N.J. CCC Standards Mathematics (Adopted 2002)

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.   4.2.  Geometry and Measurement:   ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES,  RELATIONSHIPS,  AND  MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.   4.3.  Patterns and Algebra: ALL  STUDENTS  WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES  AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.   
Grade 2 (Strands A,B,C) Grade 2 (Strands A,B,C,D,E) Grade 2 (Strands A,B,C,D)
Grade 3 (Strands A,B,C) Grade 3 (Strands A,B,C,D,E) Grade 3 (Strands A,B,C,D)
Grade 4 (Strands A,B,C) Grade 4 (Strands A,B,C,D,E) Grade 4(Strands A,B,C,D)
Grade 5 (Strands A,B,C) Grade 5 (Strands A,B,C,D,E) Grade 5 (Strands A,B,C,D)
Grade 6 (Strands A,B,C) Grade 6 (Strands A,B,C,D,E) Grade 6 (Strands A,B,C,D)
Grade 7 (Strands A,B,C) Grade 7 (Strands A,B,C,D,E) Grade 7 (Strands A,B,C,D)
Grade 8 (Strands A,B,C) Grade 8 (Strands A,B,C,D,E) Grade 8 (Strands A,B,C,D)
Grade 12 (Strands A,B,C) Grade 12 (Strands A,B,C,D,E) Grade 12 (Strands A,B,C,D)

 

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 DATA.   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 IDEAS.
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)

 

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.

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.

         

 

STANDARD 4.2     (GEOMETRY AND MEASUREMENT)     ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES,  RELATIONSHIPS,  AND  MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.

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 REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES  AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES. 

 

  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

 

Bu           4.3   Building upon  knowledge  and  skills  gained  in  preceding  grades,  by  the  end  of  Grade  12, students will: 

 

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)     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 DATA.

 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)

 

B          4.4  Building  upon  knowledge  and  skills  gained  in  preceding  grades,  by  the  end  of  Grade  12, students will:

 

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)

         

 

 

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 IDEAS.

 

Cumulative Progress Indicators

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.