in one quantity can result in a change in another, using
pictures, graphs, charts, and equations.
2.
Use patterns, relations, symbolic
algebra, and linear functions to model situations.
·
Using
concrete materials
(manipulatives), tables,
graphs, verbal
rules, algebraic
expressions/equations/inequalities
·
Growth
situations, such
as population
growth and
compound interest,
using recursive (e.g., NOW-NEXT) formulas (cf.
science standard 5.5 and social studies standard 6.6)
D.
Procedures
1.
Use graphing techniques on a
number line.
·
Absolute value
·
Arithmetic
operations represented
by vectors
(arrows) (e.g.,
.-3 + 6. is .left 3, right
6.)
2.
Solve
simple linear
equations informally,
graphically, and
using formal
algebraic methods.
·
Multi-step, integer
coefficients only (although answers may not be integers)
·
Using
paper-and-pencil, calculators, graphing calculators, spreadsheets, and other
technology
3.
Solve simple linear inequalities.
4.
Create, evaluate, and simplify
algebraic expressions involving variables.
·
Distributive property
·
Substitution of a number
for a variable
·
Order of operations,
including appropriate use of parentheses
·
Translation of a verbal
phrase or sentence into an algebraic expression, equation, or inequality, and
vice versa
5.
Understand and apply the
properties of operations, numbers, equations, and inequalities.
·
Additive inverse
·
Multiplicative inverse
·
Addition and
multiplication properties of equality
·
Addition and
multiplication properties of inequalities
Strand A.
Patterns
1.
Use models and algebraic formulas
to represent and analyze sequences and series.
·
Explicit formulas for nth
terms
·
Sums of finite arithmetic
series
·
Sums of finite and
infinite geometric series
2.
Develop an informal notion of
limit.
3.
Use inductive reasoning to form
generalizations.
Strand B.
Functions and
Relationships
1.
Understand relations and
functions and select, convert flexibly among, and use various representations
for them, including equations or inequalities, tables, and graphs.
2.
Analyze
and explain
the general
properties and
behavior of
functions of
one variable, using appropriate graphing technologies.
·
Slope of a line or curve
·
Domain and range
·
Intercepts
·
Continuity
·
Maximum/minimum
·
Estimating roots of
equations
·
Intersecting points as
solutions of systems of equations
·
Rates of change
3.
Understand and perform
transformations on commonly-used functions.
·
Translations, reflections,
dilations
·
Effects on linear and
quadratic graphs of parameter changes in equations
·
Using graphing calculators
or computers for more complex functions
4.
Understand
and compare
the properties
of classes
of functions,
including exponential,
polynomial, rational, and trigonometric functions.
·
Linear vs. non-linear
·
Symmetry
·
Increasing/decreasing on
an interval
C.
Modeling
1.
Use functions to model real-world
phenomena and solve problems that involve varying quantities.
·
Linear, quadratic,
exponential, periodic (sine and cosine), and step functions (e.g., price of
mailing a first-class letter over the past 200 years)
·
Direct and inverse
variation
·
Absolute value
·
Expressions, equations and
inequalities
·
Same function can model
variety of phenomena
·
Growth/decay and change in
the natural world
·
Applications in
mathematics, biology, and economics (including compound interest)
2.
Analyze
and describe
how a
change in
an independent variable
leads to change in a dependent one.
3.
Convert recursive formulas to
linear or exponential functions (e.g., Tower of Hanoi and doubling).
D.
Procedures
1.
Evaluate and simplify
expressions.
·
Add and subtract
polynomials
·
Multiply a polynomial by a
monomial or binomial
·
Divide a polynomial by a
monomial
2.
Select and use appropriate
methods to solve equations and inequalities.
·
Linear equations .
algebraically
·
Quadratic
equations .
factoring (when
the coefficient
of x2
is 1) and using the quadratic formula
·
All types of equations
using graphing, computer, and graphing calculator techniques
3.
Judge the meaning, utility, and
reasonableness of the results of symbol manipulations, including those
carried out by technology.
STANDARD
4.4
(DATA ANALYSIS, PROBABILITY, AND DISCRETE
MATHEMATICS)
Cumulative
Progress Indicators
By
4.4
By the end of Grade
2, students will:
Strand A.
Data Analysis
1.
Collect,
generate, record,
and organize
data in response to questions, claims, or curiosity.
·
Data collected from
students. everyday experiences
·
Data generated from chance
devices, such as spinners and dice
2.
Read, interpret, construct, and
analyze displays of data.
·
Pictures, tally chart,
pictograph, bar graph, Venn diagram
·
Smallest to largest, most
frequent (mode)
Strand B.
Probability
1.
Use chance devices like spinners
and dice to explore concepts of probability.
·
Certain, impossible
·
More likely, less likely,
equally likely
2.
Provide probability of specific
outcomes.
·
Probability
of getting
specific outcome
when coin
is tossed,
when die
is rolled, when spinner is
spun (e.g., if spinner has five
equal sectors, then probability of getting a particular sector is one out of
five)
·
When picking a marble from
a bag with three red marbles and four blue marbles, the probability of getting a
red marble is three out of seven
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Sort and classify objects
according to attributes.
·
Venn diagrams
2.
Generate
all possibilities
in simple
counting situations (e.g., all outfits
involving two shirts and
three pants).
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Follow simple sets of directions
(e.g., from one location to another, or from a recipe).
2.
Color simple maps with a small
number of colors.
3.
Play simple two-person games
(e.g., tic-tac-toe) and informally explore the idea of what the outcome should
be.
4.
Explore concrete models of
vertex-edge graphs (e.g. vertices as .islands. and edges as .bridges.).
·
Paths from one vertex to
another
Buil
4.4
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 3, students will:
Strand A.
Data Analysis
1.
Collect, generate, organize, and
display data in response to
questions, claims,
or curiosity.
·
Data collected from the
classroom environment
2.
Read, interpret, construct,
analyze, generate questions about, and draw inferences from displays of data.
·
Pictograph, bar graph,
table
Strand B.
Probability
1.
Use
everyday events
and chance
devices, such
as dice,
coins, and
unevenly divided spinners,
to explore concepts of probability.
·
Likely, unlikely, certain,
impossible
·
More likely, less likely,
equally likely
2.
Predict probabilities in a
variety of situations (e.g., given the number of items of each color in a bag,
what is the probability that an item picked will have a particular color).
·
What students think will
happen (intuitive)
·
Collect data and use that
data to predict the probability (experimental)
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Represent
and classify
data according
to attributes,
such as
shape or
color, and
relationships.
·
Venn diagrams
·
Numerical and alphabetical
order
2.
Represent all possibilities for a
simple counting situation in an organized way and draw conclusions from this
representation.
·
Organized lists, charts
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Follow,
devise, and
describe practical
sets of
directions (e.g.,
to add
two 2-digit numbers).
2.
Explore vertex-edge graphs
·
Vertex, edge
·
Path
3.
Find the smallest number of
colors needed to color a map.
Buildi 4.4
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 4, students will:
Strand A.
Data Analysis
1.
Collect, generate, organize, and
display data in response to
questions, claims,
or curiosity.
·
Data collected from the
school environment
2.
Read, interpret, construct,
analyze, generate questions about, and draw inferences from displays of data.
·
Pictograph, bar graph,
line plot, line graph, table
·
Average (mean), most
frequent (mode), middle term (median)
Strand B.
Probability
1.
Use
everyday events
and chance
devices, such
as dice,
coins, and
unevenly divided spinners,
to explore concepts of probability.
·
Likely, unlikely, certain,
impossible, improbable, fair, unfair
·
More likely, less likely,
equally likely
·
Probability of tossing
.heads. does not depend on outcomes of previous tosses
2.
Determine probabilities of simple
events based on equally likely outcomes and express them as fractions.
3.
Predict probabilities in a
variety of situations (e.g., given the number of items of each color in a bag,
what is the probability that an item picked will have a particular color).
·
What students think will
happen (intuitive)
·
Collect data and use that
data to predict the probability (experimental)
·
Analyze all possible
outcomes to find the probability (theoretical)
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Represent
and classify
data according
to attributes,
such as
shape or
color, and relationships.
·
Venn diagrams
·
Numerical and alphabetical
order
2.
Represent all possibilities for a
simple counting situation in an organized way and draw conclusions from this
representation.
·
Organized lists, charts,
tree diagrams
·
Dividing into categories
(e.g., to find the total number of rectangles in a grid, find the number of
rectangles of each size and add the results)
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Follow,
devise, and
describe practical
sets of
directions (e.g.,
to add
two 2-digit numbers).
2.
Play
two-person games
and devise
strategies for
winning the
games (e.g.,
.make 5" where players alternately
add 1
or 2 and
the person
who reaches
5, or another designated number, is the winner).
3.
Explore vertex-edge graphs and
tree diagrams.
·
Vertex, edge,
neighboring/adjacent, number of neighbors
·
Path, circuit (i.e., path
that ends at its starting point)
4.
Find the smallest number of
colors needed to color a map or a graph.
Buildi
4.4
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 5, students will:
Strand A.
Data Analysis
1.
Collect, generate, organize, and
display data.
·
Data generated from
surveys
2.
Read, interpret, select,
construct, analyze, generate questions about, and draw inferences from displays
of data.
·
Bar graph, line graph,
circle graph, table
·
Range, median, and mean
3.
Respond to questions about data
and generate their own questions and hypotheses.
Strand B.
Probability
1.
Determine probabilities of
events.
·
Event, probability of an
event
·
Probability of certain
event is 1 and of impossible event is 0
2.
Determine probability using
intuitive, experimental, and theoretical methods (e.g., using model of picking
items of different colors from a bag).
·
Given numbers
of various types
of items
in a
bag, what
is the
probability that
an item of one type will be picked
·
Given
data obtained
experimentally, what
is the likely distribution
of items
in the bag
3.
Model
situations involving
probability using
simulations (with
spinners, dice)
and theoretical models.
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Solve counting problems and
justify that all possibilities have been enumerated without
duplication.
·
Organized lists, charts,
tree diagrams, tables
2.
Explore the multiplication
principle of counting in simple situations by representing all possibilities in
an organized way (e.g., you can make 3 x
4 = 12 outfits using 3 shirts and 4 skirts).
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Devise strategies for winning
simple games (e.g., start with two piles of objects, each of two players in turn
removes any number of objects from a single pile, and the person to take the
last group of objects wins) and express those strategies as sets of directions.
Buildi 4.4
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 6, students will:
Strand A.
Data Analysis
1.
Collect, generate, organize, and
display data.
·
Data generated from
surveys
2.
Read, interpret, select,
construct, analyze, generate questions about, and draw inferences from displays
of data.
·
Bar graph, line graph,
circle graph, table, histogram
·
Range, median, and mean
·
Calculators and computers
used to record and process information
3.
Respond
to questions
about data,
generate their own questions and hypotheses, and formulate strategies for
answering their questions and testing their hypotheses.
Strand B.
Probability
1.
Determine probabilities of
events.
·
Event, complementary
event, probability of an event
·
Multiplication rule for
probabilities
·
Probability of certain
event is 1 and of impossible event is 0
·
Probabilities of event and
complementary event add up to 1
2.
Determine probability using
intuitive, experimental, and theoretical methods (e.g., using model of picking
items of different colors from a bag).
·
Given numbers
of various types
of items
in a
bag, what
is the
probability that
an item of one type will be picked
·
Given
data obtained
experimentally, what
is the likely distribution
of items
in the bag
3.
Explore compound events.
4.
Model
situations involving
probability using
simulations (with
spinners, dice)
and theoretical models.
5.
Recognize and understand the
connections among the concepts of
independent outcomes, picking at random, and fairness.
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Solve counting problems and
justify that all possibilities have been enumerated without duplication.
·
Organized lists, charts,
tree diagrams, tables
·
Venn diagrams
2.
Apply the multiplication
principle of counting.
·
Simple situations (e.g.,
you can make 3 x
4 = 12 outfits using 3 shirts and 4 skirts).
·
Number of ways a specified
number of items can be arranged in order (concept of permutation)
·
Number
of ways
of selecting
a slate
of officers
from a
class (e.g.,
if there
are 23 students and 3
officers, the number is 23 x
22 x 21)
3.
List the possible combinations of
two elements chosen from a given set (e.g., forming a committee of two from a
group of 12 students, finding how many handshakes there will be among ten people
if everyone shakes each other person.s hand once).
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Devise strategies for winning
simple games (e.g., start with two piles of objects, each of two players in turn
removes any number of objects from a single pile, and the person to take the
last group of objects wins) and express those strategies as sets of directions.
2.
Analyze vertex-edge graphs and
tree diagrams.
·
Can a picture or a
vertex-edge graph be drawn with a single line? (degree of vertex)
·
Can you get from any
vertex to any other vertex? (connectedness)
3.
Use vertex-edge graphs to find
solutions to practical problems.
·
Delivery route that stops
at specified sites but involves least travel
·
Shortest route from one
site on a map to another
4.4
Building
upon knowledge and skills gained in preceding grades, by the end of
Grade 7, students will:
Strand A.
Data Analysis
1.
Select
and use
appropriate representations
for sets of data, and measures of central tendency (mean, median, and mode).
·
Type of display most
appropriate for given data
·
Box-and-whisker plot,
upper quartile, lower quartile
·
Scatter plot
·
Calculators and computer
used to record and process information
2.
Make inferences and formulate and
evaluate arguments based on displays and analysis of data.
Strand B.
Probability
1.
Interpret probabilities as
ratios, percents, and decimals.
2.
Model
situations involving
probability with simulations
(using spinners, dice,
calculators and
computers) and theoretical models.
·
Frequency, relative
frequency
3.
Estimate
probabilities and
make predictions
based on
experimental and
theoretical probabilities.
4.
Play
and analyze
probability-based games, and
discuss the
concepts of
fairness and expected value.
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Apply the multiplication
principle of counting.
·
Permutations:
ordered situations with replacement (e.g., number of possible license
plates) vs. ordered situations without replacement (e.g., number of possible
slates of 3 class officers from a 23 student class)
2.
Explore
counting problems
involving Venn
diagrams with
three attributes (e.g., there are 15,
20, and
25 students
respectively in the chess
club, the
debating team,
and the engineering society;
how many different students belong to the three clubs if there are 6 students in
chess and debating, 7 students in chess and
engineering, 8 students in debating and engineering, and 2 students in
all three?).
3.
Apply techniques of systematic
listing, counting, and reasoning in a variety of different contexts.
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Use vertex-edge graphs to
represent and find solutions to practical problems.
·
Finding the shortest
network connecting specified sites
·
Finding the shortest route
on a map from one site to another
·
Finding the shortest
circuit on a map that makes a tour of specified sites
Buildi
4.4
Building upon knowledge and skills gained in preceding grades, by the end of
Grade 8, students will:
Strand A.
Data Analysis
1.
Select
and use
appropriate representations
for sets of data, and measures of central tendency (mean, median, and mode).
·
Type of display most
appropriate for given data
·
Box-and-whisker plot,
upper quartile, lower quartile
·
Scatter plot
·
Calculators and computer
used to record and process information
·
Finding the median and
mean (weighted average) using frequency data.
·
Effect of additional data
on measures of central tendency
2.
Make inferences and formulate and
evaluate arguments based on displays and analysis of data.
3.
Estimate lines of best fit and
use them to interpolate within the range of the data.
4.
Use surveys and sampling
techniques to generate data and draw conclusions about large groups.
Strand B.
Probability
1.
Interpret probabilities as
ratios, percents, and decimals.
2.
Determine probabilities of
compound events.
3.
Explore the probabilities of
conditional events (e.g., if there are seven marbles in a bag, three red and
four green, what is the probability that two marbles picked from the bag,
without replacement, are both red).
4.
Model
situations involving
probability with simulations
(using spinners, dice,
calculators and
computers) and theoretical models.
·
Frequency, relative
frequency
5.
Estimate
probabilities and
make predictions
based on
experimental and
theoretical probabilities.
6.
Play
and analyze
probability-based games, and
discuss the
concepts of
fairness and expected value.
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Apply the multiplication
principle of counting.
·
Permutations:
ordered situations with replacement (e.g., number of possible license
plates) vs. ordered situations without replacement (e.g., number of possible
slates of 3 class officers from a 23 student class)
·
Factorial notation
·
Concept of combinations
(e.g., number of possible
delegations of 3 out of 23 students)
2.
Explore
counting problems
involving Venn
diagrams with
three attributes (e.g., there are 15,
20, and
25 students
respectively in the chess
club, the
debating team,
and the engineering society;
how many different students belong to the three clubs if there are 6 students in
chess and debating, 7 students in chess and
engineering, 8 students in debating and engineering, and 2 students in
all three?).
3.
Apply techniques of systematic
listing, counting, and reasoning in a variety of different contexts.
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Use
vertex-edge graphs
and algorithmic
thinking to
represent and
find solutions
to practical problems.
·
Finding the shortest
network connecting specified sites
·
Finding a minimal route
that includes every street (e.g., for trash pick-up)
·
Finding the shortest route
on a map from one site to another
·
Finding the shortest
circuit on a map that makes a tour of specified sites
·
Limitations of computers
(e.g., the number of routes for a delivery truck visiting n sites is n!,
so finding the shortest circuit by examining all circuits would overwhelm the
capacity of any computer, now or in the future, even if n is less than 100)
Strand A.
Data Analysis
1.
Use surveys and sampling
techniques to generate data and draw conclusions about large groups.
·
Advantages/disadvantages
of sample selection
methods (e.g.,
convenience sampling, responses to survey, random sampling)
2.
Evaluate the use of data in
real-world contexts.
·
Accuracy and
reasonableness of conclusions drawn
·
Bias in conclusions drawn
(e.g., influence of how data is displayed)
·
Statistical claims based
on sampling
3.
Design a statistical experiment,
conduct the experiment, and interpret and communicate the outcome.
4.
Estimate
or determine
lines of
best fit
(or curves
of best
fit if
appropriate) with
technology, and use them to interpolate within the range of the data.
5.
Analyze data using technology,
and use statistical terminology to describe conclusions.
·
Measures of dispersion:
variance, standard deviation, outliers
·
Correlation coefficient
·
Normal
distribution (e.g., approximately
95% of the sample lies between two standard deviations on either side of the
mean)
Strand B.
Probability
1.
Calculate the expected value of a
probability-based game, given the probabilities and payoffs of the various
outcomes, and determine whether the game is fair.
2.
Use concepts and formulas of area
to calculate geometric probabilities.
3.
Model
situations involving
probability with simulations
(using spinners, dice,
calculators
and computers)
and theoretical
models, and
solve problems
using these models.
4.
Determine probabilities in
complex situations.
·
Conditional events
·
Complementary events
·
Dependent and independent
events
5.
Estimate
probabilities and
make predictions
based on
experimental and
theoretical probabilities.
6.
Understand
and use
the .law
of large
numbers. (that
experimental results tend
to approach theoretical probabilities after a large number of trials).
C.
Discrete
Mathematics.Systematic Listing and Counting
1.
Calculate combinations with
replacement (e.g., the number of possible ways of tossing a coin 5 times and
getting 3 heads) and without replacement (e.g., number of possible delegations
of 3 out of 23 students).
2.
Apply
the multiplication
rule of
counting in
complex situations,
recognize the difference
between situations with replacement and without replacement, and recognize the
difference between ordered and unordered counting situations.
3.
Justify solutions to counting
problems.
4.
Recognize and explain
relationships involving combinations and Pascal.s Triangle, and apply those
methods to situations involving probability.
D.
Discrete
Mathematics.Vertex-Edge Graphs and Algorithms
1.
Use
vertex-edge graphs
and algorithmic
thinking to
represent and
solve practical problems.
·
Circuits that include
every edge in a graph
·
Circuits that include
every vertex in a graph
·
Scheduling
problems (e.g.,
when project
meetings should
be scheduled
to avoid conflicts) using
graph coloring
·
Applications to science
(e.g., who-eats-whom graphs, genetic
trees, molecular structures)
2.
Explore strategies for making
fair decisions.
·
Combining individual
preferences into a group decision (e.g., determining winner of an election or
selection process)
·
Determining
how many
Student Council
representatives each
class (9th,
10th, 11th,
and 12th
grade) gets when the classes have unequal sizes (apportionment)
At
4.5 At
each grade
level, with respect to content appropriate for that grade level, students will:
Strand A.
Problem Solving
1.
Learn
mathematics through problem solving, inquiry, and discovery.
2.
Solve
problems that arise in mathematics and in other contexts (cf. workplace
readiness standard 8.3).
·
Open-ended problems
·
Non-routine problems
·
Problems with multiple
solutions
·
Problems that can be
solved in several ways
3.
Select and apply a variety of
appropriate problem-solving strategies (e.g., .try a simpler problem. or
.make a diagram.) to solve problems.
4.
Pose problems of various types
and levels of difficulty.
5.
Monitor their progress and
reflect on the process of their problem solving activity.
Strand B.
Communication
1.
Use communication to organize and
clarify their mathematical thinking.
·
Reading and writing
·
Discussion, listening, and
questioning
2.
Communicate their mathematical
thinking coherently and clearly to peers, teachers, and others, both orally and
in writing.
3.
Analyze and evaluate the
mathematical thinking and strategies of others.
4.
Use the language of mathematics
to express mathematical ideas precisely.
C.
Connections
1.
Recognize
recurring themes
across mathematical
domains (e.g.,
patterns in
number, algebra, and geometry).
2.
Use
connections among
mathematical ideas to explain
concepts (e.g., two linear equations have
a unique
solution because
the lines they represent
intersect at a single point).
3.
Recognize that mathematics is
used in a variety of contexts outside of mathematics.
4.
Apply mathematics in practical
situations and in other disciplines.
5.
Trace
the development
of mathematical
concepts over
time and
across cultures (cf. world languages and social studies standards).
6.
Understand how mathematical ideas
interconnect and build on one another to produce as coherent whole.
D.
Reasoning
1.
Recognize that mathematical
facts, procedures, and claims must be justified.
2.
Use reasoning to support their
mathematical conclusions and problem solutions.
3.
Select and use various types of
reasoning and methods of proof.
4.
Rely on reasoning, rather than
answer keys, teachers, or peers, to check the correctness of their problem
solutions.
5.
Make and investigate mathematical
conjectures.
·
Counterexamples as a means
of disproving conjectures
·
Verifying conjectures
using informal reasoning or proofs.
6.
Evaluate examples of mathematical
reasoning and determine whether they are valid.
E.
Representations
1.
Create
and use
representations to organize,
record, and
communicate mathematical
ideas.
·
Concrete representations
(e.g., base-ten blocks or algebra tiles)
·
Pictorial representations
(e.g., diagrams, charts, or tables)
·
Symbolic representations
(e.g., a formula)
·
Graphical representations
(e.g., a line graph)
2.
Select, apply, and translate
among mathematical representations to solve problems.
3.
Use
representations to
model and
interpret physical, social, and mathematical
phenomena.
F.
Technology
1.
Use technology to gather,
analyze, and communicate mathematical information.
2.
Use
computer spreadsheets,
software, and
graphing utilities
to organize
and display quantitative
information.
3.
Use
graphing calculators
and computer
software to
investigate properties of
functions and their graphs.
4.
Use calculators as
problem-solving tools (e.g., to explore patterns, to validate solutions).
5.
Use computer software to make and
verify conjectures about geometric objects.
6.
Use
computer-based laboratory
technology for
mathematical applications in the sciences.