Numeration and Order: Counting

Correct number sequences through rote counting and counting objects. Counting on from a number, backward, skip counting. Experiences with objects of various sizes, shapes, and arrangements.
 

• Counting: oral and rote to 115, sequence, counting on and back from a number, interrupted counts, skip counts by 2, 5, and 10, tally counting, counting with an action, rational counting with various objects
• Calculator use: count-on, count by ones, count backwards, skip count
• One more/one less
• Identify small numbers without counting
• Estimate quantity
• Concept of zero

Read and write numbers; use the term "digit".
 

• How numbers are used: telephone numbers, number books, number stories, number models, games, addition and subtraction equations
• One to One correspondence
• Number line - 1 to 180 - relate to sets of objects
• Read, write, compare, order, and sequence numbers to 180.
• Understand each 'ten" number as 10 + a digit
• Explore the quantity "100"
• Arrays
• Fractions: stories, one half, one third, one fourth
• Make exchanges
• Calculator: place value, how many, next number; add and subtract; with number grid
• Ordinal numbers

Rules; dealing with more than one element; pairs and collections of things (objects and numbers); relationships; patterns.
 

• Patterns: Create, reproduce, copy, verbalize, categorize, sort
• Introduce functions: find the rule
• Relations: input/output based on a defined rule

Weight; linear measure; area. Customary and metric measurement units. Nonstandard units. Designate units; record results.
 

• Use vocabulary of weight measurement (weight, the same as, heavy, heavier, lighter, less, more, equal to, equivalent)
• Estimate measures/weights
• Nonstandard measurements
• Compare weights/lengths
• Order weights/lengths
• Area: estimation
• Volume: full, empty, half-full, more, less
• Measuring Tools:  Ruler, yardstick, meterstick, tape measure

Names and value of money; coins and bill, underlying tool for teaching place value, fractions, equivalence, and the decimal system.
 

• Value of money
• Coins: Recognize and use: penny, nickel, dime, quarter, $1 $10; Compare, find equivalent values, exchanges, count past $1, estimate
• Equivalent fractional terms of money: quarter is one-fourth of a dollar

Clocks; calendars; number line. Understanding the duration and relationship of time units. Ordering units of time
 

• Use vocabulary of time: before, after, early, late, yesterday, today, tomorrow, slow, fast, morning, afternoon, next week, week, o'clock, minute, hour, half-hour, halfway between.
• Calendar Activities
• Clock and time: tell time to the hour, parts of clock, minutes, estimate time, elapsed time, Time lines and order of events

Informal and intuitive awareness about geometric shapes
 

• Shapes:  Square, rectangle, circle, triangle, rhombus, triangle, quadrilaterals -describe, create, compare, locate
• Symmetry

Collect, organize, and display data. Tables and graphs. Make counts and comparisons of data.
 

• Bar Graphs
• Pictographs
• Parts of a graph
• Reading a graph
• Interpreting results
• Probability
• Coordinates

Operations

Concrete problems that arise in children's daily life. Story problems; reversibility; relation vocabulary. The things we "do" with numbers.
 

• Use relations vocabulary such as:  bigger, smaller, more, less, the same, fewer, over, under, higher, lower, on the same level, equivalent
• Addition and Subtraction (with objects, stories, models), symbolic language
• Estimation on number line
• Number families
• Equivalent numbers
• Missing addends
• Counting-on
• Fractions:  one-half
• Calculator use: solutions and plus and minus keys

Numeration and Order: Counting

Correct number sequences through rote counting and counting objects.
Counting on from a number, backward, skip counting. Verbal counting patterns.
Experiences with objects of various sizes, shapes, and arrangements.
 

• Verbal counts by 1,2, 3, 4, 5, 6, 10, and 25
• Count to 1000 forward and backward
• Count by 10 from any number
• Tally counts

Read and write numbers. Whole numbers, fractions, decimals. Place value.
 

• Read, write, and order numbers to 1000
• Equivalencies
• Calculator: Key in numbers; add, subtract;( =/R); place value
• Zero as a number
• Ten digits in our number system
• Place value: ones, tens, thousands
• Number patterns
• Fractions
• Negative numbers
• Decimals and money

Measures of length, area, capacity, weight, temperature, and elapsed time. Coin and currency recognition and exchanges. Clocks, calendars, rules, timelines, thermometers, and ordinal numbers.
 

• Nonstandard linear measures
• Standard units of measure (tape measure, ruler)
• Measurement terms (inch, foot, meter, centimeter)
• Weight (estimation, ordering by weight, use different types of scales)
• Money: penny, nickel, dime, quarter, dollar - count, show, and write amounts using cent and dollar notation - coin equivalencies,make exchanges, count change
• Number line concepts
• Ordinal numbers and patterns
• Time: Clocks (digital and analog), estimate time, use a.m. and p.m, notate time, tell time
• to the minute, sixty minutes in an hour, determine intervals between two times.

Memorize number fact. Games and exercises for skills in arithmetic. Informal work with properties.
 

• Greater than, less than, equal to and their symbols
• A total as the sum of two parts
• Number line
• Addition and Subtraction Facts
• Fact Families
• Vertical and horizontal formats for writing facts
• Fact Shortcuts - turnaround rule, doubles, plus zero, plus one
• Inverse relationships between addition facts and subtraction facts
• Add 10; add 100 to any number
• Add and subtract 0 and 1 to/from 2 - or 3 - digit numbers
• Extend addition and subtraction facts to add and subtract tens
• Mentally add and subtract tens to any number
• Add and subtract numbers of differing digits
• Add and subtract coin amounts

Exploring Data and Chance

Collect, order, and display data. Tables, charts, and graphs.
 

• Tally counts
• Order data - Highest/lowest numbers in a data set
• Range (variation)
• Typical Value (Middle value)
• Bar graphs
• Collect, analyze, and interpret real life data

Explore 2- and 3- dimensional shapes.
 

• Identify~ and draw - triangle, quadrangle, square, rhombus, trapezoid, hexagon, circle, pentagon, heptagon, octagon
• Identify and construct- sphere, cone, cylinder, rectangular prism, cube, pyramid (rectangular and triangular)
• Faces and corners of shapes
• Regular and irregular polygons
• Parts of polygons - side, vertex, angle
• Name points, line segments, figures
• Introduce line symmetry

Attributes, patterns, sequences, relations, and functions.
 

• Patterns - create and recognize by shape, color, rotation - predict Find the Rule

Numeration and Order - Counting

Correct number sequences through rote counting and counting objects; counting on, back, skip counting; verbal counting patterns; experiences with objects of various sizes, shapes, and arrangements.
 

•  Count to 10,000
•  Skip counting by 2s, 5s, 10s
•  Count forward and backward by 2, 5, 10 from 1000
•  Count with: calculator, number grid, number line

Numeration and Order: Numeration

Read and write whole numbers, fractions, and decimals: place value.
 

• Whole Numbers:
• Read and write numbers to IO, 000
• Place value to 10,000 (base-10 system)
• Equivalent names for numbers
• Compare numbers: base-10 blocks; <,>,=; order 4-digit numbers
• Context of numbers (label or unit)
• Estimation
• Fractions: regions; equal parts; parts of one; parts of sets; equivalent; relations among fractions
• Decimals: notation for money; names for money

Exploring Data and Chance

Collect, order, and display data: tables, charts, and graphs
 

• Collect, organize, compare and analyze real-life data
• Construct and interpret bar graphs
• Use tally marks
• Construct data tables
• Frequency table; frequency distributions
• Median and mode

Geometry and Spatial Sense

Explore 2- and 3-dimensional shapes.
 

• Classify geometric shapes
• Similarities and differences between 3-dimensional shapes; prisms,  pyramids, cylinders, spheres, cones
• Explore regular polyhedrons; construct a tetrahedron
• Work with polygons: triangles, squares, rectangles, parallelograms,  rhombuses, trapezoids, pentagons, hexagons, heptagons, octagons
• Perimeter and area of polygons
• Define points; define and name line segments
• Symmetry

Patterns, Functions, and Sequences

Attributes, patterns, sequences, relations, and functions.
 

• Patterns in numbers; on number grids; in doubles and halves; in multiples; on  add/subtract fact tables
• Odd and even numbers
• Number sequences
• Sort attribute blocks to match a rule

Measures and Measurement; Reference Frames

Measures of length, area, capacity, weight, temperature, and elapsed time; coin and currency; clocks. calendars, rulers, timeliness thermometers, and ordinal numbers.
 

• Customary and metric measure
• Measure length: inches, feet, yards, miles, millimeters, centimeters, meters,
• half-inch, half centimeter
• Measure weight
• Measure capacity
• Estimate measures
• Equivalent units of measure
• Money: value of coins and bills to $1 00; exchange; make change; estimate;  decimal notation
• Time: hour, half hour, quarter hour, 5-minute, minute; analog clock; estimate;  time lines; calendar
• Temperature: Fahrenheit, Celsius; compare temperatures

Memorize number facts; develop skills in arithmetic; informal work with properties.
 

• Master basic addition and subtraction facts
• Relationship between addition/subtraction; fact families
• Addition: "Parts-and-totals" problems; 1-. 2-, and 3-digit numbers; missing   addends; 3 addends
• Subtraction: compare quantities; I-, 2-, and 3-digit numbers;complements of 10
• Multiplication: introduce; equal groups; arrays
• Division: introduce; diagrams
• Relationship between multiplication/division
• Parentheses to specify order of operations
• Mental arithmetic

Numeration and Order

Whole numbers, integers, fractions, and decimals.

• Numbers and their uses
• Read, write, and order numbers to 1,000,000
• Calendar facts
• Fractions: equal parts, notation, writing fractions, finding a fraction of a number,  equivalent, quantities greater than one, equivalent
• Money notation
• Decimal place value to thousandths
• Positive and negative numbers
• Calculator use: skip counting, money amounts, percents as hundredths

Measures, Measurement and Reference Frames

Measures of length, area, capacity, weight, and elapsed time. Clocks, calendars, timeliness thermometers, and ordinal numbers. Money.

• Estimation in measurement
• Nonstandard linear units of measure
• Standard units of length(customary and metric)
• Perimeter
• Area
• Volume(customary and metric)
• Weight
• Diameter and circumference
• Compare, order, and convert measures
• Time to the minute
• Elapsed time
• Estimated distances
• Temperature
• Coordinate grids; ordered pairs
• Estimated and exact costs
• Make change
• Calculate sales tax

Games and exercises for arithmetic concepts, skills, and procedures.
 

• Addition and subtraction facts
• Addition of three 1 -digit numbers
• Addition with three or more addends
• Addition and subtraction number families
• Multiplication arrays
• Equal shares and equal groups
• Multiplication and division facts
• Multiplication and division number families
• Square numbers through 1 00
• Estimate products of I -digit numbers by dollars and cents of 2-digit numbers
• Multidigit multiplication
• Multiples of 1 0, 1 00, and 1 000 (multiplying and dividing)
• Mental arithmetic
• Explore division procedures

Exploring Data and Chance

Collect, order, and display data. Tables, charts, and graphs.
 

• Collect, record, display, analyze, and compare data
• Construct tally charts, bar graphs, line graphs, and frequency tables
• Frequency distributions
• Median, mode, mean
• Language of chance events
• Probability

Geometry and Spatial Sense

Explore 2- and 3-dimensional shapes and other geometric concepts.
 

• Identify and name polygons
• Construct 3-dimensional shapes
• Identify and notate points, segments, rays, or lines
• Relation of two or more line segments, rays, or lines intersecting, parallel, forming   angles, and segments as sides of polygons
• Rotations or turns
• Construct triangles, quadrilaterals, and other polygons
• Construct pyramids and prisms
• Construct angles and label accurately
• Measure angles and estimate angle measure

Patterns, sequences, attributes, relations, and functions
 

• Patterns in four-digit numbers
• Number patterns (frames and arrows)
• Number sequences
• Common attributes among figures in a set

Algebra and Uses of Variables

Equivalent names, number models, parenthesis, missing addends, missing factors.
 

• Number families
• Function machine(input/output)
• Number stories with diagram use
• Number models for area
• Relationship between addition and subtraction
• Relationship between multiplication and division
• Number models with parenthesis
• Factors of whole number

Numeration and Order

Place value; read and write whole numbers and decimals; scientific notation; large and small numbers; integers; fractions; equivalence classes; relations (<, > =).

• Uses of numbers
• Base- IO place value system for naming whole numbers to millions
• Read and write numbers in the millions
• Rounding numbers to the nearest million
• Meaning of decimals
• Decimal notation and place value to 3-place decimals'
• Terminating and repeating decimals
• Meaning and use of fractions
• Fractional parts in a collection of objects
• Comparing fractional parts
• Equivalent fractions
• Renaming fractions as decimals
• Introduction to powers of ten with scientific notation
• Introduction to negative numbers
• Links between fraction, decimal, and percent notation
• Percents
• Equivalent names for percents

Measures and Measurement and Reference Frames

Basic measures such as length, weight(mass), capacity, elapsed time, angles; compound measures such as area, volume, density, calories. Reference frames including ordinal numbers, time of day, dates, time lines, thermometers.

• Personal references to estimate lengths in metric and standard units
• Metric linear measurement: millimeter, centimeter, decimeter, meter
• Convert measurements to other metric units
• Inch, foot, yard, mile
• Ounce, pound, kilogram
• Cup, pint, quart, half-gallon, gallon
• Perimeter
• Area
• Volume
• Equivalent units of measure
• Grid coordinates, parallel/meridian partitioning, latitude and longitude, ordered pairs • Time to the nearest minute, time zones, time comparisons

Mental Arithmetic and Number Systems

Arithmetic concepts (including meaning of operations), skills, and procedures; whole number facts and extensions (addition, subtraction, multiplication, division); estimation; number theory (including primes, composites, divisibility); properties.

•  Mastery of addition and subtraction facts
• Alternative algorithms for addition and subtraction of multidigit numbers
• Estimate sums
• Add and subtract positive and negative numbers
• Add and subtract two-place decimals
• Relationships between multiplication and division facts
• Multiplication/division diagrams
• Multiply by 2-digit numbers
• Divide by two digit divisors
• Alternative division algorithms
• Mental arithmetic with multiplication and division
• Estimate products
• Multiples of IO, I 00, I 000
• Multiply by powers of ten
• Find a fractional part of a number
• Missing digit problems
• Ratios as fractions
• Meaning of percents(ratios)
• Interpreting percents
• Uses of percents and rates
• Percent of a region
• Find percent of a number

Find or collect data, landmarks; display data in table, charts, graphs, probability.

• Collect, organize, display, and analyze a set of data
• Maximum, minimum, range, mean, median, mode
• Basic concepts and vocabulary for chance events
• Find, predict, perform experiments, analyze, and compare results of chance events
• Read, interpret, and compare data in pie graphs, scale drawings, tables, and maps
• Take, tabulate, analyze, and predict survey results
• Post credits and debits
• Evaluate reliability of data

Geometry and Spatial Sense

Notations, definitions, and basic relationships for 2- and 3-dimensional figures; basics metric properties of plane figures; congruence, similarity, transformations of geometric figures.

• Notations, definitions, and basic relationships for 2- and 3-dimensional figures; basic metric properties of plane figures; congruence, similarity, transformations of geometric figures.
• Vocabulary and notation for line segments, lines, rays
• Properties of angles, triangles, quadrangles, polygons
• Classify quadrangles
• Meaning of parallel, classify parallelograms
• Define polygons
• Circles and compass constructions
• Protractor use
• Congruent transformations, reflection, rotation, translation, similarity
• Figures in a coordinate plane
• Symmetry
• Construct geometric solids: surface, face, edge, vertex, base, volume, interior
• Classify angles: acute, obtuse, reflex, right, straight

Patterns, Functions, and Sequences

Synopsis:  Number and visual sequences, formula models using variables.

• Skip counting on a calculator, find missing whole numbers on a number line
• Transformations as design tools
• Patterns: 9-patch, frieze, tile, multiplication, data displays
• Equivalent unit rate
• Rate tables
• Equivalent fractions by multiplying numerator and denominator by the same number

Synopsis: Properties of number systems; operations with positive and negative numbers; parenthesis, order of operations; write and solve equations(open sentences); relations on number lines; function models(rules, tables, and coordinate grids).

• Missing addend problems
• Number sentences and relation symbols
• Parenthesis use
• Open sentences vocabulary and solutions
• Formulas

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Numeration and Order

Place value, read and write whole numbers and decimals, scientific notation; large and small numbers; integers, fractions; equivalence classes; relations (<, >$ =.)

• Read, write, and compare numbers in the millions, billions, trillions
• Prime and composite numbers
• Square numbers: representations and properties
• Compare numbers in exponential notation
• Use negative numbers
• Place value from trillions to thousandths
• Meaning of fractions and fraction notation
• Equivalent fractions
• Rename fractions as decimals
• Rename fractions as mixed numbers and mixed numbers as fractions
• Compare and order fractions (like and unlike denominators)
• Uses of fractions
• Repeating decimals
• Percent as ratio: percents to compare fractions with unlike denominators
• Rename percents as decimals and fractions
• Ratio: meaning, notation, equivalent forms
• Pi as a ratio

Measures and Measurement and Reference Frames

Basic measures such as length, weight (mass) capacity, elapsed time, angles; compound measures such as area, volume, density, calories. Reference frames including ordinal numbers, time of day, dates, time lines, thermometers.

• Estimation challenges
• Fractions and linear measure
• Convert fractions of hours to minutes
• Equivalent measures
• Highest/lowest elevations
• Measure in fractions of inches: one-sixteenth inch and millimeter
• Fractions, decimals on the number line
• Measure to nearest gram
• Area and notation for square units
• Perimeter, circumference
• Perimeter, circumference
• Reasonable measures of volume and capacity
• Volume and notation for cubic units
• Time equivalents
• Fractions and equivalent time
• Convert units of time: minute, hour, day, week, year, decade, century
• Temperature change and negative numbers
• Celsius/Fahrenheit equivalents

Operations: Mental Arithmetic and Number Systems

Arithmetic concepts including meaning of operations, skills and procedure, whole number facts and extensions (addition, subtraction, multiplication, division); estimation; number theory including primes, composites, divisibility; properties.
 

Operations: Algorithms and Procedures

Multidigit procedures for whole numbers and decimals; fraction operation procedures; procedures with scientific calculators.

• Addition algorithms: opposite-change; partial sums
• Rules to add and subtract positive and negative numbers on a calculator
• Subtraction algorithms: same-change; partial- differences
• Multiplication algorithms: partial-products, lattice method
• Division algorithm: successive estimation
• Divide with one- and two-digit divisors
• Square key and square root key on a calculator
• Fractions to decimals using a calculator
• Powers key on the calculator
• Scientific notation on the calculator
• Multiplication facts (for mastery)
• Factors, products, factor pairs
• Multiples of a number
• Prime factorization
• Simplify fractions
• Least common multiples
• Mental arithmetic activities
• Divisibility: tests for 2, 3, 4, 5, 6, 9, 1 0
• Division facts
• Square roots of square numbers
• Fractional parts of whole numbers
• Fraction percent equivalencies
• Add and subtract fractions (like and unlike denominators)
• Add and subtract mixed numbers
• Multiplication of fractions, whole numbers, and mixed numbers
• Round decimals: up, down, to nearest 0.5
• Add and subtract decimals
• Estimate with percents
• Find percent of a number
• Scientific notation
• Estimate sums and distances

Exploring Data and Chance

Find or collect data; landmarks; display data in tables, charts, graphs;  probability.

• Collect and organize data
• Display data: bar graphs, line graphs, circle graphs, tables, charts
•  Find: mean, median, mode, maximum, minimum, range
• Analyze data
• Read and interpret graphs
•  Analyze fairness of a game
• Chance events: certain, likely, 50-50, unlikely, impossible
• Probabilities expressed as words and percents
• Predictions using tables of values, formulas, and line graphs
• Use ratios to examine trends in data
• Interpret and make color-coded maps

Geometry and Spatial Sense

Notations, definitions, and basic relationships for 2- and 3-dimensional figures; basic metric properties of plane figures; congruence, similarity, transformations of geometric figures.

• Properties of regular polygons, parallel lines, perpendicular lines
• Tessellation with regular polygons, quadrangles, and a special polygon
• Names of geometric solids
• Properties of geometric solids: edge, face, curved surface, flat, vertex, apex, base, height, regular Two-dimensional shapes
• Kites, parallelograms, perimeter
• Scale drawings of sports playing fields
• Radius and diameter
• Use a compass to draw a circle; radius, diameter, circumference
• Draw angles with half-circle and full-circle protractors
• Measure angles with a protractor
• Equilateral, isosceles, and scalene triangles
•  Acute, obtuse, right, reflex, straight, vertical, and adjacent angles
• Copy a triangle: with a protractor, with compass, with straightedge
• Copy line segments
• Sum of angle and measure of each angle of regular polygons
• Congruent figure
• Transformation, translation, reflections on a coordinate grid
• Base and height of rectangles, parallelograms, and triangles
• Rectangle method to find area on nonrectangular figures; use substitution
• Estimate and calculate volume of a cylinder
• Area of circles, rectangles, triangles, parallelograms
• Surface area of a cylinder

Patterns, Functions, and Sequences

Number and visual sequences; formula models using variables.

• Use the Sieve of Eratosthenes to find prime numbers; 1-200
• Patterns
• Use a rule; find a rule
• Formula for area of rectangle, triangle, parallelogram
• Pick's formula for area of polygons
• Formulas for the circumference of an area and circle
• Formulas for the volume of a prism, sphere, cylinder
• Formula for the relationships among faces, vertices, and edges of a polyhedron
• Effect of enlargements and reductions of perimeter and area
• Rate relationships with tables of values, formulas, and line graphs
• Relationship between volume of prisms and pyramids and the volume of cylinders and cones      Match event to graphs
• Rate tables

Properties of number systems; operation with positive and negative numbers; parenthesis, order of operations; write and solve equations (open sentences); relationships on number lines; function models (rules, tables, and coordinate graphs.)

• Missing factors
• Variables in problems and formulas
• Expressions with variables
• Evaluate expressions
• Algebraic expressions to represent number stories
• Determine if number sentences are true or false
• Number sentences with parenthesis
• Ordered number pairs for points on a coordinate grid (four quadrants)
• Number models with positive and negative numbers
• Plot ordered numbers on a coordinate grid
• Pan balance equations: add or remove same number of objects of equal weight from  both pans; two equations with two unknowns

Numeration and Order

Place value, read and write whole numbers and decimals; scientific notation;
integers; fractions; equivalence classes; relations.

• Compare, order, read, and write numbers through the billions
• Compare and order positive and negative numbers
• Positive and negative numbers
• Positive and negative powers to 10
• Add and subtract positive and negative numbers
• Place value; whole numbers, decimals
• Scientific notation
• Exponential notation
• Properties of number systems
• Properties of rational numbers
• The real number line
• Prime and composite numbers
• Square roots
• Fractions/percents equivalencies
• Equivalent fractions, mixed numbers, decimals, percents
• Compare and order fractions and mixed numbers
• Rename improper fractions as mixed numbers or whole numbers
• Rename fractions as percents
• Rename mixed numbers as fractions
• Fractions to express probabilities
• Rates as comparisons
• Rename percents as degrees for sectors of circle graphs
• Rate, rate units, rate notation, per-unit rates
• Scales (ratio comparisons) for models and maps
• Equivalent ratios

Basic measure; length, weight (mass) capacity, elapsed time, angles; compound
measures; area, volume, density, calories; reference frames; ordinal numbers,
times, dates, time lines; thermometers.

• Compare U.S. customary and metric units of measure
• Measures to nearest one-half, one4ourth, one-eight, one-sixteenth inch
• Measure to the nearest tenth of a centimeter
• Convert measurements to different units
• Estimate lengths, weights, or volumes, and then measure to check the estimates for
• a variety of objects
• Perimeter and areas of rectangles
• Largest area for a given perimeter
• Largest perimeter for a given area
• Area problems
• Perimeter, circumference, area, and volume formulas
• Radius and diameter relationship
• Scale drawings and models
• Map scales
• Draw and measure angles with full-circle and half-circle protractors
• Estimate measures of angles
• Plot ordered number pairs
• Graph on coordinate grids to solve problems
• Plot and name points on a coordinate grid
• Relationship between coordinates of endpoints and midpoints of line segments
• Name coordinates

Arithmetic concepts (Meanings of operations), Skills, Whole number facts and
extensions, estimation, number theory; properties.
 

• Estimate products of whole numbers and decimals
• Estimate products of decimals
• Estimate sums of fractions
• Estimate percents and fractions of a circle
• Estimate percent equivalents for fractions
• Estimate a fraction and a percent of a whole
• Estimate products and quotients of fractions with positive and negative numbers
• Estimate products of decimals

Multi digit procedures for whole numbers and decimals; fractions operation
procedures with scientific calculators

• Extend multiplication and division facts
• Multiples of numbers
• Multiply with fractions, decimals, and percents
• Multiplication property of one; reciprocals property
• Multiplication and division with integers
• Size-change factors and similar figures(multiplication application
• Fractions in simplest form
• Rate tables and finding per-unit rates
• Ratios and areas of rectangles
• Ratios to find lengths of corresponding sides of similar polygons
• Multiply whole numbers; decimals; by powers of 10
• Division algorithm; use successive estimates
• Divide by 1 and 2-digit divisors
• Add and subtract positive and negative numbers with a calculator
• Computations and the distributive property
• Common denominators
• Add and subtract fractions with like and unlike denominators
• Add and subtract mixed numbers with like and unlike denominators
• Fraction multiplication algorithm
• Multiply mixed numbers, fractions and mixed numbers
• Division of fractions algorithm
• Divide mixed numbers
• Divide fractions
• Use a percent circle
• Percent of a number
• Percent one number is of another
• Rename fractions as decimals
• Scientific notation
• Convert numbers in exponential to standard notation
• Order of operations

Collect data; landmarks; tables; charts, graphs, probability.

• Organize data from pa personal survey
• Collect, analyze, display data from a recall survey
• Use data to evaluate a diagram and a table
• Graph data from rate tables
• Construct line plots
• Construct, read, and interpret bar graphs, broken-line graphs, step graphs, and time
• graphs
• Read and interpret graphs taken from magazine and newspaper articles
• Construct, read, and interpret circle graphs
• Graph data on a coordinate grid
• Venn diagrams
• Identify and use landmarks: minimum, maximum, mean, median, mode, range
• Probability of an event
• Tree diagrams
• Computer spreadsheets

Notation, definitions, and basic relationships for 2- and 3-dimensional figures;
basic metric properties of plane figures; congruence, similarity, transformations of
geometric figures.

• Circle: interior, sector, radius
• Rectangles
• Descriptions of polygons on the Geometry Template
• Diameter of a sphere
• Ellipses
• Properties of polygons
• Congruence
• Compass and straight-edge constructions
• Types of triangles and quadrilaterals
• Ratios and corresponding side and angles of similar polygons
• Golden rectangles
• Triangles, rectangles, parallologram, circles, polygons
• Rectangular prisms, cylinders, spheres
• Pythagorean Theorem: meaning and proof
• Parallel and perpendicular lines
• Transversals
• Acute and reflex angles
• Sum of the measures of interior angles of a triangle
• Adjacent, vertical, consecutive, supplementary angles
• Straight angles
• Tessellations: regular, semi regular, Escher-type translations
• Isometry transformations: reflections, translations, rotations
• Symmetry: rotation, point, order of rotation symmetry
• Cross sections of solids: cube, cylinder, cone, pyramid, prism, torus
• Topological transformations: genus, properties, equivalence

Number and visual sequences, formula models using variables.
 

• Find patterns in relationships between area and perimeter
• Find a pattern and continue a number sequence
• General patterns and special cases
• Multiplication patterns
• Patterns with positive and negative numbers
• Complete a table given a rule
• Devise a formula
• Computer spreadsheets using formulas
• Formulas for volume: rectangular prisms, cylinders, spheres
• Formulas to solve problems
• Solve problems using formulas for area and volumes of trapezoids and cones
• Formulas, tables, and coordinate grids

Properties of number systems, operations with integers; parenthesis, order of operations; equations; relations of numbers lines; function models (mles, tables,
coordinate grids).
 

• Applications of variables in generalizations, number patterns (properties), formulas
• and spreadsheets
• Variables to state fraction multiplication and division algorithm
• Variables to state reciprocal property
• Variables in operations with integers
• Variables to state distributive property of multiplication over addition and subtraction
• Solve simple number sentences
• Write number sentences to solve problems
• Solve simple open sentences
• Substitute a given value and evaluate expressions
• Number sentences: relation, operation, and grouping symbols
• Equivalent-equation method of solving equations
• Order of operations
• Solution sets of inequalities

Numeration and Order

Place value; read and write whole numbers and decimals; scientific notation, integers; fractions; equivalence classes; relations

• Equivalent names for numbers.
• Compare, order, read, and write numbers from millionths through trillions.
• Compare and order integers.
• Real numbers and the real number line.
• Scientific notation.
• Exponential notation
• Properties of the number system.
• Equivalence and the appropriate use of decimals, fractions, and percents.
• Unit rates.
• Ratio, proportion and percent.
• Golden Ratio.

Basic measure; length; weight(mass) capacity; elapsed time; angles; compound measures; area, volume, time lines; thermometers.

• Choose and use appropriate customary units and metric units of measurement.
• Convert within each system.
• Convert units of time, and compute elapsed time and time zones.
• Largest perimeter for a given area.
• Largest area for a given perimeter.
• Estimate identify and use formulas for lengths, weights, area, perimeter, volume,and circumference.
• Maximize the volume of a pnsm
• Know what a change in the dimensions of a prism will do to the surface area and volume.
• Recognize that two figures with the same volume can have different surface areas.
• Parallel and perpendicular lines their properties, and angle relationships.
• Area of trapezoid, in addition to rectangles3 parallelograms, circles1 squares and composite figures.
• Surface areas and volumes of prisms, pyramids and cylinders.
•  Angle classification and construction.

Operations

Arithmetic of whole numbers, fractions, decimals, percents, integers,estimation and mental arithmetic

• Multiply or divide by powers of 10.
• Estimate and round, sums, differences, products, and quotients of whole numbers, decimals, fractions, percents and integers.
• Reciprocal.
• Size change factors.
• Divisibility.
• Factors and multiples.
• Percents greater than 100 and less than 1
• Percent of a number.
• Percent one number is of another.
• Find number when percent is known.
• Percent of change.
• Applications of percent: markup, sales tax, discount commission, simple interest
• Order of operations.
• Use of scientific calculators.

Collect and analyze data in charts, tables, graphs, probability

• Collect, summarize and display data.
• Draw inferences from data.
• Measures of central tendency, variability, and correlation.
• Use of technology.
• Venn diagrams.
• Choose an appropriate graphical representation.
• Line plots.
• Frequency tables.
• Histograms
• Stem and leaf plots.
• Scatterplots.
• Box and whiskers plot.
• Quartile.
• Outlier.
• Misleading graphs.
• Calculate; mean, median, mode, range, minimum, and maximum.
• Probability Independent and dependent events.
• Permutations and Combinations.
• Counting Principle.
• Experimental probability.

Notation, definitions, and basic relationships for 2- and 3- dimensional figures.

• Polygons and their properties.
•  Classify quadrilaterals.
•  Classify triangles.
•  Name the parts of a prism and pyramid.
•  Draw two dimensional view of a three dimensional figure.
•  Rotations, Reflections and Translations.
•  Tessellations.
•  Pythagorean theorem
•  Isomorphisms
•  Symmetry and Congruence.
•  Similarity
•  Angle measurement, construction and classification.
•  Scale Factors.
•  Coordinate graphing.

Number and visual sequences, formula models using variables.
 

•  Patterns in relationships among area~ perimeter and circumference.
• Formulas, tables, and coordinate grids.
•  Geometric patterns.
•  Arithmetic patterns.
•  Pascal's Triangle
•  Fibonnaci sequence.
•  Sequences and series.
•  Set theory
•  Functions: linear, absolute value, exp~nentiaI, quadratic1 and step.
•  Fractals.

Properties of number system, operations with integers, order of operations, equations, relations functions.

• Write number sentences to solve problems.
•  Write, evaluate,and simplify algebraic expressions.
•  Identify appropriate input and output values for a function.
•  Model a function with tables, graphs,or equations.
• Use formulas to find values and to make graphs.
•  Simplify, solve, and graph one-step equations and inequalities.
•  Simplify, solve, and graph two-step equations.
•  Write and graph inequalities on a number line including intervals with open and closed upper and lower bounds.
•  Choose appropriate scales on axes of a graph.
•  Absolute value equations.
•  Rational expressions.
•  Systems of equations.
•  Slope of a line.
•  Use of technology

Numeration and Order

Place value; read and write whole numbers and decimals; scientific notation, integers; fractions; equivalence classes; relations.
 

• Compare and order all real numbers and the number line.
• Notation: scientific and exponential.
• Properties of the real number system.
• Equivalence and the appropriate use of decimals, fractions, and percents.
• Ratio, proportion and percent.
• Rational and Irrational numbers.
• Absolute value.
• Complex numbers.
• Square roots.
• Scientific notation.
• Exponents and rules of exponents.

Basic measure; length; weight(mass) capacity; elapsed time; angles; compound measures; area, volume, time lines; thermometers.

• Choose and use appropriate customary units and metric units of measurement.
• Convert within each system.
• Convert units of time, and compute elapsed time and time zones.
• Estimate identify and use formulas for lengths, weights, area, perimeter, volume,and circumference.
• Areas of polygons and composite figures.
• Surface areas and volumes of prisms, pyramids, cylinders and cones.
• Maximize the volume of a prism.
• Compare volume to surface area ratios for different sizes and shapes of cylinders and prisms.
• Compare changes in area, perimeter, surface area and volume when an object’s dimensions are changed.
• Area Vs. Perimeter relationships.
• Surface area vs. Volume relationships.
• Tangent ratio.
• Parallel and perpendicular lines , their properties, and angle relationships.
• Angle classification and construction

Operations

Arithmetic of whole numbers, fractions, decimals, percents, integers,estimation  and mental arithmetic

• Estimate and round, sums, differences, products, and quotients of whole numbers, decimals, fractions, percents and integers.
• Inverse operations.
• Size change factors.
• Factors and multiples.
• Percents greater than 100 and less than 1.
• Percent of a number.
• Percent one number is of another.
• Find number when percent is known.
• Percent of change.
• Applications of percent: markup, sales tax, discount commission, simple interest
• Order of operations.
• Use of scientific calculators and technology.

Collect and analyze data in charts, tables, graphs, probability.

• Collect, summarize and display data.
• Draw inferences from data and make predictions.
• Interpret coordinate graphs without scales or labels.
• Draw a graph to match a  written description of a situation.
• Match tables and graphs to a verbal description of a situation.
• Make and test hypotheses.
• Measures of central tendency, variability, and correlation.
• Use of technology.
• Use of various plots:  line plots, stem-and-leaf plots, box and whiskers plots, scatterplots.
• Visually fit a line to data points on a scatterplot and use it to make predictions.
• Frequency tables.
• Histograms, multiple-bar and line graphs..
• Probability distribution.
• Permutations and Combinations.
• Counting Principle.
• Exponential growth and decay.

Geometry

Notation, definitions, and basic relationships for 2- and 3- dimensional figures.

• Polygons and their properties.
• Find sums of interior angles of polygons.
• Classify polygons.
• Pythagorean theorem.
• Isomorphisms: translations, rotations, reflections.
• Symmetry, Congruence and Similarity.
• Scale Factors, ratio and proportion.
• Coordinate graphing.
• Analytic geometry.
• Construct nets for pyramids, prisms and cylinders.
• Identify faces, vertices, and edges of three-dimensional figures.
• Draw two-dimensional and three-dimensional views.
• Identify transversal and the appropriate angle formations.

Number and visual sequences, formula models using variables.

• Patterns in relationships among area, perimeter and circumference.
• Formulas, tables, and coordinate grids.
• Arithmetic, geometric, and algebraic patterns.
• Pascal’s Triangle
• Fibonnaci sequence.
• Sequences and series.
• Set theory.
• Functions: linear, absolute value, exponential, quadratic, trigonometric, polynomial, radical and step.
• Inverse function.
• Fractals.
• Use of technology.

Algebra and Use of Variables

Properties of number system, operations with integers, order of operations, equations, relations functions.

• Write number sentences to solve problems.
• Write, evaluate,and simplify algebraic expressions.
• Simplify, solve, and graph one-step equations and inequalities.
• Simplify, solve, and graph two-step equations and inequalities.
• Absolute value equations.
• Rational expressions.
• Systems of equations.
• Slope and y-intercept of various lines.
• Graph linear and non-linear functions.
• Use tables, graphs and equations to model relationships.
• Draw a graph to match a written description of a situation.
• Exponential growth and decay.
• Use of the distributive property.
• Describe input and output of a function and understand that output depends upon input.
• Use a variable expression to write a rule for a sequence.
• Factoring.