• Use simple drawings for accurate representation of the problem.

• 2E
• 2F

• Use a variety of manipulatives to accurately represent the problem.

• 2E
• 2F

• Restate problems in own words to demonstrate understanding.

• 2E
• 2F

• Translate from pictures of objects or manipulatives to numerical expressions.
• Use words to describe a problem.
• Translate from numerical expressions to pictures or manipulatives.

• 2E
• 2F

• Use manipulatives to represent the problem.

• 2F
• 2G

• Identify a pattern.
• Choose an operation.
• Use trial and error.
• Use manipulatives.
• Evaluate methods of solution.
• Draw a picture or diagram.
• Work backwards.
• Solve a simpler problem.
• Choose appropriate operation.
• Identify key words to identify appropriate strategy.
• Identify questions to be answered.

• 2G
• 2H

• Explain to others through drawing, speaking, writing and/or manipulatives.

• 2E
• 2F

• Translate from pictures of objects or manipulatives to numerical expressions.
• Use words to describe a problem.
• Translate from numerical expressions to pictures or manipulatives.

• 2E
• 2F

• Make organized lists or tables of information necessary.
• Identify important information.
• Choose appropriate operation.
• Identify key words to identify appropriate strategy.
• Identify questions to be answered.

• 2E
• 2F
• 2G
• 2H

• Identify a pattern.
• Choose an operation.
• Use trial and error.
• Use manipulatives.
• Evaluate methods of solution.
• Draw a picture or diagram.
• Work backwards.
• Solve a simpler problem.
• Choose appropriate operation.
• Identify key words to identify appropriate strategy.
• Identify questions to be answered.
• Solve multi-step problems.

• 2E
• 2F
• 5A
• 6B
• 6C

• Identify the problem.
• Recognize key information.
• Recognize key words to determine operation needed.
• Interpret data (e.g., diagram, array,
• organized list, table, T-chart, pictures).
• Select the information to solve the problem.

• Explain operations/ strategies used (e.g., draw a picture, make a model, guess and check, make a list, make a table, use a pattern, work backwards, solve a simpler problem, draw a diagram).
• Interpret remainders.
• Identify too much or too little information.
• Determine whether to over or under estimate.
• Justify your answer.

Use a variety of problem-solving strategies.

• Choose and apply the correct strategy (work backwards, guess and check, draw a picture, make a chart/table, find a pattern).
• Choose appropriate operation.
• Identify key words to identify appropriate strategy.
• Identify question(s) to be answered.
• Solve multi-step problems.

Understand basic valid and invalid arguments.

• Process matches strategy.
• Analyze too much/too little     information.
• Determine the reasonableness of answer.

• Choose and apply the correct strategy.
• Choose appropriate operation.
• Display information in an organized and efficient way.
• Recognize multiple ways to solve a problem.
• Use appropriate labels.

Use a variety of measuring processes to model and to solve problems.

• Choose and apply an appropriate measurement strategy ( perimeter, circumference, area, volume, time and money).

• Identify key words to select  operation.

• Choose and apply an appropriate problem solving strategy (guess and check, draw a picture, make a model, make a list, make a table, find a pattern, work backwards, solve a simpler problem, draw a diagram).

• Use estimation to determine if solutions to word problems are reasonable.
• Determine whether a given estimate is an overestimate or underestimate.

Use a variety of strategies to understand problem-solving situations and processes.

• Choose appropriate operation.
• Use correct order of operations.
• Choose an appropriate strategy.
• Identify key words.

• Recognize patterns.
• Describe a pattern.
• Make predictions based on patterns.
• Communicate arguments or thought processes.

• Recognize and articulate a problem.

• Translate words to a mathematical expression or equation.
• Recognize and use symbols for absolute value, square root, scientific notation, permutations, and combinations.

• Identify key words.
• Design a simpler problem.

• Check for a reasonable answer.
• Communicate steps and processes used to arrive at answer.

• Use deductive reasoning to write two-column, paragraph, or flow chart proofs.
• Verifying statements by a theorem, postulate, corollary, or definition.
• Justify statements using algebraic properties.

• Create a math model.
• Uses a math model to solve problems.
• Develop a two column proof.
• Use software to solve problems.
• Create Venn diagrams.

• Use counterexamples.
• Use estimation to check if answers are reasonable.

• Identify keywords.
• Translate phrases to algebraic expressions.

• Use order of operations.
• Identify and use of grouping symbols.

Use ordinal numbers appropriately (1st, 2nd, 3rd).

• Apply ordinal numbers to drawings, manipulatives or real life situations.

Count objects to 20.

• Apply one-to-one correspondence.
• Orally rote count to 20.
• Apply a method for organizing objects when counting.

• Recognize written numerals.
• Identify the number of objects in sets.
• Read and understand number lines.

• Compare sets that have more, less or equal amounts using manipulatives.
• Determine one more or one less than a number using manipulatives.
• Put numbers in order from least to greatest or greatest to least using manipulatives.

• Apply ordinal numbers to drawings, manipulatives or real life situations.

• Apply one-to-one correspondence.
• Orally rote count to 100.
• Apply a method for organizing objects when counting.
• Count by 2’s, 5’s and/or 10’s.

• Recognize written numerals.
• Identify the number of objects in sets.
• Read and understand number lines and hundreds charts.
• Count on or count back from a given number.
• Determine greater than, less than or equal.
• Put numbers in order from least to greatest or greatest to least.

Understand the relationship of fractional parts, 1/3,1/2 and 1/4 to a whole.

• Use models to show basic fractions as part of a whole.

• Skip count by tens.
• When given a group of manipulatives organized by tens and ones, students can determine the number using the place value.
• Identify the number of tens in a given number using manipulatives.

Count numbers 0-1000 (e.g., by 1’s, 2’s, 5’s, 10’s, 100’s).

• Apply one-to-one correspondence.
• Orally rote count to 1000.
• Apply a method for organizing objects when counting.
• Count by 2’s, 5’s, 10’s and/or 100’s.

• Recognize written numerals.
• Read and understand number lines.
• Given a picture or representation of a number, determine the corresponding numeral.
• Given a numeral, relate the numeral to a picture or representation.
• Relate written numerals to the corresponding word.
• Relate objects in sets to the corresponding numeral.

Read and write numerals 0-1000 (e.g., standard and expanded form).

• Recognize written numerals.
• State numerals.
• Write standard and expanded form using manipulatives.
• Write numbers up to 1000.

• Determine greater than, less than or equal.
• Put numbers in order from least to greatest or greatest to least.
• Apply ordinal numbers, such as “first” through “twentieth” correctly.
• Accurately use symbols [<, >, =] and language, such as “between”, “less than”, “greater than” or “equal to”.

• Recognize that the denominator tells the number of parts in a whole and the numerator tells how many parts are represented.
• Identify halves, thirds, fourths and eighths.
• Represent  using manipulatives or pictures halves, thirds, fourths and eighths.
• Write fractions naming halves, thirds, fourths and eighths.

• Determine the number using the place value when given a group of manipulatives organized by ones,  tens and hundreds.
• Identify numbers in terms of place value.
• Count on or count back from a given number by 1’s, 10’s and 100’s.

• 1A
• 1F

• Determine the number using the place value given a group of manipulatives organized by ones, tens and hundreds.
• Identify the number of tens, hundreds and thousands in a given number using manipulatives.
• Show representations of numbers with manipulatives verbally, drawing, and in writing.
• Identify numbers in terms of place value.
• Count on or count back from a given number by 1’s, 10’s, 100’s, 1000’s and 10,000’s.

• Recognize that even numbers end in 0,2,4,6,8.
• Recognize that odd numbers end in 1,3,5,7.

• Use manipulatives/coins to express numbers through the hundredths place.
• Show representations of decimals with manipulatives and/or money verbally, drawing, and in writing.

• Determine greater than, less than or equal.
• Put numbers in order from least to greatest or greatest to least.
• Accurately use symbols [<, >, =] and language, such as “between”, “less than”, “greater than”, or “equal to”.
• Skip count by 10’s, 100’s and 1000’s.

• Identify fractions with like denominators, students look at the numerators and the larger the numerator the larger the fraction.
• Identify fractions with like numerators, students look at the denominator and the larger the denominator the smaller the fraction.
• Use benchmark fractions like 0, 1/2, or 1 to compare fractions.

• Use decimal place value to determine size of decimal.
• Use money to compare amounts.

• Recognize written numerals.
• State and write numerals from 1,000 to 999,999.
• Write standard and expanded form.

• 1a
• 1b
• 1e
• 1f

• Identify and understand the place value and value of each digit in numbers through the millions.
• Places the commas in the correct position.
• Reads and writes whole numbers.
• Reads and writes whole numbers in word form, standard form, and expanded form.
• Orders and compares numbers through millions.

• 1b
• 1f

• Recognizes that the number to the right of said place value determines whether you round up or stays the same. ( ≥5 rounds up and <5 stays the same.
• Recognizes that when rounding, numbers to the left of the said place value stays the same.
• Recognizes that when rounding, numbers to the right of the said place value changes to zero.

• 1d
• 1e

• Recognizes equivalent fractions.
• Multiplies or divides numerator and denominator by the same number to find common denominator.

• 1a
• 1c

• Reads, writes, and recognizes fractions.
• Reads, writes, and recognizes decimals.
• Reads, writes, and recognizes mixed numbers.
• Reads, writes, and recognizes whole numbers.

• Convert fractions to decimals to percents.
• Convert a mixed number to an improper fraction and vice versa.
• Write and read fractions, decimals, and percents.
• Understand equivalence.
• Understand the relationship between fractions, decimals and percents.

• Understand place value.
• Understand periods (ones period, thousands period, etc.).
• Read and write numbers in standard form, expanded form, and word form.

• Understand and use rules of divisibility.
• Identify the greatest common factor of two whole numbers by listing the factors or prime factorization.
• Identify the least common multiple of two whole numbers by listing multiples or prime factorization.
• Identify the least common denominator for fractions by using least common multiple.

• 1i
• 2h

Understand the characteristics and applications of scientific notation and exponential notation.

• Understand when scientific notation/exponential notation should be used.
• Change a number from standard form to scientific notation and vice versa.
• Understand the meaning of base, exponent, and power.

• 1c
• 1d

Understand the relationships among fractions,

decimals, and percents.

• Move fluently between fractions, decimals, and percents.
• Compare and order fractions, decimals, and percents.

Understand the concept of prime and composite numbers.

• Recognize the difference between prime and composite numbers.
• Recognize that 0 and 1 are not prime or composite.

• 1e
• 1f
• 1i

• Move fluently among the various representations of rational numbers.
• Compare and order rational numbers.

Understand the basic laws of exponents.

• Add exponents when multiplying like bases.
• Multiply exponents when a power is raised to another power.
• Subtract exponents when dividing like bases.
• Identify like terms when simplifying expressions.

• Solve a linear system using matrices.
• Understand the terminology that is used with matrices, sequences and finite graphs (linear programming, step and piecewise functions).
• Perform operations with matrices.
• Use different types of sequences including geometric, arithmetic, explicit and recursive.
• Discover and continue patterns and sequences.
• Classify and interpret data from line, bar, circle, step, trigonometric graphs etc.

• Classifying numbers in their correct subsystem(s) (rational, irrational, integer, natural, whole)
• Distinguish between rational and irrational numbers.
• Define each subsystem of the real number system.

• Know and use the laws of exponents, including scientific notation.
• Define i.
• Distinguish between different  types of notation. 
• Simplify expressions involving exponents and roots.
• Solve equations involving exponents and roots.
• Use order of operations involving exponents and roots.
• Convert between exponential and logarithmic expressions.
• Use logarithms to solve exponential equations.

• Compare/convert decimals and percents.
• Use proportions to solve problems involving equations and similarity.
• Find a percent of a certain number.
• Solve decrease and increase problems.
• Distinguish between different notations of ratios.
• Identify place value of numbers in decimals.
• Know appropriate uses for decimals, fractions or percentages to solve problems.
• Be able to define ratio and proportion.
• Write a trig ratio and use to solve a problem.
• Use Laws of Sines and Cosines to solve problems.

• Use manipulatives to represent addition problems.

• Use manipulatives to represent  subtraction problems.

• Compare sets that have more, less or equal using manipulatives.

• Use strategies (counting on, reasoning from a known fact, doubles, doubles +1, ten frames, etc.) to solve addition facts to ten.

• Use strategies (counting down, counting up, reasoning from a known fact, doubles, ten frames, etc.) to solve subtraction facts to ten.

• Use a variety of strategies to add two digit numbers with no regrouping. 
• Use the process of adding two-digit numbers with no regrouping.

• Use a variety of strategies to solve a problem without regrouping.
• Use the process of subtracting with two-digit numbers with no regrouping.

• Use strategies (counting on, reasoning from a known fact, doubles, doubles +1, ten frames, etc.) to solve addition facts through 20.

• Use strategies (counting down, counting up, reasoning from a known fact, doubles, ten frames, etc.) to solve subtraction facts through 20.

• Use a variety of strategies to solve two-digit  addition problems with and without  regrouping.
• Define the term sum.

• Use a variety of strategies to solve two-digit subtraction problems with and without  regrouping.
• Define the term difference.

• 1F
• 2B

• Know subtraction is the opposite of addition.
• Use strategies such as fact families and fact triangles.

• Use strategies to compute basic facts (X2 doubles the number, X3 doubles and adds one more, X4 double double, skip counting, X9 pattern, and reason from a known fact).

• Apply traditional rules for rounding (≥5 rounds up and <5 stays the same).
• Use front-end estimation when appropriate.
• Determine when to overestimate or underestimate.
• Use benchmark numbers to estimate.

• Choose appropriate operation
• Identify key words to identify appropriate strategy.
• Identify questions to be answered.
• Identify the problem.
• Recognize key information.
• Recognize key words to determine operation needed.
• Interpret data (e.g., diagram, array, organized list, table, T-chart, pictures)
• Select the information to solve the problem.
• Use parentheses.

• Apply strategies for basic addition and subtraction facts.
• Rename numbers  (e.g., regroup, borrow, trade, etc.).
• Apply mental math skills when appropriate.
• Estimate to check reasonableness of answer.

• 2 B
• 2C

• Know multiplication basic facts 0-9.
• Use the process of multiplying by a 2-digit number.
• Identify the terms: product and factor.

• 2B
• 2C
• 2D
• 3B

Use all four operations with money.

• Line up decimal points when adding and subtracting.
• Place decimal point and dollar sign correctly.
• Know basic facts for all operations.
• Place the decimal in the correct place value when multiplying and dividing.

• Know division is the opposite of multiplication.
• Use multiplication basic facts to help divide.
• Identify the terms: divisor, dividend and the quotient.

• Recognize division is the opposite of multiplication.
• Use multiplication to check division.
• Recognize the division steps when dividing by a 1-digit divisor with or without a remainder.
• Identify divisor, dividend and the quotient .
• Know multiplication facts 0-9.
• Know division facts 0-9.

• 1A
• 1B

• Recognize addition and subtraction are inverse operations.
• Recognize multiplication and division are inverse operations.
• Identify multiplication as repeated addition.
• Use greater than and less than symbols correctly.
• Use equal and not equal symbols correctly.

• 1B
• 2A

Use estimation strategies (rounding/front end).

• Estimate to check reasonableness of problems by rounding.
• Understand the rules of rounding (0-4 stays the same, 5 + rounds up).
• Estimate sums, differences, products, and quotients.
• Understand when to estimate to determine the appropriate estimation.
• Use the highest place value in front end estimation.

• Know basic facts.
• Recognize compatible numbers.
• Understand and apply the process of estimating. 
• Understand the properties of addition and multiplication.
• Understand the process of rounding.

• Know basic facts.
• Add multi-digit numbers with regrouping.
• Subtract multi-digit numbers with regrouping.
• Multiply a 2 or 3 digit by a 2 or 3 digit with and without regrouping.
• Divide by 1 or 2 digit divisors with or without remainders.
• Appropriately place zeros in the quotient.
• Know vocabulary terms (dividend, divisor, quotient, sum, addend,  difference, factor, product)
• Interpret remainders.

• Know basic facts.
• Line up the decimals when adding and subtracting.
• Count decimal place values when multiplying.
• Move the decimal when multiplying or dividing by 10,100, or 1000.
• Divide a decimal by a whole number.

• Apply divisibility rules to simplify fractions.
• Add the numerators and the denominator stays the same.
• Understand equivalence.

• Apply divisibility rules to simplify fractions.
• Add or subtract the numerator, and the  denominator stays the same.
• Understand equivalence.
• Understand and  apply multiples and factors of a number in fraction conversion.
• Find a common denominator and convert the fraction.

• Use the estimate to determine reasonableness of results.
• Use inverse operations to check answers.
• Identify compatible numbers.
• Identify key words.
• Use number sense.

• Use powers of ten, compatible numbers, breaking numbers apart, front-end estimation, rounding.
• Estimate and judge reasonableness of answer.

6.3.2a Add, subtract, multiply, and divide whole numbers.

• Add and subtract multi-digit numbers.
• Use multiplication as a check for division.
• Multiply and divide a multi-digit number by a 1-digit number.
• Divide a 4-digit number by a 2-digit number.
• Develop computational fluency with all operations.

6.3.2b Add, subtract, multiply, and divide, fractions and mixed numbers.

• Add and subtract mixed numbers with unlike denominators with regrouping.
• Add and subtract fractions with like and unlike denominators; change improper fractions to mixed numbers.
• Multiply and divide a whole  number by a fraction.
• Divide a mixed number by a whole number or a fraction.
• Divide a whole number, fraction or mixed number by a mixed number.
• Change a fractional numeral to its simplest form (lowest terms).

6.3.2c Add, subtract, multiply, and divide decimals.

• Add and subtract decimals through hundred-thousandths.
• Multiply a decimal by a decimal.
• Divide a decimal by a whole number and by a decimal.
• Multiply a decimal by multiples of 10, 100, or 1000.

Understand the correct order of operations for performing arithmetic computations.

• Evaluate expressions using the order of operations (may include parentheses or exponents).
• Solve equations involving more than one operation.

• Use possible estimation strategies (front-end estimating, rounding, compatible numbers and clustering).

• 2b
• 2c
• 2d
• 2e
• 2f (integers)

• Use appropriate algorithm to compute.
• (i.e.  lining up decimal to add and subtract, using common denominators, putting decimal in appropriate place when multiplying and dividing).

• 2d
• 2e
• 2g

• Move fluently among fractions, decimals, and percents when computing.

• Use possible estimation strategies  (front-end estimation, rounding, compatible numbers and clustering).

• 2b
• 2c
• 2d
• 2e
• 2f
• 2h

• Use appropriate algorithm to  compute.

• Choose and use appropriate algorithm.

• Calculate discounts, markups, tax, tip, and commissions.
• Recognize benchmark percentages (25%,50%,75%,100%, etc.).

• Use order of operations to simplify expressions involving positive and negative numbers and variables.
• Define and combine like terms.
• Use the distributive property to multiply polynomials.  (FOIL for binomials).
• Perform long division and/or synthetic division of polynomials.
• Use factoring to simplify rational expressions.

• Use order of operations to simplify expressions involving ALL operations.
• Use reciprocals and roots when necessary.
• Find the absolute value of a number.
• Simplify rational expressions involving roots.
• Simplify an expression with a negative or zero exponent.
• Apply the properties of exponents.

• Find percents of numbers.
• Find percent increase and percent decrease.
• Apply interpreted data from tables and graphs to solve problems.
• Use proportions to create scale drawings.

• Compute radical expressions that contain positive rational numbers.
• Know how to simplify radical expressions (rationalize the denominator/conjugate).

9-12.3.5

• Use guess and check.
• Check the answer by calculating an estimate.
• “Just stop and think about it!” approach.

• State the name and value of a coin when given a model.

• State time shown on an analog clock to the nearest  hour. 

• Use a calendar to identify days of the week, sequence of days, months and seasons.

• Use non-standard measuring tools to compare objects by size and weight.

• Identify and write time shown on an analog clock to the hour- and half-hour.

• Use manipulatives or graphic representations to count combinations of coins (pennies, nickels and dimes).

• Use non-standard tools (unifix cubes, links, inchworms) to measure weight and length.

• Skip count by 5s  Identify and write time shown on an analog and digital clock by five minute intervals.

• Use  models to count to a $1.00 given quarters and half dollars.

• Read thermometers in both Celsius and Fahrenheit degrees.

• Use standard units to measure length, height and weight.
• Compare and order numbers.

• Identify and write time shown on an analog clock to the nearest minute  Identify and write time according to a.m. and p.m. 
• Recognize what time of the day a.m. and p.m. represent.

• Use models to count, write and compute money up to $5.00. 
• Skip count by 5s, 10s, and 25s  Count on by 10s and 5s.

• Identify length, weight and volume to the nearest whole unit.
• Explain how to calculate perimeter, weight and volume. 
• Calculate perimeter and volume given a model.

• Identify 60 minutes in an hour  Identify five minute sections on a clock. 
• Identify 30 minutes and 15 minutes on a clock.
• Count on or back by hours and minutes until final elapsed time is calculated. 
• Subtract time by converting hours and minutes.

• Count coin values correctly. 
• Count dollar values correctly.

• Count coin values correctly  Subtract money as decimals. 
• Count back change by starting with the smallest value of money.

• Know how to find perimeter, volume, and area using standard units. 
• Choose the appropriate tool and unit to measure using standard units. 
• Measure correctly.

• Know how to find perimeter, volume, and area using metric units. 
• Choose the appropriate tool and unit to measure using metric units. 
• Measure correctly.

• 3a
• 3f

• Decide the appropriate unit (size) to use for measurement.
• Know appropriate units for length, weight,  capacity.
• Convert units of length, weight, and capacity in the standard system.
• Convert units of length, weight, and capacity in the metric system.

• 3c
• 3d

Understand and apply the basic measures of perimeter, area, volume.

• Differentiate between perimeter, area, and volume.
• Find the perimeter of any polygon including the correct  label.
• Find the area of squares and rectangles including the correct label.
• Find the volume of a rectangular solid including the correct  label.

Understand and manipulate the concept of time (add, subtract, convert).

• Add, subtract, and convert seconds, minutes, hours, days, weeks, months, and years.
• Calculate elapsed time.

Select and use appropriate tools for given measurement situations.

• Select the appropriate unit of measure for length, area, volume and mass.
• Convert measurements in the customary system.
• Convert measurements in the metric system.
• Measure length with metric measures.
• Measure length with customary measures.
• Measure angles using a protractor.

Understand and apply measures of perimeter, area, volume, and circumference.

• Find the perimeter of the square or rectangle by using the formula.
• Calculate the area of irregular shapes.
• Calculate the area of a parallelogram and rectangle.
• Calculate the circumference of a circle using the formula.

• 3c
• 3d
• 3f

• Choose appropriate tools for obtaining measurements to calculate surface area, perimeter, volume and area.
• Express measurements using appropriate units.

• 3a
• 3f

• Convert within a system of measurement (e.g. inches to feet, centimeters to kilometers, etc.).

• 3c
• 3d
• 3f

• Choose appropriate tools for obtaining measurements to calculate area, perimeter, volume and circumference.
• Express measurements using appropriate units.

• Find perimeter and area of basic polygons, circles, and irregular figures. (2D)
• Find volumes and surface areas of cylinders, cones, prisms, spheres, pyramids, and irregular solids. (3D)
• Identify measurements needed to solve for perimeter, area, or volume in all figures.
• Identify and name 2-D and 3-D figures.
• Use correct units of measure.
• Sketch a figure/draw a diagram.

• Understand the relationship between position, velocity, and acceleration.
• Use correct labels.
• Use and analyze graphs to assist in problem solving process.
• Use reciprocal rates to solve problems.
• Solve motion problems using formulas, charts, tables, and technology.
• Work with derivatives to find the position, velocity and acceleration of an object.

• Define precision and accuracy using the correct measuring tools. 
• Express answer using significant digits and/or scientific notation with the correct unit of measure.

• Use appropriate tools to determine measurement.
• Accurately read the measurement units on these tools.
• Choose reasonable units to measure in both the metric and standard systems of measure.
• Represent angle measurements in degrees and radians.

• Uses proportions to assist in the conversion process.
• Convert different units dealing with distance, speed, mass etc.
• Convert to common units to solve multi-dimensional problems.

• Identify similarities/differences among basic plane figures: circles, square, rectangle and triangle. 
• Identify and continue patterns.

• Use words such as inside, outside, between, above, below and behind.

• Using pictures and manipulatives students can identify, label and continue patterns.

• Identify properties/descriptors of plane figures (e.g., square has four equal sides, circle is a closed figure).
• Describe and create two dimensional shapes using properties (number of sides, length of sides, and numbers of corners).

• Explain properties of spheres, cubes, rectangular prisms, cylinders, pyramids, and cones and basic geometric terms (e.g., sides, edges, and corners). 
• Sort plane figures by properties. 
• Count faces, vertices and edges.

• Define symmetry and draw lines of symmetry.   
• Identify examples and non-examples of symmetry.

• Identify parallelograms, trapezoids, spheres and cubes when shown a model or a picture.

• Draw the line of symmetry for a given shape or picture. 
• Identify congruent figures. 
• Draw a figure that is congruent  to a given figure.

• Identify parallel, perpendicular, and/or intersecting lines. 
• Identify right, acute and/or obtuse angles. 
• Identify a line, ray and/or line segment. 
• Draw a line, ray and/or line segment. 
• Draw parallel, perpendicular and intersecting lines. 
• Draw a right, acute and / or obtuse angle.

• Classify polygons by the number of faces, vertices, edges and angles  Identify lines of symmetry in two-dimensional figures. 
• Identify lines of symmetry in three- dimensional figures. 
• Sort and classify two and three dimensional figures by properties.

• Know basic geometric language for describing and naming shapes (e.g., trapezoid, parallelogram, cube, sphere and polygon).

• Identify similar and congruent figures. 
• Perform slides, flips and turns. 
• Given two figures, determine if slides, flips or turns were performed. 
• Identify lines of symmetry in two-dimensional shapes. 
• Draw a similar and congruent figure. 
• Draw lines of symmetry in two-dimensional figures.

• 4a
• 4b
• 4c

• Recognize and understand the vocabulary.
• Classify types of angles as acute, obtuse, and right.
• Classify polygons by sides and angles.
• Identify symmetrical figures.
• Identify congruent  figures.
• Provide real world examples of the figures.

• Measure angles by using a protractor
• Identify points, line segments, rays, angles and lines.
• Classify types of angles as acute, obtuse, and right.
• Classify types of polygons as triangle, quadrilateral, pentagon, hexagon, and octagon.
• Identify the diameter, radius, chord, and circumference of a circle.
• Determine congruency and similarity between lines, angles and polygons.
• Classify types of quadrilaterals as square, rectangle, rhombus, trapezoid, and parallelogram.
• Find the missing angle in a triangle when two angles are given.
• Identify when two intersecting lines are perpendicular.
• Identify when two lines are parallel.
• Use correct vocabulary.

• 4c
• 4d

• Names basic characteristics of polygons (e.g.  can name a polygon based on the number of sides).
• Measure and describe angles.
• Recognize parallel and perpendicular lines.
• Use proper notation to identify angles, lines, and parts of lines.
• Recognize congruent figures and use proper notation to identify them.

• 4c
• 4d

Understand the characteristics of (classification) and relationships among quadrilaterals, triangles, and circles.

• Name basic characteristics of polygons (e.g. a rectangle has 4 right angles, opposite sides parallel and congruent).
• Use proper notation to identify angles, lines, and parts of lines.
• Recognize types of triangles according to angles and size.

• Discover relationships between lines (slope, parallel, coincident and intersecting lines) and angles (angles formed by parallel lines cut by a transversal, classification of angles by size).

• Solve a right triangle.
• Apply the Pythagorean Theorem to real-world situations.
• Identify the parts of a right triangle.
• Use the converse of the Pythagorean Theorem to decide if a triangle is a right triangle.

• Know relationships among the sides and areas of 30-60-90 and 45-45-90 right triangles.
• Use properties to develop the trig functions.
• Relate special triangles to the unit circle.

• Use sector, area, circumference and arc length in circles to solve problems.
• Solve problems involving angular and linear velocity.
• Interpret scale drawings and vice versa.

• Construct basic points of concurrency, (centroid, circumcenter, orthocenter, incenter).
• Construct basic shapes such as squares, equilateral triangles, parallel lines, perpendicular lines, midpoint, etc.

• Convert between forms of linear equations such as standard form, slope-intercept and point-slope.
• Graph linear equations.

To understand and use geometric transformations and their properties to solve problems. [Geometry, Pre-Calculus]

• Use dilations and scale factors when solving problems dealing with similar shapes and figures.
• Perform reflections, rotations, and translations.
• Create tessellations using transformations.
• Identify symmetry including points and lines.

• Find triangle segment lengths using basic trig functions.
• Find angle measurements using basic trig functions.
• Define angle of elevation and depression.

• Determine slopes of parallel and perpendicular lines.
• Distinguish between no slope and undefined slope.
• Use scale factors and congruency statements to find segment lengths and angle measurements.

Understand the characteristics and uses of vectors (e.g., representations of velocity ad force) and basic operations on vectors (e.g., vector addition, scalar multiplication). [Pre-Calculus]

• Solve problems involving velocity and force by applying vectors.
• Perform vector addition and scalar multiplication.

9-12.

• Recognize shapes generated by the intersections of differing objects.
• Use a variety of methods to determine area and lengths of shapes that are generated.

• Make simple graphs and charts with pictures and/or simple objects.

• Answer questions based on the graph. 
• State observations and comparisons based on the graph.

• Collect  data. 
• Organize data in charts and/or tables. 
• Make graphs with pictures and/or simple objects.

• Understand and use the method of tallying for gathering data. 
• Organize data into an appropriate numbering system for the graph, table or chart. 
• Use the given data to create a graph, table or chart.  
• Use an appropriate title and labels for the axis.
• Make simple graphs and charts with pictures, simple objects and graph paper.

• Answer questions based on the data in the graph. 
• State observations and comparisons based on the graph.

• Understand and use the method of tallying for gathering data. 
• Organize data into an appropriate numbering system for the graph, table or chart. 
• Use the given data to create a graph, table or chart. 
• Use an appropriate title and labels for the axis.

• Recognize and understand the number system used. 
• Use a key to interpret data. 
• Compare data to answer questions.  
• State observations and comparisons based on the graph.

• Understand that data represents specific pieces of information from the real world. 
• Identify least, greatest, most often  and least often.   
• Make predictions based on the represented data. 
• Make comparisons. 

• Collect  data using observations, measurements, surveys or experiments. 
• Organize and construct line, circle or bar graphs.  
• Create a key to indicate quantity represented.

• 5a
• 5b

Use graphs and plots to display information.

• Use information (given or collected) to create double bar graphs, line graphs, pictographs, line plots, stem and leaf plots.
• Understand how to use the parts of a graph  (key, title, axis labels, scale, intervals).

Read and interpret tables, graphs, and charts.

• Interpret information from different graphs including picture, pie, bar, and line.
• Use a table to solve algebraic problems.
• Explain and justify conclusions drawn from tables, charts and graphs.

Understand appropriate measures of statistics, (e.g. mean, median, mode, and range).

• Compute mean, median, mode and range.
• Choose the measure that best describes a set of data.

• 5a
• 5b

Make comparisons, predictions and inferences from data in a variety of formats.

• Gather information from charts, tables and graphs.
• Use information to draw conclusions and/or make predictions.

• 5a
• 5b

• Choose the best method for displaying data that is clear and concise.

• 5a
• 5b

• Recognize how data can be manipulated/displayed to influence others, both positively and negatively.

• Collect data.
• Create an appropriate display such as stem-leaf charts, bar and line graphs, box diagrams, histograms, frequency distributions, etc.
• Calculate measures of central tendency such as mean, median, and mode.
• Calculate 5 number summary (minimum, 1st quartile, median, 3rd quartile, maximum).

• Use data analysis to determine right and left skew and a bell curve.
• Calculate a regression equation.
• Calculate and use standard deviation.

• 6d
• 5b

• Find lines of best fit and prediction equations. 
• Know when it is appropriate to interpolate and extrapolate.

• 2k
• 5a
• 6d

• Interpret problem and decide on best graphing technique.
• Organize information and represent it with a contingency table.

• Collect and analyze data.
• Calculate the measures of central tendency and determine which is most appropriate in the given context.

• Introduce the likelihood of certain events.

• Discuss the likelihood of certain events.

• Predict the likelihood of certain events.

• Recognize that some events cannot be predicted fairly well because we might not know all of the circumstances.

• Determine if an event  is certain, possible or impossible to happen.  
• Students will understand the terms likely and unlikely.

• Use the term chance correctly when describing the likelihood of an event. 
• Use the terms of probability to describe the fairness or unfairness of a game.

• Compute the probability or chance of something happening compared to the number of possible outcomes. 
• Collect and record data.

• Conduct an experiment and analyze the data. 
• Recognize the difference between experimental probability and theoretical probability. 
• Predict future outcomes from the sample.

• Understand a fair versus an unfair game.
• Show probability as a fraction or a ratio.
• Understand and create a tree diagram, outcome grid, sample space and read it to determine probability.

• Predict results of a given situation of probability using coins or a spinner.

• Perform a probability experiment.
• Record and analyze the results.

• 5b
• 5c

• Calculate theoretical probability and use results to make predictions.

• Find the number of ways a set of items can be arranged both when order matters and when it doesn’t.

• Understand how trends in current data are used to make predictions.

• 5b
• 5c

• Understand how to collect a random sample.
• Understand that a sample size affects the validity of  the outcome.
• Recognize the importance of current data.

• 5b
• 5c

• Calculate theoretical probability and use results to make predictions.

• Use tree diagrams or lists to solve probabilities.
• Collect and use experimental data to evaluate probabilities.
• Use formulas such the multiplication counting principle to find probabilities.
• Use permutations and combinations.
• Compute odds.

• Find geometric probabilities.
• Use compliments to find the probability.
• Analyze binomial distributions.
• Use normal distributions to approximate probabilities.
• Find expected values of outcomes.

• Use permutations to count the number of arrangements.
• Use combinations to count the number of ways an event can happen.
• Use the addition and multiplication rules to determine the probability of multiple events.
• Differentiate between independent and dependent events.

• Identify, create and extend patterns with physical objects, geometric shapes, manipulatives and pictures.

• Identify, create and extend patterns with physical objects, geometric shapes, manipulatives and pictures.   
• Label the pattern and identify the pattern unit.

• Use fact families and various strategies to solve number sentences such as 3 + ___ = 10.

• Identify patterns in events, designs, shapes, sets of numbers, etc.

• Identify, create and extend patterns both linear and non-linear, with physical objects, geometric shapes, manipulatives and pictures.

• Identify, create and extend number patterns (e.g., skip counting, function machines, etc.).

• Use fact families and various strategies to solve number sentences 3 + N = 10. 
• Use number sentences such as 3 + N = 10 to solve problems. 
• Use number sentences to describe patterns. 
• Define variable.

• Solve sentences involving one or both operations, using a order of operations strategy.

• Understand overall concepts of multiplication and division. 
• Apply basic multiplication and division facts.

• Understand the vocabulary in regards to patterning. 
• Use terms correctly when discussing patterns. 
• Extend a pattern.

• Employ T-charts, function rules  (input- output), number charts, etc.

• Know what  the equal sign means. 
• Find the missing number in an equation. 
• Know that both sides of the equal  sign must balance.

• Apply basic facts. 
• Solve for a variable by recognizing a number sentence balances when the value of the left side equals the right.

• Apply basic facts. 
• Compare and order numbers. 
• Read number expressions using symbols of equality and inequality.

• Create tessellations. 
• Identify the commonalities or rule of the pattern.  

• Identify a horizontal and vertical axis. 
• Identify the x and y axis. 
• Plot points on a coordinate plane. 
• Understand ordered pairs (over, up).

• 6a
• 6b

Identify patterns and explain the rule that the pattern is generated from.

• Extend a pattern or sequence.
• Identify the rule for input/output tables.
• Know the vocabulary term “variable”.
• Evaluate an algebraic expression.
• Explain the rule.

Solve simple (addition, subtraction) equations with one variable using informal or formal methods.

• Solve for variable using inverse operation.
• Use algebraic expression to solve for missing digits.
• Check answer for appropriateness.

• 6b
• 6c

Understand basic algebraic terms and symbols (e.g. equation, inequality, variable, exponent).

• Convert word expressions to algebraic expressions.

Use a coordinate grid for a variety of representations (e.g., number, figures, points, lines).

• Graph in all 4 quadrants.

• 6c
• 6f

Solve simple inequalities with rational number solutions, using concrete and informal methods.

• Manipulate the numbers to solve the inequalities.
• Explain the solution.

Solve two-step equations of one variable using informal and formal methods.

• Show the use of inverse operations to find the solution.
• Use algebra tiles or other models to determine the solution.

• Use multiple methods to find slope.
• Find the x-/y-intercepts.
• Make a table of values.
• Match a graph with its corresponding equation and/or table.
• Express coordinates as ordered pairs, graphed points, mapping, or table.
• Interpret information from the graph.
• Find the line of best fit using technology.
• Find the maximum and minimum coordinates.
• Graph equations using various given information, such as two points, point and slope, etc.

• Recognize the parent functions: constant, linear, quadratic, cubic, absolute value, and exponential.
• Determine if a graph is a function and/or one to one.
• Determine the vertex, x and y intercepts, and local minimums and maximums.
• Understand how a graph is translated, and its shape. (stretch or shrink)

• Recognize the basic functions (constant, linear, absolute value, quadratic, cubic, step, piecewise, circles, etc.)
• Recognize properties of the basic functions.

• Know basic ratios of the 6 trig functions. 
• Know the relationships between the 6 trig functions.
• Graph the 6 trig functions.
• Find and explain amplitude, phase shift, and period for each graph.

• Find horizontal, vertical, and slant asymptotes of a function.
• Recognize general shapes of the graphs of the advanced functions.

Use a variety of methods (e.g. with graphs, algebraic methods, and matrices) to solve linear and quadratic equations [Algebra I, II, Pre-Calculus

• Find roots or zeros with various methods: factoring, completing the square, using the quadratic formula.

• Execute graphing and linear programming.
• Use test points to determine shading.

• Solve systems of equations with graphs, elimination, substitution, Cramer’s Rule, inverse matrix, and augmented matrix.

• Find complex roots.
• Perform operations with imaginary numbers.
• Graph complex numbers using an Argand diagram.
• Represent a complex answer in a+bi form.

• Apply the Binomial Theorem.
• Understand the concept of limits and asymptotes.
• Use delta notation as it applies to slope, difference equations, and derivatives.
• Understand the meaning of f(x).