Virginia Math Standards

1594 standards, Kindergarten through AP.

Grade K

K.CE Computation and Estimation
K.CE.1 The student will model and solve single-step contextual problems using addition and subtraction with whole numbers within 10.
K.CE.1.aUse objects, drawings, words, or numbers to compose and decompose numbers less than or equal to 5 in multiple ways.
K.CE.1.bRecognize and describe with fluency part-part-whole relationships for numbers up to 5 in a variety of configurations.
K.CE.1.cModel and identify the number that makes 5 when added to a given number less than or equal to 5.
K.CE.1.dUse objects, drawings, words, or numbers to compose and decompose numbers less than or equal to 10 in multiple ways.
K.CE.1.eModel and identify the number that makes 10 when added to a given number less than or equal to 10.
K.CE.1.fModel and solve single-step contextual problems (join, separate, and part-part-whole) using 10 or fewer concrete objects.
K.MG Measurement and Geometry
K.MG.1 The student will reason mathematically by making direct comparisons between two objects or events using the attributes of length, height, weight, volume, and time.
K.MG.1.a Use direct comparisons to compare, describe, and justify the:
K.MG.1.a.ilengths of two objects using the terms longer or shorter;
K.MG.1.a.iiheights of two objects using the terms taller or shorter;
K.MG.1.a.iiiweights of two objects using the terms heavier or lighter;
K.MG.1.a.ivvolumes of two containers using the terms more or less; and
K.MG.1.a.vamount of time spent on two events using the terms longer or shorter.
K.MG.2 The student will identify, describe, name, compare, and construct plane figures (circles, triangles, squares, and rectangles).
K.MG.2.aIdentify and name concrete and pictorial representations of circles, triangles, squares, and rectangles regardless of their orientation in space.
K.MG.2.bDescribe triangles, squares, and rectangles to include the number of sides and number of vertices.
K.MG.2.cDescribe a circle using terms such as round and curved.
K.MG.2.dDistinguish between examples and nonexamples of identified plane figures (circles, triangles, squares, and rectangles).
K.MG.2.eCompare and contrast two plane figures using characteristics to describe similarities and differences.
K.MG.2.fConstruct plane figures (circles, triangles, squares, and rectangles) using a variety of materials (e.g., straws, sticks, pipe cleaners).
K.MG.3 The student will describe the units of time represented in a calendar.
K.MG.3.aIdentify a calendar as a tool used to measure time.
K.MG.3.bName the days of the week and state that there are seven days in one week.
K.MG.3.cDetermine the day before and after a given day (e.g., yesterday, today, tomorrow).
K.MG.3.dName the twelve months of the year and state that there are twelve months in one year.
K.MG.3.eDistinguish between days of the week and months of the year.
K.NS Number and Number Sense
K.NS.1 The student will utilize flexible counting strategies to determine and describe quantities up to 100.
K.NS.1.aUse one-to-one correspondence to determine how many are in a given set containing 30 or fewer concrete objects (e.g., cubes, pennies, balls), and describe the last number named as the total number of objects counted.
K.NS.1.bRecognize and explain that the number of objects remains the same regardless of the arrangement or the order in which the objects are counted.
K.NS.1.cRepresent forward counting by ones using a variety of tools, including five-frames, ten-frames, and number paths (a prelude to number lines).
K.NS.1.dCount forward orally by ones from 0 to 100.
K.NS.1.eCount forward orally by ones, within 100, starting at any given number.
K.NS.1.fCount backward orally by ones when given any number between 1 and 20.
K.NS.1.gState the number after, without counting, when given any number between 0 and 30.
K.NS.1.hState the number before, without counting, when given any number between 1 and 20.
K.NS.1.iUse objects, drawings, words, or numbers to compose and decompose numbers 11-19 into a ten and some ones.
K.NS.1.jGroup a collection of up to 100 objects (e.g., counters, pennies, cubes) into sets of ten and count by tens to determine the total (e.g., there are 3 groups of ten and 6 leftovers, 36 total objects).
K.NS.2 The student will identify, represent, and compare quantities up to 30.
K.NS.2.aRead, write, and identify the numerals 0 through 30.
K.NS.2.bConstruct a set of objects that corresponds to a given numeral within 30, including an empty set.
K.NS.2.cDetermine and write the numeral that corresponds to the total number of objects in a given set of 30 or fewer concrete objects or pictorial models.
K.NS.2.dGiven a set of up to 30 objects, construct another set which has more, fewer, or the same number of objects using concrete or pictorial models.
K.NS.2.eGiven a numeral up to 30, construct a set which has more, fewer, or the same number of objects using concrete or pictorial models.
K.NS.2.fCompare two sets containing up to 30 concrete objects or pictorial models, using the terms more, fewer, or the same as (equal to).
K.NS.2.gCompare numbers up to 30, to the benchmarks of 5 and to the benchmark of 10 using various models (e.g., five frames, ten frames, number paths [a prelude to number lines], beaded racks, hands) using the terms greater than, less than, or the same as (equal to).
K.PFA Patterns, Functions, and Algebra
K.PFA.1 The student will identify, describe, extend, and create simple repeating patterns using various representations.
K.PFA.1.aIdentify and describe the core found in repeating patterns.
K.PFA.1.bExtend a repeating pattern by adding at least two complete repetitions of the core to the pattern.
K.PFA.1.cCreate and describe a repeating pattern using objects, colors, sounds, movements, or pictures.
K.PS Probability and Statistics
K.PS.1 The student will apply the data cycle (pose questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on object graphs and picture graphs.
K.PS.1.aSort and classify concrete objects into appropriate subsets (categories) based on one attribute (e.g., size, shape, color, thickness).
K.PS.1.bDescribe and label attributes (e.g., size, color, shape) of a set of objects (e.g., coins, counters, buttons) that has been sorted.
K.PS.1.cPose questions, given a predetermined context, that require the collection of data (limited to 25 or fewer data points for no more than four categories).
K.PS.1.dDetermine the data needed to answer a posed question, and collect the data using various methods (e.g., counting objects, drawing pictures).
K.PS.1.eOrganize and represent a data set (vertically or horizontally) by sorting concrete objects into organized groups to form a simple object graph.
K.PS.1.fOrganize and represent a data set (vertically or horizontally) using pictures to form a simple picture graph.
K.PS.1.g Analyze data represented in object graphs and picture graphs and communicate results:
K.PS.1.g.iask and answer questions about the data represented in object graphs and picture graphs (e.g., how many in each category, which categories have the greatest, least, or the same amount of data); and
K.PS.1.g.iidraw conclusions about the data and make predictions based on the data.

Grade 1

1.CE Computation and Estimation
1.CE.1 The student will recall with automaticity addition and subtraction facts within 10 and represent, solve, and justify solutions to single-step problems, including those in context, using addition and subtraction with whole numbers within 20.
1.CE.1.aRecognize and describe with fluency part-part-whole relationships for numbers up to 10 in a variety of configurations.
1.CE.1.bDemonstrate fluency with addition and subtraction within 10 by applying reasoning strategies (e.g., count on/count back, one more/one less, doubles, make ten).
1.CE.1.cRecall with automaticity addition and subtraction facts within 10.
1.CE.1.dInvestigate, recognize, and describe part-part-whole relationships for numbers up to 20 in a variety of configurations (e.g., beaded racks, double ten frames).
1.CE.1.eSolve addition and subtraction problems within 20 using various strategies (e.g., inverse relationships: if 9 + 3 = 12 then 12 – 3 = 9; decomposition using known sums/differences: 9 + 7 can be thought of as 9 decomposed into 2 and 7, then use doubles, 7 + 7 = 14; 14 + 2 = 16 or decompose the 7 into 1 and 6; make a ten: 1 + 9 = 10; 10 + 6 = 16).
1.CE.1.fRepresent, solve, and justify solutions to single-step addition and subtraction problems (join, separate, and part-part-whole) within 20, including those in context, using words, objects, drawings, or numbers.
1.CE.1.gDetermine the unknown whole number that will result in a sum or difference of 10 or 20 (e.g., 14 – __ = 10 or 15 + __ = 20).
1.CE.1.hIdentify and use (+) as a symbol for addition and (–) as a symbol for subtraction.
1.CE.1.iDescribe the equal symbol (=) as a balance representing an equivalent relationship between expressions on either side of the equal symbol (e.g., 6 and 1 is the same as 4 and 3; 6 + 1 is balanced with 4 + 3; 6 + 1 = 4 + 3).
1.CE.1.jUse concrete materials to model, identify, and justify when two expressions are not equal (e.g., 10 – 3 is not equal to 3 + 5).
1.CE.1.kUse concrete materials to model an equation that represents the relationship of two expressions of equal value.
1.CE.1.lWrite an equation that could be used to represent the solution to an oral, written, or picture problem.
1.DS Data in Context
1.DS.1 The student will identify specific examples of real-world problems that can be effectively addressed using data science.
1.DS.1.aIdentify and explain characteristics that best lend themselves to a data driven approach to problem solving.
1.DS.1.bFormulate questions based on context.
1.DS.1.cUnderstand the type of data relevant to the context of the question at hand.
1.DS.1.dDefine relationships between variables and constant relationships.
1.DS.1.eCreate a hypothesis of interest in terms of measurable data.
1.DS.1.fDefine the stages of the data cycle and how each stage is related to the other.
1.DS.1.gIdentify and explain constraints of the data-driven approach.
1.DS.2 The student will be able to formulate a top-down plan for data collection and analysis, with quantifiable results, based on the context of a problem.
1.DS.2.a Design a data project plan, which is aligned with the data science cycle, that includes the following components:
1.DS.2.a.idefinition of the goal of the project as it pertains to a real-world problem;
1.DS.2.a.iiidentification of the various parameters of the problem and stakeholders;
1.DS.2.a.iiia timeline for the project with deliverables;
1.DS.2.a.ivKey Performance Indicators (KPI) for the successful data project deliverables;
1.DS.2.a.vresource needs and tools for the project;
1.DS.2.a.vibias considerations for the sampling process of the project; and
1.DS.2.a.viilimitations of the project.
1.DS.2.b Given the context and parameters of a problem, choose from among various sampling techniques, which may include
1.DS.2.b.isimple random;
1.DS.2.b.iisystematic;
1.DS.2.b.iiistratified; and
1.DS.2.b.ivcluster; to justify the sampling methodology of the project design and implementation.
1.MG Measurement and Geometry
1.MG.1 The student will reason mathematically using nonstandard units to measure and compare objects by length, weight, and volume.
1.MG.1.a Use nonstandard units to measure the:
1.MG.1.a.ilengths of two objects (units laid end to end with no gaps or overlaps) and compare the measurements using the terms longer/shorter, taller/shorter, or the same as;
1.MG.1.a.iiweights of two objects (using a balance scale or a pan scale) and compare the measurements using the terms lighter, heavier, or the same as; and
1.MG.1.a.iiivolumes of two containers and compare the measurements using the terms more, less, or the same as.
1.MG.1.bMeasure the length, weight, or volume of the same object or container with two different units and describe how and why the measurements differ.
1.MG.2 The student will describe, sort, draw, and name plane figures (circles, triangles, squares, and rectangles), and compose larger plane figures by combining simple plane figures.
1.MG.2.aDescribe triangles, squares, and rectangles using the terms sides, vertices, and angles. Describe a circle using terms such as round and curved.
1.MG.2.bSort plane figures based on their characteristics (e.g., number of sides, vertices, angles, curved).
1.MG.2.cDraw and name the plane figure (circle, square, rectangle, triangle) when given information about the number of sides, vertices, and angles.
1.MG.2.dIdentify, name, and describe representations of circles, squares, rectangles, and triangles, regardless of orientation, in different environments and explain reasoning.
1.MG.2.eRecognize and name the angles found in rectangles and squares as right angles.
1.MG.2.fCompose larger plane figures by combining two or three simple plane figures (triangles, squares, and/or rectangles).
1.MG.3 The student will demonstrate an understanding of the concept of passage of time (to the nearest hour and half-hour) and the calendar.
1.MG.3.aIdentify different tools to measure time including clocks (analog and digital) and calendar.
1.MG.3.bDescribe the units of time represented on a clock as minutes and hours.
1.MG.3.cTell time to the hour and half-hour, using analog and digital clocks.
1.MG.3.dDescribe the location of the hour hand relative to time to the hour and half-hour on an analog clock.
1.MG.3.eDescribe the location of the minute hand relative to time to the hour and half-hour on an analog clock.
1.MG.3.fMatch the time shown on a digital clock to an analog clock to the hour and half-hour.
1.MG.3.gIdentify specific days/dates on a calendar (e.g., What date is Saturday? How many Fridays are in October?).
1.MG.3.hUse ordinal numbers first through tenth to describe the relative position of specific days/dates (e.g., What is the first Monday in October? What day of the week is May 6th?).
1.MG.3.iDetermine the day/date before and after a given day/date (e.g., Today is the 8th, so yesterday was the ?), and a date that is a specific number of days/weeks in the past or future (e.g., Tim’s birthday is in 10 days, what will be the date of his birthday?).
1.NS Number and Number Sense
1.NS.1 The student will utilize flexible counting strategies to determine and describe quantities up to 120.
1.NS.1.aCount forward orally by ones from 0 to 120 starting at any number between 0 and 120.
1.NS.1.bCount backward orally by ones when given any number between 1 and 30.
1.NS.1.cRepresent forward counting patterns when counting by groups of 5 and groups of 10 up to 120 using a variety of tools (e.g., objects, coins, 120 chart).
1.NS.1.dRepresent forward counting patterns when counting by groups of 2 up to at least 30 using a variety of tools (e.g., beaded number strings, number paths [a prelude to number lines], 120 chart).
1.NS.1.eGroup a collection of up to 120 objects into tens and ones, and count to determine the total (e.g., 5 groups of ten and 6 ones is equal to 56 total objects).
1.NS.1.fIdentify a penny, nickel, and dime by their attributes and describe the number of pennies equivalent to a nickel and a dime.
1.NS.1.gCount by ones, fives, or tens to determine the value of a collection of like coins (pennies, nickels, or dimes), whose total value is 100 cents or less.
1.NS.2 The student will represent, compare, and order quantities up to 120.
1.NS.2.aRead and write numerals 0-120 in sequence and out of sequence.
1.NS.2.bEstimate the number of objects (up to 120) in a given collection and justify the reasonableness of an answer.
1.NS.2.cCreate a concrete or pictorial representation of a number using tens and ones and write the corresponding numeral up to 120 (e.g., 47 can be represented as 47 ones or it can be grouped into 4 tens with 7 ones left over).
1.NS.2.dDescribe the number of groups of tens and ones when given a two-digit number and justify reasoning.
1.NS.2.eCompare two numbers between 0 and 120 represented pictorially or with concrete objects using the terms greater than, less than, or equal to.
1.NS.2.fOrder three sets, each set containing up to 120 objects, from least to greatest, and greatest to least.
1.NS.3 The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into two and four equal-sized parts.
1.NS.3.aRepresent equal shares of a whole with two or four sharers, when given a contextual problem.
1.NS.3.bRepresent and name halves and fourths of a whole, using a region/area model (e.g., pie pieces, pattern blocks, paper folding, drawings) and a set model (e.g., eggs, marbles, counters) limited to two or four items.
1.NS.3.cDescribe and justify how shares are equal pieces or equal parts of the whole (limited to halves, fourths) when given a contextual problem.
1.PFA Patterns, Functions, and Algebra
1.PFA.1 The student will identify, describe, extend, create, and transfer repeating patterns and increasing patterns using various representations.
1.PFA.1.aIdentify and describe repeating and increasing patterns.
1.PFA.1.bAnalyze a repeating or increasing pattern and generalize the change to extend the pattern using objects, colors, movements, pictures, or geometric figures.
1.PFA.1.cCreate a repeating or increasing pattern using objects, pictures, movements, colors, or geometric figures.
1.PFA.1.dTransfer a repeating or increasing pattern from one form to another.
1.PS Probability and Statistics
1.PS.1 The student will apply the data cycle (pose questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on object graphs, picture graphs, and tables.
1.PS.1.aSort and classify concrete objects into appropriate subsets (categories) based on one or two attributes, such as size, shape, color, and/or thickness (e.g., sort a set of objects that are both red and thick).
1.PS.1.bDescribe and label attributes of a set of objects that has been sorted.
1.PS.1.cPose questions, given a predetermined context, that require the collection of data (limited to 25 or fewer data points for no more than four categories).
1.PS.1.dDetermine the data needed to answer a posed question and collect the data using various methods (e.g., counting objects, drawing pictures, tallying).
1.PS.1.eOrganize and represent a data set by sorting the collected data using various methods (e.g., tallying, T-charts).
1.PS.1.fRepresent a data set (vertically or horizontally) using object graphs, picture graphs, and tables.
1.PS.1.g Analyze data represented in object graphs, picture graphs, and tables and communicate results:
1.PS.1.g.iask and answer questions about the data represented in object graphs, picture graphs, and tables (e.g., total number of data points represented, how many in each category, how many more or less are in one category than another); and
1.PS.1.g.iidraw conclusions about the data and make predictions based on the data.

Grade 2

2.CE Computation and Estimation
2.CE.1 The student will recall with automaticity addition and subtraction facts within 20 and estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers where addends or minuends do not exceed 100.
2.CE.1.aApply strategies (e.g., rounding to the nearest 10, compatible numbers, other number relationships) to estimate a solution for single-step addition or subtraction problems, including those in context, where addends and minuends do not exceed 100.
2.CE.1.bApply strategies (e.g., the use of concrete and pictorial models, place value, properties of addition, the relationship between addition and subtraction) to determine the sum or difference of two whole numbers where addends or minuends do not exceed 100.
2.CE.1.cRepresent, solve, and justify solutions to single-step and multistep contextual problems (e.g., join, separate, part-part-whole, comparison) involving addition or subtraction of whole numbers where addends or minuends do not exceed 100.
2.CE.1.dDemonstrate fluency with addition and subtraction within 20 by applying reasoning strategies (e.g., doubles, near doubles, make-a-ten, compensations, inverse relationships).
2.CE.1.eRecall with automaticity addition and subtraction facts within 20.
2.CE.1.fUse patterns, models, and strategies to make generalizations about the algebraic properties for fluency (e.g., 4 + 3 is equal to 3 + 4; 0 + 8 = 8).
2.CE.1.gDetermine the missing number in an equation (number sentence) through modeling and justification with addition and subtraction within 20 (e.g., 3 + __ = 5 or __ + 2 = 5; 5 – __ = 3 or 5 – 2 = __).
2.CE.1.hUse inverse relationships to write all related facts connected to a given addition or subtraction fact model within 20 (e.g., given a model for 3 + 4 = 7, write 4 + 3 = 7, 7 – 4 = 3, and 7 – 3 = 4).
2.CE.1.iDescribe the not equal symbol (≠) as representing a relationship where expressions on either side of the not equal symbol represent different values and justify reasoning.
2.CE.1.jRepresent and justify the relationship between values and expressions as equal or not equal using appropriate models and/or symbols (e.g., 9 + 24 = 10 + 23; 45 – 9 = 46 – 10; 15 + 16 ≠ 31 + 15).
2.DS Data Bias
2.DS.3 The student will recognize the importance of data literacy and develop an awareness of how the analysis of data can be used in problem solving to effect change and create innovative solutions.
2.DS.3.aFormulate relevant/clarifying questions to identify potential data biases presented in existing analyses/visualizations.
2.DS.3.bEffectively read data summaries and visualizations and explain/translate into nontechnical terms in proper context.
2.DS.3.cIdentify potential data biases in terms of data presented and discuss the potential effects of such biases in terms of how they could affect data analysis and decision making.
2.DS.3.dIdentify privacy and consumer protection issues that might be a result of how data is presented.
2.DS.3.eDescribe the types of data that business, industry, and government entities collect and possible ways the data is used.
2.DS.4 The student will be able to identify data biases in the data collection process and understand the implications and privacy issues surrounding data collection and processing.
2.DS.4.aIdentify data biases in the data collection process that include, but are not limited to, confirmation, selection, outliers, overfitting / under fitting, and confounding and describe mitigation strategies for these biases.
2.DS.4.bProvide examples of sampling biases in terms of data collection and the potential effects.
2.DS.4.cIdentify and describe data biases as a producer as well as a consumer/decision maker of data.
2.DS.4.dDescribe how the data collection process should be focused, relevant, and limited to the scope of the data project plan.
2.DS.4.eDescribe privacy considerations in the collection of data as both a consumer and producer.
2.MG Measurement and Geometry
2.MG.1 The student will reason mathematically using standard units (U.S. Customary) with appropriate tools to estimate, measure, and compare objects by length, weight, and liquid volume to the nearest whole unit.
2.MG.1.a Explain the purpose of various measurement tools and how to use them appropriately by:
2.MG.1.a.iidentifying a ruler as an instrument to measure length;
2.MG.1.a.iiidentifying different types of scales as instruments to measure weight; and
2.MG.1.a.iiiidentifying different types of measuring cups as instruments to measure liquid volume.
2.MG.1.b Use U.S. Customary units to estimate, measure, and compare the two for reasonableness:
2.MG.1.b.ithe length of an object to the nearest inch, using a ruler;
2.MG.1.b.iithe weight of an object to the nearest pound, using a scale; and
2.MG.1.b.iiithe liquid volume of a container to the nearest cup, using a measuring cup.
2.MG.2 The student will demonstrate an understanding of the concept of time to the nearest five minutes, using analog and digital clocks.
2.MG.2.aIdentify the number of minutes in an hour (60 minutes) and the number of hours in a day (24 hours).
2.MG.2.bDetermine the unit of time (minutes, hours, days, or weeks) that is most appropriate when measuring a given activity or context and explain reasoning (e.g., Would you measure the time it takes to brush your teeth in minutes or hours?).
2.MG.2.cShow, tell, and write time to the nearest five minutes, using analog and digital clocks.
2.MG.2.dMatch a written time (e.g., 1:35, 6:20, 9:05) to the time shown on an analog clock to the nearest five minutes.
2.MG.3 The student will identify, describe, and create plane figures (including circles, triangles, squares, and rectangles) that have at least one line of symmetry and explain its relationship with congruency.
2.MG.3.aExplore a figure using a variety of tools (e.g., paper folding, geoboards, drawings) to show and justify a line of symmetry, if one exists.
2.MG.3.bCreate figures with at least one line of symmetry using various concrete and pictorial representations.
2.MG.3.cDescribe the two resulting figures formed by a line of symmetry as being congruent (having the same shape and size).
2.MG.4 The student will describe, name, compare, and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms).
2.MG.4.aTrace faces of solid figures (cubes and rectangular prisms) to create the set of plane figures related to the solid figure.
2.MG.4.bCompare and contrast models and nets (cutouts) of cubes and rectangular prisms (e.g., number and shapes of faces, edges, vertices).
2.MG.4.cGiven a concrete or pictorial model, name and describe the solid figure (sphere, cube, and rectangular prism) by its characteristics (e.g., number of edges, number of vertices, shapes of faces).
2.MG.4.dCompare and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms) according to their characteristics (e.g., number and shapes of their faces, edges, vertices).
2.NS Number and Number Sense
2.NS.1 The student will utilize flexible counting strategies to determine and describe quantities up to 200.
2.NS.1.aRepresent forward counting patterns when counting by groups of 2 up to at least 50, starting at various multiples of 2 and using a variety of tools (e.g., objects, number lines, hundreds charts).
2.NS.1.bRepresent forward counting patterns created when counting by groups of 5s, 10s, and 25s starting at various multiples up to at least 200 using a variety of tools (e.g., objects, number lines, hundreds charts).
2.NS.1.cDescribe and use patterns in skip counting by multiples of 2 (to at least 50), and multiples of 5, 10, and 25 (to at least 200) to justify the next number in the counting sequence.
2.NS.1.dRepresent forward counting patterns when counting by groups of 100 up to at least 1,000 starting at 0 using a variety of tools (e.g., objects, number lines, calculators, one thousand charts).
2.NS.1.eRepresent backward counting patterns when counting by groups of 10 from 200 or less using a variety of tools including objects, number lines, calculators, and hundreds charts.
2.NS.1.fDescribe and use patterns in skip counting backwards by 10s (from at least 200) to justify the next number in the counting sequence.
2.NS.1.gChoose a reasonable estimate up to 1,000 when given a contextual problem (e.g., What would be the best estimate for the number of students in our school – 5, 50, or 500?).
2.NS.1.hRepresent even numbers (up to 50) with concrete objects, using two equal groups or two equal addends.
2.NS.1.iRepresent odd numbers (up to 50) with concrete objects, using two equal groups with one leftover or two equal addends plus 1.
2.NS.1.jDetermine whether a number (up to 50) is even or odd using concrete objects and justify reasoning (e.g., dividing collections of objects into two equal groups, pairing objects).
2.NS.2 The student will demonstrate an understanding of the ten-to-one relationships of the base 10 number system to represent, compare, and order whole numbers up to 999.
2.NS.2.aWrite the three-digit whole number represented by a given model (e.g., concrete objects, pictures of base 10 blocks).
2.NS.2.bRead, write, and represent three-digit numbers in standard form, expanded form, and word form, using concrete or pictorial representations.
2.NS.2.cApply patterns within the base 10 system to determine and communicate, orally and in written form, the place (ones, tens, hundreds) and value of each digit in a three-digit whole number (e.g., in 352, the 5 represents 5 tens and its value is 50).
2.NS.2.dInvestigate and explain the ten-to-one relationships among ones, tens, and hundreds, using models.
2.NS.2.eCompose and decompose whole numbers up to 200 by making connections between a variety of models (e.g., base 10 blocks, place value cards, presented orally, in expanded or standard form) and counting strategies (e.g., 156 can be 1 hundred, 5 tens, 6 ones; 1 hundred, 4 tens, 16 ones; 15 tens, 6 ones).
2.NS.2.fPlot and justify the position of a given number up to 100 on a number line with pre-marked benchmarks of 1s, 2s, 5s, 10s, or 25s.
2.NS.2.gCompare two whole numbers, each 999 or less, represented concretely, pictorially, or symbolically, using words (greater than, less than, or equal to) and symbols (>, <, or =). Justify reasoning orally, in writing, or with a model.
2.NS.2.hOrder up to three whole numbers, each 999 or less, represented concretely, pictorially, or symbolically from least to greatest and greatest to least.
2.NS.3 The student will use mathematical reasoning and justification to solve contextual problems that involve partitioning models into equal-sized parts (halves, fourths, eighths, thirds, and sixths).
2.NS.3.aModel and describe fractions as representing equal-size parts of a whole.
2.NS.3.bDescribe the relationship between the number of fractional parts needed to make a whole and the size of the parts (i.e., as the whole is divided into more parts, each part becomes smaller).
2.NS.3.cCompose the whole for a given fractional part and its value (in context) for halves, fourths, eighths, thirds, and sixths (e.g., when given 1/4, determine how many pieces would be needed to make 4/4).
2.NS.3.dUsing same-size fraction pieces, from a region/area model, count by unit fractions up to two wholes (e.g., zero one-fourths, one one-fourth, two one-fourths, three one-fourths, four one-fourths, five one-fourths; or zero-fourths, one-fourth, two-fourths, three-fourths, four-fourths, five-fourths).
2.NS.3.e Given a context, represent, name, and write fractional parts of a whole for halves, fourths, eighths, thirds, and sixths using:
2.NS.3.e.iregion/area models (e.g., pie pieces, pattern blocks, geoboards);
2.NS.3.e.iilength models (e.g., paper fraction strips, fraction bars, rods, number lines); and
2.NS.3.e.iiiset models (e.g., chips, counters, cubes).
2.NS.3.fCompare unit fractions for halves, fourths, eighths, thirds, and sixths using words (greater than, less than or equal to) and symbols (>, <, =), with region/area and length models.
2.NS.4 The student will solve problems that involve counting and representing money amounts up to $2.00.
2.NS.4.aIdentify a quarter and its value and determine multiple ways to represent the value of a quarter using pennies, nickels, and/or dimes.
2.NS.4.bCount by ones, fives, tens, and twenty-fives to determine the value of a collection of mixed coins and one-dollar bills whose total value is $2.00 or less.
2.NS.4.cConstruct a set of coins and/or bills to total a given amount of money whose value is $2.00 or less.
2.NS.4.dRepresent the value of a collection of coins and one-dollar bills (limited to $2.00 or less) using the cent (¢) and dollar ($) symbols and decimal point (.).
2.PFA Patterns, Functions, and Algebra
2.PFA.1 The student will describe, extend, create, and transfer repeating and increasing patterns (limited to addition of whole numbers) using various representations.
2.PFA.1.aIdentify and describe repeating and increasing patterns.
2.PFA.1.bAnalyze a repeating or increasing pattern and generalize the change to extend the pattern using objects, pictures, and numbers.
2.PFA.1.cCreate a repeating or increasing pattern using various representations (e.g., objects, pictures, numbers).
2.PFA.1.dTransfer a given repeating or increasing pattern from one form to another (e.g., objects, pictures, numbers) and explain the connection between the two patterns.
2.PS Probability and Statistics
2.PS.1 The student will apply the data cycle (pose questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on pictographs and bar graphs.
2.PS.1.aPose questions, given a predetermined context, that require the collection of data (limited to 25 or fewer data points for no more than six categories).
2.PS.1.bDetermine the data needed to answer a posed question and collect the data using various methods (e.g., voting; creating lists, tables, or charts; tallying).
2.PS.1.cOrganize and represent a data set using a pictograph where each symbol represents up to 2 data points. Determine and use a key to assist in the analysis of the data.
2.PS.1.dOrganize and represent a data set using a bar graph with a title and labeled axes (limited to 25 or fewer data points for up to six categories, and limit increments of scale to multiples of 1 or 2).
2.PS.1.e Analyze data represented in pictographs and bar graphs and communicate results:
2.PS.1.e.iask and answer questions about the data represented in pictographs and bar graphs (e.g., total number of data points represented, how many in each category, how many more or less are in one category than another). Pictograph keys will be limited to symbols representing 1, 2, 5, or 10 pieces of data and bar graphs will be limited to scales with increments in multiples of 1, 2, 5, or 10; and
2.PS.1.e.iidraw conclusions about the data and make predictions based on the data.

Grade 3

3.CE Computation and Estimation
3.CE.1 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers where addends and minuends do not exceed 1,000.
3.CE.1.aDetermine and justify whether an estimate or an exact answer is appropriate when solving single-step and multistep contextual problems involving addition and subtraction, where addends and minuends do not exceed 1,000.
3.CE.1.bApply strategies (e.g., rounding to the nearest 10 or 100, using compatible numbers, using other number relationships) to estimate a solution for single-step or multistep addition or subtraction problems, including those in context, where addends or minuends do not exceed 1,000.
3.CE.1.cApply strategies (e.g., place value, properties of addition, other number relationships) and algorithms, including the standard algorithm, to determine the sum or difference of two whole numbers where addends and minuends do not exceed 1,000.
3.CE.1.dIdentify and use the appropriate symbol to distinguish between expressions that are equal and expressions that are not equal (e.g., 256 – 13 = 220 + 23; 457 + 100 ≠ 557 + 100).
3.CE.1.eRepresent, solve, and justify solutions to single-step and multistep contextual problems involving addition and subtraction with whole numbers where addends and minuends do not exceed 1,000.
3.CE.2 The student will recall with automaticity multiplication and division facts through 10 × 10; and represent, solve, and justify solutions to single-step contextual problems using multiplication and division with whole numbers.
3.CE.2.aRepresent multiplication and division of whole numbers through 10 × 10, including in a contextual situation, using a variety of approaches and models (e.g., repeated addition/subtraction, equal-sized groups/sharing, arrays, equal jumps on a number line, using multiples to skip count).
3.CE.2.bUse inverse relationships to write the related facts connected to a given model for multiplication and division of whole numbers through 10 × 10.
3.CE.2.cApply strategies (e.g., place value, the properties of multiplication and/or addition) when multiplying and dividing whole numbers.
3.CE.2.dDemonstrate fluency with multiplication facts through 10 × 10 by applying reasoning strategies (e.g., doubling, add-a-group, subtract-a-group, near squares, and inverse relationships).
3.CE.2.eRepresent, solve, and justify solutions to single-step contextual problems that involve multiplication and division of whole numbers through 10 × 10.
3.CE.2.fRecall with automaticity the multiplication facts through 10 × 10 and the corresponding division facts.
3.CE.2.gCreate an equation to represent the mathematical relationship between equivalent expressions using multiplication and/or division facts through 10 × 10 (e.g., 4 × 3 = 14 – 2, 35 ÷ 5 = 1 × 7).
3.DS Data and Communication
3.DS.5 The student will use storytelling as a strategy to effectively communicate with data.
3.DS.5.aDefine storytelling and explain the importance of storytelling as a strategy to communicate the idea behind and results of a data science project effectively.
3.DS.5.bExplain the steps involved in data storytelling and how it relates to the data cycle.
3.DS.5.cEffectively identify a story worth telling based on the data (looking for trends, correlations, outliers) and by asking a question or forming a hypothesis based on insight and audience.
3.DS.5.dEffectively select visualizations that simplify the information, highlight the most important data, and communicate key points quickly.
3.DS.5.eEffectively simplify the information presented to make it more concise and focus the audience's attention on the key parameters that support the student’s hypothesis.
3.DS.5.fEffectively form a narrative based on data available to provide context, insight, and interpretation to make the analysis more relevant to a given audience.
3.DS.5.gExplain how data storytelling should include complete and accurate information, and consistent visuals for effective communication.
3.DS.6 The student will justify the design, use, and effectiveness of different forms of data visualizations.
3.DS.6.aConduct exploratory data analysis using visualization.
3.DS.6.bFormulate questions from exploration of a data set to consider how data will communicate a story.
3.DS.6.cDetermine the effectiveness of different data visualization choices based on the data context from conventional statistical charts to unconventional/emerging data visualizations to more complex visualizations.
3.DS.6.dCreate a visualization of a data set and summarize the representation using the context of the data.
3.DS.6.eCompare two or more different representations to ensure the design communicates the features and behavior of data sets.
3.DS.6.fJustify design choices (based on data set type, size, context, and audience) of data visualizations to highlight important features, trends, and insights.
3.MG Measurement and Geometry
3.MG.1 The student will reason mathematically using standard units (U.S. Customary and metric) with appropriate tools to estimate and measure objects by length, weight/mass, and liquid volume to the nearest half or whole unit.
3.MG.1.aJustify whether an estimate or an exact measurement is needed for a contextual situation and choose an appropriate unit.
3.MG.1.b Estimate and measure:
3.MG.1.b.ilength of an object to the nearest U.S. Customary unit (1/2 inch, inch, foot, yard) and metric unit (centimeter, meter);
3.MG.1.b.iiweight/mass of an object to the nearest U.S. Customary unit (pound) and metric unit (kilogram); and
3.MG.1.b.iiiliquid volume to the nearest U.S. Customary unit (cup, pint, quart, gallon) and metric unit (liter).
3.MG.1.cCompare estimates of length, weight/mass, or liquid volume with the actual measurements.
3.MG.2 The student will use multiple representations to estimate and solve problems, including those in context, involving area and perimeter (in both U.S. Customary and metric units).
3.MG.2.a Solve problems, including those in context, involving area:
3.MG.2.a.idescribe and give examples of area as a measurement in contextual situations; and
3.MG.2.a.iiestimate and determine the area of a given surface by counting the number of square units, describe the measurement (using the number and unit) and justify the measurement.
3.MG.2.b Solve problems, including those in context, involving perimeter:
3.MG.2.b.idescribe and give examples of perimeter as a measurement in contextual situations;
3.MG.2.b.iiestimate and measure the distance around a polygon (with no more than six sides) to determine the perimeter and justify the measurement; and
3.MG.2.b.iiigiven the lengths of all sides of a polygon (with no more than six sides), determine its perimeter and justify the measurement.
3.MG.3 The student will demonstrate an understanding of the concept of time to the nearest minute and solve single-step contextual problems involving elapsed time in one-hour increments within a 12-hour period.
3.MG.3.aTell and write time to the nearest minute, using analog and digital clocks.
3.MG.3.bMatch a written time (e.g., 4:38, 7:09, 12:51) to the time shown on analog and digital clocks to the nearest minute.
3.MG.3.c Solve single-step contextual problems involving elapsed time in one-hour increments, within a 12-hour period (within a.m. or within p.m.) when given:
3.MG.3.c.ithe starting time and the ending time, determine the amount of time that has elapsed;
3.MG.3.c.iithe starting time and amount of elapsed time in one-hour increments, determine the ending time; or
3.MG.3.c.iiithe ending time and the amount of elapsed time in one-hour increments, determine the starting time.
3.MG.4 The student will identify, describe, classify, compare, combine, and subdivide polygons.
3.MG.4.aDescribe a polygon as a closed plane figure composed of at least three line segments that do not cross.
3.MG.4.bClassify figures as polygons or not polygons and justify reasoning.
3.MG.4.cIdentify and describe triangles, quadrilaterals, pentagons, hexagons, and octagons in various orientations, with and without contexts.
3.MG.4.dIdentify and name examples of polygons (triangles, quadrilaterals, pentagons, hexagons, octagons) in the environment.
3.MG.4.eClassify and compare polygons (triangles, quadrilaterals, pentagons, hexagons, octagons).
3.MG.4.fCombine no more than three polygons, where each has three or four sides, and name the resulting polygon (triangles, quadrilaterals, pentagons, hexagons, octagons).
3.MG.4.gSubdivide a three-sided or four-sided polygon into no more than three parts and name the resulting polygons.
3.NS Number and Number Sense
3.NS.1 The student will use place value understanding to read, write, and determine the place and value of each digit in a whole number, up to six digits, with and without models.
3.NS.1.aRead and write six-digit whole numbers in standard form, expanded form, and word form.
3.NS.1.bApply patterns within the base 10 system to determine and communicate, orally and in written form, the place and value of each digit in a six-digit whole number (e.g., in 165,724, the 5 represents 5 thousands and its value is 5,000).
3.NS.1.cCompose, decompose, and represent numbers up to 9,999 in multiple ways, according to place value (e.g., 256 can be 1 hundred, 14 tens, 16 ones, but also 25 tens, 6 ones), with and without models.
3.NS.2 The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to 9,999.
3.NS.2.aCompare two whole numbers, each 9,999 or less, using symbols (>, <, =, ≠) and/or words (greater than, less than, equal to, not equal to), with and without models.
3.NS.2.bOrder up to three whole numbers, each 9,999 or less, represented with and without models, from least to greatest and greatest to least.
3.NS.3 The student will use mathematical reasoning and justification to represent and compare fractions (proper and improper) and mixed numbers with denominators of 2, 3, 4, 5, 6, 8, and 10), including those in context.
3.NS.3.a Represent, name, and write a given fraction (proper or improper) or mixed number with denominators of 2, 3, 4, 5, 6, 8, and 10 using:
3.NS.3.a.iregion/area models (e.g., pie pieces, pattern blocks, geoboards);
3.NS.3.a.iilength models (e.g., paper fraction strips, fraction bars, rods, number lines); and
3.NS.3.a.iiiset models (e.g., chips, counters, cubes).
3.NS.3.bIdentify a fraction represented by a model as the sum of unit fractions.
3.NS.3.cUse a model of a fraction greater than one to count the fractional parts to name and write it as an improper fraction and as a mixed number (e.g., 1/4, 2/4, 3/4, 4/4, 5/4 = 1 1/4).
3.NS.3.dCompose and decompose fractions (proper and improper) with denominators of 2, 3, 4, 5, 6, 8, and 10 in multiple ways (e.g., 7/4 = 4/4 + 3/4 or 4/6 = 3/6 + 1/6 = 2/6 + 2/6) with models.
3.NS.3.eCompare a fraction, less than or equal to one, to the benchmarks of 0, 1/2, and 1 using area/region models, length models, and without models.
3.NS.3.fCompare two fractions (proper or improper) and/or mixed numbers with like numerators of 2, 3, 4, 5, 6, 8, and 10 (e.g., 2/3 > 2/8) using words (greater than, less than, equal to) and/or symbols (>, <, =), using area/region models, length models, and without models.
3.NS.3.gCompare two fractions (proper or improper) and/or mixed numbers with like denominators of 2, 3, 4, 5, 6, 8, and 10 (e.g., 3/6 < 4/6) using words (greater than, less than, equal to) and/or symbols (>, <, =), using area/region models, length models, and without models.
3.NS.3.hRepresent equivalent fractions with denominators of 2, 3, 4, 5, 6, 8, or 10, using region/area models and length models.
3.NS.4 The student will solve problems, including those in context, that involve counting, comparing, representing, and making change for money amounts up to $5.00.
3.NS.4.aDetermine the value of a collection of bills and coins whose total is $5.00 or less.
3.NS.4.bConstruct a set of bills and coins to total a given amount of money whose value is $5.00 or less.
3.NS.4.cCompare the values of two sets of coins or two sets of bills and coins, up to $5.00, with words (greater than, less than, equal to) and/or symbols (>, <, =) using concrete or pictorial models.
3.NS.4.dSolve contextual problems to make change from $5.00 or less by using counting on or counting back strategies with concrete or pictorial models.
3.PFA Patterns, Functions, and Algebra
3.PFA.1 The student will identify, describe, extend, and create increasing and decreasing patterns (limited to addition and subtraction of whole numbers), including those in context, using various representations.
3.PFA.1.aIdentify and describe increasing and decreasing patterns using various representations (e.g., objects, pictures, numbers, number lines).
3.PFA.1.bAnalyze an increasing or decreasing pattern and generalize the change to extend the pattern or identify missing terms using various representations.
3.PFA.1.cSolve contextual problems that involve identifying, describing, and extending patterns.
3.PFA.1.dCreate increasing and decreasing patterns using objects, pictures, numbers, and number lines.
3.PFA.1.eInvestigate and explain the connection between two different representations of the same increasing or decreasing pattern.
3.PS Probability and Statistics
3.PS.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on pictographs and bar graphs.
3.PS.1.aFormulate questions that require the collection or acquisition of data.
3.PS.1.bDetermine the data needed to answer a formulated question and collect or acquire existing data (limited to 30 or fewer data points for no more than eight categories) using various methods (e.g., polls, observations, tallies).
3.PS.1.cOrganize and represent a data set using pictographs that include an appropriate title, labeled axes, and key. Each pictograph symbol should represent 1, 2, 5 or 10 data points.
3.PS.1.dOrganize and represent a data set using bar graphs with a title and labeled axes, with and without the use of technology tools. Determine and use an appropriate scale (increments limited to multiples of 1, 2, 5 or 10).
3.PS.1.e Analyze data represented in pictographs and bar graphs, and communicate results orally and in writing:
3.PS.1.e.idescribe the categories of data and the data as a whole (e.g., data were collected on preferred ways to cook or prepare eggs - scrambled, fried, hard boiled, and egg salad);
3.PS.1.e.iiidentify parts of the data that have special characteristics, including categories with the greatest, the least, or the same (e.g., most students prefer scrambled eggs);
3.PS.1.e.iiimake inferences about data represented in pictographs and bar graphs;
3.PS.1.e.ivuse characteristics of the data to draw conclusions about the data and make predictions based on the data (e.g., it is unlikely that a third grader would like hard boiled eggs); and
3.PS.1.e.vsolve one- and two-step addition and subtraction problems using data from pictographs and bar graphs.

Grade 4

4.CE Computation and Estimation
4.CE.1 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction with whole numbers.
4.CE.1.aDetermine and justify whether an estimate or an exact answer is appropriate when solving contextual problems involving addition and subtraction with whole numbers. Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than.
4.CE.1.bApply strategies (e.g., rounding to the nearest 100 or 1,000, using compatible numbers, other number relationships) to estimate a solution for single-step or multistep addition or subtraction problems with whole numbers, where addends or minuends do not exceed 10,000.
4.CE.1.cApply strategies (e.g., place value, properties of addition, other number relationships) and algorithms, including the standard algorithm, to determine the sum or difference of two whole numbers, where addends and minuends do not exceed 10,000.
4.CE.1.dEstimate, represent, solve, and justify solutions to single-step and multistep contextual problems involving addition and subtraction with whole numbers where addends and minuends do not exceed 1,000,000.
4.CE.2 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using multiplication with whole numbers, and single-step problems, including those in context, using division with whole numbers; and recall with automaticity the multiplication facts through 12 × 12 and the corresponding division facts.
4.CE.2.aDetermine and justify whether an estimate or an exact answer is appropriate when solving contextual problems involving multiplication and division of whole numbers. Refine estimates by adjusting the final amount, using terms such as closer to, between, and a little more than.
4.CE.2.bRecall with automaticity the multiplication facts through 12 × 12 and the corresponding division facts.
4.CE.2.cCreate an equation using addition, subtraction, multiplication, and division to represent the relationship between equivalent mathematical expressions (e.g., 4 × 3 = 2 × 6; 10 + 8 = 36 ÷ 2; 12 × 4 = 60 –12).
4.CE.2.dIdentify and use the appropriate symbol to distinguish between expressions that are equal and expressions that are not equal, using addition, subtraction, multiplication, and division (e.g., 4 × 12 = 8 × 6 and 64 ÷ 8 ≠ 8 × 8).
4.CE.2.eDetermine all factor pairs for a whole number 1 to 100, using concrete, pictorial, and numerical representations.
4.CE.2.fDetermine common factors and the greatest common factor of no more than three numbers.
4.CE.2.g Apply strategies (e.g., rounding, place value, properties of multiplication and/or addition) and algorithms, including the standard algorithm, to estimate and determine the product of two whole numbers when given:
4.CE.2.g.ia two-digit factor and a one-digit factor;
4.CE.2.g.iia three-digit factor and a one-digit factor; or
4.CE.2.g.iiia two-digit factor and a two-digit factor.
4.CE.2.hEstimate, represent, solve, and justify solutions to single-step and multistep contextual problems that involve multiplication with whole numbers.
4.CE.2.iApply strategies (e.g., rounding, compatible numbers, place value) and algorithms, including the standard algorithm, to estimate and determine the quotient of two whole numbers, given a one-digit divisor and a two- or three-digit dividend, with and without remainders.
4.CE.2.jEstimate, represent, solve, and justify solutions to single-step contextual problems involving division with whole numbers.
4.CE.2.kInterpret the quotient and remainder when solving a contextual problem.
4.CE.3 The student will estimate, represent, solve, and justify solutions to single-step problems, including those in context, using addition and subtraction of fractions (proper, improper, and mixed numbers with like denominators of 2, 3, 4, 5, 6, 8, 10, and 12), with and without models; and solve single-step contextual problems involving multiplication of a whole number (12 or less) and a unit fraction, with models.
4.CE.3.aEstimate and determine the sum or difference of two fractions (proper or improper) and/or mixed numbers, having like denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12 (e.g., 3/8 + 3/8, 2 1/5 + 4/5, 7/4 – 5/4) and simplify the resulting fraction. Addition and subtraction with fractions may include regrouping.
4.CE.3.bEstimate, represent, solve, and justify solutions to single-step contextual problems using addition and subtraction with fractions (proper or improper) and/or mixed numbers, having like denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fraction. Addition and subtraction with fractions may include regrouping.
4.CE.3.cSolve single-step contextual problems involving multiplication of a whole number, limited to 12 or less, and a unit fraction (e.g., 6 × 1/3, 1/5 × 8, 2 × 1/10), with models.
4.CE.3.dApply the inverse property of multiplication in models (e.g., use a visual fraction model to represent 4/4 or 1 as the product of 4 × 1/4).
4.CE.4 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction of decimals through the thousandths, with and without models.
4.CE.4.a Apply strategies (e.g., rounding to the nearest whole number, using compatible numbers) and algorithms, including the standard algorithm, to estimate and determine the sum or difference of two decimals through the thousandths, with and without models, in which:
4.CE.4.a.idecimals do not exceed the thousandths; and
4.CE.4.a.iiaddends, subtrahends, and minuends are limited to four digits.
4.CE.4.bEstimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition and subtraction of decimals through the thousandths.
4.DS Data Modeling
4.DS.10 The student will be able to summarize and interpret data represented in both conventional and emerging visualizations.
4.DS.10.aApply descriptive statistics to explain measures of central tendency and measures of variability/dispersion to describe center and spread in visualizations of distributions.
4.DS.10.b Define emerging visualizations and describe summarization of characteristics and relationships based on audience and purpose which may include:
4.DS.10.b.ia heat map, which uses color to show changes and magnitude of a third variable to a two-dimensional plot; and
4.DS.10.b.iia bubble chart, which is a multivariate graph that is both a scatterplot and a proportional area chart. Typically, each plotted point then represents a third variable by the area of its circle.
4.DS.10.cInterpret various emerging visualizations by describing patterns, trends, and relationships between and among the variables.
4.DS.11 The student will select statistical models and use goodness of fit testing to extract actionable knowledge directly from data.
4.DS.11.aCalculate the theoretical probability of random events and compare them to the observed frequencies.
4.DS.11.bDescribe the normal curve determined by the mean and standard deviation of a univariate data set.
4.DS.11.cFit nonlinear models to data sets and use these models to predict unobserved data values.
4.DS.11.dSelect pairs of variables that identify meaningful clusters of data.
4.DS.11.e Select an appropriate statistical distribution and test its goodness of fit based on the context of the data being analyzed. Statistical distributions may include, but are not limited to
4.DS.11.e.iNormal;
4.DS.11.e.iiBinomial; and
4.DS.11.e.iiiPoisson.
4.DS.7 The student will be able to assess reliability of source data in preparation for mathematical modeling.
4.DS.7.aExplain why determining the reliability of big data sources is a key skill that data scientists use to build data trust across an organization.
4.DS.7.bDescribe the difference between reliability of a data source compared to statistical reliability and validity in research analysis. Assess processing source data for reliability based on validity, completeness, and uniqueness.
4.DS.8 The student will be able to acquire and prepare big data sets for modeling and analysis.
4.DS.8.aExplain the pros and cons of collecting data versus acquiring it from existing sources.
4.DS.8.b Apply matrix operations using algebraic methods (with the support of technology tools) to:
4.DS.8.b.iwrangle the data (sort, select, filter, and replace);
4.DS.8.b.iiclean the data;
4.DS.8.b.iiiformat and enrich the data; and
4.DS.8.b.ivcombine and store the data.
4.DS.8.cRead data from different sources for preparation and analysis.
4.DS.8.dIdentify important parameters about a big data set based on the context of data collected/acquired.
4.DS.8.e Define and document the process of ingesting, formatting, and cleaning data for future decision making by:
4.DS.8.e.imaking data more easily understood by a wider audience; and
4.DS.8.e.iiconnecting data with existing contextual data.
4.DS.9 The student will select and analyze data models to make predictions, while assessing accuracy and sources of uncertainty.
4.DS.9.aIdentify factors that contribute to the overall behavior of a data set (e.g., true values, bias, and noise).
4.DS.9.bFit models based on the behavior of the data, (e.g., models of univariate and bivariate data), in order to make predictions.
4.DS.9.cDistinguish between linear and nonlinear associations between variables using visualizations.
4.DS.9.dIdentify models that are overly complex and therefore fitting to random noise which decreases their predictive accuracy.
4.DS.9.eUse regression techniques to perform selection of optimal features.
4.DS.9.fRecognize the potential implications of removing features.
4.DS.9.gSelect the optimal model for a data set from among a large collection of models, using technological tools.
4.MG Measurement and Geometry
4.MG.1 The student will reason mathematically to solve problems, including those in context, that involve length, weight/mass, and liquid volume using U.S. Customary and metric units.
4.MG.1.a Determine an appropriate unit of measure to use when measuring:
4.MG.1.a.ilength in both U.S. Customary (inch, foot, yard, mile) and metric units (millimeter, centimeter, meter);
4.MG.1.a.iiweight/mass in both U.S. Customary (ounce, pound) and metric units (gram, kilogram); and
4.MG.1.a.iiiliquid volume in both U.S. Customary (cup, pint, quart, gallon) and metric units (milliliter, liter).
4.MG.1.b Estimate and measure:
4.MG.1.b.ilength of an object to the nearest U.S. Customary unit (1/2 inch, 1/4 inch, 1/8 inch, foot, yard) and nearest metric unit (millimeter, centimeter, or meter);
4.MG.1.b.iiweight/mass of an object to the nearest U.S. Customary unit (ounce, pound) and nearest metric unit (gram, kilogram); and
4.MG.1.b.iiiliquid volume to the nearest U.S. Customary unit (cup, pint, quart, gallon) and nearest metric unit (milliliter, liter).
4.MG.1.cCompare estimates of length, weight/mass, or liquid volume with the actual measurements.
4.MG.1.d Given the equivalent measure of one unit, solve problems, including those in context, by determining the equivalent measures within the U.S. Customary system for:
4.MG.1.d.ilength (inches and feet, feet and yards, inches and yards);
4.MG.1.d.iiweight/mass (ounces and pounds); and
4.MG.1.d.iiiliquid volume (cups, pints, quarts, and gallons).
4.MG.2 The student will solve single-step and multistep contextual problems involving elapsed time (limited to hours and minutes within a 12-hour period).
4.MG.2.a Solve single-step and multistep contextual problems involving elapsed time in hours and minutes, within a 12-hour period (within a.m., within p.m., and across a.m. and p.m.) when given:
4.MG.2.a.ithe starting time and the ending time, determine the amount of time that has elapsed in hours and minutes;
4.MG.2.a.iithe starting time and amount of elapsed time in hours and minutes, determine the ending time; or
4.MG.2.a.iiithe ending time and the amount of elapsed time in hours and minutes, determine the starting time.
4.MG.3 The student will use multiple representations to develop and use formulas to solve problems, including those in context, involving area and perimeter limited to rectangles and squares (in both U.S. Customary and metric units).
4.MG.3.aUse concrete materials and pictorial models to develop a formula for the area and perimeter of a rectangle (including a square).
4.MG.3.bDetermine the area and perimeter of a rectangle when given the measure of two adjacent sides (in whole number units), with and without models.
4.MG.3.cDetermine the area and perimeter of a square when given the measure of one side (in whole number units), with and without models.
4.MG.3.dUse concrete materials and pictorial models to explore the relationship between area and perimeter of rectangles.
4.MG.3.eIdentify and represent rectangles with the same perimeter and different areas or with the same area and different perimeters.
4.MG.3.fSolve contextual problems involving area and perimeter of rectangles and squares.
4.MG.4 The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.
4.MG.4.aIdentify and describe points, lines, line segments, rays, and angles, including endpoints and vertices.
4.MG.4.bDescribe endpoints and vertices in relation to lines, line segments, rays, and angles.
4.MG.4.cDraw representations of points, line segments, rays, angles, and lines, using a ruler or straightedge.
4.MG.4.dIdentify parallel, perpendicular, and intersecting lines and line segments in plane and solid figures, including those in context.
4.MG.4.eUse symbolic notation to name points, lines, line segments, rays, angles, and to describe parallel and perpendicular lines.
4.MG.5 The student will classify and describe quadrilaterals (parallelograms, rectangles, squares, rhombi, and/or trapezoids) using specific properties and attributes.
4.MG.5.aDevelop definitions for parallelograms, rectangles, squares, rhombi, and trapezoids through the exploration of properties and attributes.
4.MG.5.bIdentify and describe points, line segments, angles, and vertices in quadrilaterals.
4.MG.5.cIdentify and describe parallel, intersecting, perpendicular, and congruent sides in quadrilaterals.
4.MG.5.d Compare, contrast, and classify quadrilaterals (parallelograms, rectangles, squares, rhombi, and/or trapezoids) based on the following properties and attributes:
4.MG.5.d.iparallel sides;
4.MG.5.d.iiperpendicular sides;
4.MG.5.d.iiicongruence of sides; and
4.MG.5.d.ivnumber of right angles.
4.MG.5.eDenote properties of quadrilaterals and identify parallel sides, congruent sides, and right angles by using geometric markings.
4.MG.5.fUse symbolic notation to name line segments and angles in quadrilaterals.
4.MG.6 The student will identify, describe, compare, and contrast plane and solid figures according to their characteristics (number of angles, vertices, edges, and the number and shape of faces), with and without models.
4.MG.6.aIdentify concrete models and pictorial representations of solid figures (cube, rectangular prism, square pyramid, sphere, cone, and cylinder).
4.MG.6.bIdentify and describe solid figures (cube, rectangular prism, square pyramid, and sphere) according to their characteristics (number of angles, vertices, edges, and by the number and shape of faces).
4.MG.6.cCompare and contrast plane and solid figures (limited to circles, squares, triangles, rectangles, spheres, cubes, square pyramids, and rectangular prisms) according to their characteristics (number of sides, angles, vertices, edges, and the number and shape of faces).
4.NS Number and Number Sense
4.NS.1 The student will use place value understanding to read, write, and identify the place and value of each digit in a nine-digit whole number.
4.NS.1.aRead nine-digit whole numbers, presented in standard form, and represent the same number in written form.
4.NS.1.bWrite nine-digit whole numbers in standard form when the numbers are presented orally or in written form.
4.NS.1.cApply patterns within the base 10 system to determine and communicate, orally and in written form, the place and value of each digit in a nine-digit whole number (e.g., in 568,165,724, the 8 represents 8 millions and its value is 8,000,000).
4.NS.2 The student will demonstrate an understanding of the base 10 system to compare and order whole numbers up to seven digits.
4.NS.2.aCompare two whole numbers up to seven digits each, using words (greater than, less than, equal to, not equal to) and/or using symbols (>, <, =, ≠).
4.NS.2.bOrder up to four whole numbers up to seven digits each, from least to greatest or greatest to least.
4.NS.3 The student will use mathematical reasoning and justification to represent, compare, and order fractions (proper, improper, and mixed numbers with denominators 12 or less), with and without models.
4.NS.3.aCompare and order no more than four fractions (proper or improper), and/or mixed numbers, with like denominators by comparing the number of parts (numerators) using fractions with denominators of 12 or less (e.g., 1/5 < 3/5). Justify comparisons orally, in writing, or with a model.
4.NS.3.bCompare and order no more than four fractions (proper or improper), and/or mixed numbers, with like numerators and unlike denominators by comparing the size of the parts using fractions with denominators of 12 or less (e.g., 3/8 < 3/5). Justify comparisons orally, in writing, or with a model.
4.NS.3.cUse benchmarks (e.g., 0, 1/2, or 1) to compare and order no more than four fractions (proper or improper), and/or mixed numbers, with like and unlike denominators of 12 or less. Justify comparisons orally, in writing, or with a model.
4.NS.3.dCompare two fractions (proper or improper) and/or mixed numbers using fractions with denominators of 12 or less, using the symbols >, <, and = (e.g., 2/3 > 1/7). Justify comparisons orally, in writing, or with a model.
4.NS.3.eRepresent equivalent fractions with denominators of 12 or less, with and without models.
4.NS.3.fCompose and decompose fractions (proper and improper) and/or mixed numbers with denominators of 12 or less, in multiple ways, with and without models.
4.NS.3.gRepresent the division of two whole numbers as a fraction given a contextual situation and a model (e.g., 3/5 means the same as 3 divided by 5 or 3/5 represents the amount of muffin each of five children will receive when sharing three muffins equally).
4.NS.4 The student will use mathematical reasoning and justification to represent, compare, and order decimals through thousandths, with and without models.
4.NS.4.aInvestigate and describe the ten-to-one place value relationship for decimals through thousandths, using concrete models (e.g., place value mats/charts, decimal squares, base 10 blocks).
4.NS.4.bRepresent and identify decimals expressed through thousandths, using concrete, pictorial, and numerical representations.
4.NS.4.cRead and write decimals expressed through thousandths, using concrete, pictorial, and numerical representations.
4.NS.4.dIdentify and communicate, both orally and in written form, the place and value of each digit in a decimal through thousandths (e.g., given 0.385, the 8 is in the hundredths place and has a value of 0.08).
4.NS.4.eCompare using symbols (<, >, =) and/or words (greater than, less than, equal to) and order (least to greatest and greatest to least), a set of no more than four decimals expressed through thousandths, using multiple strategies (e.g., benchmarks, place value, number lines). Justify comparisons with a model, orally, and in writing.
4.NS.5 The student will reason about the relationship between fractions and decimals (limited to halves, fourths, fifths, tenths, and hundredths) to identify and represent equivalencies.
4.NS.5.aRepresent fractions (proper or improper) and/or mixed numbers as decimals through hundredths, using multiple representations, limited to halves, fourths, fifths, tenths, and hundredths.
4.NS.5.bIdentify and model equivalent relationships between fractions (proper or improper) and/or mixed numbers and decimals, using halves, fourths, fifths, tenths, and hundredths.
4.NS.5.cWrite the decimal and fraction equivalent for a given model (e.g., 1/4 = 0.25 or 0.25 = 1/4; 1.25 = 5/4 or 1 1/4; 1.02 = 102/100 or 1 2/100).
4.PFA Patterns, Functions, and Algebra
4.PFA.1 The student will identify, describe, extend, and create increasing and decreasing patterns (limited to addition, subtraction, and multiplication of whole numbers), including those in context, using various representations.
4.PFA.1.aIdentify, describe, extend, and create increasing and decreasing patterns using various representations (e.g., objects, pictures, numbers, number lines, input/output tables, and function machines).
4.PFA.1.bAnalyze an increasing or decreasing single-operation numerical pattern found in lists, input/output tables, or function machines and generalize the change to identify the rule, extend the pattern, or identify missing terms.
4.PFA.1.cGiven a rule, create increasing and decreasing patterns using numbers and input/output tables (including function machines).
4.PFA.1.dSolve contextual problems that involve identifying, describing, and extending increasing and decreasing patterns using single-operation input and output rules.
4.PS Probability and Statistics
4.PS.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on line graphs.
4.PS.1.aFormulate questions that require the collection or acquisition of data.
4.PS.1.bDetermine the data needed to answer a formulated question and collect or acquire existing data (limited to 10 or fewer data points) using various methods (e.g., observations, measurements, experiments).
4.PS.1.cOrganize and represent a data set using line graphs with a title and labeled axes with whole number increments, with and without the use of technology tools.
4.PS.1.d Analyze data represented in line graphs and communicate results orally and in writing:
4.PS.1.d.idescribe the characteristics of the data represented in a line graph and the data as a whole (e.g., the time period when the temperature increased the most);
4.PS.1.d.iiidentify parts of the data that have special characteristics and explain the meaning of the greatest, the least, or the same (e.g., the highest temperature shows the warmest day);
4.PS.1.d.iiimake inferences about data represented in line graphs;
4.PS.1.d.ivdraw conclusions about the data and make predictions based on the data to answer questions; and
4.PS.1.d.vsolve single-step and multistep addition and subtraction problems using data from line graphs.
4.PS.2 The student will model and determine the probability of an outcome of a simple event.
4.PS.2.aDescribe probability as the degree of likelihood of an outcome occurring using terms such as impossible, unlikely, equally likely, likely, and certain.
4.PS.2.bModel and determine all possible outcomes of a given simple event where there are no more than 24 possible outcomes, using a variety of manipulatives (e.g., coins, two-sided counters, number cubes, spinners).
4.PS.2.cWrite the probability of a given simple event as a fraction between 0 and 1, where there are no more than 24 possible outcomes.
4.PS.2.dDetermine the likelihood of an event occurring and relate it to its whole number or fractional representation (e.g., impossible or zero; equally likely; certain or one).
4.PS.2.eCreate a model or contextual problem to represent a given probability.

Grade 5

5.CE Computation and Estimation
5.CE.1 The student will estimate, represent, solve, and justify solutions to single-step and multistep contextual problems using addition, subtraction, multiplication, and division with whole numbers.
5.CE.1.aEstimate the sum, difference, product, and quotient of whole numbers in contextual problems.
5.CE.1.b Represent, solve, and justify solutions to single-step and multistep contextual problems by applying strategies (e.g., estimation, properties of addition and multiplication) and algorithms, including the standard algorithm, involving addition, subtraction, multiplication, and division of whole numbers, with and without remainders, in which:
5.CE.1.b.isums, differences, and products do not exceed five digits;
5.CE.1.b.iifactors do not exceed two digits by three digits;
5.CE.1.b.iiidivisors do not exceed two digits; or
5.CE.1.b.ivdividends do not exceed four digits.
5.CE.1.cInterpret the quotient and remainder when solving a contextual problem.
5.CE.2 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition and subtraction of fractions with like and unlike denominators (with and without models), and solve single-step contextual problems involving multiplication of a whole number and a proper fraction, with models.
5.CE.2.aDetermine the least common multiple of two numbers to find the least common denominator for two fractions.
5.CE.2.bEstimate and determine the sum or difference of two fractions (proper or improper) and/or mixed numbers, having like and unlike denominators limited to 2, 3, 4, 5, 6, 8, 10, and 12 (e.g., 5/8 + 1/4, 4/5 – 2/3, 3 3/4 + 2 5/12), and simplify the resulting fraction.
5.CE.2.cEstimate and solve single-step and multistep contextual problems involving addition and subtraction with fractions (proper or improper) and/or mixed numbers having like and unlike denominators, with and without models. Denominators should be limited to 2, 3, 4, 5, 6, 8, 10, and 12. Answers should be expressed in simplest form.
5.CE.2.dSolve single-step contextual problems involving multiplication of a whole number, limited to 12 or less, and a proper fraction (e.g., 9 × 2/3, 8 × 3/4), with models. The denominator will be a factor of the whole number and answers should be expressed in simplest form.
5.CE.3 The student will estimate, represent, solve, and justify solutions to single-step and multistep problems, including those in context, using addition, subtraction, multiplication, and division with decimal numbers.
5.CE.3.aApply estimation strategies (e.g., rounding to the nearest whole number, tenth or hundredth; compatible numbers, place value) to determine a reasonable solution for single-step and multistep contextual problems involving addition, subtraction, and multiplication of decimals, and single-step contextual problems involving division of decimals.
5.CE.3.b Estimate and determine the product of two numbers using strategies and algorithms, including the standard algorithm, when given:
5.CE.3.b.ia two-digit factor and a one-digit factor (e.g., 2.3 × 4; 0.08 × 0.9; .16 × 5);
5.CE.3.b.iia three-digit factor and a one-digit factor (e.g., 0.156 × 4, 3.28 × 7, 8.09 × 0.2); and
5.CE.3.b.iiia two-digit factor and a two-digit factor (e.g., 0.85 × 3.7, 14 × 1.6, 9.2 × 3.5).
5.CE.3.c Estimate and determine the quotient of two numbers using strategies and algorithms, including the standard algorithm, in which:
5.CE.3.c.iquotients do not exceed four digits with or without a decimal point;
5.CE.3.c.iiquotients may include whole numbers, tenths, hundredths, or thousandths;
5.CE.3.c.iiidivisors are limited to a single digit whole number or a decimal expressed as tenths; and
5.CE.3.c.ivno more than one additional zero will need to be annexed.
5.CE.3.dSolve single-step and multistep contextual problems involving addition, subtraction, and multiplication of decimals by applying strategies (e.g., estimation, modeling) and algorithms, including the standard algorithm.
5.CE.3.eSolve single-step contextual problems involving division with decimals by applying strategies (e.g., estimation, modeling) and algorithms, including the standard algorithm.
5.CE.4 The student will simplify numerical expressions with whole numbers using the order of operations.
5.CE.4.a Use order of operations to simplify numerical expressions with whole numbers, limited to addition, subtraction, multiplication, and division in which:
5.CE.4.a.iexpressions may contain no more than one set of parentheses;
5.CE.4.a.iisimplification will be limited to five whole numbers and four operations in any combination of addition, subtraction, multiplication, or division;
5.CE.4.a.iiiwhole numbers will be limited to two digits or less; and
5.CE.4.a.ivexpressions should not include braces, brackets, or fraction bars.
5.CE.4.bGiven a whole number numerical expression involving more than one operation, describe which operation is completed first, which is second, and which is third.
5.DS Data and Computing
5.DS.12 The student will be able to select and utilize appropriate technological tools and functions within those tools to process and prepare data for analysis.
5.DS.12.aUtilize technology tools to be able to access data effectively from multiple sources (e.g., tables, column separated values, spreadsheets, documents, databases).
5.DS.12.bUtilize tools and functions (in tools) to effectively explore the data for issues and errors before beginning to process it.
5.DS.12.cDefine the (tools and technological) process to optimally ingest data and to export data after processing.
5.DS.12.d Utilize tools and their functions to clean and validate data by:
5.DS.12.d.iremoving data that are incomplete, incorrect, or duplicated;
5.DS.12.d.iiremoving extraneous data or outliers; and
5.DS.12.d.iiistandardizing data to conform to contextual norms (e.g., privacy, sensitive data).
5.DS.12.e Utilize tools and their functions to combine and store data by:
5.DS.12.e.imerging multiple data sets for efficiency purposes; and
5.DS.12.e.iioptimizing storage of data based on volume, velocity, and variety.
5.DS.12.fUtilize tools to format and store the data appropriately to allow for effective analysis.
5.DS.13 The student will be able to select and utilize appropriate technological tools and functions within those tools to analyze and communicate data effectively.
5.DS.13.aSelect and utilize technology tools to effectively generate conventional and unconventional visualizations of data to explore patterns and/or analyze a large data set.
5.DS.13.bUtilize specific functions in technology tools to perform descriptive and inferential statistical analysis.
5.DS.13.cUtilize coding to store and extract data more effectively for data analysis.
5.DS.13.dSelect and apply features of technology tools effectively to organize, summarize and gain insight from data.
5.DS.13.eSelect the appropriate visualization based on context and audience and create it using technology tools to effectively communicate an idea.
5.MG Measurement and Geometry
5.MG.1 The student will reason mathematically to solve problems, including those in context, that involve length, mass, and liquid volume using metric units.
5.MG.1.a Determine the most appropriate unit of measure to use in a contextual problem that involves metric units:
5.MG.1.a.ilength (millimeters, centimeters, meters, and kilometers);
5.MG.1.a.iimass (grams and kilograms); and
5.MG.1.a.iiiliquid volume (milliliters and liters).
5.MG.1.b Estimate and measure to solve contextual problems that involve metric units:
5.MG.1.b.ilength (millimeters, centimeters, and meters);
5.MG.1.b.iimass (grams and kilograms); and
5.MG.1.b.iiiliquid volume (milliliters and liters).
5.MG.1.c Given the equivalent metric measure of one unit, in a contextual problem, determine the equivalent measurement within the metric system:
5.MG.1.c.ilength (millimeters, centimeters, meters, and kilometers);
5.MG.1.c.iimass (grams and kilograms); and
5.MG.1.c.iiiliquid volume (milliliters and liters).
5.MG.2 The student will use multiple representations to solve problems, including those in context, involving perimeter, area, and volume.
5.MG.2.aInvestigate and develop a formula for determining the area of a right triangle.
5.MG.2.bEstimate and determine the area of a right triangle, with diagrams, when the base and the height are given in whole number units, in metric or U.S. Customary units, and record the solution with the appropriate unit of measure (e.g., 16 square inches).
5.MG.2.cDescribe volume as a measure of capacity and give examples of volume as a measurement in contextual situations.
5.MG.2.dInvestigate and develop a formula for determining the volume of rectangular prisms using concrete objects.
5.MG.2.eSolve problems, including those in context, to estimate and determine the volume of a rectangular prism using concrete objects, diagrams, and formulas when the length, width, and height are given in whole number units. Record the solution with the appropriate unit of measure (e.g., 12 cubic inches).
5.MG.2.fIdentify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation.
5.MG.2.gSolve contextual problems that involve perimeter, area, and volume in standard units of measure.
5.MG.3 The student will classify and measure angles and triangles, and solve problems, including those in context.
5.MG.3.aClassify angles as right, acute, obtuse, or straight and justify reasoning.
5.MG.3.bClassify triangles as right, acute, or obtuse and equilateral, scalene, or isosceles and justify reasoning.
5.MG.3.cIdentify congruent sides and right angles using geometric markings to denote properties of triangles.
5.MG.3.dCompare and contrast the properties of triangles.
5.MG.3.eIdentify the appropriate tools (e.g., protractor, straightedge, angle ruler, available technology) to measure and draw angles.
5.MG.3.fMeasure right, acute, obtuse, and straight angles, using appropriate tools, and identify measures in degrees.
5.MG.3.gUse models to prove that the sum of the interior angles of a triangle is 180 degrees and use the relationship to determine an unknown angle measure in a triangle.
5.MG.3.hSolve addition and subtraction contextual problems to determine unknown angle measures on a diagram.
5.NS Number and Number Sense
5.NS.1 The student will use reasoning and justification to identify and represent equivalency between fractions (with denominators that are thirds, eighths, and factors of 100) and decimals; and compare and order sets of fractions (proper, improper, and/or mixed numbers having denominators of 12 or less) and decimals (through thousandths).
5.NS.1.aUse concrete and pictorial models to represent fractions with denominators that are thirds, eighths, and factors of 100 in their equivalent decimal form.
5.NS.1.bUse concrete and pictorial models to represent decimals in their equivalent fraction form (thirds, eighths, and factors of 100).
5.NS.1.cIdentify equivalent relationships between decimals and fractions with denominators that are thirds, eighths, and factors of 100 in their equivalent decimal form, with and without models.
5.NS.1.dCompare (using symbols <, >, =) and order (least to greatest and greatest to least) a set of no more than four decimals and fractions (proper, improper) and/or mixed numbers using multiple strategies (e.g., benchmarks, place value, number lines). Justify solutions orally, in writing, or with a model.
5.NS.2 The student will demonstrate an understanding of prime and composite numbers, and determine the prime factorization of a whole number up to 100.
5.NS.2.aGiven a whole number up to 100, create a concrete or pictorial representation to demonstrate whether the number is prime or composite, and justify reasoning.
5.NS.2.bClassify, compare, and contrast whole numbers up to 100 using the characteristics prime and composite.
5.NS.2.cDetermine the prime factorization for a whole number up to 100.
5.PFA Patterns, Functions, and Algebra
5.PFA.1 The student will identify, describe, extend, and create increasing and decreasing patterns with whole numbers, fractions, and decimals, including those in context, using various representations.
5.PFA.1.aIdentify, describe, extend, and create increasing and decreasing patterns using various representations (e.g., objects, pictures, numbers, number lines, input/output tables, function machines).
5.PFA.1.bAnalyze an increasing or decreasing single-operation numerical pattern found in lists, input/output tables, and function machines, and generalize the change to identify the rule, extend the pattern, or identify missing terms. (Patterns will be limited to addition, subtraction, multiplication, and division of whole numbers; addition and subtraction of fractions with like denominators of 12 or less; and addition and subtraction of decimals expressed in tenths or hundredths).
5.PFA.1.cSolve contextual problems that involve identifying, describing, and extending increasing and decreasing patterns using single-operation input and output rules (limited to addition, subtraction, multiplication, and division of whole numbers; addition and subtraction of fractions with like denominators of 12 or less; and addition and subtraction of decimals expressed in tenths or hundredths).
5.PFA.2 The student will investigate and use variables in contextual problems.
5.PFA.2.aDescribe the concept of a variable (presented as a box, letter, or other symbol) as a representation of an unknown quantity.
5.PFA.2.bWrite an equation (with a single variable that represents an unknown quantity and one operation) from a contextual situation, using addition, subtraction, multiplication, or division.
5.PFA.2.cUse an expression with a variable to represent a given verbal expression involving one operation (e.g., “5 more than a number” can be represented by 𝑦 + 5).
5.PFA.2.dCreate and write a word problem to match a given equation with a single variable and one operation.
5.PS Probability and Statistics
5.PS.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on line plots (dot plots) and stem-and-leaf plots.
5.PS.1.aFormulate questions that require the collection or acquisition of data.
5.PS.1.bDetermine the data needed to answer a formulated question and collect or acquire existing data (limited to 30 or fewer data points) using various methods (e.g., polls, observations, measurements, experiments).
5.PS.1.cOrganize and represent a data set using a line plot (dot plot) with a title, labeled axes, and a key, with and without the use of technology tools. Line plots (dot plots) may contain whole numbers, fractions, or decimals.
5.PS.1.dOrganize and represent numerical data using a stem-and-leaf plot with a title and key, where the stems are listed in ascending order and the leaves are in ascending order, with or without commas between the leaves.
5.PS.1.e Analyze data represented in line plots (dot plots) and stem-and-leaf plots and communicate results orally and in writing:
5.PS.1.e.idescribe the characteristics of the data represented in a line plot (dot plot) and stem-and-leaf plot as a whole (e.g., the shape and spread of the data);
5.PS.1.e.iimake inferences about data represented in line plots (dot plots) and stem-and-leaf plots (e.g., based on a line plot (dot plot) of the number of books students in a bus line have in their backpack, every student will have from two to four books in their backpack);
5.PS.1.e.iiiidentify parts of the data that have special characteristics and explain the meaning of the greatest, the least, or the same (e.g., the stem-and-leaf plot shows that the same number of students scored in the 90s as scored in the 70s);
5.PS.1.e.ivdraw conclusions about the data and make predictions based on the data to answer questions; and
5.PS.1.e.vsolve single-step and multistep addition and subtraction problems using data from line plots (dot plots) and stem-and-leaf plots.
5.PS.2 The student will solve contextual problems using measures of center and the range.
5.PS.2.aDescribe mean as fair share.
5.PS.2.bDescribe and determine the mean of a set of data values representing data from a given context as a measure of center.
5.PS.2.cDescribe and determine the median of a set of data values representing data from a given context as a measure of center.
5.PS.2.dDescribe and determine the mode of a set of data values representing data from a given context as a measure of center.
5.PS.2.eDescribe and determine the range of a set of data values representing data from a given context as a measure of spread.
5.PS.3 The student will determine the probability of an outcome by constructing a model of a sample space and using the Fundamental (Basic) Counting Principle.
5.PS.3.aDetermine the probability of an outcome by constructing a sample space (with a total of 24 or fewer equally likely possible outcomes), using a tree diagram, list, or chart to represent and determine all possible outcomes.
5.PS.3.bDetermine the number of possible outcomes by using the Fundamental (Basic) Counting Principle.

Grade 6

6.CE Computation and Estimation
6.CE.1 The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.
6.CE.1.aDemonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.
6.CE.1.bMultiply and divide fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.
6.CE.1.cInvestigate and explain the effect of multiplying or dividing a fraction, whole number, or mixed number by a number between zero and one.
6.CE.1.dEstimate, determine, and justify the solution to single-step and multistep problems in context that involve addition and subtraction with fractions (proper or improper) and mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. Answers are expressed in simplest form.
6.CE.1.eEstimate, determine, and justify the solution to single-step and multistep problems in context that involve multiplication and division with fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.
6.CE.2 The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.
6.CE.2.aDemonstrate/model addition, subtraction, multiplication, and division of integers using pictorial representations or concrete manipulatives.
6.CE.2.bAdd, subtract, multiply, and divide two integers.
6.CE.2.cSimplify an expression that contains absolute value bars | | and an operation with two integers (e.g., –|5 – 8| or |–12|/8) and represent the result on a number line.
6.CE.2.dEstimate, determine, and justify the solution to one and two-step contextual problems, involving addition, subtraction, multiplication, and division with integers.
6.MG Measurement and Geometry
6.MG.1 The student will identify the characteristics of circles and solve problems, including those in context, involving circumference and area.
6.MG.1.aIdentify and describe chord, diameter, radius, circumference, and area of a circle.
6.MG.1.b Investigate and describe the relationship between:
6.MG.1.b.idiameter and radius;
6.MG.1.b.iiradius and circumference; and
6.MG.1.b.iiidiameter and circumference.
6.MG.1.cDevelop an approximation for pi (3.14) by gathering data and comparing the circumference to the diameter of various circles, using concrete manipulatives or technological models.
6.MG.1.dDevelop the formula for circumference using the relationship between diameter, radius, and pi.
6.MG.1.eSolve problems, including those in context, involving circumference and area of a circle when given the length of the diameter or radius.
6.MG.2 The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles and parallelograms.
6.MG.2.aDevelop the formula for determining the area of parallelograms and triangles using pictorial representations and concrete manipulatives (e.g., two-dimensional diagrams, grid paper).
6.MG.2.bSolve problems, including those in context, involving the perimeter and area of triangles and parallelograms.
6.MG.3 The student will describe the characteristics of the coordinate plane and graph ordered pairs.
6.MG.3.aIdentify and label the axes, origin, and quadrants of a coordinate plane.
6.MG.3.bIdentify and describe the location (quadrant or the axis) of a point given as an ordered pair. Ordered pairs will be limited to coordinates expressed as integers.
6.MG.3.cGraph ordered pairs in the four quadrants and on the axes of a coordinate plane. Ordered pairs will be limited to coordinates expressed as integers.
6.MG.3.dIdentify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane. Ordered pairs will be limited to coordinates expressed as integers.
6.MG.3.eRelate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers.
6.MG.3.fDraw polygons in the coordinate plane given coordinates for the vertices; use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate. Ordered pairs will be limited to coordinates expressed as integers. Apply these techniques in the context of solving contextual and mathematical problems.
6.MG.4 The student will determine congruence of segments, angles, and polygons.
6.MG.4.aIdentify regular polygons.
6.MG.4.bDraw lines of symmetry to divide regular polygons into two congruent parts.
6.MG.4.cDetermine the congruence of segments, angles, and polygons given their properties.
6.MG.4.dDetermine whether polygons are congruent or noncongruent according to the measures of their sides and angles.
6.NS Number and Number Sense
6.NS.1 The student will reason and use multiple strategies to express equivalency, compare, and order numbers written as fractions, mixed numbers, decimals, and percents.
6.NS.1.aEstimate and determine the percent represented by a given model (e.g., number line, picture, verbal description), including percents greater than 100% and less than 1%.
6.NS.1.bRepresent and determine equivalencies among decimals (through the thousandths place) and percents incorporating the use of number lines, and concrete and pictorial models.
6.NS.1.cRepresent and determine equivalencies among fractions (proper or improper) and mixed numbers that have denominators that are 12 or less or factors of 100 and percents incorporating the use of number lines, and concrete and pictorial models.
6.NS.1.dRepresent and determine equivalencies among decimals, percents, fractions (proper or improper), and mixed numbers that have denominators that are 12 or less or factors of 100 incorporating the use of number lines, and concrete and pictorial models.
6.NS.1.eUse multiple strategies (e.g., benchmarks, number line, equivalency) to compare and order no more than four positive rational numbers expressed as fractions (proper or improper), mixed numbers, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less or factors of 100) with and without models. Justify solutions orally, in writing or with a model. Ordering may be in ascending or descending order.
6.NS.2 The student will reason and use multiple strategies to represent, compare, and order integers.
6.NS.2.aRepresent integers (e.g., number lines, concrete materials, pictorial models), including models derived from contextual situations, and identify an integer represented by a point on a number line.
6.NS.2.bCompare and order integers using a number line.
6.NS.2.cCompare integers, using mathematical symbols (<, >, =).
6.NS.2.dIdentify and describe the absolute value of an integer as the distance from zero on the number line.
6.NS.3 The student will recognize and represent patterns with whole number exponents and perfect squares.
6.NS.3.aRecognize and represent patterns with bases and exponents that are whole numbers.
6.NS.3.bRecognize and represent patterns of perfect squares not to exceed 20^2, by using concrete and pictorial models.
6.NS.3.cJustify if a number between 0 and 400 is a perfect square through modeling or mathematical reasoning.
6.NS.3.dRecognize and represent powers of 10 with whole number exponents by examining patterns in place value.
6.PFA Patterns, Functions, and Algebra
6.PFA.1 The student will use ratios to represent relationships between quantities, including those in context.
6.PFA.1.aRepresent a relationship between two quantities using ratios.
6.PFA.1.bRepresent a relationship in context that makes a comparison by using the notations 𝑎/𝑏, 𝑎:𝑏, and 𝑎 to 𝑏.
6.PFA.1.cRepresent different comparisons within the same quantity or between different quantities (e.g., part to part, part to whole, whole to whole).
6.PFA.1.dCreate a relationship in words for a given ratio expressed symbolically.
6.PFA.1.eCreate a table of equivalent ratios to represent a proportional relationship between two quantities, when given a ratio.
6.PFA.1.fCreate a table of equivalent ratios to represent a proportional relationship between two quantities, when given a contextual situation.
6.PFA.2 The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values).
6.PFA.2.aIdentify the unit rate of a proportional relationship represented by a table of values, a contextual situation, or a graph.
6.PFA.2.bDetermine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate.
6.PFA.2.cDetermine whether a proportional relationship exists between two quantities, when given a table of values, context, or graph.
6.PFA.2.dWhen given a contextual situation representing a proportional relationship, find the unit rate and create a table of values or a graph.
6.PFA.2.eMake connections between and among multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs.
6.PFA.3 The student will write and solve one-step linear equations in one variable, including contextual problems that require the solution of a one-step linear equation in one variable.
6.PFA.3.aIdentify and develop examples of the following algebraic vocabulary: equation, variable, expression, term, and coefficient.
6.PFA.3.bRepresent and solve one-step linear equations in one variable, using a variety of concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles, weights on a balance scale).
6.PFA.3.cApply properties of real numbers and properties of equality to solve a one-step equation in one variable. Coefficients are limited to integers and unit fractions. Numeric terms are limited to integers.
6.PFA.3.dConfirm solutions to one-step linear equations in one variable using a variety of concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles, weights on a balance scale).
6.PFA.3.eWrite a one-step linear equation in one variable to represent a verbal situation, including those in context.
6.PFA.3.fCreate a verbal situation in context given a one-step linear equation in one variable.
6.PFA.4 The student will represent a contextual situation using a linear inequality in one variable with symbols and graphs on a number line.
6.PFA.4.aGiven the graph of a linear inequality in one variable on a number line, represent the inequality in two equivalent ways (e.g., 𝑥 < –5 or –5 > 𝑥) using symbols. Symbols include <, >, ≤, ≥.
6.PFA.4.bWrite a linear inequality in one variable to represent a given constraint or condition in context or given a graph on a number line.
6.PFA.4.cGiven a linear inequality in one variable, create a corresponding contextual situation or create a number line graph.
6.PFA.4.dUse substitution or a number line graph to justify whether a given number in a specified set makes a linear inequality in one variable true.
6.PFA.4.eIdentify a numerical value(s) that is part of the solution set of a given inequality in one variable.
6.PS Probability and Statistics
6.PS.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on circle graphs.
6.PS.1.aFormulate questions that require the collection or acquisition of data with a focus on circle graphs.
6.PS.1.bDetermine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).
6.PS.1.cDetermine the factors that will ensure that the data collected is a sample that is representative of a larger population.
6.PS.1.dOrganize and represent data using circle graphs, with and without the use of technology tools. The number of data values should be limited to allow for comparisons that have denominators of 12 or less or those that are factors of 100 (e.g., in a class of 20 students, 7 choose apples as a favorite fruit, so the comparison is 7 out of 20, 7/20, or 35%).
6.PS.1.eAnalyze data represented in a circle graph by making observations and drawing conclusions.
6.PS.1.fCompare data represented in a circle graph with the same data represented in other graphs, including but not limited to bar graphs, pictographs, and line plots (dot plots), and justify which graphical representation best represents the data.
6.PS.2 The student will represent the mean as a balance point and determine the effect on statistical measures when a data point is added, removed, or changed.
6.PS.2.aRepresent the mean of a set of data graphically as the balance point represented in a line plot (dot plot).
6.PS.2.bDetermine the effect on measures of center when a single value of a data set is added, removed, or changed.
6.PS.2.cObserve patterns in data to identify outliers and determine their effect on mean, median, mode, or range.

Grade 7

7.CE Computation and Estimation
7.CE.1 The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.
7.CE.1.aEstimate, solve, and justify solutions to contextual problems involving addition, subtraction, multiplication, and division with rational numbers expressed as integers, fractions (proper or improper), mixed numbers, and decimals. Fractions may be positive or negative. Decimals may be positive or negative and are limited to the thousandths place.
7.CE.2 The student will solve problems, including those in context, involving proportional relationships.
7.CE.2.aGiven a proportional relationship between two quantities, create and use a ratio table to determine missing values.
7.CE.2.bWrite and solve a proportion that represents a proportional relationship between two quantities to find a missing value, including problems in context.
7.CE.2.cApply proportional reasoning to solve problems in context, including converting units of measurement, when given the conversion factor.
7.CE.2.dEstimate and determine the percentage of a given whole number, including but not limited to the use of benchmark percentages.
7.MG Measurement and Geometry
7.MG.1 The student will investigate and determine the volume formula for right cylinders and the surface area formulas for rectangular prisms and right cylinders and apply the formulas in context.
7.MG.1.aDevelop the formulas for determining the volume of right cylinders and solve problems, including those in contextual situations, using concrete objects, diagrams, and formulas.
7.MG.1.bDevelop the formulas for determining the surface area of rectangular prisms and right cylinders and solve problems, including those in contextual situations, using concrete objects, two-dimensional diagrams, nets, and formulas.
7.MG.1.cDetermine if a problem in context, involving a rectangular prism or right cylinder, represents the application of volume or surface area.
7.MG.1.dDescribe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a factor of 1/4, 1/3, 1/2, 2, 3, or 4, including those in contextual situations.
7.MG.1.eDescribe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a factor of 1/2 or 2, including those in contextual situations.
7.MG.2 The student will solve problems and justify relationships of similarity using proportional reasoning.
7.MG.2.aIdentify corresponding congruent angles of similar quadrilaterals and triangles, through the use of geometric markings.
7.MG.2.bIdentify corresponding sides of similar quadrilaterals and triangles.
7.MG.2.cGiven two similar quadrilaterals or triangles, write similarity statements using symbols.
7.MG.2.dWrite proportions to express the relationships between the lengths of corresponding sides of similar quadrilaterals and triangles.
7.MG.2.eRecognize and justify if two quadrilaterals or triangles are similar using the ratios of corresponding side lengths.
7.MG.2.fSolve a proportion to determine a missing side length of similar quadrilaterals or triangles.
7.MG.2.gGiven angle measures in a quadrilateral or triangle, determine unknown angle measures in a similar quadrilateral or triangle.
7.MG.2.hApply proportional reasoning to solve problems in context including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths.
7.MG.3 The student will compare and contrast quadrilaterals based on their properties and determine unknown side lengths and angle measures of quadrilaterals.
7.MG.3.a Compare and contrast properties of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid:
7.MG.3.a.iparallel/perpendicular sides and diagonals;
7.MG.3.a.iicongruence of angle measures, side, and diagonal lengths; and
7.MG.3.a.iiilines of symmetry.
7.MG.3.b Sort and classify quadrilaterals as parallelograms, rectangles, trapezoids, rhombi, and/or squares based on their properties:
7.MG.3.b.iparallel/perpendicular sides and diagonals;
7.MG.3.b.iicongruence of angle measures, side, and diagonal lengths; and
7.MG.3.b.iiilines of symmetry.
7.MG.3.cGiven a diagram, determine an unknown angle measure in a quadrilateral, using properties of quadrilaterals.
7.MG.3.dGiven a diagram, determine an unknown side length in a quadrilateral using properties of quadrilaterals.
7.MG.4 The student will apply dilations of polygons in the coordinate plane.
7.MG.4.aGiven a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been dilated. Scale factors are limited to 1/4, 1/2, 2, 3, or 4. The center of the dilation will be the origin.
7.MG.4.bSketch the image of a dilation of a polygon limited to a scale factor of 1/4, 1/2, 2, 3, or 4. The center of the dilation will be the origin.
7.MG.4.cIdentify and describe dilations in context including, but not limited to, scale drawings and graphic design.
7.NS Number and Number Sense
7.NS.1 The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.
7.NS.1.aInvestigate and describe powers of 10 with negative exponents by examining patterns.
7.NS.1.bRepresent a power of 10 with a negative exponent in fraction and decimal form.
7.NS.1.cConvert between numbers greater than 0 written in scientific notation and decimals.
7.NS.1.dCompare and order no more than four numbers greater than 0 written in scientific notation. Ordering may be in ascending or descending order.
7.NS.2 The student will reason and use multiple strategies to compare and order rational numbers.
7.NS.2.aUse multiple strategies (e.g., benchmarks, number line, equivalency) to compare (using symbols <, >, =) and order (a set of no more than four) rational numbers expressed as integers, fractions (proper or improper), mixed numbers, decimals, and percents. Fractions and mixed numbers may be positive or negative. Decimals may be positive or negative and are limited to the thousandths place. Ordering may be in ascending or descending order. Justify solutions orally, in writing or with a model.
7.NS.3 The student will recognize and describe the relationship between square roots and perfect squares.
7.NS.3.aDetermine the positive square root of a perfect square from 0 to 400.
7.NS.3.bDescribe the relationship between square roots and perfect squares.
7.PFA Patterns, Functions, and Algebra
7.PFA.1 The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in 𝑦 = 𝑚𝑥 form, and graphs, including problems in context.
7.PFA.1.aDetermine the slope, 𝑚, as the rate of change in a proportional relationship between two quantities given a table of values, graph, or contextual situation and write an equation in the form 𝑦 = 𝑚𝑥 to represent the direct variation relationship. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).
7.PFA.1.bIdentify and describe a line with a slope that is positive, negative, or zero (0), given a graph.
7.PFA.1.cGraph a line representing a proportional relationship, between two quantities given an ordered pair on the line and the slope, 𝑚, as rate of change. Slope may include positive or negative values.
7.PFA.1.dGraph a line representing a proportional relationship between two quantities given the equation of the line in the form 𝑦 = 𝑚𝑥, where 𝑚 represents the slope as rate of change. Slope may include positive or negative values.
7.PFA.1.eMake connections between and among representations of a proportional relationship between two quantities using problems in context, tables, equations, and graphs. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).
7.PFA.2 The student will simplify numerical expressions, simplify and generate equivalent algebraic expressions in one variable, and evaluate algebraic expressions for given replacement values of the variables.
7.PFA.2.aUse the order of operations and apply the properties of real numbers to simplify numerical expressions. Exponents are limited to 1, 2, 3, or 4 and bases are limited to positive integers. Expressions should not include braces { } but may include brackets [ ] and absolute value bars | |. Square roots are limited to perfect squares.
7.PFA.2.bRepresent equivalent algebraic expressions in one variable using concrete manipulatives and pictorial representations (e.g., colored chips, algebra tiles).
7.PFA.2.cSimplify and generate equivalent algebraic expressions in one variable by applying the order of operations and properties of real numbers. Expressions may require combining like terms to simplify. Expressions will include only linear and numeric terms. Coefficients and numeric terms may be positive or negative rational numbers.
7.PFA.2.dUse the order of operations and apply the properties of real numbers to evaluate algebraic expressions for given replacement values of the variables. Exponents are limited to 1, 2, 3, or 4 and bases are limited to positive integers. Expressions should not include braces { } but may include brackets [ ] and absolute value bars | |. Square roots are limited to perfect squares. Limit the number of replacements to no more than three per expression. Replacement values may be positive or negative rational numbers.
7.PFA.3 The student will write and solve two-step linear equations in one variable, including problems in context, that require the solution of a two-step linear equation in one variable.
7.PFA.3.aRepresent and solve two-step linear equations in one variable using a variety of concrete materials and pictorial representations.
7.PFA.3.bApply properties of real numbers and properties of equality to solve two-step linear equations in one variable. Coefficients and numeric terms will be rational.
7.PFA.3.cConfirm algebraic solutions to linear equations in one variable.
7.PFA.3.dWrite a two-step linear equation in one variable to represent a verbal situation, including those in context.
7.PFA.3.eCreate a verbal situation in context given a two-step linear equation in one variable.
7.PFA.3.fSolve problems in context that require the solution of a two-step linear equation.
7.PFA.4 The student will write and solve one- and two-step linear inequalities in one variable, including problems in context, that require the solution of a one- and two-step linear inequality in one variable.
7.PFA.4.aApply properties of real numbers and the addition, subtraction, multiplication, and division properties of inequality to solve one- and two-step inequalities in one variable. Coefficients and numeric terms will be rational.
7.PFA.4.bInvestigate and explain how the solution set of a linear inequality is affected by multiplying or dividing both sides of the inequality statement by a rational number less than zero.
7.PFA.4.cRepresent solutions to one- or two-step linear inequalities in one variable algebraically and graphically using a number line.
7.PFA.4.dWrite one- or two-step linear inequalities in one variable to represent a verbal situation, including those in context.
7.PFA.4.eCreate a verbal situation in context given a one or two-step linear inequality in one variable.
7.PFA.4.fSolve problems in context that require the solution of a one- or two-step inequality.
7.PFA.4.gIdentify a numerical value(s) that is part of the solution set of as given one- or two-step linear inequality in one variable.
7.PFA.4.hDescribe the differences and similarities between solving linear inequalities in one variable and linear equations in one variable.
7.PS Probability and Statistics
7.PS.1 The student will use statistical investigation to determine the probability of an event and investigate and describe the difference between the experimental and theoretical probability.
7.PS.1.aDetermine the theoretical probability of an event.
7.PS.1.bGiven the results of a statistical investigation, determine the experimental probability of an event.
7.PS.1.cDescribe changes in the experimental probability as the number of trials increases.
7.PS.1.dInvestigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.
7.PS.2 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on histograms.
7.PS.2.aFormulate questions that require the collection or acquisition of data with a focus on histograms.
7.PS.2.bDetermine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).
7.PS.2.cDetermine how sample size and randomness will ensure that the data collected is a sample that is representative of a larger population.
7.PS.2.dOrganize and represent numerical data using histograms with and without the use of technology.
7.PS.2.eInvestigate and explain how using different intervals could impact the representation of the data in a histogram.
7.PS.2.fCompare data represented in histograms with the same data represented in other graphs, including but not limited to line plots (dot plots), circle graphs, and stem-and-leaf plots, and justify which graphical representation best represents the data.
7.PS.2.gAnalyze data represented in histograms by making observations and drawing conclusions. Determine how histograms reveal patterns in data that cannot be easily seen by looking at the corresponding given data set.

Grade 8

8.CE Computation and Estimation
8.CE.1 The student will estimate and apply proportional reasoning and computational procedures to solve contextual problems.
8.CE.1.aEstimate and solve contextual problems that require the computation of one discount or markup and the resulting sale price.
8.CE.1.bEstimate and solve contextual problems that require the computation of the sales tax, tip and resulting total.
8.CE.1.cEstimate and solve contextual problems that require the computation of the percent increase or decrease.
8.MG Measurement and Geometry
8.MG.1 The student will use the relationships among pairs of angles that are vertical angles, adjacent angles, supplementary angles, and complementary angles to determine the measure of unknown angles.
8.MG.1.aIdentify and describe the relationship between pairs of angles that are vertical, adjacent, supplementary, and complementary.
8.MG.1.bUse the relationships among supplementary, complementary, vertical, and adjacent angles to solve problems, including those in context, involving the measure of unknown angles.
8.MG.2 The student will investigate and determine the surface area of square-based pyramids and the volume of cones and square-based pyramids.
8.MG.2.aDetermine the surface area of square-based pyramids by using concrete objects, nets, diagrams, and formulas.
8.MG.2.bDetermine the volume of cones and square-based pyramids, using concrete objects, diagrams, and formulas.
8.MG.2.cExamine and explain the relationship between the volume of cones and cylinders, and the volume of rectangular prisms and square based pyramids.
8.MG.2.dSolve problems in context involving volume of cones and square-based pyramids and the surface area of square-based pyramids.
8.MG.3 The student will apply translations and reflections to polygons in the coordinate plane.
8.MG.3.aGiven a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been translated vertically, horizontally, or a combination of both.
8.MG.3.bGiven a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been reflected over the 𝑥- or 𝑦-axis.
8.MG.3.cGiven a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been translated and reflected over the 𝑥- or 𝑦-axis or reflected over the 𝑥- or 𝑦-axis and then translated.
8.MG.3.dSketch the image of a polygon that has been translated vertically, horizontally, or a combination of both.
8.MG.3.eSketch the image of a polygon that has been reflected over the 𝑥- or 𝑦-axis.
8.MG.3.fSketch the image of a polygon that has been translated and reflected over the 𝑥- or 𝑦-axis, or reflected over the 𝑥- or 𝑦-axis and then translated.
8.MG.3.gIdentify and describe transformations in context (e.g., tiling, fabric, wallpaper designs, art).
8.MG.4 The student will apply the Pythagorean Theorem to solve problems involving right triangles, including those in context.
8.MG.4.aVerify the Pythagorean Theorem using diagrams, concrete materials, and measurement.
8.MG.4.bDetermine whether a triangle is a right triangle given the measures of its three sides.
8.MG.4.cIdentify the parts of a right triangle (the hypotenuse and the legs) given figures in various orientations.
8.MG.4.dDetermine the measure of a side of a right triangle, given the measures of the other two sides.
8.MG.4.eApply the Pythagorean Theorem, and its converse, to solve problems involving right triangles in context.
8.MG.5 The student will solve area and perimeter problems involving composite plane figures, including those in context.
8.MG.5.aSubdivide a plane figure into triangles, rectangles, squares, trapezoids, parallelograms, circles, and semicircles. Determine the area of subdivisions and combine to determine the area of the composite plane figure.
8.MG.5.bSubdivide a plane figure into triangles, rectangles, squares, trapezoids, parallelograms, and semicircles. Use the attributes of the subdivisions to determine the perimeter of the composite plane figure.
8.MG.5.cApply perimeter, circumference, and area formulas to solve contextual problems involving composite plane figures.
8.NS Number and Number Sense
8.NS.1 The student will compare and order real numbers and determine the relationships between real numbers.
8.NS.1.aEstimate and identify the two consecutive natural numbers between which the positive square root of a given number lies and justify which natural number is the better approximation. Numbers are limited to natural numbers from 1 to 400.
8.NS.1.bUse rational approximations (to the nearest hundredth) of irrational numbers to compare, order, and locate values on a number line. Radicals may include both positive and negative square roots of values from 0 to 400 yielding an irrational number.
8.NS.1.cUse multiple strategies (e.g., benchmarks, number line, equivalency) to compare and order no more than five real numbers expressed as integers, fractions (proper or improper), decimals, mixed numbers, percents, numbers written in scientific notation, radicals, and 𝜋. Radicals may include both positive and negative square roots of values from 0 to 400. Ordering may be in ascending or descending order. Justify solutions orally, in writing or with a model.
8.NS.2 The student will investigate and describe the relationship between the subsets of the real number system.
8.NS.2.aDescribe and illustrate the relationships among the subsets of the real number system by using representations (e.g., graphic organizers, number lines). Subsets include rational numbers, irrational numbers, integers, whole numbers, and natural numbers.
8.NS.2.bClassify and explain why a given number is a member of a particular subset or subsets of the real number system.
8.NS.2.cDescribe each subset of the set of real numbers and include examples and non-examples.
8.PFA Patterns, Functions, and Algebra
8.PFA.1 The student will represent, simplify, and generate equivalent algebraic expressions in one variable.
8.PFA.1.aRepresent algebraic expressions using concrete manipulatives or pictorial representations (e.g., colored chips, algebra tiles), including expressions that apply the distributive property.
8.PFA.1.bSimplify and generate equivalent algebraic expressions in one variable by applying the order of operations and properties of real numbers. Expressions may need to be expanded (using the distributive property) or require combining like terms to simplify. Expressions will include only linear and numeric terms. Coefficients and numeric terms may be rational.
8.PFA.2 The student will determine whether a given relation is a function and determine the domain and range of a function.
8.PFA.2.aDetermine whether a relation, represented by a set of ordered pairs, a table, or a graph of discrete points is a function. Sets are limited to no more than 10 ordered pairs.
8.PFA.2.bIdentify the domain and range of a function represented as a set of ordered pairs, a table, or a graph of discrete points.
8.PFA.3 The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the 𝑦-intercept (𝑏) and the coordinates of the ordered pairs in graphs will be limited to integers).
8.PFA.3.aDetermine how adding a constant (𝑏) to the equation of a proportional relationship 𝑦 = 𝑚𝑥 will translate the line on a graph.
8.PFA.3.bDescribe key characteristics of linear functions including slope (𝑚), 𝑦-intercept (𝑏), and independent and dependent variables.
8.PFA.3.cGraph a linear function given a table, equation, or a situation in context.
8.PFA.3.dCreate a table of values for a linear function given a graph, equation in the form of 𝑦 = 𝑚𝑥 + 𝑏, or context.
8.PFA.3.eWrite an equation of a linear function in the form 𝑦 = 𝑚𝑥 + 𝑏, given a graph, table, or a situation in context.
8.PFA.3.fCreate a context for a linear function given a graph, table, or equation in the form 𝑦 = 𝑚𝑥 + 𝑏.
8.PFA.4 The student will write and solve multistep linear equations in one variable, including problems in context that require the solution of a multistep linear equation in one variable.
8.PFA.4.aRepresent and solve multistep linear equations in one variable with the variable on one or both sides of the equation (up to four steps) using a variety of concrete materials and pictorial representations.
8.PFA.4.bApply properties of real numbers and properties of equality to solve multistep linear equations in one variable (up to four steps). Coefficients and numeric terms will be rational. Equations may contain expressions that need to be expanded (using the distributive property) or require combining like terms to solve.
8.PFA.4.cWrite a multistep linear equation in one variable to represent a verbal situation, including those in context.
8.PFA.4.dCreate a verbal situation in context given a multistep linear equation in one variable.
8.PFA.4.eSolve problems in context that require the solution of a multistep linear equation.
8.PFA.4.fInterpret algebraic solutions in context to linear equations in one variable.
8.PFA.4.gConfirm algebraic solutions to linear equations in one variable.
8.PFA.5 The student will write and solve multistep linear inequalities in one variable, including problems in context that require the solution of a multistep linear inequality in one variable.
8.PFA.5.aApply properties of real numbers and properties of inequality to solve multistep linear inequalities (up to four steps) in one variable with the variable on one or both sides of the inequality. Coefficients and numeric terms will be rational. Inequalities may contain expressions that need to be expanded (using the distributive property) or require combining like terms to solve.
8.PFA.5.bRepresent solutions to inequalities algebraically and graphically using a number line.
8.PFA.5.cWrite multistep linear inequalities in one variable to represent a verbal situation, including those in context.
8.PFA.5.dCreate a verbal situation in context given a multistep linear inequality in one variable.
8.PFA.5.eSolve problems in context that require the solution of a multistep linear inequality in one variable.
8.PFA.5.fIdentify a numerical value(s) that is part of the solution set of a given inequality.
8.PFA.5.gInterpret algebraic solutions in context to linear inequalities in one variable.
8.PS Probability and Statistics
8.PS.1 The student will use statistical investigation to determine the probability of independent and dependent events, including those in context.
8.PS.1.aDetermine whether two events are independent or dependent and explain how replacement impacts the probability.
8.PS.1.bCompare and contrast the probability of independent and dependent events.
8.PS.1.cDetermine the probability of two independent events.
8.PS.1.dDetermine the probability of two dependent events.
8.PS.2 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on boxplots.
8.PS.2.aFormulate questions that require the collection or acquisition of data with a focus on boxplots.
8.PS.2.bDetermine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).
8.PS.2.cDetermine how statistical bias might affect whether the data collected from the sample is representative of the larger population.
8.PS.2.dOrganize and represent a numeric data set of no more than 20 items, using boxplots, with and without the use of technology.
8.PS.2.eIdentify and describe the lower extreme (minimum), upper extreme (maximum), median, upper quartile, lower quartile, range, and interquartile range given a data set, represented by a boxplot.
8.PS.2.fDescribe how the presence of an extreme data point (outlier) affects the shape and spread of the data distribution of a boxplot.
8.PS.2.gAnalyze data represented in a boxplot by making observations and drawing conclusions.
8.PS.2.hCompare and analyze two data sets represented in boxplots.
8.PS.2.iGiven a contextual situation, justify which graphical representation (e.g., pictographs, bar graphs, line graphs, line plots/dot plots, stem-and-leaf plots, circle graphs, histograms, and boxplots) best represents the data.
8.PS.2.jIdentify components of graphical displays that can be misleading.
8.PS.3 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on scatterplots.
8.PS.3.aFormulate questions that require the collection or acquisition of data with a focus on scatterplots.
8.PS.3.bDetermine the data needed to answer a formulated question and collect the data (or acquire existing data) of no more than 20 items using various methods (e.g., observations, measurement, surveys, experiments).
8.PS.3.cOrganize and represent numeric bivariate data using scatterplots with and without the use of technology.
8.PS.3.dMake observations about a set of data points in a scatterplot as having a positive linear relationship, a negative linear relationship, or no relationship.
8.PS.3.eAnalyze and justify the relationship of the quantitative bivariate data represented in scatterplots.
8.PS.3.fSketch the line of best fit for data represented in a scatterplot.

Algebra 1

A.EI Equations and Inequalities
A.EI.1 The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.
A.EI.1.aWrite a linear equation or inequality in one variable to represent a contextual situation.
A.EI.1.bSolve multistep linear equations in one variable, including those in contextual situations, by applying the properties of real numbers and/or properties of equality.
A.EI.1.cSolve multistep linear inequalities in one variable algebraically and graph the solution set on a number line, including those in contextual situations, by applying the properties of real numbers and/or properties of inequality.
A.EI.1.dRearrange a formula or literal equation to solve for a specified variable by applying the properties of equality.
A.EI.1.eDetermine if a linear equation in one variable has one solution, no solution, or an infinite number of solutions.
A.EI.1.fVerify possible solution(s) to multistep linear equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.
A.EI.2 The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables.
A.EI.2.aCreate a system of two linear equations in two variables to represent a contextual situation.
A.EI.2.bApply the properties of real numbers and/or properties of equality to solve a system of two linear equations in two variables, algebraically and graphically.
A.EI.2.cDetermine whether a system of two linear equations has one solution, no solution, or an infinite number of solutions.
A.EI.2.dCreate a linear inequality in two variables to represent a contextual situation.
A.EI.2.eRepresent the solution of a linear inequality in two variables graphically on a coordinate plane.
A.EI.2.fCreate a system of two linear inequalities in two variables to represent a contextual situation.
A.EI.2.gRepresent the solution set of a system of two linear inequalities in two variables, graphically on a coordinate plane.
A.EI.2.hVerify possible solution(s) to a system of two linear equations, a linear inequality in two variable, or a system of two linear inequalities algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.
A.EI.3 The student will represent, solve, and interpret the solution to a quadratic equation in one variable.
A.EI.3.aSolve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems.
A.EI.3.bDetermine and justify if a quadratic equation in one variable has no real solutions, one real solution, or two real solutions.
A.EI.3.cVerify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
A.EO Expressions and Operations
A.EO.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.
A.EO.1.aTranslate between verbal quantitative situations and algebraic expressions, including contextual situations.
A.EO.1.bEvaluate algebraic expressions which include absolute value, square roots, and cube roots for given replacement values to include rational numbers, without rationalizing the denominator.
A.EO.2 The student will perform operations on and factor polynomial expressions in one variable.
A.EO.2.aDetermine sums and differences of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models.
A.EO.2.bDetermine the product of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models, the application of the distributive property, and the use of area models. The factors should be limited to five or fewer terms (e.g., (4𝑥 + 2)(3𝑥 + 5) represents four terms and (𝑥 + 1)(2𝑥² + 𝑥 + 3) represents five terms).
A.EO.2.cFactor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors.
A.EO.2.dDetermine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.
A.EO.2.eRepresent and demonstrate equality of quadratic expressions in different forms (e.g., concrete, verbal, symbolic, and graphical).
A.EO.3 The student will derive and apply the laws of exponents.
A.EO.3.aDerive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.
A.EO.3.bSimplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.
A.EO.4 The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.
A.EO.4.aSimplify and determine equivalent radical expressions involving the square root of a whole number in simplest form.
A.EO.4.bSimplify and determine equivalent radical expressions involving the cube root of an integer.
A.EO.4.cAdd, subtract, and multiply radicals, limited to numeric square and cube root expressions.
A.EO.4.dGenerate equivalent numerical expressions and justify their equivalency for radicals using rational exponents, limited to rational exponents of 1/2 and 1/3 (e.g., √5 = 5 1/2; ³√8 = 8 1/3 = (2³) 1/3 = 2).
A.F Functions
A.F.1 The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.
A.F.1.aDetermine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.
A.F.1.bInvestigate and explain how transformations to the parent function 𝑦 = 𝑥 affect the rate of change (slope) and the 𝑦-intercept of a linear function.
A.F.1.cWrite equivalent algebraic forms of linear functions, including slope-intercept form, standard form, and point-slope form, and analyze and interpret the information revealed by each form.
A.F.1.d Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations:
A.F.1.d.igiven the graph of a line;
A.F.1.d.iigiven two points on the line whose coordinates are integers;
A.F.1.d.iiigiven the slope and a point on the line whose coordinates are integers;
A.F.1.d.ivvertical lines as 𝑥 = 𝑎; and
A.F.1.d.vhorizontal lines as 𝑦 = 𝑐.
A.F.1.eWrite the equation of a line parallel or perpendicular to a given line through a given point.
A.F.1.fGraph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.
A.F.1.gFor any value, 𝑥, in the domain of 𝑓, determine 𝑓(𝑥), and determine 𝑥 given any value 𝑓(𝑥) in the range of 𝑓, given an algebraic or graphical representation of a linear function.
A.F.1.hCompare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations.
A.F.2 The student will investigate, analyze, and compare characteristics of functions, including quadratic, and exponential functions, and model quadratic and exponential relationships.
A.F.2.aDetermine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function; for relations that are functions, determine the domain and range.
A.F.2.bGiven an equation or graph, determine key characteristics of a quadratic function including 𝑥-intercepts (zeros), 𝑦-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable.
A.F.2.cGraph a quadratic function, 𝑓(𝑥), in two variables using a variety of strategies, including transformations 𝑓(𝑥) + 𝑘 and 𝑘𝑓(𝑥), where 𝑘 is limited to rational values.
A.F.2.dMake connections between the algebraic (standard and factored forms) and graphical representation of a quadratic function.
A.F.2.eGiven an equation or graph of an exponential function in the form 𝑦 = 𝑎𝑏ˣ (where 𝑏 is limited to a natural number), interpret key characteristics, including 𝑦-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable.
A.F.2.fGraph an exponential function, 𝑓(𝑥), in two variables using a variety of strategies, including transformations 𝑓(𝑥) + 𝑘 and 𝑘𝑓(𝑥), where 𝑘 is limited to rational values.
A.F.2.gFor any value, 𝑥, in the domain of 𝑓, determine 𝑓(𝑥) of a quadratic or exponential function. Determine 𝑥 given any value 𝑓(𝑥) in the range of 𝑓 of a quadratic function. Explain the meaning of 𝑥 and 𝑓(𝑥) in context.
A.F.2.hCompare and contrast the key characteristics of linear functions (𝑓(𝑥) = 𝑥), quadratic functions (𝑓(𝑥) = 𝑥²), and exponential functions (𝑓(𝑥) = 𝑏ˣ) using tables and graphs.
A.ST Statistics
A.ST.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear and quadratic functions.
A.ST.1.aFormulate investigative questions that require the collection or acquisition of bivariate data.
A.ST.1.bDetermine what variables could be used to explain a given contextual problem or situation or answer investigative questions.
A.ST.1.cDetermine an appropriate method to collect a representative sample, which could include a simple random sample, to answer an investigative question.
A.ST.1.dGiven a table of ordered pairs or a scatterplot representing no more than 30 data points, use available technology to determine whether a linear or quadratic function would represent the relationship, and if so, determine the equation of the curve of best fit.
A.ST.1.eUse linear and quadratic regression methods available through technology to write a linear or quadratic function that represents the data where appropriate and describe the strengths and weaknesses of the model.
A.ST.1.fUse a linear model to predict outcomes and evaluate the strength and validity of these predictions, including through the use of technology.
A.ST.1.gInvestigate and explain the meaning of the rate of change (slope) and 𝑦-intercept (constant term) of a linear model in context.
A.ST.1.hAnalyze relationships between two quantitative variables revealed in a scatterplot.
A.ST.1.iMake conclusions based on the analysis of a set of bivariate data and communicate the results.

Algebra 2

A2.EI Equations and Inequalities
A2.EI.1 The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.
A2.EI.1.aCreate an absolute value equation in one variable to model a contextual situation.
A2.EI.1.bSolve an absolute value equation in one variable algebraically and verify the solution graphically.
A2.EI.1.cCreate an absolute value inequality in one variable to model a contextual situation.
A2.EI.1.dSolve an absolute value inequality in one variable and represent the solution set using set notation, interval notation, and using a number line.
A2.EI.1.eVerify possible solution(s) to absolute value equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
A2.EI.2 The student will represent, solve, and interpret the solution to quadratic equations in one variable over the set of complex numbers and solve quadratic inequalities in one variable.
A2.EI.2.aCreate a quadratic equation or inequality in one variable to model a contextual situation.
A2.EI.2.bSolve a quadratic equation in one variable over the set of complex numbers algebraically.
A2.EI.2.cDetermine the solution to a quadratic inequality in one variable over the set of real numbers algebraically.
A2.EI.2.dVerify possible solution(s) to quadratic equations or inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
A2.EI.3 The student will solve a system of equations in two variables containing a quadratic expression.
A2.EI.3.aCreate a linear-quadratic or quadratic-quadratic system of equations to model a contextual situation.
A2.EI.3.bDetermine the number of solutions to a linear-quadratic and quadratic-quadratic system of equations in two variables.
A2.EI.3.cSolve a linear-quadratic and quadratic-quadratic system of equations algebraically and graphically, including situations in context.
A2.EI.3.dVerify possible solution(s) to linear-quadratic or quadratic-quadratic system of equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
A2.EI.4 The student will represent, solve, and interpret the solution to an equation containing rational algebraic expressions.
A2.EI.4.aCreate an equation containing a rational expression to model a contextual situation.
A2.EI.4.bSolve rational equations with real solutions containing factorable algebraic expressions algebraically and graphically. Algebraic expressions should be limited to linear and quadratic expressions.
A2.EI.4.cVerify possible solution(s) to rational equations algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
A2.EI.4.dJustify why a possible solution to an equation containing a rational expression might be extraneous.
A2.EI.5 The student will represent, solve, and interpret the solution to an equation containing a radical expression.
A2.EI.5.aSolve an equation containing no more than one radical expression algebraically and graphically.
A2.EI.5.bVerify possible solution(s) to radical equations algebraically, graphically, and with technology, to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.
A2.EI.5.cJustify why a possible solution to an equation with a square root might be extraneous.
A2.EI.6 The student will represent, solve, and interpret the solution to a polynomial equation.
A2.EI.6.aDetermine a factored form of a polynomial equation, of degree three or higher, given its zeros or the 𝑥-intercepts of the graph of its related function.
A2.EI.6.bDetermine the number and type of solutions (real or imaginary) of a polynomial equation of degree three or higher.
A2.EI.6.cSolve a polynomial equation over the set of complex numbers.
A2.EI.6.dVerify possible solution(s) to polynomial equations of degree three or higher algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions in context.
A2.EO Expressions and Operations
A2.EO.1 The student will perform operations on and simplify rational expressions.
A2.EO.1.aAdd, subtract, multiply, or divide rational algebraic expressions, simplifying the result.
A2.EO.1.bJustify and determine equivalent rational algebraic expressions with monomial and binomial factors. Algebraic expressions should be limited to linear and quadratic expressions.
A2.EO.1.cRecognize a complex algebraic fraction and simplify it as a product or quotient of simple algebraic fractions.
A2.EO.1.dRepresent and demonstrate equivalence of rational expressions written in different forms.
A2.EO.2 The student will perform operations on and simplify radical expressions.
A2.EO.2.aSimplify and determine equivalent radical expressions that include numeric and algebraic radicands.
A2.EO.2.bAdd, subtract, multiply, and divide radical expressions that include numeric and algebraic radicands, simplifying the result. Simplification may include rationalizing the denominator.
A2.EO.2.cConvert between radical expressions and expressions containing rational exponents.
A2.EO.3 The student will perform operations on polynomial expressions and factor polynomial expressions in one and two variables.
A2.EO.3.aDetermine sums, differences, and products of polynomials in one and two variables.
A2.EO.3.bFactor polynomials completely in one and two variables with no more than four terms over the set of integers.
A2.EO.3.cDetermine the quotient of polynomials in one and two variables, using monomial, binomial, and factorable trinomial divisors.
A2.EO.3.dRepresent and demonstrate equality of polynomial expressions written in different forms and verify polynomial identities including the difference of squares, sum and difference of cubes, and perfect square trinomials.
A2.EO.4 The student will perform operations on complex numbers.
A2.EO.4.aExplain the meaning of 𝑖.
A2.EO.4.bIdentify equivalent radical expressions containing negative rational numbers and expressions in 𝑎 + 𝑏𝑖 form.
A2.EO.4.cApply properties to add, subtract, and multiply complex numbers.
A2.F Functions
A2.F.1 The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.
A2.F.1.aDistinguish between the graphs of parent functions for square root, cube root, rational, exponential, and logarithmic function families.
A2.F.1.bWrite the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including 𝑓(𝑥) + 𝑘; 𝑓(𝑘𝑥); 𝑓(𝑥 + 𝑘); and 𝑘𝑓(𝑥), where 𝑘 is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation.
A2.F.1.cGraph a square root, cube root, rational, exponential, and logarithmic function, given the equation, using transformations of the parent function including 𝑓(𝑥) + 𝑘; 𝑓(𝑘𝑥); 𝑓(𝑥 + 𝑘); and 𝑘𝑓(𝑥), where 𝑘 is limited to rational values. Use technology to verify transformations of the functions.
A2.F.1.dDetermine when two variables are directly proportional, inversely proportional, or neither, given a table of values. Write an equation and create a graph to represent a direct or inverse variation, including situations in context.
A2.F.1.eCompare and contrast the graphs, tables, and equations of square root, cube root, rational, exponential, and logarithmic functions.
A2.F.2 The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.
A2.F.2.aDetermine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.
A2.F.2.bCompare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions.
A2.F.2.cDetermine the intervals on which the graph of a function is increasing, decreasing, or constant.
A2.F.2.dDetermine the location and value of absolute (global) maxima and absolute (global) minima of a function.
A2.F.2.eDetermine the location and value of relative (local) maxima or relative (local) minima of a function.
A2.F.2.fFor any value, 𝑥, in the domain of 𝑓, determine 𝑓(𝑥) using a graph or equation. Explain the meaning of 𝑥 and 𝑓(𝑥) in context, where applicable.
A2.F.2.gDescribe the end behavior of a function.
A2.F.2.hDetermine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic).
A2.F.2.iDetermine the inverse of a function algebraically and graphically, given the equation of a linear or quadratic function (linear, quadratic, and square root). Justify and explain why two functions are inverses of each other.
A2.F.2.jGraph the inverse of a function as a reflection over the line 𝑦 = 𝑥.
A2.F.2.kDetermine the composition of two functions algebraically and graphically.
A2.ST Statistics
A2.ST.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on univariate quantitative data represented by a smooth curve, including a normal curve.
A2.ST.1.aFormulate investigative questions that require the collection or acquisition of a large set of univariate quantitative data or summary statistics of a large set of univariate quantitative data and investigate questions using a data cycle.
A2.ST.1.bCollect or acquire univariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires.
A2.ST.1.cExamine the shape of a data set (skewed versus symmetric) that can be represented by a histogram, and sketch a smooth curve to model the distribution.
A2.ST.1.dIdentify the properties of a normal distribution.
A2.ST.1.eDescribe and interpret a data distribution represented by a smooth curve by analyzing measures of center, measures of spread, and shape of the curve.
A2.ST.1.fCalculate and interpret the 𝑧-score for a value in a data set.
A2.ST.1.gCompare two data points from two different distributions using 𝑧-scores.
A2.ST.1.hDetermine the solution to problems involving the relationship of the mean, standard deviation, and 𝑧-score of a data set represented by a smooth or normal curve.
A2.ST.1.iApply the Empirical Rule to answer investigative questions.
A2.ST.1.jCompare multiple data distributions using measures of center, measures of spread, and shape of the distributions.
A2.ST.2 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear, quadratic, exponential, or a combination of these functions.
A2.ST.2.aFormulate investigative questions that require the collection or acquisition of bivariate data and investigate questions using a data cycle.
A2.ST.2.bCollect or acquire bivariate data through research, or using surveys, observations, scientific experiments, polls, or questionnaires.
A2.ST.2.cRepresent bivariate data with a scatterplot using technology.
A2.ST.2.dDetermine whether the relationship between two quantitative variables is best approximated by a linear, quadratic, exponential, or a combination of these functions.
A2.ST.2.eDetermine the equation(s) of the function(s) that best models the relationship between two variables using technology. Curves of best fit may include a combination of linear, quadratic, or exponential (piecewise-defined) functions.
A2.ST.2.fUse the correlation coefficient to designate the goodness of fit of a linear function using technology.
A2.ST.2.gMake predictions, decisions, and critical judgments using data, scatterplots, or the equation(s) of the mathematical model.
A2.ST.2.hEvaluate the reasonableness of a mathematical model of a contextual situation.
A2.ST.3 The student will compute and distinguish between permutations and combinations.
A2.ST.3.aCompare and contrast permutations and combinations to count the number of ways that events can occur.
A2.ST.3.bCalculate the number of permutations of 𝑛 objects taken 𝑟 at a time.
A2.ST.3.cCalculate the number of combinations of 𝑛 objects taken 𝑟 at a time.
A2.ST.3.dUse permutations and combinations as counting techniques to solve contextual problems.
A2.ST.3.eCalculate and verify permutations and combinations using technology.

Algebra, Functions, and Data Analysis

AFDA.AF Algebra and Functions
AFDA.AF.1 The student will investigate, analyze, and compare linear, quadratic, and exponential function families, algebraically and graphically, using transformations.
AFDA.AF.1.aIdentify graphs and equations of parent functions for linear, quadratic, and exponential function families.
AFDA.AF.1.bDescribe the transformation from the parent function given the equation or the graph of the function.
AFDA.AF.1.cDetermine and analyze whether a linear, quadratic, or exponential function best models a given representation, including those in context.
AFDA.AF.1.dWrite the equation of a linear, quadratic, or exponential function, given a graph, using transformations of the parent function.
AFDA.AF.1.eUse a graphical or algebraic representation of a function to solve problems within a context, graphically and algebraically, when appropriate.
AFDA.AF.1.fGraph a function given the equation of a function, using transformations of the parent function. Use technology to verify transformations of functions.
AFDA.AF.1.gCompare and contrast linear, quadratic, and exponential functions using multiple representations (e.g., graphs, tables, equations, verbal descriptions).
AFDA.AF.2 The student will investigate and analyze characteristics of the graphs of linear, quadratic, exponential, and piecewise-defined functions.
AFDA.AF.2.aDetermine the domain and range of a function given a graphical representation, including those limited by contexts.
AFDA.AF.2.bIdentify intervals on a graph for which a function is increasing, decreasing, or constant.
AFDA.AF.2.cGiven a graph, identify the location and value of the absolute maximum and absolute minimum of a function over the domain of a function.
AFDA.AF.2.dGiven a graph, determine the zeros and intercepts of a function.
AFDA.AF.2.eDescribe and recognize the connection between points on the graph and the value of a function.
AFDA.AF.2.fDescribe the end behavior of a function given its graph.
AFDA.AF.2.gIdentify horizontal and/or vertical asymptotes from the graph of a function, if they exist.
AFDA.AF.2.hDescribe and relate the characteristics of the graphs of linear, quadratic, exponential, and piecewise-defined functions, including those in contextual situations.
AFDA.AF.3 The student will represent and interpret contextual situations with constraints that require optimization using linear programming techniques, including systems of linear equations or inequalities, solving graphically and when appropriate, algebraically.
AFDA.AF.3.aRepresent and interpret contextual problems requiring optimization with systems of linear equations or inequalities.
AFDA.AF.3.bSolve systems of no more than four equations or inequalities graphically and when appropriate, algebraically.
AFDA.AF.3.cIdentify the feasible region of a system of linear inequalities.
AFDA.AF.3.dIdentify the coordinates of the vertices of a feasible region.
AFDA.AF.3.eDetermine and describe the maximum or minimum value for the function defined over a feasible region.
AFDA.AF.3.fInterpret the validity of possible solution(s) algebraically, graphically, using technology, and in context and justify the reasonableness of the answer(s) or the solution method in context.
AFDA.DA Data Analysis
AFDA.DA.1 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear, quadratic, and exponential functions.
AFDA.DA.1.aFormulate investigative questions that require the collection or acquisition of bivariate data, where exactly two of the variables are quantitative.
AFDA.DA.1.bCollect or acquire bivariate data from a representative sample to answer an investigative question.
AFDA.DA.1.cRepresent bivariate data with a scatterplot using technology and describe how the variables are related in terms of the given context.
AFDA.DA.1.dMake predictions, decisions, and critical judgments using data, scatterplots, or the equation(s) of the mathematical model.
AFDA.DA.2 The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on the design and implementation of an experiment and/or observational study.
AFDA.DA.2.aFormulate questions that can be addressed with data and assess the type of data relevant to the question (e.g., quantitative versus categorical).
AFDA.DA.2.bInvestigate, describe, and determine best sampling techniques, such as simple random sampling, stratified sampling, and cluster sampling.
AFDA.DA.2.cPlan and conduct an experiment and/or observational study. The experimental design should address control, randomization, and minimization of experimental error.
AFDA.DA.2.dCollect or acquire data to answer a statistical question.
AFDA.DA.2.eRecognize that data may contain errors, have missing values, or may be biased, and make decisions about how to account for these issues.
AFDA.DA.2.fIdentify biased sampling methods.
AFDA.DA.2.gGiven a plan for an observational study, identify possible sources of bias, and describe ways to reduce bias.
AFDA.DA.2.hSelect, create, and use appropriate visual representations of data to brainstorm solutions.
AFDA.DA.2.iUse appropriate statistical methods to analyze data.
AFDA.DA.2.jCommunicate the description of an experiment and/or observational study, the resulting data, analysis, and the validity of the conclusions.
AFDA.DA.3 The student will calculate and interpret probabilities, including those in contextual situations.
AFDA.DA.3.aAnalyze, interpret, and make predictions based on theoretical probability.
AFDA.DA.3.bCalculate conditional probabilities for dependent, independent, and mutually exclusive events.
AFDA.DA.3.cRepresent and calculate probabilities using Venn diagrams, probability trees, organized lists, two-way tables, simulations, or other probability models.
AFDA.DA.3.dInterpret probabilities from simulations or experiments to make informed decisions and justify the rationale.
AFDA.DA.3.eDefine and give contextual examples of complementary, dependent, independent, and mutually exclusive events.
AFDA.DA.3.fGiven two or more events in a problem setting, determine whether the events are complementary, dependent, independent, and/or mutually exclusive.
AFDA.DA.3.gCompare and contrast permutations and combinations, including those in contextual situations.
AFDA.DA.3.hCalculate the number of permutations of 𝑛 objects taken 𝑟 at a time, without repetition.
AFDA.DA.3.iCalculate the number of combinations of 𝑛 objects taken 𝑟 at a time, without repetition.
AFDA.DA.4 The student will describe and apply the properties of normal distribution, including those in contextual situations.
AFDA.DA.4.aIdentify and describe the properties of a normal distribution.
AFDA.DA.4.bDetermine when the normal distribution is a reasonable representation of the data.
AFDA.DA.4.cDescribe how the mean and the standard deviation affect the graph of the normal distribution.
AFDA.DA.4.dCalculate and interpret the 𝑧-score for a data point, given the mean and the standard deviation.
AFDA.DA.4.eCompare two sets of normally distributed data using a standard normal distribution and 𝑧-scores, given the mean and the standard deviation.
AFDA.DA.4.fRepresent probability as the area under the curve of a standard normal distribution.
AFDA.DA.4.gDetermine probabilities associated with areas under the standard normal curve, using technology or a table of Standard Normal Probabilities.
AFDA.DA.4.hInvestigate, represent, and determine relationships between a normally distributed data set and its descriptive statistics.

Computer Mathematics

CM.AP Applications of Programming
CM.AP.1 The student will write and implement programs using sequencing, selection, and iteration to perform a specific task or solve a problem, including those arising from mathematical and interdisciplinary contexts.
CM.AP.1.aDetermine what components of programming are needed to implement a step-by-step plan to perform a specific task or solve a problem.
CM.AP.1.bWrite a computer program that includes sequencing, selection (conditionals), and iteration (loops).
CM.AP.1.c Write and implement computer programs to solve mathematical problems using
CM.AP.1.c.iformulas and equations;
CM.AP.1.c.iifunctions;
CM.AP.1.c.iiiprobability and statistics; and
CM.AP.1.c.ivdata-analysis.
CM.AP.2 The student will create documentation using written comments to annotate the intended purpose of the components of a user-created program.
CM.AP.2.a Create documentation using written comments to:
CM.AP.2.a.idescribe the overall purpose of a program;
CM.AP.2.a.iialign a previously created step-by-step plan to a written program;
CM.AP.2.a.iiidescribe pre-conditions and post-conditions; and
CM.AP.2.a.ivimprove the readability of a program.
CM.AP.3 The student will verify how programs access and process variables.
CM.AP.3.aVerify that the variable types are aligned to the purpose of the algorithm.
CM.AP.3.bVerify that global variables are set to constant values before run time.
CM.AP.3.cDifferentiate between the scopes of variables (e.g., global scope versus local scope) and verify the intended use.
CM.AP.4 The student will translate a mathematical expression or statement into computer code.
CM.AP.4.aDeclare, initialize, and assign variables to represent mathematical expressions or statements.
CM.AP.4.bImplement order of operations, including logical and relational operators.
CM.AP.4.cTranslate a mathematical expression or statement into a programming statement(s).
CM.AP.5 The student will trace existing code to interpret the intended purpose.
CM.AP.5.a Trace existing code of an algorithm to
CM.AP.5.a.iidentify values at each stage of an algorithm; and
CM.AP.5.a.iipredict return values of functions given specific arguments.
CM.AP.5.bUse tracing to describe the intended purpose of existing code for an algorithm.
CM.CP Components of Programming
CM.CP.1 The student will design a step-by-step plan to perform a task or solve a problem, including those arising from mathematical or interdisciplinary contexts.
CM.CP.1.aDesign a step-by-step plan to perform a task or solve a problem using a flowchart or pseudocode that outlines the subtasks needed.
CM.CP.1.bDefine the variables needed to perform a task or solve a problem.
CM.CP.1.cDefine the constraints of a task or problem (e.g., pre-conditions, post-conditions) to determine the desired input and output.
CM.CP.2 The student will construct Boolean expressions and implement conditional statements.
CM.CP.2.aWrite and implement Boolean expressions using logical and relational operators (e.g., !, &&, ||, ==, <, >, >=, <=, !=).
CM.CP.2.bWrite and implement “if” conditional statements.
CM.CP.2.cWrite and implement “if/else” conditional statements.
CM.CP.2.dWrite and implement compound conditional statements (e.g., nested conditionals, chained conditional statements).
CM.CP.2.eDetermine which parts of an algorithm are executed based on a condition being true or false.
CM.CP.3 The student will perform iteration with loops.
CM.CP.3.aWrite and implement “while” and “for” loops.
CM.CP.3.bDifferentiate between loops that run a fixed number of times and loops that run an indefinite number of times (e.g., stopping dependent on variable conditions).
CM.CP.3.cIdentify conditions that cause infinite loops.
CM.CP.3.dDetermine the outcome of code segments that include loops.
CM.CP.4 The student will write and implement the output phase of a computer program.
CM.CP.4.a Write and implement the output phase of a computer program, which may include:
CM.CP.4.a.iformatting output in text-based environments;
CM.CP.4.a.iidisplaying output through a graphical user interface; and
CM.CP.4.a.iiisending output to a physical device (e.g., speakers, robots, LED lights).
CM.CP.4.bWrite output to a file.
CM.CP.5 The student will write and implement the input phase of a computer program.
CM.CP.5.aWrite and implement input statements to store user given values into a program.
CM.CP.5.bValidate input data using exception coding (e.g., using a “while” loop to control valid input by a user).
CM.CP.5.cDetermine what output a program will produce given a specific input.
CM.CP.6 The student will implement library functions.
CM.CP.6.aImplement library functions to process data.
CM.CP.6.bImplement library functions to perform mathematical operations (e.g., random, absolute value, square root, power).
CM.CP.6.cImplement void library functions and return library functions.
CM.CP.6.dImplement overloaded library functions.
CM.CP.7 The student will write and implement user-defined functions.
CM.CP.7.aWrite and implement a user-defined function to complete a task or sub-task.
CM.CP.7.bWrite and implement void functions and return functions.
CM.CP.7.cWrite and implement functions that accept parameters.
CM.CP.8 The student will implement pre-defined algorithms, including search routines and sort routines.
CM.CP.8.aDifferentiate between types of search routines.
CM.CP.8.bDifferentiate between types of sort routines.
CM.CP.8.cImplement pre-defined algorithms.
CM.CP.8.dImplement a search routine on a one-dimensional list or an array, including sequential search and binary search.
CM.CP.8.eImplement a sort routine on a one-dimensional list or an array (e.g., selection sort, insertion sort, merge sort).
CM.DRS Data Representation and Storage
CM.DRS.1 The student will represent data and convert data between different number systems.
CM.DRS.1.aRepresent data in different number systems, including binary, decimal, and hexadecimal.
CM.DRS.1.bConvert data between number systems (e.g., binary to decimal, decimal to hexadecimal).
CM.DRS.2 The student will differentiate between variable data types based upon their characteristics.
CM.DRS.2.a Describe the characteristics of different variable data types, including
CM.DRS.2.a.iBoolean;
CM.DRS.2.a.iicharacter;
CM.DRS.2.a.iiiinteger;
CM.DRS.2.a.ivdecimal (double/float); and
CM.DRS.2.a.vstring.
CM.DRS.2.bDifferentiate between variable data types to determine the data type needed based upon intended use (e.g., character versus string, integer versus double/float).
CM.DRS.3 The student will represent data using appropriate data structures.
CM.DRS.3.aGiven a specific task or problem, determine the appropriate data structure (e.g., lists, arrays, objects) to represent data.
CM.DRS.3.b Perform tasks related to lists or arrays (one-dimensional or two-dimensional), including
CM.DRS.3.b.ideclare a list or array (one-dimensional or two-dimensional);
CM.DRS.3.b.iichoose an appropriate data type for a list or an array; and
CM.DRS.3.b.iiifill the list or array with data.
CM.DRS.3.cAccess and manipulate a particular element of a list or an array.
CM.DRS.3.dImplement predefined objects to consolidate related information of different data types.
CM.EP Evaluation of Programming
CM.EP.1 The student will test a program to match a sample output, using a set of data.
CM.EP.1.aProduce a given output by entering a data set.
CM.EP.1.bTest a program including boundary cases and inaccurate data types to verify the intended outcomes.
CM.EP.2 The student will identify errors and debug a program using various techniques.
CM.EP.2.aDifferentiate among syntax errors, runtime errors, and logic errors.
CM.EP.2.b Debug a program using various techniques:
CM.EP.2.b.iinterpret syntax and runtime error messages;
CM.EP.2.b.iiplace controlled breaks;
CM.EP.2.b.iiioutput intermediate results;
CM.EP.2.b.ivdisable a section of code by converting it into a comment;
CM.EP.2.b.vtrace code to identify logic errors; and
CM.EP.2.b.viuse debugging tools available in the programming environment.
CM.EP.3 The student will compare and contrast the efficiency of computer programs.
CM.EP.3.a Compare and contrast the efficiency of computer programs in terms of
CM.EP.3.a.icomplexity of algorithms with the same intended outcomes;
CM.EP.3.a.iimemory space used; and
CM.EP.3.a.iiirun time.

Discrete Mathematics

DM.CM Computational Methods
DM.CM.1 The student will describe and apply sorting and searching algorithms used in processing and communicating information.
DM.CM.1.aSelect and apply a sorting algorithm, such as a bubble sort, merge sort, or network sort.
DM.CM.1.bDescribe the advantages and disadvantages of various sorting algorithms.
DM.CM.1.cAnalyze the knapsack and bin-packing problems.
DM.CM.1.dSelect and apply search algorithms to analyze problems.
DM.CM.1.eDetermine the average, best-case, and worst-case reasoning for different searches.
DM.CM.2 The student will use recursive processes.
DM.CM.2.aCompare and contrast iterative and recursive processes.
DM.CM.2.bUse recursive processes to model growth and decay.
DM.CM.2.cUse recursive processes to create fractals.
DM.CM.2.dUse recursive processes to generate the Fibonacci sequence.
DM.CM.2.eDetermine if a recursive solution is more efficient than an iterative solution.
DM.CM.3 The student will identify and apply cryptographic methods.
DM.CM.3.aCompare and contrast ciphers and codes.
DM.CM.3.bDescribe the evolution of cipher systems.
DM.CM.3.cIdentify the Fundamental Theorem of Arithmetic.
DM.CM.3.dDescribe how the complexity of prime factorization is used in cryptography.
DM.CM.3.eDescribe modular arithmetic in context (e.g., clocks, days of the week, measures of time).
DM.CM.3.fAnalyze the relationship between divisibility and modulus.
DM.CM.3.gDetermine congruence within modular arithmetic.
DM.CM.3.hPerform operations within modular arithmetic.
DM.CM.3.iApply modular arithmetic to problems in context (e.g., cryptography, International Standard Book Number (ISBN), International Bank Account Number (IBAN)).
DM.CM.4 The student will analyze the limitations of algorithms and their contextual relationships in computing.
DM.CM.4.aDescribe maximum complexity of an algorithm using Big O notation.
DM.CM.4.bDescribe Turing machines and how they are used to test the limits of computation.
DM.CM.4.cDescribe the halting problem and explain how it characterizes the fundamental limitations of computation and undecidability.
DM.CM.4.dExplain the P versus NP problem and defend a justification for equality, inequality, or undecidability.
DM.CM.4.eAnalyze how the equivalence of P- and NP-class problems might impact society.
DM.GT Graph Theory
DM.GT.1 The student will represent problems using vertex-edge graphs. The concepts of degree, connectedness, paths, planarity, and directed graphs will be analyzed.
DM.GT.1.aIllustrate the basic terminology of graph theory (e.g., vertex, edge, graph, degree of a vertex).
DM.GT.1.bUse graphs to map situations in which the vertices represent objects, and edges represent a particular relationship between objects.
DM.GT.1.cIdentify and describe degree and connectedness.
DM.GT.1.dDetermine whether a graph is planar or nonplanar.
DM.GT.1.eAnalyze the relationship between faces, edges, and vertices using Euler’s formula (𝐹 = 𝐸 – 𝑉 + 2).
DM.GT.1.fUse directed graphs (digraphs) to represent situations with restrictions in traversal possibilities.
DM.GT.1.gDetermine when graphs are trees.
DM.GT.2 The student will solve problems through analysis and application of circuits, cycles, Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Optimal solutions will be determined using existing algorithms and student-created algorithms.
DM.GT.2.aDetermine whether a graph has an Euler circuit or path, and determine the circuit or path, if it exists.
DM.GT.2.bDetermine whether a graph has a Hamilton circuit or path, and determine the circuit or path, if it exists.
DM.GT.2.cCount the number of Hamilton circuits for a complete graph with 𝑛 vertices.
DM.GT.2.dUse an Euler circuit algorithm to solve optimization problems.
DM.GT.3 The student will apply graphs to conflict-resolution problems, such as graph coloring, scheduling, matching, and optimization.
DM.GT.3.aModel projects consisting of several subtasks, using a graph.
DM.GT.3.bUse graphs to resolve conflicts that arise in scheduling.
DM.GT.3.cUse graph coloring to determine the chromatic number of a graph.
DM.GT.4 The student will recognize and apply algorithms to solve configuration, conflict resolution, and sorting problems.
DM.GT.4.aRecognize algorithms such as nearest neighbor, brute force, and cheapest link as they apply to graphs.
DM.GT.4.bUse Kruskal’s algorithm to determine the shortest spanning tree of a connected graph.
DM.GT.4.cUse Prim’s algorithm to determine the shortest spanning tree of a connected graph.
DM.GT.4.dUse Dijkstra’s algorithm to determine the shortest spanning tree of a connected graph.
DM.GT.5 The student will use algorithms to schedule tasks to determine a minimum project time.
DM.GT.5.aSpecify in a digraph the order in which tests are to be performed.
DM.GT.5.bIdentify the critical path to determine the earliest completion time (minimum project time).
DM.GT.5.cUse the list-processing algorithm to determine an optimal schedule.
DM.GT.5.dCreate and test scheduling algorithms.
DM.LR Logical Reasoning
DM.LR.1 The student will use reasoning to develop and apply logical arguments.
DM.LR.1.aUse Venn diagrams to codify and solve logic problems.
DM.LR.1.bExpress logical statements in symbolic form.
DM.LR.1.cRepresent a conditional statement as its converse, inverse, and contrapositive.
DM.LR.1.dDescribe how symbolic logic can be used to map the processes of computer applications.
DM.LR.1.eConstruct a truth table to display all possible input combinations and their outputs.
DM.LR.1.fIdentify the rules of inference and model basic logical statements including De Morgan’s Law.
DM.LR.1.gApply logical reasoning to model contextual situations and make decisions.
DM.LR.2 The student will apply logic and proof techniques in the construction of a sound argument.
DM.LR.2.aApply informal logical reasoning to contextual problems (e.g., predicting the behavior of software, solving puzzles).
DM.LR.2.bOutline the basic structure of a proof technique (e.g., direct proof, proof by contradiction, induction).
DM.LR.2.cDeduce the best type of proof for a given problem.
DM.LR.2.dUse the rules of inference to construct direct proofs and proofs by contradiction.
DM.LR.2.eConstruct induction proofs involving summations and inequalities.
DM.LR.2.fUse a truth table to prove the logical equivalence of statements.
DM.LR.3 The student will apply Boolean algebra to represent and analyze the function of logical gates and circuits.
DM.LR.3.aExplain basic properties of Boolean algebra: duality, complements, and standard forms.
DM.LR.3.bRepresent verbal statements as Boolean expressions.
DM.LR.3.cApply Boolean algebra to prove identities and simplify expressions.
DM.LR.3.dGenerate truth tables that encode the truth and falsity of two or more statements.
DM.LR.3.eExplain the operation of discrete logic gates.
DM.LR.3.fDescribe the relationship between Boolean algebra and electronic circuits.
DM.LR.3.gAnalyze a combinational network using Boolean expressions.
DM.LR.3.hDesign simple combinational networks that use NAND (AND followed by NOT), NOR (OR followed by NOT), and XOR (exclusive-OR) gates.
DM.LR.4 The student will use mathematical induction to prove formulas and mathematical statements.
DM.LR.4.aCompare and contrast inductive and deductive reasoning.
DM.LR.4.bExplain the relationship between weak and strong induction.
DM.LR.4.cConstruct induction proofs involving a divisibility argument.
DM.LR.4.dProve the Binomial Theorem through mathematical induction.
DM.SNT Set and Number Theory
DM.SNT.1 The student will identify and use the properties of sets and set operations.
DM.SNT.1.aCompare and contrast sets, relations, and functions.
DM.SNT.1.bExpress relationships between sets using Venn diagrams.
DM.SNT.1.cDescribe a set using set-builder notation.
DM.SNT.1.dConstruct new sets using the set operations intersection, union, difference, and complement.
DM.SNT.1.eIdentify the laws of set theory (e.g., associative, commutative, distributive, De Morgan’s Law).
DM.SNT.1.fUse the principle of inclusion and exclusion to determine the size of a set.
DM.SNT.1.gUse the properties of set operations to prove set equality.
DM.SNT.2 The student will apply the formulas of combinatorics.
DM.SNT.2.aCreate a tree diagram to represent relationships between independent events.
DM.SNT.2.bUse the Fundamental (Basic) Counting Principle to determine the number of possible outcomes of an event.
DM.SNT.2.cDetermine the number of combinations possible when subsets of 𝑟 elements are selected from a set of 𝑛 elements without regard to order.
DM.SNT.2.dDetermine the number of permutations possible when 𝑟 objects selected from 𝑛 objects are ordered.
DM.SNT.2.eUse the pigeonhole principle to solve packing problems to facilitate proofs.
DM.SNT.2.fConstruct a proof by induction using principles of combinatorics.
DM.SNT.3 The student will use Pascal’s Triangle to analyze numerical patterns and relationships.
DM.SNT.3.aConstruct Pascal’s Triangle.
DM.SNT.3.bExpand binomials having positive integral exponents, using the Binomial Theorem and Pascal’s Triangle.
DM.SNT.3.cCompare the binomial coefficient to the calculation of combinations.
DM.SNT.3.dIdentify the Fibonacci numbers within Pascal’s Triangle.

Data Science

Geometry

G.DF Two- and Three-Dimensional Figures
G.DF.1 The student will create models and solve problems, including those in context, involving surface area and volume of rectangular and triangular prisms, cylinders, cones, pyramids, and spheres.
G.DF.1.aIdentify the shape of a two-dimensional cross section of a three-dimensional figure.
G.DF.1.bCreate models and solve problems, including those in context, involving surface area of three-dimensional figures, as well as composite three-dimensional figures.
G.DF.1.cSolve multistep problems, including those in context, involving volume of three-dimensional figures, as well as composite three-dimensional figures.
G.DF.1.dDetermine unknown measurements of three-dimensional figures using information such as length of a side, area of a face, or volume.
G.DF.2 The student will determine the effect of changing one or more dimensions of a three-dimensional geometric figure and describe the relationship between the original and changed figure.
G.DF.2.aDescribe how changes in one or more dimensions of a figure affect other derived measures (perimeter, area, total surface area, and volume) of the figure.
G.DF.2.bDescribe how changes in surface area and/or volume of a figure affect the measures of one or more dimensions of the figure.
G.DF.2.cSolve problems, including those in context, involving changing the dimensions or derived measures of a three-dimensional figure.
G.DF.2.dCompare ratios between side lengths, perimeters, areas, and volumes of similar figures.
G.DF.2.eRecognize when two- and three-dimensional figures are similar and solve problems, including those in context, involving attributes of similar geometric figures.
G.PC Polygons and Circles
G.PC.1 The student will prove and justify theorems and properties of quadrilaterals, and verify and use properties of quadrilaterals to solve problems, including the relationships between the sides, angles, and diagonals.
G.PC.1.aSolve problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.
G.PC.1.bProve and justify that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the slope formula, the distance formula, and the midpoint formula.
G.PC.1.cProve and justify theorems and properties of quadrilaterals using deductive reasoning.
G.PC.1.dUse congruent segment, congruent angle, angle bisector, perpendicular line, and/or parallel line constructions to verify properties of quadrilaterals.
G.PC.2 The student will verify relationships and solve problems involving the number of sides and measures of angles of convex polygons.
G.PC.2.aSolve problems involving the number of sides of a regular polygon given the measures of the interior and exterior angles of the polygon.
G.PC.2.bJustify the relationship between the sum of the measures of the interior and exterior angles of a convex polygon and solve problems involving the sum of the measures of the angles.
G.PC.2.cJustify the relationship between the measure of each interior and exterior angle of a regular polygon and solve problems involving the measures of the angles.
G.PC.3 The student will solve problems, including those in context, by applying properties of circles.
G.PC.3.aDetermine the proportional relationship between the arc length or area of a sector and other parts of a circle.
G.PC.3.bSolve for arc measures and angles in a circle formed by central angles.
G.PC.3.cSolve for arc measures and angles in a circle involving inscribed angles.
G.PC.3.dCalculate the length of an arc of a circle.
G.PC.3.eCalculate the area of a sector of a circle.
G.PC.3.fApply arc length or sector area to solve for an unknown measurement of the circle including the radius, diameter, arc measure, central angle, arc length, or sector area.
G.PC.4 The student will solve problems in the coordinate plane involving equations of circles.
G.PC.4.aDerive the equation of a circle of given the center and radius using the Pythagorean Theorem.
G.PC.4.b Solve problems in the coordinate plane involving equations of circles:
G.PC.4.b.igiven a graph or the equation of a circle in standard form, identify the coordinates of the center of the circle;
G.PC.4.b.iigiven the coordinates of the endpoints of a diameter of a circle, determine the coordinates of the center of the circle.
G.PC.4.b.iiigiven a graph or the equation of a circle in standard form, identify the length of the radius or diameter of the circle.
G.PC.4.b.ivgiven the coordinates of the endpoints of the diameter of a circle, determine the length of the radius or diameter of the circle.
G.PC.4.b.vgiven the coordinates of the center and the coordinates of a point on the circle, determine the length of the radius or diameter of the circle; and
G.PC.4.b.vigiven the coordinates of the center and length of the radius of a circle, identify the coordinates of a point(s) on the circle.
G.PC.4.c Determine the equation of a circle given:
G.PC.4.c.ia graph of a circle with a center with coordinates that are integers;
G.PC.4.c.iicoordinates of the center and a point on the circle;
G.PC.4.c.iiicoordinates of the center and the length of the radius or diameter; and
G.PC.4.c.ivcoordinates of the endpoints of a diameter.
G.RLT Reasoning, Lines and Transformations
G.RLT.1 The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams.
G.RLT.1.aTranslate propositional statements and compound statements into symbolic form, including negations (~𝑝, read “not 𝑝”), conjunctions (𝑝 ∧ 𝑞, read “𝑝 and 𝑞”), disjunctions (𝑝 ∨ 𝑞, read “𝑝 or 𝑞”), conditionals (𝑝 → 𝑞, read “if 𝑝 then 𝑞”), and biconditionals (𝑝 ↔ 𝑞, read “𝑝 if and only if 𝑞”), including statements representing geometric relationships.
G.RLT.1.bIdentify and determine the validity of the converse, inverse, and contrapositive of a conditional statement, and recognize the connection between a biconditional statement and a true conditional statement with a true converse, including statements representing geometric relationships.
G.RLT.1.cUse Venn diagrams to represent set relationships, including union, intersection, subset, and negation.
G.RLT.1.dInterpret Venn diagrams, including those representing contextual situations.
G.RLT.2 The student will analyze, prove, and justify the relationships of parallel lines cut by a transversal.
G.RLT.2.a Prove and justify angle pair relationships formed by two parallel lines and a transversal, including:
G.RLT.2.a.icorresponding angles;
G.RLT.2.a.iialternate interior angles;
G.RLT.2.a.iiialternate exterior angles;
G.RLT.2.a.ivsame-side (consecutive) interior angles; and
G.RLT.2.a.vsame-side (consecutive) exterior angles.
G.RLT.2.bProve two or more lines are parallel given angle measurements expressed numerically or algebraically.
G.RLT.2.cSolve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal.
G.RLT.3 The student will solve problems, including contextual problems, involving symmetry and transformation.
G.RLT.3.aLocate, count, and draw lines of symmetry given a figure, including figures in context.
G.RLT.3.bDetermine whether a figure has point symmetry, line symmetry, both, or neither, including figures in context.
G.RLT.3.c Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include:
G.RLT.3.c.itranslations;
G.RLT.3.c.iireflections over any horizontal or vertical line or the lines 𝑦 = 𝑥 or 𝑦 = –𝑥;
G.RLT.3.c.iiiclockwise or counterclockwise rotations of 90°, 180°, 270°, or 360° on a coordinate grid where the center of rotation is limited to the origin; and
G.RLT.3.c.ivdilations, from a fixed point on a coordinate grid.
G.TR Triangles
G.TR.1 The student will determine the relationships between the measures of angles and lengths of sides in triangles, including problems in context.
G.TR.1.aGiven the lengths of three segments, determine whether a triangle could be formed.
G.TR.1.bGiven the lengths of two sides of a triangle, determine the range in which the length of the third side must lie.
G.TR.1.cOrder the sides of a triangle by their lengths when given information about the measures of the angles.
G.TR.1.dOrder the angles of a triangle by their measures when given information about the lengths of the sides.
G.TR.1.eSolve for interior and exterior angles of a triangle, when given two angles.
G.TR.2 The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.
G.TR.2.aUse definitions, postulates, and theorems (including Side-Side-Side (SSS); Side-Angle-Side (SAS); Angle-Side-Angle (ASA); Angle-Angle-Side (AAS); and Hypotenuse-Leg (HL)) to prove and justify two triangles are congruent.
G.TR.2.bUse algebraic methods to prove that two triangles are congruent.
G.TR.2.cUse coordinate methods, such as the slope formula and the distance formula, to prove two triangles are congruent.
G.TR.2.dGiven a triangle, use congruent segment, congruent angle, and/or perpendicular line constructions to create a congruent triangle (SSS, SAS, ASA, AAS, and HL).
G.TR.3 The student will, given information in the form of a figure or statement, prove and justify two triangles are similar using direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar triangles.
G.TR.3.aUse definitions, postulates, and theorems (including Side-Angle-Side (SAS); Side-Side-Side (SSS); and Angle-Angle (AA)) to prove and justify that triangles are similar.
G.TR.3.bUse algebraic methods to prove that triangles are similar.
G.TR.3.cUse coordinate methods, such as the slope formula and the distance formula, to prove two triangles are similar.
G.TR.3.dDescribe a sequence of transformations that can be used to verify similarity of triangles located in the same plane.
G.TR.3.eSolve problems, including those in context involving attributes of similar triangles.
G.TR.4 The student will model and solve problems, including those in context, involving trigonometry in right triangles and applications of the Pythagorean Theorem.
G.TR.4.aDetermine whether a triangle formed with three given lengths is a right triangle.
G.TR.4.bFind and verify trigonometric ratios using right triangles.
G.TR.4.cModel and solve problems, including those in context, involving right triangle trigonometry (sine, cosine, and tangent ratios).
G.TR.4.dSolve problems using the properties of special right triangles.
G.TR.4.eSolve for missing lengths in geometric figures, using properties of 45°-45°-90° triangles, where rationalizing denominators may be necessary.
G.TR.4.fSolve for missing lengths in geometric figures, using properties of 30°-60°-90° triangles, where rationalizing denominators may be necessary.
G.TR.4.gSolve problems, including those in context, involving right triangles using the Pythagorean Theorem and its converse, including recognizing Pythagorean Triples.

Mathematical Analysis

MA.AG Analytic Geometry
MA.AG.1 The student will identify and analyze the properties of conic sections and sketch a graph given an equation.
MA.AG.1.aGiven a translation or rotation matrix, determine an equation for the transformed function or conic section.
MA.AG.1.bConvert between standard and general forms of conic equations by completing the square.
MA.AG.1.cGraph conic sections from equations written in general or standard form using transformations.
MA.AG.1.dIdentify characteristics of conic sections including center, vertices, axes, symmetry, foci, directrix, eccentricity, and asymptotes.
MA.AG.1.eRepresent applications of conic sections.
MA.AG.2 The student will use parametric equations to model and solve problems in context.
MA.AG.2.aGraph and analyze parametric equations and use the graph to determine solutions.
MA.AG.2.bUse parametric equations to model contextual problems, including motion over time.
MA.AG.3 The student will perform operations with vectors in the coordinate plane.
MA.AG.3.aUse vector notation.
MA.AG.3.bPerform the operations of addition, subtraction, and scalar multiplication, graphically and algebraically on vectors.
MA.AG.3.cFind the dot (inner) product of two vectors and use it to determine the angle between two vectors.
MA.AG.3.dDetermine if two vectors are orthogonal.
MA.AG.3.eExpress complex numbers in vector notation.
MA.AG.3.fVerify properties of the dot product.
MA.AG.3.gDetermine the components of a vector.
MA.AG.3.hDetermine the norm (magnitude) of a vector.
MA.AG.3.iFind a unit vector in the same direction of a given vector.
MA.AG.3.jApply vectors to problems in context.
MA.AG.4 The student will investigate and identify the characteristics of the graphs of polar equations.
MA.AG.4.aClassify polar equations (rose, cardioid, limaçon, lemniscate, spiral, and circle), given the graph or the equation.
MA.AG.4.bDetermine the effects of changes in the parameters of polar equations on the graph, using graphing technology.
MA.AG.4.cConvert between polar and rectangular forms of coordinates.
MA.AG.4.dConvert between complex numbers written in rectangular form and polar form.
MA.AG.4.eConvert equations between polar and rectangular forms.
MA.AG.4.fDetermine and verify the intersection of the graphs of two polar equations, using graphing technology.
MA.AG.5 The student will use matrices to organize data and will add and subtract matrices, multiply matrices, multiply matrices by a scalar, and use matrices to solve systems of equations.
MA.AG.5.aMultiply matrices by a scalar.
MA.AG.5.bAdd, subtract, and multiply matrices.
MA.AG.5.cRepresent problems with a system of no more than three linear equations.
MA.AG.5.dExpress a system of linear equations as a matrix equation.
MA.AG.5.eSolve a system of equations using matrices.
MA.AG.5.fDetermine the inverse of a two-by-two or three-by-three matrix using paper and pencil.
MA.AG.5.gVerify two matrices are inverses using matrix multiplication.
MA.AG.5.hVerify the commutative and associative properties for matrix addition and multiplication.
MA.CF Characteristics of Functions
MA.CF.1 The student will identify and analyze the properties of polynomial, rational, piecewise-defined, absolute value, radical, and step functions and sketch the graphs of the functions.
MA.CF.1.aUse mathematical reasoning to identify polynomial, rational, piecewise-defined, absolute value, radical, and step functions, given an equation or graph.
MA.CF.1.b Given multiple representations of a polynomial, rational, piecewise-defined, absolute value, radical, and step function, analyze:
MA.CF.1.b.idomain and range;
MA.CF.1.b.iiroots (including complex roots);
MA.CF.1.b.iiiintercepts;
MA.CF.1.b.ivsymmetry (including even and odd functions);
MA.CF.1.b.ixrelative and/or absolute maximum and minimum points.
MA.CF.1.b.vasymptotes (horizontal, vertical, and oblique/slant;
MA.CF.1.b.vipoints of discontinuity;
MA.CF.1.b.viiintervals for which the function is increasing, decreasing or constant;
MA.CF.1.b.viiiend behavior; and
MA.CF.1.cSketch the graph of a polynomial, rational, piecewise-defined, absolute value, radical, and step function.
MA.CF.2 The student will determine the limit of a function if it exists.
MA.CF.2.aVerify estimates about the limit of a function using graphing technology.
MA.CF.2.bDetermine the limit of a function algebraically and verify with graphing technology.
MA.CF.2.cDetermine the limit of a function numerically and verify with graphing technology.
MA.CF.2.dUse proper limit notation, including when describing the end behavior of a function.
MA.CF.2.e As the variable approaches a finite number,
MA.CF.2.e.idetermine the limit of a function numerically by direct substitution;
MA.CF.2.e.iidetermine the limit of a function using algebraic manipulation;
MA.CF.2.e.iiiestimate the limit of a function using a table; and
MA.CF.2.e.ivdetermine the limit of a function from a given graph.
MA.CF.2.fAs the variable approaches positive or negative infinity, analyze the limit of a function to describe the end behavior.
MA.CF.3 The student will analyze and describe the continuity of functions.
MA.CF.3.aDescribe continuity of a function.
MA.CF.3.bUse mathematical notation to communicate and describe the continuity of functions including polynomial, rational, piecewise, absolute value, radical, and step function, using graphical and algebraic methods.
MA.CF.3.cProve continuity at a point, using the definition.
MA.CF.3.dClassify types of discontinuity based on which condition of continuity is violated.
MA.FR Functional Relationships
MA.FR.1 The student will analyze compositions of functions to determine and verify inverses of functions.
MA.FR.1.aConstruct the composition of functions algebraically and graphically.
MA.FR.1.bDetermine the domain and range of composite functions algebraically and graphically.
MA.FR.1.cDevelop the inverse of a function algebraically and graphically.
MA.FR.1.dCompare the domain and range of the inverse of a function with the original function, both algebraically and graphically.
MA.FR.1.eUse mathematical reasoning to generalize and communicate the criteria for an inverse function to exist.
MA.FR.2 The student will analyze the characteristics of exponential and logarithmic functions, and sketch the graphs of the functions.
MA.FR.2.aGeneralize characteristics of exponential and logarithmic functions from an equation or a graph.
MA.FR.2.bDefine 𝑒 and estimate its value.
MA.FR.2.cConvert between equations written in logarithmic and exponential form.
MA.FR.2.dUse laws of exponents and properties of logarithms to solve equations and simplify expressions.
MA.FR.2.eRepresent contextual problems, using exponential and logarithmic functions, to include common and natural logarithms.
MA.FR.2.fSketch the graph of exponential and logarithmic functions and identify asymptotes, end behavior, intercepts, domain, and range.
MA.FR.3 The student will analyze sequences and finite series, and model and solve problems in context using sequences and series.
MA.FR.3.aUse and interpret the notation: ∑, 𝑛, 𝑛ᵗʰ, and 𝑎ₙ.
MA.FR.3.bDerive the formulas associated with arithmetic and geometric sequences and series.
MA.FR.3.cDetermine the 𝑛ᵗʰ term, 𝑎ₙ, for an arithmetic or geometric sequence.
MA.FR.3.dDetermine the sum, 𝑆ₙ, if it exists, of an arithmetic or geometric series.
MA.FR.3.eModel and solve problems in context, using sequences and series.
MA.FR.3.fDistinguish between a convergent and divergent series.
MA.FR.3.gDescribe convergent series in relation to the concept of a limit.

Probability and Statistics

PS.DC Data in Context
PS.DC.1 The student will use a statistical cycle to formulate questions, describe types of data, data sources, and constraints within the context of a problem.
PS.DC.1.aDefine the stages of the statistical cycle and how each stage relates to the others.
PS.DC.1.bFormulate questions and conclusions based on context.
PS.DC.1.cUnderstand the type of data relevant to the question at hand (e.g., quantitative versus categorical).
PS.DC.1.dCompare and contrast population and sample, and parameter and statistic.
PS.DC.1.eIdentify and explain constraints of the statistical approach.
PS.DC.2 The student will compare and contrast data collection methods to plan and conduct an observational study.
PS.DC.2.aInvestigate and describe sampling techniques (e.g., simple random sampling, stratified sampling, systematic sampling, cluster sampling).
PS.DC.2.bDetermine which sampling technique is best, given a particular context.
PS.DC.2.cInvestigate and explain biased influences inherent within sampling methods and various forms of response bias.
PS.DC.2.dUse the statistical cycle to plan and conduct an observational study to answer a question or address a problem.
PS.DC.3 The student will utilize the principles of experimental design to plan and conduct a well-designed experiment.
PS.DC.3.a Describe the principles of experimental design, including:
PS.DC.3.a.itreatment/control groups;
PS.DC.3.a.iiblinding/placebo effects;
PS.DC.3.a.iiiexperimental units/subjects; and
PS.DC.3.a.ivblocking/matched pairs and completely randomized designs.
PS.DC.3.bEvaluate the principles of experimental design to address comparison, randomization, replication, and control within the context of the problem.
PS.DC.3.cCompare and contrast controlled experiments and observational studies and the conclusions that may be drawn from each.
PS.DC.3.dUse the statistical cycle to plan and conduct a well-designed experiment to answer a question or address a problem.
PS.DC.3.eSelect a data collection method appropriate for a given context.
PS.DS Descriptive Statistics
PS.DS.1 The student will represent and analyze data visualizations of univariate quantitative data, including dot plots, stemplots, boxplots, cumulative frequency graphs, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers, within the context of a problem.
PS.DS.1.aCreate and interpret graphical displays of data, including dot plots, stemplots, boxplots, cumulative frequency graphs, and histograms, using appropriate technology.
PS.DS.1.b Examine the graphs within the context of the problem by analyzing:
PS.DS.1.b.ishape;
PS.DS.1.b.iimeasures of center;
PS.DS.1.b.iiispread; and
PS.DS.1.b.ivunusual features of the data (e.g., outliers, clusters, gaps).
PS.DS.2 The student will represent and analyze numerical characteristics of univariate quantitative data sets to describe patterns and departures from patterns within the context of a problem.
PS.DS.2.aInterpret measures of central tendency: mean, median, and mode.
PS.DS.2.bInterpret measures of spread: range, interquartile range, variance, and standard deviation.
PS.DS.2.cIdentify possible outliers, using an algorithm.
PS.DS.2.dInvestigate and explain the influence of outliers on a univariate data set.
PS.DS.2.eInvestigate and explain ways in which standard deviation addresses variability by examining the formula for standard deviation.
PS.DS.3 The student will represent, compare, and analyze distributions of two or more univariate quantitative data sets, numerically and graphically.
PS.DS.3.aCreate graphical displays of data, including back-to-back stemplots, parallel dot plots, parallel boxplots, and histograms, using appropriate technology.
PS.DS.3.b Compare and contrast two or more univariate data sets, numerically and graphically, within the context of a problem by analyzing:
PS.DS.3.b.ishape;
PS.DS.3.b.iimeasures of center;
PS.DS.3.b.iiimeasures of spread; and
PS.DS.3.b.ivunusual features of the data (e.g., clusters, gaps, outliers).
PS.DS.4 The student will represent and analyze categorical data, using two-way tables and other graphical displays, to describe patterns and relationships.
PS.DS.4.aCreate and interpret graphical displays of univariate categorical data, including bar graphs within the context of the problem, using appropriate technology.
PS.DS.4.bCreate and interpret graphical displays comparing distributions of two or more univariate categorical data sets including segmented and side-by-side bar graphs within the context of the problem, using appropriate technology.
PS.DS.4.cGenerate and interpret a two-way table as a summary of the information obtained from two categorical variables.
PS.DS.4.dCalculate and interpret marginal, relative, and conditional frequencies to analyze data in a two-way table within the context of a problem.
PS.DS.5 The student will represent and analyze quantitative bivariate data with scatterplots to identify and describe the relationship between two variables.
PS.DS.5.aCreate scatterplots, using appropriate technology.
PS.DS.5.b Examine and interpret scatterplots in the context of the problem by analyzing:
PS.DS.5.b.ithe form of relationship for linear and nonlinear trends;
PS.DS.5.b.iithe direction of the relationship for positive, negative, or no association;
PS.DS.5.b.iiithe strength of the relationship such as strong, moderate, or weak; and
PS.DS.5.b.ivthe presence of unusual features within the data (e.g., clusters, gaps, influential points, outliers).
PS.DS.6 The student will create and interpret a linear model using the least squares regression method to assess the relationship between two quantitative variables.
PS.DS.6.aCreate the least squares regression model using technology to interpret the contextual meaning of the slope and 𝑦-intercept.
PS.DS.6.bUsing technology, calculate and interpret the correlation coefficient, 𝑟, within the context of a problem.
PS.DS.6.cUsing technology, calculate and interpret the coefficient of determination, 𝑟², within the context of a problem.
PS.DS.6.dUse regression lines to make predictions, and identify the limitations of the predictions, such as extrapolation.
PS.DS.6.eCalculate and interpret a residual to understand the error of a prediction.
PS.DS.6.fUsing technology, calculate and interpret the standard deviation of the residuals, 𝑠.
PS.IS Inferential Statistics
PS.IS.1 The student will apply properties of sampling distributions and inference procedures to make decisions about population proportions.
PS.IS.1.aDescribe the shape, center, and spread of the sampling distribution of a proportion within the context of a problem.
PS.IS.1.b Given a problem, construct a one sample 𝑧 confidence interval:
PS.IS.1.b.iidentify the basic conditions for inference: random sample, independence, and normality;
PS.IS.1.b.iicalculate a confidence interval using technology; and
PS.IS.1.b.iiiinterpret the interval within the context of the problem.
PS.IS.1.cExplain how changes in confidence level and sample size affect width of the confidence interval and margin of error.
PS.IS.1.dCalculate and interpret a point estimate and margin of error of a confidence interval for a proportion within the context of the problem.
PS.IS.1.eExplain how and why the hypothesis testing procedure allows one to reach a statistical decision.
PS.IS.1.f Given a problem, apply the one sample 𝑧 hypothesis testing procedures:
PS.IS.1.f.iconstruct appropriate null and alternate hypotheses;
PS.IS.1.f.iiidentify the basic conditions for inference: random sample; independence, and normality;
PS.IS.1.f.iiicalculate and interpret the 𝑝-value using technology;
PS.IS.1.f.ivdetermine and justify whether to reject the null hypothesis; and
PS.IS.1.f.vinterpret the results within the context of the problem.
PS.IS.1.gUse the statistical cycle to plan and conduct a statistical study about a proportion to answer a question or address a problem with inference.
PS.IS.2 The student will apply properties of sampling distributions and inference procedures to make decisions about populations.
PS.IS.2.aDescribe the shape, center, and spread of the sampling distribution of a mean within the context of a problem.
PS.IS.2.bCalculate and interpret a point estimate and a margin of error for a confidence interval of a mean within the context of a problem.
PS.IS.2.cDescribe the use of the Central Limit Theorem in satisfying the assumptions and conditions for inference about a mean.
PS.IS.2.dIdentify the properties of a 𝑡 distribution.
PS.IS.2.e Given a problem, construct a one sample 𝑡 confidence interval:
PS.IS.2.e.iidentify the basic conditions for inference: random sample, independence, and approximate normality;
PS.IS.2.e.iicalculate a confidence interval using technology; and
PS.IS.2.e.iiiinterpret the interval within the context of the problem.
PS.IS.2.f Given a problem, apply the one sample 𝑡 hypothesis testing procedures:
PS.IS.2.f.iconstruct appropriate null and alternate hypotheses;
PS.IS.2.f.iiidentify the basic conditions for inference: random sample, independence, and approximate normality;
PS.IS.2.f.iiicalculate and interpret the 𝑝 value using technology;
PS.IS.2.f.ivdetermine and justify whether to reject the null hypothesis; and
PS.IS.2.f.vinterpret the results within the context of the problem.
PS.P Probability
PS.P.1 The student will organize information and apply probability rules to compute probabilities of events within the context of a problem.
PS.P.1.aGiven two or more events, determine whether the events are complementary, dependent, independent, and/or mutually exclusive, and compute the probability of those events.
PS.P.1.bRepresent and calculate probabilities using Venn diagrams, tree diagrams, and two-way tables.
PS.P.1.cApply the addition rule, the multiplication rule, and complementary rule to calculate probabilities.
PS.P.1.dCalculate conditional probabilities to determine the association or independence of two events.
PS.P.2 The student will represent and interpret situations using discrete random distributions, including binomial distributions.
PS.P.2.aIdentify discrete random variables and create a table to represent valid discrete probability distributions within the context of a problem.
PS.P.2.bCalculate and interpret the mean (expected value) and standard deviation for a discrete random variable within the context of a problem.
PS.P.2.cDetermine if a discrete random variable satisfies the conditions for a binomial distribution.
PS.P.2.dDesign and conduct a simulation of a binomial distribution.
PS.P.2.eCalculate and interpret probabilities from a binomial distribution within the context of a problem.
PS.P.2.fCalculate the mean and standard deviation for binomial distributions.
PS.P.2.gDescribe the center, shape, and spread of a discrete random variable within the context of a problem.
PS.P.3 The student will represent and interpret situations using normal distributions.
PS.P.3.aCompare and contrast discrete and continuous distributions.
PS.P.3.bRepresent probability as the area under the curve of a normal distribution using the Empirical Rule and graphing technology.
PS.P.3.cDescribe the center, shape, and spread of normal distributions within the context of a problem.
PS.P.3.dCompare and contrast two or more sets of normally distributed data using 𝑧-scores, percentiles, or probabilities within the context of a problem.
PS.P.3.eStandardize a data value from a normal distribution and interpret the 𝑧-score within the context of a problem.
PS.P.3.fCalculate and interpret probabilities of a normal distribution using technology within the context of a problem.

Trigonometry

T.CT Circular Trigonometry
T.CT.1 The student will determine the degree and radian measure of angles; sketch angles in standard position on a coordinate plane; and determine the sine, cosine, tangent, cosecant, secant, and cotangent of an angle, given a point on the terminal side of an angle in standard position or the value of a trigonometric function of the angle.
T.CT.1.aDefine a radian as a unit of angle measure and determine the relationship between the radian measure of an angle and the length of the intercepted arc in a circle.
T.CT.1.bDetermine the degree and radian measure of angles to include both negative and positive rotations in the coordinate plane.
T.CT.1.cFind both positive and negative coterminal angles for a given angle.
T.CT.1.dIdentify the quadrant or axis in/on which the terminal side of an angle lies.
T.CT.1.eDraw a reference right triangle when given a point on the terminal side of an angle in standard position.
T.CT.1.fDraw a reference right triangle when given the value of a trigonometric function of an angle (sine, cosine, tangent, cosecant, secant, and cotangent).
T.CT.1.gDetermine the value of any trigonometric function (sine, cosine, tangent, cosecant, secant, and cotangent) when given a point on the terminal side of an angle in standard position.
T.CT.1.hGiven one trigonometric function value, determine the other five trigonometric function values.
T.CT.1.iCalculate the length of an arc of a circle in radians.
T.CT.1.jCalculate the area of a sector of a circle.
T.CT.2 The student will develop and apply the properties of the unit circle in degrees and radians.
T.CT.2.aConvert between radian and degree measure of special angles of the unit circle without the use of technology.
T.CT.2.bDefine the six circular trigonometric functions of an angle in standard position on the unit circle.
T.CT.2.cApply knowledge of right triangle trigonometry, special right triangles, and the properties of the unit circle to determine trigonometric functions values of special angles (0°, 30°, 45°, 60°, and 90°) and their related angles in degree and radians without the use of technology.
T.GT Graphs of Trigonometric Functions
T.GT.1 The student will graph and analyze trigonometric functions and apply trigonometric functions to represent periodic phenomena.
T.GT.1.aSketch the graph of the six parent trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for at least a two-period interval.
T.GT.1.bDetermine the domain and range, amplitude, period, and asymptote locations for a trigonometric function, given a graph or an equation.
T.GT.1.cDescribe the effects of changing the parameters (𝐴, 𝐵, 𝐶, or 𝐷 in the standard form of a trigonometric equation) on the graph of the function using graphing technology.
T.GT.1.dSketch the graph of a transformed sine, cosine, and tangent function written in standard form by using transformations for at least a two-period interval, including both positive and negative values for the domain.
T.GT.1.eApply trigonometric functions and their graphs to represent periodic phenomena.
T.GT.2 The student will graph the six inverse trigonometric functions.
T.GT.2.aDetermine the domain and range of the inverse trigonometric functions.
T.GT.2.bUse the restrictions on the domain of an inverse trigonometric function to determine a value of the inverse trigonometric function.
T.GT.2.cGraph inverse trigonometric functions.
T.IE Identities and Equations
T.IE.1 The student will evaluate expressions involving the six trigonometric functions and the inverse sine, cosine, and tangent functions.
T.IE.1.aDetermine the values of trigonometric functions, with and without graphing technology.
T.IE.1.bDetermine angle measures by using the inverse trigonometric functions, with and without a graphing technology.
T.IE.1.cEvaluate composite functions that involve trigonometric functions and inverse trigonometric functions.
T.IE.2 The student will use basic trigonometric identity substitutions to simplify and verify trigonometric identities.
T.IE.2.a Use trigonometric identities to make algebraic substitutions to simplify and verify trigonometric identities. The basic trigonometric identities include:
T.IE.2.a.ireciprocal identities;
T.IE.2.a.iiPythagorean identities;
T.IE.2.a.iiisum and difference identities;
T.IE.2.a.ivdouble-angle identities; and
T.IE.2.a.vhalf-angle identities.
T.IE.2.bApply the sum, difference, and half-angle identities to evaluate trigonometric function values of angles that are not integer multiples of the special angles to solve problems, including contextual situations.
T.IE.3 The student will solve trigonometric equations and inequalities.
T.IE.3.aSolve trigonometric equations with and without restricted domains algebraically and graphically.
T.IE.3.bSolve trigonometric inequalities algebraically and graphically.
T.IE.3.cVerify and justify algebraic solutions to trigonometric equations and inequalities, using graphing technology.
T.TT Triangle Trigonometry
T.TT.1 The student will determine the sine, cosine, tangent, cotangent, secant, and cosecant of the acute angles in a right triangle and use these ratios to solve for missing sides and angle measures, including application in contextual problems.
T.TT.1.aDefine and represent the six triangular trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in a right triangle.
T.TT.1.bDescribe the relationships between side lengths in special right triangles (30°-60°-90° and 45°-45°-90°).
T.TT.1.cUse the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines to solve contextual problems.
T.TT.1.dRepresent and solve contextual problems involving right triangles, including problems involving angles of elevation and depression.
T.TT.2 The student will find the area of any triangle and solve for the lengths of the sides and measures of the angles in a non-right triangle using the Law of Sines and the Law of Cosines.
T.TT.2.aApply the Law of Sines, and the Law of Cosines, as appropriate, to find missing sides and angles in non-right triangles.
T.TT.2.bRecognize the ambiguous case when applying the Law of Sines and the potential for two triangle solutions in some situations.
T.TT.2.cSolve problems that integrate the use of the Law of Sines and the Law of Cosines and the triangle area formula (Area = 1/2 𝑎𝑏 sin 𝐶, where 𝑎 and 𝑏 are triangle sides and 𝐶 is the included angle) to find the area of any triangle, including those in contextual problems.

AP Calculus Learning Objectives

CB.Calc.LIM Limits
CB.Calc.LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.
CB.Calc.LIM-1.ARepresent limits analytically using correct notation.
CB.Calc.LIM-1.BInterpret limits expressed in analytic notation.
CB.Calc.LIM-1.CEstimate limits of functions.
CB.Calc.LIM-1.DDetermine the limits of functions using limit theorems.
CB.Calc.LIM-1.EDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.
CB.Calc.LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.
CB.Calc.LIM-2.AJustify conclusions about continuity at a point using the definition.
CB.Calc.LIM-2.BDetermine intervals over which a function is continuous.
CB.Calc.LIM-2.CDetermine values of x or solve for parameters that make discontinuous functions continuous, if possible.
CB.Calc.LIM-2.DInterpret the behavior of functions using limits involving infinity.
CB.Calc.LIM-3 The derivative of a function is defined as the limit of a difference quotient.
CB.Calc.LIM-3.AInterpret a limit as a definition of a derivative.
CB.Calc.LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.
CB.Calc.LIM-4.ADetermine limits of functions that result in indeterminate forms.
CB.Calc.LIM-5 Definite integrals can be approximated using geometric and numerical methods.
CB.Calc.LIM-5.AApproximate a definite integral using geometric and numerical methods.
CB.Calc.LIM-5.BInterpret the limiting case of the Riemann sum as a definite integral.
CB.Calc.LIM-5.CRepresent the limiting case of the Riemann sum as a definite integral.
CB.Calc.LIM-6 Improper integrals can be determined using limits of definite integrals.
CB.Calc.LIM-6.AEvaluate an improper integral or determine that the integral diverges.
CB.Calc.LIM-7 The behavior of a convergent or divergent series can be determined or estimated.
CB.Calc.LIM-7.ADetermine whether a series converges or diverges.
CB.Calc.LIM-7.BApproximate the sum of a series.
CB.Calc.LIM-8 Power series and Taylor series can represent functions.
CB.Calc.LIM-8.ARepresent a function at a point as a Taylor polynomial.
CB.Calc.LIM-8.BApproximate function values using a Taylor polynomial.
CB.Calc.LIM-8.CDetermine the error bound associated with a Taylor polynomial approximation.
CB.Calc.LIM-8.DDetermine the radius of convergence and interval of convergence for a power series.
CB.Calc.LIM-8.ERepresent a function as a Taylor series or a Maclaurin series.
CB.Calc.LIM-8.FInterpret Taylor series and Maclaurin series.
CB.Calc.LIM-8.GRepresent a given function as a power series.
CB.Calc.CHA Change
CB.Calc.CHA-1 The derivative of a function is the instantaneous rate of change of the function.
CB.Calc.CHA-1.AInterpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.
CB.Calc.CHA-2 Derivatives can be computed using definitions and rules.
CB.Calc.CHA-2.ADetermine average rates of change using difference quotients.
CB.Calc.CHA-2.BRepresent the derivative of a function as the limit of a difference quotient.
CB.Calc.CHA-2.CDetermine the equation of a line tangent to a curve at a given point.
CB.Calc.CHA-2.DEstimate derivatives.
CB.Calc.CHA-3 Derivatives can be used to model rates of change in applied contexts.
CB.Calc.CHA-3.AInterpret the meaning of a derivative in context.
CB.Calc.CHA-3.BCalculate rates of change in applied contexts.
CB.Calc.CHA-3.CInterpret rates of change in applied contexts.
CB.Calc.CHA-3.DCalculate related rates in applied contexts.
CB.Calc.CHA-3.EInterpret related rates in applied contexts.
CB.Calc.CHA-3.FApproximate a value on a curve using the equation of a tangent line.
CB.Calc.CHA-3.GCalculate derivatives of parametric functions.
CB.Calc.CHA-3.HCalculate derivatives of vector-valued functions.
CB.Calc.CHA-4 Integrals can be used to model accumulation and net change in applied contexts.
CB.Calc.CHA-4.AInterpret the meaning of areas associated with the graph of a rate of change in context.
CB.Calc.CHA-4.BDetermine the average value of a function using definite integrals.
CB.Calc.CHA-4.CDetermine values for positions and rates of change using definite integrals in problems involving rectilinear motion.
CB.Calc.CHA-4.DInterpret the meaning of a definite integral in accumulation problems.
CB.Calc.CHA-4.EDetermine net change using definite integrals in applied contexts.
CB.Calc.CHA-5 Definite integrals can be used to calculate areas, volumes, and lengths.
CB.Calc.CHA-5.ACalculate areas in the plane using the definite integral.
CB.Calc.CHA-5.BCalculate volumes of solids with known cross sections using definite integrals.
CB.Calc.CHA-5.CCalculate volumes of solids of revolution using definite integrals.
CB.Calc.CHA-5.DCalculate areas of regions defined by polar curves using definite integrals.
CB.Calc.CHA-6 Arc length can be calculated using definite integrals.
CB.Calc.CHA-6.ADetermine the length of a curve in the plane defined by a function, using a definite integral.
CB.Calc.CHA-6.BDetermine the length of a curve in the plane defined by parametric functions, using a definite integral.
CB.Calc.FUN Analysis of Functions
CB.Calc.FUN-1 Existence theorems can be used to draw conclusions about the behavior of functions.
CB.Calc.FUN-1.AExplain the behavior of a function on an interval using the Intermediate Value Theorem.
CB.Calc.FUN-1.BJustify conclusions about functions by applying the Mean Value Theorem over an interval.
CB.Calc.FUN-1.CJustify conclusions about functions by applying the Extreme Value Theorem.
CB.Calc.FUN-2 A function's derivative, when it exists, can be used to determine whether the function is increasing or decreasing.
CB.Calc.FUN-2.AExplain the relationship between differentiability and continuity.
CB.Calc.FUN-3 Derivatives can be calculated for various types of functions.
CB.Calc.FUN-3.ACalculate derivatives of familiar functions.
CB.Calc.FUN-3.BCalculate derivatives of products and quotients of differentiable functions.
CB.Calc.FUN-3.CCalculate derivatives of compositions of differentiable functions.
CB.Calc.FUN-3.DCalculate derivatives of implicitly defined functions.
CB.Calc.FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.
CB.Calc.FUN-3.FDetermine higher order derivatives of a function.
CB.Calc.FUN-3.GCalculate derivatives of functions written in polar coordinates.
CB.Calc.FUN-4 A function's derivative can be used to understand function behavior.
CB.Calc.FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.
CB.Calc.FUN-4.BCalculate minimum and maximum values in applied contexts or analysis of functions.
CB.Calc.FUN-4.CInterpret minimum and maximum values calculated in applied contexts.
CB.Calc.FUN-4.DDetermine critical points of implicit relations.
CB.Calc.FUN-4.EJustify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.
CB.Calc.FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.
CB.Calc.FUN-5.ARepresent accumulation functions using definite integrals.
CB.Calc.FUN-6 Recognizing and applying techniques for antidifferentiation.
CB.Calc.FUN-6.ACalculate a definite integral using areas and properties of definite integrals.
CB.Calc.FUN-6.BEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.
CB.Calc.FUN-6.CDetermine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.
CB.Calc.FUN-6.DUse substitution and rearrangements into equivalent forms, such as long division and completing the square, for integrands requiring these techniques.
CB.Calc.FUN-6.EUse integration by parts for integrands requiring this technique.
CB.Calc.FUN-6.FUse integration by linear partial fractions for integrands requiring this technique.
CB.Calc.FUN-7 Solving differential equations allows us to model rates of change.
CB.Calc.FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.
CB.Calc.FUN-7.BVerify solutions to differential equations.
CB.Calc.FUN-7.CEstimate solutions to differential equations.
CB.Calc.FUN-7.DDetermine general solutions to differential equations.
CB.Calc.FUN-7.EDetermine particular solutions to differential equations.
CB.Calc.FUN-7.FInterpret the meaning of a differential equation and its variables in context.
CB.Calc.FUN-7.GDetermine general and particular solutions for problems involving differential equations in context.
CB.Calc.FUN-7.HInterpret the meaning of the logistic growth model in context.
CB.Calc.FUN-8 Solutions to differential equations can model motion in the plane.
CB.Calc.FUN-8.ADetermine a particular solution given a rate vector and initial conditions.
CB.Calc.FUN-8.BDetermine values for positions and rates of change in problems involving planar motion.

AP Statistics Learning Objectives

CB.Stats.VAR Variation and Distribution
CB.Stats.VAR-1 Exploring data through asking and answering statistical questions.
CB.Stats.VAR-1.AIdentify questions to be answered, based on variation in one-variable data.
CB.Stats.VAR-1.BIdentify variables in a set of data.
CB.Stats.VAR-1.CClassify types of variables.
CB.Stats.VAR-1.DIdentify questions to be answered about possible relationships in data.
CB.Stats.VAR-1.EIdentify questions to be answered about data collection methods.
CB.Stats.VAR-1.FIdentify questions suggested by patterns in data.
CB.Stats.VAR-1.GIdentify questions suggested by variation in statistics for samples collected from the same population.
CB.Stats.VAR-1.HIdentify questions suggested by variation in the shapes of distributions of samples taken from the same population.
CB.Stats.VAR-1.IIdentify questions suggested by probabilities of errors in statistical inference.
CB.Stats.VAR-1.JIdentify questions suggested by variation between observed and expected counts in categorical data.
CB.Stats.VAR-1.KIdentify questions suggested by variation in scatter plots.
CB.Stats.VAR-2 Normal distributions and comparing data.
CB.Stats.VAR-2.ACompare a data distribution to the normal distribution model.
CB.Stats.VAR-2.BDetermine proportions and percentiles from a normal distribution.
CB.Stats.VAR-2.CCompare measures of relative position in data sets.
CB.Stats.VAR-3 Designing and analyzing experiments.
CB.Stats.VAR-3.AIdentify the components of an experiment.
CB.Stats.VAR-3.BDescribe elements of a well-designed experiment.
CB.Stats.VAR-3.CCompare experimental designs and methods.
CB.Stats.VAR-3.DExplain why a particular experimental design is appropriate.
CB.Stats.VAR-3.EInterpret the results of a well-designed experiment.
CB.Stats.VAR-4 Probability rules and calculations.
CB.Stats.VAR-4.ACalculate probabilities for events and their complements.
CB.Stats.VAR-4.BInterpret probabilities for events.
CB.Stats.VAR-4.CExplain why two events are (or are not) mutually exclusive.
CB.Stats.VAR-4.DCalculate conditional probabilities.
CB.Stats.VAR-4.ECalculate probabilities for independent events and for the union of two events.
CB.Stats.VAR-5 Discrete random variables and probability distributions.
CB.Stats.VAR-5.ARepresent the probability distribution for a discrete random variable.
CB.Stats.VAR-5.BInterpret a probability distribution.
CB.Stats.VAR-5.CCalculate parameters for a discrete random variable.
CB.Stats.VAR-5.DInterpret parameters for a discrete random variable.
CB.Stats.VAR-5.ECalculate parameters for linear combinations of random variables.
CB.Stats.VAR-5.FDescribe the effects of linear transformations of parameters of random variables.
CB.Stats.VAR-6 Normal distributions and hypothesis testing for proportions.
CB.Stats.VAR-6.ACalculate the probability that a particular value lies in a given interval of a normal distribution.
CB.Stats.VAR-6.BDetermine the interval associated with a given area in a normal distribution.
CB.Stats.VAR-6.CDetermine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.
CB.Stats.VAR-6.DIdentify the null and alternative hypotheses for a population proportion.
CB.Stats.VAR-6.EIdentify an appropriate testing method for a population proportion.
CB.Stats.VAR-6.FVerify the conditions for making statistical inferences when testing a population proportion.
CB.Stats.VAR-6.GCalculate an appropriate test statistic and p-value for a population proportion.
CB.Stats.VAR-6.HIdentify the null and alternative hypotheses for a difference of two population proportions.
CB.Stats.VAR-6.IIdentify an appropriate testing method for the difference of two population proportions.
CB.Stats.VAR-6.JVerify the conditions for making statistical inferences when testing a difference of two population proportions.
CB.Stats.VAR-6.KCalculate an appropriate test statistic for the difference of two population proportions.
CB.Stats.VAR-7 t-distributions and hypothesis testing for means and regression.
CB.Stats.VAR-7.ADescribe t-distributions.
CB.Stats.VAR-7.BIdentify an appropriate testing method for a population mean with unknown sigma, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.CIdentify the null and alternative hypotheses for a population mean with unknown sigma, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.DVerify the conditions for the test for a population mean, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.ECalculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.FIdentify an appropriate selection of a testing method for a difference of two population means.
CB.Stats.VAR-7.GIdentify the null and alternative hypotheses for a difference of two population means.
CB.Stats.VAR-7.HVerify the conditions for the significance test for the difference of two population means.
CB.Stats.VAR-7.ICalculate an appropriate test statistic for a difference of two means.
CB.Stats.VAR-7.JIdentify the appropriate selection of a testing method for a slope of a regression model.
CB.Stats.VAR-7.KIdentify appropriate null and alternative hypotheses for a slope of a regression model.
CB.Stats.VAR-7.LVerify the conditions for the significance test for the slope of a regression model.
CB.Stats.VAR-7.MCalculate an appropriate test statistic for the slope of a regression model.
CB.Stats.VAR-8 Chi-square distributions and tests.
CB.Stats.VAR-8.ADescribe chi-square distributions.
CB.Stats.VAR-8.BIdentify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.
CB.Stats.VAR-8.CIdentify an appropriate testing method for a distribution of proportions in a set of categorical data.
CB.Stats.VAR-8.DCalculate expected counts for the chi-square test for goodness of fit.
CB.Stats.VAR-8.EVerify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.
CB.Stats.VAR-8.FCalculate the appropriate statistic for the chi-square test for goodness of fit.
CB.Stats.VAR-8.GDetermine the p-value for the chi-square test for goodness of fit.
CB.Stats.VAR-8.HCalculate expected counts for two-way tables of categorical data.
CB.Stats.VAR-8.IIdentify the null and alternative hypotheses for a chi-square test for homogeneity or independence.
CB.Stats.VAR-8.JIdentify an appropriate testing method for comparing distributions in two-way tables of categorical data.
CB.Stats.VAR-8.KVerify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.
CB.Stats.VAR-8.LCalculate the appropriate statistic for a chi-square test for homogeneity or independence.
CB.Stats.VAR-8.MDetermine the p-value for a chi-square test for independence or homogeneity.
CB.Stats.UNC Uncertainty and Inference
CB.Stats.UNC-1 Representing and describing categorical and quantitative data.
CB.Stats.UNC-1.ARepresent categorical data using frequency or relative frequency tables.
CB.Stats.UNC-1.BDescribe categorical data represented in frequency or relative tables.
CB.Stats.UNC-1.CRepresent categorical data graphically.
CB.Stats.UNC-1.DDescribe categorical data represented graphically.
CB.Stats.UNC-1.ECompare multiple sets of categorical data.
CB.Stats.UNC-1.FClassify types of quantitative variables.
CB.Stats.UNC-1.GRepresent quantitative data graphically.
CB.Stats.UNC-1.HDescribe the characteristics of quantitative data distributions.
CB.Stats.UNC-1.ICalculate measures of center and position for quantitative data.
CB.Stats.UNC-1.JCalculate measures of variability for quantitative data.
CB.Stats.UNC-1.KExplain the selection of a particular measure of center and/or variability for describing a set of quantitative data.
CB.Stats.UNC-1.LRepresent summary statistics for quantitative data graphically.
CB.Stats.UNC-1.MDescribe summary statistics of quantitative data represented graphically.
CB.Stats.UNC-1.NCompare graphical representations for multiple sets of quantitative data.
CB.Stats.UNC-1.OCompare summary statistics for multiple sets of quantitative data.
CB.Stats.UNC-1.PCompare numerical and graphical representations for two categorical variables.
CB.Stats.UNC-1.QCalculate statistics for two categorical variables.
CB.Stats.UNC-1.RCompare statistics for two categorical variables.
CB.Stats.UNC-1.SRepresent bivariate quantitative data using scatterplots.
CB.Stats.UNC-2 Probability and simulation.
CB.Stats.UNC-2.AEstimate probabilities using simulation.
CB.Stats.UNC-3 Random variables and sampling distributions.
CB.Stats.UNC-3.AEstimate probabilities of binomial random variables using data from a simulation.
CB.Stats.UNC-3.BCalculate probabilities for a binomial distribution.
CB.Stats.UNC-3.CCalculate parameters for a binomial distribution.
CB.Stats.UNC-3.DInterpret probabilities and parameters for a binomial distribution.
CB.Stats.UNC-3.ECalculate probabilities for geometric random variables.
CB.Stats.UNC-3.FCalculate parameters of a geometric distribution.
CB.Stats.UNC-3.GInterpret probabilities and parameters for a geometric distribution.
CB.Stats.UNC-3.HEstimate sampling distributions using simulation.
CB.Stats.UNC-3.IExplain why an estimator is or is not unbiased.
CB.Stats.UNC-3.JCalculate estimates for a population parameter.
CB.Stats.UNC-3.KDetermine parameters of a sampling distribution for sample proportions.
CB.Stats.UNC-3.LDetermine whether a sampling distribution for a sample proportion can be described as approximately normal.
CB.Stats.UNC-3.MInterpret probabilities and parameters for a sampling distribution for a sample proportion.
CB.Stats.UNC-3.NDetermine parameters of a sampling distribution for a difference in sample proportions.
CB.Stats.UNC-3.ODetermine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.
CB.Stats.UNC-3.PInterpret probabilities and parameters for a sampling distribution for a difference in proportions.
CB.Stats.UNC-3.QDetermine parameters of a sampling distribution for a sample mean.
CB.Stats.UNC-3.RDetermine whether a sampling distribution of a sample mean can be described as approximately normal.
CB.Stats.UNC-3.SInterpret probabilities and parameters for a sampling distribution for a sample mean.
CB.Stats.UNC-3.TDetermine parameters of a sampling distribution for a difference in sample means.
CB.Stats.UNC-3.UDetermine whether a sampling distribution of a difference in sample means can be described as approximately normal.
CB.Stats.UNC-3.VInterpret probabilities and parameters for a sampling distribution for a difference in sample means.
CB.Stats.UNC-4 Confidence intervals for proportions and means.
CB.Stats.UNC-4.AIdentify an appropriate confidence interval procedure for a population proportion.
CB.Stats.UNC-4.BVerify the conditions for calculating confidence intervals for a population proportion.
CB.Stats.UNC-4.CDetermine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.
CB.Stats.UNC-4.DCalculate an appropriate confidence interval for a population proportion.
CB.Stats.UNC-4.ECalculate an interval estimate based on a confidence interval for a population proportion.
CB.Stats.UNC-4.FInterpret a confidence interval for a population proportion.
CB.Stats.UNC-4.GJustify a claim based on a confidence interval for a population proportion.
CB.Stats.UNC-4.HIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.
CB.Stats.UNC-4.IIdentify an appropriate confidence interval procedure for a comparison of population proportions.
CB.Stats.UNC-4.JVerify the conditions for calculating confidence intervals for a difference between population proportions.
CB.Stats.UNC-4.KCalculate an appropriate confidence interval for a comparison of population proportions.
CB.Stats.UNC-4.LCalculate an interval estimate based on a confidence interval for a difference of proportions.
CB.Stats.UNC-4.MInterpret a confidence interval for a difference of proportions.
CB.Stats.UNC-4.NJustify a claim based on a confidence interval for a difference of proportions.
CB.Stats.UNC-4.OIdentify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.PVerify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.QDetermine the margin of error for a given sample size for a one-sample t-interval.
CB.Stats.UNC-4.RCalculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.SInterpret a confidence interval for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.TJustify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.UIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.
CB.Stats.UNC-4.VIdentify an appropriate confidence interval procedure for a difference of two population means.
CB.Stats.UNC-4.WVerify the conditions for calculating confidence intervals for the difference of two population means.
CB.Stats.UNC-4.XDetermine the margin of error for the difference of two population means.
CB.Stats.UNC-4.YCalculate an appropriate confidence interval for a difference of two population means.
CB.Stats.UNC-4.ZInterpret a confidence interval for a difference of population means.
CB.Stats.UNC-5 Errors in statistical inference.
CB.Stats.UNC-5.AIdentify Type I and Type II errors.
CB.Stats.UNC-5.BCalculate the probability of a Type I and Type II errors.
CB.Stats.UNC-5.CIdentify factors that affect the probability of errors in significance testing.
CB.Stats.UNC-5.DInterpret Type I and Type II errors.
CB.Stats.DAT Data Analysis
CB.Stats.DAT-1 Bivariate data analysis and linear regression.
CB.Stats.DAT-1.ADescribe the characteristics of a scatter plot.
CB.Stats.DAT-1.BDetermine the correlation for a linear relationship.
CB.Stats.DAT-1.CInterpret the correlation for a linear relationship.
CB.Stats.DAT-1.DCalculate a predicted response value using a linear regression model.
CB.Stats.DAT-1.ERepresent differences between measured and predicted responses using residual plots.
CB.Stats.DAT-1.FDescribe the form of association of bivariate data using residual plots.
CB.Stats.DAT-1.GEstimate parameters for the least-squares regression line model.
CB.Stats.DAT-1.HInterpret coefficients for the least-squares regression line model.
CB.Stats.DAT-1.IIdentify influential points in regression.
CB.Stats.DAT-1.JCalculate a predicted response using a least-squares regression line for a transformed data set.
CB.Stats.DAT-2 Studies, sampling, and sources of bias.
CB.Stats.DAT-2.AIdentify the type of a study.
CB.Stats.DAT-2.BIdentify appropriate generalizations and determinations based on observational studies.
CB.Stats.DAT-2.CIdentify a sampling method, given a description of a study.
CB.Stats.DAT-2.DExplain why a particular sampling method is or is not appropriate for a given situation.
CB.Stats.DAT-2.EIdentify potential sources of bias in sampling methods.
CB.Stats.DAT-3 Interpreting and justifying conclusions from significance tests.
CB.Stats.DAT-3.AInterpret the p-value of a significance test for a population proportion.
CB.Stats.DAT-3.BJustify a claim about the population based on the results of a significance test for a population proportion.
CB.Stats.DAT-3.CInterpret the p-value of a significance test for a difference of population proportions.
CB.Stats.DAT-3.DJustify a claim about the population based on the results of a significance test for a difference of population proportions.
CB.Stats.DAT-3.EInterpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.
CB.Stats.DAT-3.FJustify a claim about the population based on the results of a significance test for a population mean.
CB.Stats.DAT-3.GInterpret the p-value of a significance test for a difference of population means.
CB.Stats.DAT-3.HJustify a claim about the population based on the results of a significance test for a difference of two population means in context.
CB.Stats.DAT-3.IInterpret the p-value for the chi-square test for goodness of fit.
CB.Stats.DAT-3.JJustify a claim about the population based on the results of a chi-square test for goodness of fit.
CB.Stats.DAT-3.KInterpret the p-value for the chi-square test for homogeneity or independence.
CB.Stats.DAT-3.LJustify a claim about the population based on the results of a chi-square test for homogeneity or independence.
CB.Stats.DAT-3.MInterpret the p-value of a significance test for the slope of a regression model.
CB.Stats.DAT-3.NJustify a claim about the population based on the results of a significance test for the slope of a regression model.

Standards on this page are sourced from published state frameworks. Goblins aligns tutoring to these standards.