Texas Math Standards

1022 standards, Kindergarten through AP.

Kindergarten

K.2 Number and operations. The student applies mathematical process standards to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system.

K.2.ACount forward and backward to at least 20 with and without objects.

K.2.BRead, write, and represent whole numbers from 0 to at least 20 with and without objects or pictures.

K.2.CCount a set of objects up to at least 20 and demonstrate that the last number said tells the number of objects in the set regardless of their arrangement or order.

K.2.DRecognize instantly the quantity of a small group of objects in organized and random arrangements.

K.2.EGenerate a set using concrete and pictorial models that represents a number that is more than, less than, and equal to a given number up to 20.

K.2.FGenerate a number that is one more than or one less than another number up to at least 20.

K.2.GCompare sets of objects up to at least 20 in each set using comparative language.

K.2.HUse comparative language to describe two numbers up to 20 presented as written numerals.

K.2.ICompose and decompose numbers up to 10 with objects and pictures.

K.3 Number and operations. The student applies mathematical process standards to develop an understanding of addition and subtraction situations in order to solve problems.

K.3.AModel the action of joining to represent addition and the action of separating to represent subtraction.

K.3.BSolve word problems using objects and drawings to find sums up to 10 and differences within 10.

K.3.CExplain the strategies used to solve problems involving adding and subtracting within 10 using spoken words, concrete and pictorial models, and number sentences.

K.4 Number and operations. The student applies mathematical process standards to identify coins in order to recognize the need for monetary transactions.

K.4.AIdentify U.S. coins by name, including pennies, nickels, dimes, and quarters.

K.5 Algebraic reasoning. The student applies mathematical process standards to identify the pattern in the number word list.

K.5.ARecite numbers up to at least 100 by ones and tens beginning with any given number.

K.6 Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties.

K.6.AIdentify two-dimensional shapes, including circles, triangles, rectangles, and squares as special rectangles.

K.6.BIdentify three-dimensional solids, including cylinders, cones, spheres, and cubes, in the real world.

K.6.CIdentify two-dimensional components of three-dimensional objects.

K.6.DIdentify attributes of two-dimensional shapes using informal and formal geometric language interchangeably.

K.6.EClassify and sort a variety of regular and irregular two- and three-dimensional figures regardless of orientation or size.

K.6.FCreate two-dimensional shapes using a variety of materials and drawings.

K.7 Geometry and measurement. The student applies mathematical process standards to directly compare measurable attributes.

K.7.AGive an example of a measurable attribute of a given object, including length, capacity, and weight.

K.7.BCompare two objects with a common measurable attribute to see which object has more of/less of the attribute and describe the difference.

K.8 Data analysis. The student applies mathematical process standards to collect and organize data to make it useful for interpreting information.

K.8.ACollect, sort, and organize data into two or three categories.

K.8.BUse data to create real-object and picture graphs.

K.8.CDraw conclusions from real-object and picture graphs.

K.9 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security.

K.9.AIdentify ways to earn income.

K.9.BDifferentiate between money received as income and money received as gifts.

K.9.CList simple skills required for jobs.

K.9.DDistinguish between wants and needs and identify income as a source to meet one's wants and needs.

Grade 1

1.2 Number and operations. The student applies mathematical process standards to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system related to place value.

1.2.ARecognize instantly the quantity of structured arrangements.

1.2.BUse concrete and pictorial models to compose and decompose numbers up to 120 in more than one way as so many hundreds, so many tens, and so many ones.

1.2.CUse objects, pictures, and expanded and standard forms to represent numbers up to 120.

1.2.DGenerate a number that is greater than or less than a given whole number up to 120.

1.2.EUse place value to compare whole numbers up to 120 using comparative language.

1.2.FOrder whole numbers up to 120 using place value and open number lines.

1.2.GRepresent the comparison of two numbers to 100 using the symbols >, <, or =.

1.3 Number and operations. The student applies mathematical process standards to develop and use strategies for whole number addition and subtraction computations in order to solve problems.

1.3.AUse concrete and pictorial models to determine the sum of a multiple of 10 and a one-digit number in problems up to 99.

1.3.BUse objects and pictorial models to solve word problems involving joining, separating, and comparing sets within 20 and unknowns as any one of the terms in the problem such as 2 + 4 = [ ]; 3 + [ ] = 7; and 5 = [ ] - 3.

1.3.CCompose 10 with two or more addends with and without concrete objects.

1.3.DApply basic fact strategies to add and subtract within 20, including making 10 and decomposing a number leading to a 10.

1.3.EExplain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.

1.3.FGenerate and solve problem situations when given a number sentence involving addition or subtraction of numbers within 20.

1.4 Number and operations. The student applies mathematical process standards to identify coins, their values, and the relationships among them in order to recognize the need for monetary transactions.

1.4.AIdentify U.S. coins, including pennies, nickels, dimes, and quarters, by value and describe the relationships among them.

1.4.BWrite a number with the cent symbol to describe the value of a coin.

1.4.CUse relationships to count by twos, fives, and tens to determine the value of a collection of pennies, nickels, and/or dimes.

1.5 Algebraic reasoning. The student applies mathematical process standards to identify and apply number patterns within properties of numbers and operations in order to describe relationships.

1.5.ARecite numbers forward and backward from any given number between 1 and 120.

1.5.BSkip count by twos, fives, and tens to determine the total number of objects up to 120 in a set.

1.5.CUse relationships to determine the number that is 10 more and 10 less than a given number up to 120.

1.5.DRepresent word problems involving addition and subtraction of whole numbers up to 20 using concrete and pictorial models and number sentences.

1.5.EUnderstand that the equal sign represents a relationship where expressions on each side of the equal sign represent the same value(s).

1.5.FDetermine the unknown whole number in an addition or subtraction equation when the unknown may be any one of the three or four terms in the equation.

1.5.GApply properties of operations to add and subtract two or three numbers.

1.6 Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties.

1.6.AClassify and sort regular and irregular two-dimensional shapes based on attributes using informal geometric language.

1.6.BDistinguish between attributes that define a two-dimensional or three-dimensional figure and attributes that do not define the shape.

1.6.CCreate two-dimensional figures, including circles, triangles, rectangles, and squares, as special rectangles, rhombuses, and hexagons.

1.6.DIdentify two-dimensional shapes, including circles, triangles, rectangles, and squares, as special rectangles, rhombuses, and hexagons and describe their attributes using formal geometric language.

1.6.EIdentify three-dimensional solids, including spheres, cones, cylinders, rectangular prisms (including cubes), and triangular prisms, and describe their attributes using formal geometric language.

1.6.FCompose two-dimensional shapes by joining two, three, or four figures to produce a target shape in more than one way if possible.

1.6.GPartition two-dimensional figures into two and four fair shares or equal parts and describe the parts using words.

1.6.HIdentify examples and non-examples of halves and fourths.

1.7 Geometry and measurement. The student applies mathematical process standards to select and use units to describe length and time.

1.7.AUse measuring tools to measure the length of objects to reinforce the continuous nature of linear measurement.

1.7.BIllustrate that the length of an object is the number of same-size units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other.

1.7.CMeasure the same object/distance with units of two different lengths and describe how and why the measurements differ.

1.7.DDescribe a length to the nearest whole unit using a number and a unit.

1.7.ETell time to the hour and half hour using analog and digital clocks.

1.8 Data analysis. The student applies mathematical process standards to organize data to make it useful for interpreting information and solving problems.

1.8.ACollect, sort, and organize data in up to three categories using models/representations such as tally marks or T-charts.

1.8.BUse data to create picture and bar-type graphs.

1.8.CDraw conclusions and generate and answer questions using information from picture and bar-type graphs.

1.9 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security.

1.9.ADefine money earned as income.

1.9.BIdentify income as a means of obtaining goods and services, oftentimes making choices between wants and needs.

1.9.CDistinguish between spending and saving.

1.9.DConsider charitable giving.

Grade 2

2.2 Number and operations. The student applies mathematical process standards to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system related to place value.

2.2.AUse concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens, and ones.

2.2.BUse standard, word, and expanded forms to represent numbers up to 1,200.

2.2.CGenerate a number that is greater than or less than a given whole number up to 1,200.

2.2.DUse place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and symbols (>, <, or =).

2.2.ELocate the position of a given whole number on an open number line.

2.2.FName the whole number that corresponds to a specific point on a number line.

2.3 Number and operations. The student applies mathematical process standards to recognize and represent fractional units and communicates how they are used to name parts of a whole.

2.3.APartition objects into equal parts and name the parts, including halves, fourths, and eighths, using words.

2.3.BExplain that the more fractional parts used to make a whole, the smaller the part; and the fewer the fractional parts, the larger the part.

2.3.CUse concrete models to count fractional parts beyond one whole using words and recognize how many parts it takes to equal one whole.

2.3.DIdentify examples and non-examples of halves, fourths, and eighths.

2.4 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations in order to solve addition and subtraction problems with efficiency and accuracy.

2.4.ARecall basic facts to add and subtract within 20 with automaticity.

2.4.BAdd up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations.

2.4.CSolve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms.

2.4.DGenerate and solve problem situations for a given mathematical number sentence involving addition and subtraction of whole numbers within 1,000.

2.5 Number and operations. The student applies mathematical process standards to determine the value of coins in order to solve monetary transactions.

2.5.ADetermine the value of a collection of coins up to one dollar.

2.5.BUse the cent symbol, dollar sign, and the decimal point to name the value of a collection of coins.

2.6 Number and operations. The student applies mathematical process standards to connect repeated addition and subtraction to multiplication and division situations that involve equal groupings and shares.

2.6.AModel, create, and describe contextual multiplication situations in which equivalent sets of concrete objects are joined.

2.6.BModel, create, and describe contextual division situations in which a set of concrete objects is separated into equivalent sets.

2.7 Algebraic reasoning. The student applies mathematical process standards to identify and apply number patterns within properties of numbers and operations in order to describe relationships.

2.7.ADetermine whether a number up to 40 is even or odd using pairings of objects to represent the number.

2.7.BUse an understanding of place value to determine the number that is 10 or 100 more or less than a given number up to 1,200.

2.7.CRepresent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem.

2.8 Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional shapes and three-dimensional solids to develop generalizations about their properties.

2.8.ACreate two-dimensional shapes based on given attributes, including number of sides and vertices.

2.8.BClassify and sort three-dimensional solids, including spheres, cones, cylinders, rectangular prisms (including cubes as special rectangular prisms), and triangular prisms, based on attributes using formal geometric language.

2.8.CClassify and sort polygons with 12 or fewer sides according to attributes, including identifying the number of sides and number of vertices.

2.8.DCompose two-dimensional shapes and three-dimensional solids with given properties or attributes.

2.8.EDecompose two-dimensional shapes such as cutting out a square from a rectangle, dividing a shape in half, or partitioning a rectangle into identical triangles and identify the resulting geometric parts.

2.9 Geometry and measurement. The student applies mathematical process standards to select and use units to describe length, area, and time.

2.9.AFind the length of objects using concrete models for standard units of length.

2.9.BDescribe the inverse relationship between the size of the unit and the number of units needed to equal the length of an object.

2.9.CRepresent whole numbers as distances from any given location on a number line.

2.9.DDetermine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes.

2.9.EDetermine a solution to a problem involving length, including estimating lengths.

2.9.FUse concrete models of square units to find the area of a rectangle by covering it with no gaps or overlaps, counting to find the total number of square units, and describing the measurement using a number and the unit.

2.9.GRead and write time to the nearest one-minute increment using analog and digital clocks and distinguish between a.m. and p.m.

2.10 Data analysis. The student applies mathematical process standards to organize data to make it useful for interpreting information and solving problems.

2.10.AExplain that the length of a bar in a bar graph or the number of pictures in a pictograph represents the number of data points for a given category.

2.10.BOrganize a collection of data with up to four categories using pictographs and bar graphs with intervals of one or more.

2.10.CWrite and solve one-step word problems involving addition or subtraction using data represented within pictographs and bar graphs with intervals of one.

2.10.DDraw conclusions and make predictions from information in a graph.

2.11 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security.

2.11.ACalculate how money saved can accumulate into a larger amount over time.

2.11.BExplain that saving is an alternative to spending.

2.11.CDistinguish between a deposit and a withdrawal.

2.11.DIdentify examples of borrowing and distinguish between responsible and irresponsible borrowing.

2.11.EIdentify examples of lending and use concepts of benefits and costs to evaluate lending decisions.

2.11.FDifferentiate between producers and consumers and calculate the cost to produce a simple item.

Grade 3

3.2 Number and operations. The student applies mathematical process standards to represent and compare whole numbers and understand relationships related to place value.

3.2.ACompose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.

3.2.BDescribe the mathematical relationships found in the base-10 place value system through the hundred thousands place.

3.2.CRepresent a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers.

3.2.DCompare and order whole numbers up to 100,000 and represent comparisons using the symbols >, <, or =.

3.3 Number and operations. The student applies mathematical process standards to represent and explain fractional units.

3.3.ARepresent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.

3.3.BDetermine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.

3.3.CExplain that the unit fraction 1/b represents the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number.

3.3.DCompose and decompose a fraction a/b with a numerator greater than zero and less than or equal to b as a sum of parts 1/b.

3.3.ESolve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8.

3.3.FRepresent equivalent fractions with denominators of 2, 3, 4, 6, and 8 using a variety of objects and pictorial models, including number lines.

3.3.GExplain that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model.

3.3.HCompare two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.

3.4 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations in order to solve problems with efficiency and accuracy.

3.4.ASolve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.

3.4.BRound to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems.

3.4.CDetermine the value of a collection of coins and bills.

3.4.DDetermine the total number of objects when equally-sized groups of objects are combined or arranged in arrays up to 10 by 10.

3.4.ERepresent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting.

3.4.FRecall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts.

3.4.GUse strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties.

3.4.HDetermine the number of objects in each group when a set of objects is partitioned into equal shares or a set of objects is shared equally.

3.4.IDetermine if a number is even or odd using divisibility rules.

3.4.JDetermine a quotient using the relationship between multiplication and division.

3.4.KSolve one-step and two-step problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.

3.5 Algebraic reasoning. The student applies mathematical process standards to analyze and create patterns and relationships.

3.5.ARepresent one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations.

3.5.BRepresent and solve one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations.

3.5.CDescribe a multiplication expression as a comparison such as 3 x 24 represents 3 times as much as 24.

3.5.DDetermine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product.

3.5.ERepresent real-world relationships using number pairs in a table and verbal descriptions.

3.6 Geometry and measurement. The student applies mathematical process standards to analyze attributes of two-dimensional geometric figures to develop generalizations about their properties.

3.6.AClassify and sort two- and three-dimensional solids, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.

3.6.BUse attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories.

3.6.CDetermine the area of rectangles with whole number side lengths in problems using multiplication related to the number of rows times the number of unit squares in each row.

3.6.DDecompose composite figures formed by rectangles into non-overlapping rectangles to determine the area of the original figure using the additive property of area.

3.6.EDecompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole and recognize that equal shares of identical wholes need not have the same shape.

3.7 Geometry and measurement. The student applies mathematical process standards to select appropriate units, strategies, and tools to solve problems involving customary and metric measurement.

3.7.ARepresent fractions of halves, fourths, and eighths as distances from zero on a number line.

3.7.BDetermine the perimeter of a polygon or a missing length when given perimeter and remaining side lengths in problems.

3.7.CDetermine the solutions to problems involving addition and subtraction of time intervals in minutes using pictorial models or tools such as a 15-minute event plus a 30-minute event equals 45 minutes.

3.7.DDetermine when it is appropriate to use measurements of liquid volume (capacity) or weight.

3.7.EDetermine liquid volume (capacity) or weight using appropriate units and tools.

3.8 Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data.

3.8.ASummarize a data set with multiple categories using a frequency table, dot plot, pictograph, or bar graph with scaled intervals.

3.8.BSolve one- and two-step problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.

3.9 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security.

3.9.AExplain the connection between human capital/labor and income.

3.9.BDescribe the relationship between the availability or scarcity of resources and how that impacts cost.

3.9.CIdentify the costs and benefits of planned and unplanned spending decisions.

3.9.DExplain that credit is used when wants or needs exceed the ability to pay and that it is the borrower's responsibility to pay it back to the lender, usually with interest.

3.9.EList reasons to save and explain the benefit of a savings plan, including for college.

3.9.FIdentify decisions involving income, spending, saving, credit, and charitable giving.

Grade 4

4.2 Number and operations. The student applies mathematical process standards to represent, compare, and order whole numbers and decimals and understand relationships related to place value.

4.2.AInterpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left.

4.2.BRepresent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals.

4.2.CCompare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =.

4.2.DRound whole numbers to a given place value through the hundred thousands place.

4.2.ERepresent decimals, including tenths and hundredths, using concrete and visual models and money.

4.2.FCompare and order decimals using concrete and visual models to the hundredths.

4.2.GRelate decimals to fractions that name tenths and hundredths.

4.2.HDetermine the corresponding decimal to the tenths or hundredths place of a specified point on a number line.

4.3 Number and operations. The student applies mathematical process standards to represent and generate fractions to solve problems.

4.3.ARepresent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b.

4.3.BDecompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations.

4.3.CDetermine if two given fractions are equivalent using a variety of methods.

4.3.DCompare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <.

4.3.ERepresent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.

4.3.FEvaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole.

4.3.GRepresent fractions and decimals to the tenths or hundredths as distances from zero on a number line.

4.4 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations and decimal sums and differences in order to solve problems with efficiency and accuracy.

4.4.AAdd and subtract whole numbers and decimals to the hundredths place using the standard algorithm.

4.4.BDetermine products of a number and 10 or 100 using properties of operations and place value understandings.

4.4.CRepresent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15.

4.4.DUse strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties.

4.4.ERepresent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations.

4.4.FUse strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor.

4.4.GRound to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers.

4.4.HSolve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.

4.5 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations.

4.5.ARepresent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.

4.5.BRepresent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence.

4.5.CUse models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w); and

4.5.DSolve problems related to perimeter and area of rectangles where dimensions are whole numbers.

4.6 Geometry and measurement. The student applies mathematical process standards to analyze geometric attributes in order to develop generalizations about their properties.

4.6.AIdentify points, lines, line segments, rays, angles, and perpendicular and parallel lines.

4.6.BIdentify and draw one or more lines of symmetry, if they exist, for a two-dimensional figure.

4.6.CApply knowledge of right angles to identify acute, right, and obtuse triangles.

4.6.DClassify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.

4.7 Geometry and measurement. The student applies mathematical process standards to solve problems involving angles less than or equal to 180 degrees.

4.7.AIllustrate the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers.

4.7.BIllustrate degrees as the units used to measure an angle, where 1/360 of any circle is one degree and an angle that "cuts" n/360 out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers.

4.7.CDetermine the approximate measures of angles in degrees to the nearest whole number using a protractor.

4.7.DDraw an angle with a given measure.

4.7.EDetermine the measure of an unknown angle formed by two non-overlapping adjacent angles given one or both angle measures.

4.8 Geometry and measurement. The student applies mathematical process standards to select appropriate customary and metric units, strategies, and tools to solve problems involving measurement.

4.8.AIdentify relative sizes of measurement units within the customary and metric systems.

4.8.BConvert measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table.

4.8.CSolve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.

4.9 Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data.

4.9.ARepresent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions.

4.9.BSolve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot.

4.10 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security.

4.10.ADistinguish between fixed and variable expenses.

4.10.BCalculate profit in a given situation.

4.10.CCompare the advantages and disadvantages of various savings options.

4.10.DDescribe how to allocate a weekly allowance among spending; saving, including for college; and sharing.

4.10.EDescribe the basic purpose of financial institutions, including keeping money safe, borrowing money, and lending.

Grade 5

5.2 Number and operations. The student applies mathematical process standards to represent, compare, and order positive rational numbers and understand relationships as related to place value.

5.2.ARepresent the value of the digit in decimals through the thousandths using expanded notation and numerals.

5.2.BCompare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.

5.2.CRound decimals to tenths or hundredths.

5.3 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for positive rational number computations in order to solve problems with efficiency and accuracy.

5.3.AEstimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division.

5.3.BMultiply with fluency a three-digit number by a two-digit number using the standard algorithm.

5.3.CSolve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm.

5.3.DRepresent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models.

5.3.ESolve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers.

5.3.FRepresent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models.

5.3.GSolve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm;

5.3.HRepresent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations.

5.3.IRepresent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.

5.3.JRepresent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models.

5.3.KAdd and subtract positive rational numbers fluently.

5.3.LDivide whole numbers by unit fractions and unit fractions by whole numbers.

5.4 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations.

5.4.AIdentify prime and composite numbers.

5.4.BRepresent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.

5.4.CGenerate a numerical pattern when given a rule in the form y = ax or y = x + a and graph.

5.4.DRecognize the difference between additive and multiplicative numerical patterns given in a table or graph.

5.4.EDescribe the meaning of parentheses and brackets in a numeric expression.

5.4.FSimplify numerical expressions that do not involve exponents, including up to two levels of grouping.

5.4.GUse concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l x w x h, V = s x s x s, and V = Bh).

5.4.HRepresent and solve problems related to perimeter and/or area and related to volume.

5.5 Geometry and measurement. The student applies mathematical process standards to classify twodimensional figures by attributes and properties.

5.5.AClassify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.

5.6 Geometry and measurement. The student applies mathematical process standards to understand, recognize, and quantify volume.

5.6.ARecognize a cube with side length of one unit as a unit cube having one cubic unit of volume and the volume of a three-dimensional figure as the number of unit cubes (n cubic units) needed to fill it with no gaps or overlaps if possible; and

5.6.BDetermine the volume of a rectangular prism with whole number side lengths in problems related to the number of layers times the number of unit cubes in the area of the base.

5.7 Geometry and measurement. The student applies mathematical process standards to understand, recognize, and quatify volume.

5.7.ASolve problems by calculating conversions within a measurement system, customary or metric.

5.8 Geometry and measurement. The student applies mathematical process standards to identify locations on a coordinate plane.

5.8.ADescribe the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and the given point (0, 0); the x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number, indicates movement parallel to the y-axis starting at the origin.

5.8.BDescribe the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane.

5.8.CGraph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table.

5.9 Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data.

5.9.ARepresent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots.

5.9.BRepresent discrete paired data on a scatterplot.

5.9.CSolve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot.

5.10 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security.

5.10.ADefine income tax, payroll tax, sales tax, and property tax.

5.10.BExplain the difference between gross income and net income.

5.10.CIdentify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments.

5.10.DDevelop a system for keeping and using financial records.

5.10.EDescribe actions that might be taken to balance a budget when expenses exceed income.

5.10.FBalance a simple budget.

Grade 6

6.2 Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms.

6.2.AClassify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.

6.2.BIdentify a number, its opposite, and its absolute value.

6.2.CLocate, compare, and order integers and rational numbers using a number line.

6.2.DOrder a set of rational numbers arising from mathematical and real-world contexts.

6.2.EExtend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.

6.3 Number and operations. The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions.

6.3.ARecognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values.

6.3.BDetermine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.

6.3.CRepresent integer operations with concrete models and connect the actions with the models to standardized algorithms.

6.3.DAdd, subtract, multiply, and divide integers fluently.

6.3.EMultiply and divide positive rational numbers fluently.

6.4 Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations.

6.4.ACompare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.

6.4.BApply qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates.

6.4.CGive examples of ratios as multiplicative comparisons of two quantities describing the same attribute.

6.4.DGive examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.

6.4.ERepresent ratios and percents with concrete models, fractions, and decimals.

6.4.FRepresent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers.

6.4.GGenerate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.

6.4.HConvert units within a measurement system, including the use of proportions and unit rates.

6.5 Proportionality. The student applies mathematical process standards to solve problems involving proportional relationships.

6.5.ARepresent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions.

6.5.BSolve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models.

6.5.CUse equivalent fractions, decimals, and percents to show equal parts of the same whole.

6.6 Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships.

6.6.AIdentify independent and dependent quantities from tables and graphs.

6.6.BWrite an equation that represents the relationship between independent and dependent quantities from a table.

6.6.CRepresent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.

6.7 Expressions, equations, and relationships. The student applies mathematical process standards to develop concepts of expressions and equations.

6.7.AGenerate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

6.7.BDistinguish between expressions and equations verbally, numerically, and algebraically.

6.7.CDetermine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.

6.7.DGenerate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

6.8 Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to represent relationships and solve problems.

6.8.AExtend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle.

6.8.BModel area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes.

6.8.CWrite equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.

6.8.DDetermine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.

6.9 Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to represent situations.

6.9.AWrite one-variable, one-step equations and inequalities to represent constraints or conditions within problems.

6.9.BRepresent solutions for one-variable, one-step equations and inequalities on number lines.

6.9.CWrite corresponding real-world problems given one-variable, one-step equations or inequalities.

6.10 Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to solve problems.

6.10.AModel and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.

6.10.BDetermine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

6.11 Measurement and data. The student applies mathematical process standards to use coordinate geometry to identify locations on a plane.

6.11.AGraph points in all four quadrants using ordered pairs of rational numbers.

6.12 Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to analyze problems.

6.12.ARepresent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots.

6.12.BUse the graphical representation of numeric data to describe the center, spread, and shape of the data distribution.

6.12.CSummarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution.

6.12.DSummarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution.

6.13 Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to solve problems.

6.13.AInterpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots.

6.13.BDistinguish between situations that yield data with and without variability.

6.14 Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor.

6.14.ACompare the features and costs of a checking account and a debit card offered by different local financial institutions.

6.14.BDistinguish between debit cards and credit cards.

6.14.CBalance a check register that includes deposits, withdrawals, and transfers.

6.14.DExplain why it is important to establish a positive credit history.

6.14.EDescribe the information in a credit report and how long it is retained.

6.14.FDescribe the value of credit reports to borrowers and to lenders.

6.14.GExplain various methods to pay for college, including through savings, grants, scholarships, student loans, and work-study.

6.14.HCompare the annual salary of several occupations requiring various levels of post-secondary education or vocational training and calculate the effects of the different annual salaries on lifetime income.

Grade 7

7.2 Numbers and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms.

7.2.AExtend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers.

7.3 Number and operations. The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions.

7.3.AAdd, subtract, multiply, and divide rational numbers fluently.

7.3.BApply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.

7.4 Proportionality. The student applies mathematical process standards to represent and solve problems involving proportional relationships.

7.4.ARepresent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.

7.4.BCalculate unit rates from rates in mathematical and real-world problems.

7.4.CDetermine the constant of proportionality (k = y/x) within mathematical and real-world problems.

7.4.DSolve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.

7.4.EConvert between measurement systems, including the use of proportions and the use of unit rates.

7.5 Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships.

7.5.AGeneralize the critical attributes of similarity, including ratios within and between similar shapes.

7.5.BDescribe π as the ratio of the circumference of a circle to its diameter.

7.5.CSolve mathematical and real-world problems involving similar shape and scale drawings.

7.6 Proportionality. The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships.

7.6.ARepresent sample spaces for simple and compound events using lists and tree diagrams.

7.6.BSelect and use different simulations to represent simple and compound events with and without technology.

7.6.CMake predictions and determine solutions using experimental data for simple and compound events;

7.6.DMake predictions and determine solutions using theoretical probability for simple and compound events.

7.6.EFind the probabilities of a simple event and its complement and describe the relationship between the two.

7.6.FUse data from a random sample to make inferences about a population.

7.6.GSolve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents.

7.6.HSolve problems using qualitative and quantitative predictions and comparisons from simple experiments.

7.6.IDetermine experimental and theoretical probabilities related to simple and compound events using data and sample spaces.

7.7 Expressions, equations, and relationships. The student applies mathematical process standards to represent linear relationships using multiple representations.

7.7.ARepresent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.

7.8 Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume.

7.8.AModel the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas.

7.8.BExplain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas.

7.8.CUse models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas.

7.9 Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems.

7.9.ASolve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids.

7.9.BDetermine the circumference and area of circles.

7.9.CDetermine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles.

7.9.DSolve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net.

7.10 Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations and inequalities to represent situations.

7.10.AWrite one-variable, two-step equations and inequalities to represent constraints or conditions within problems.

7.10.BRepresent solutions for one-variable, two-step equations and inequalities on number lines.

7.10.CWrite a corresponding real-world problem given a one-variable, two-step equation or inequality.

7.11 Expressions, equations, and relationships. The student applies mathematical process standards to solve one-variable equations and inequalities.

7.11.AModel and solve one-variable, two-step equations and inequalities.

7.11.BDetermine if the given value(s) make(s) one-variable, two-step equations and inequalities true.

7.11.CWrite and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.

7.12 Measurement and data. The student applies mathematical process standards to use statistical representations to analyze data.

7.12.ACompare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads.

7.12.BUse data from a random sample to make inferences about a population.

7.12.CCompare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations.

7.13 Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor.

7.13.ACalculate the sales tax for a given purchase and calculate income tax for earned wages.

7.13.BIdentify the components of a personal budget, including income; planned savings for college, retirement, and emergencies; taxes; and fixed and variable expenses, and calculate what percentage each category comprises of the total budget.

7.13.CCreate and organize a financial assets and liabilities record and construct a net worth statement.

7.13.DUse a family budget estimator to determine the minimum household budget and average hourly wage needed for a family to meet its basic needs in the student's city or another large city nearby.

7.13.ECalculate and compare simple interest and compound interest earnings.

7.13.FAnalyze and compare monetary incentives, including sales, rebates, and coupons.

Grade 8

8.2 Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms.

8.2.AExtend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers.

8.2.BApproximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line.

8.2.CConvert between standard decimal notation and scientific notation.

8.2.DOrder a set of real numbers arising from mathematical and real-world contexts.

8.3 Proportionality. The student applies mathematical process standards to use proportional relationships to describe dilations.

8.3.AGeneralize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation.

8.3.BCompare and contrast the attributes of a shape and its dilation(s) on a coordinate plane.

8.3.CUse an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.

8.4 Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope.

8.4.AUse similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (y2 - y1)/ (x2 - x1), is the same for any two points (x1, y1) and (x2, y2) on the same line.

8.4.BGraph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship.

8.4.CUse data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

8.5 Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions.

8.5.ARepresent linear proportional situations with tables, graphs, and equations in the form of y = kx.

8.5.BRepresent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.

8.5.CContrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation.

8.5.DUse a trend line that approximates the linear relationship between bivariate sets of data to make predictions.

8.5.ESolve problems involving direct variation.

8.5.FDistinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0.

8.5.GIdentify functions using sets of ordered pairs, tables, mappings, and graphs.

8.5.HIdentify examples of proportional and non-proportional functions that arise from mathematical and real-world problems.

8.5.IWrite an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

8.6 Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas.

8.6.ADescribe the volume formula V = Bh of a cylinder in terms of its base area and its height.

8.6.BModel the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas.

8.6.CUse models and diagrams to explain the Pythagorean theorem.

8.7 Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems.

8.7.ASolve problems involving the volume of cylinders, cones, and spheres.

8.7.BUse previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders.

8.7.CUse the Pythagorean Theorem and its converse to solve problems.

8.7.DDetermine the distance between two points on a coordinate plane using the Pythagorean Theorem.

8.8 Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations or inequalities in problem situations.

8.8.AWrite one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants.

8.8.BWrite a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants.

8.8.CModel and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants.

8.8.DUse informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

8.9 Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations.

8.9.AIdentify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.

8.10 Two-dimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts.

8.10.AGeneralize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two-dimensional shapes on a coordinate plane.

8.10.BDifferentiate between transformations that preserve congruence and those that do not.

8.10.CExplain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation.

8.10.DModel the effect on linear and area measurements of dilated two-dimensional shapes.

8.11 Measurement and data. The student applies mathematical process standards to use statistical procedures to describe data.

8.11.AConstruct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data.

8.11.BDetermine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points.

8.11.CSimulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected.

8.12 Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor.

8.12.ASolve real-world problems comparing how interest rate and loan length affect the cost of credit.

8.12.BCalculate the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest and over different periods using an online calculator.

8.12.CExplain how small amounts of money invested regularly, including money saved for college and retirement, grow over time.

8.12.DCalculate and compare simple interest and compound interest earnings.

8.12.EIdentify and explain the advantages and disadvantages of different payment methods.

8.12.FAnalyze situations to determine if they represent financially responsible decisions and identify the benefits of financial responsibility and the costs of financial irresponsibility.

8.12.GEstimate the cost of a two-year and four-year college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college.

Algebra I

A.2 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations.

A.2.ADetermine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities.

A.2.BWrite linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1), given one point and the slope and given two points.

A.2.CWrite linear equations in two variables given a table of values, a graph, and a verbal description.

A.2.DWrite and solve equations involving direct variation.

A.2.EWrite the equation of a line that contains a given point and is parallel to a given line.

A.2.FWrite the equation of a line that contains a given point and is perpendicular to a given line.

A.2.GWrite an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined.

A.2.HWrite linear inequalities in two variables given a table of values, a graph, and a verbal description.

A.2.IWrite systems of two linear equations given a table of values, a graph, and a verbal description.

A.3 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations.

A.3.ADetermine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1).

A.3.BCalculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.

A.3.CGraph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems.

A.3.DGraph the solution set of linear inequalities in two variables on the coordinate plane.

A.3.EDetermine the effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

A.3.FGraph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist.

A.3.GEstimate graphically the solutions to systems of two linear equations with two variables in real-world problems.

A.3.HGraph the solution set of systems of two linear inequalities in two variables on the coordinate plane.

A.4 Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data.

A.4.ACalculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.

A.4.BCompare and contrast association and causation in real-world problems.

A.4.CWrite, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.

A.5 Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions.

A.5.ASolve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.

A.5.BSolve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.

A.5.CSolve systems of two linear equations with two variables for mathematical and real-world problems.

A.6 Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations.

A.6.ADetermine the domain and range of quadratic functions and represent the domain and range using inequalities.

A.6.BWrite equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x - h)^2 + k), and rewrite the equation from vertex form to standard form (f(x) = ax^2 + bx + c).

A.6.CWrite quadratic functions when given real solutions and graphs of their related equations.

A.7 Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

A.7.AGraph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry.

A.7.BDescribe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions.

A.7.CDetermine the effects on the graph of the parent function f(x) = x^2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

A.8 Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

A.8.ASolve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula.

A.8.BWrite, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.

A.9 Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

A.9.ADetermine the domain and range of exponential functions of the form f(x) = ab^x and represent the domain and range using inequalities.

A.9.BInterpret the meaning of the values of a and b in exponential functions of the form f(x) = ab^x in real-world problems.

A.9.CWrite exponential functions in the form f(x) = ab^x (where b is a rational number)to describe problems arising from mathematical and real-world situations, including growth and decay.

A.9.DGraph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems.

A.9.EWrite, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems.

A.10 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions.

A.10.AAdd and subtract polynomials of degree one and degree two.

A.10.BMultiply polynomials of degree one and degree two.

A.10.CDetermine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend.

A.10.DRewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property.

A.10.EFactor, if possible, trinomials with real factors in the form ax^2 + bx + c, including perfect square trinomials of degree two.

A.10.FDecide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.

A.11 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms.

A.11.ASimplify numerical radical expressions involving square roots.

A.11.BSimplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

A.12 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions.

A.12.ADecide whether relations represented verbally, tabularly, graphically, and symbolically define a function.

A.12.BEvaluate functions, expressed in function notation, given one or more elements in their domains.

A.12.CIdentify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes.

A.12.DWrite a formula for the nth term of arithmetic and geometric sequences, given the value of several of their terms.

A.12.ESolve mathematic and scientific formulas, and other literal equations, for a specified variable.

Algebra II

A2.2 Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse.

A2.2.AGraph the functions f(x)=√x, f(x)=1/x, f(x)=x^3, f(x)=cube root of x, f(x)=b^x, f(x)=|x|, and f(x)=log base b of (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.

A2.2.BGraph and write the inverse of a function using notation such as f ^-1 (x).

A2.2.CDescribe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range.

A2.2.DUse the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

A2.3 Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.

A2.3.AFormulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic.

A2.3.BSolve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.

A2.3.CSolve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation.

A2.3.DDetermine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables.

A2.3.EFormulate systems of at least two linear inequalities in two variables.

A2.3.FSolve systems of two or more linear inequalities in two variables.

A2.3.GDetermine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

A2.4 Quadratic and square root functions, equations, and inequalities. The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions.

A2.4.AWrite the quadratic function given three specified points in the plane.

A2.4.BWrite the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.

A2.4.CDetermine the effect on the graph of f(x) = √x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c)

A2.4.DTransform a quadratic function f(x) = ax^2 + bx + c to the form f(x) = a(x - h)^2 + k to identify the different attributes of f(x).

A2.4.EFormulate quadratic and square root equations using technology given a table of data.

A2.4.FSolve quadratic and square root equations.

A2.4.GIdentify extraneous solutions of square root equations.

A2.4.HSolve quadratic inequalities.

A2.5 Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems.

A2.5.ADetermine the effects on the key attributes on the graphs of f(x) = b^x and f(x) = log base b of x where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d.

A2.5.BFormulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation.

A2.5.CRewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations.

A2.5.DSolve exponential equations of the form y = ab^x where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions.

A2.5.EDetermine the reasonableness of a solution to a logarithmic equation.

A2.6 Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions.

A2.6.AAnalyze the effect on the graphs of f(x) = x^3 and f(x) = cube root of x when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d.

A2.6.BSolve cube root equations that have real roots.

A2.6.CAnalyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d.

A2.6.DFormulate absolute value linear equations.

A2.6.ESolve absolute value linear equations.

A2.6.FSolve absolute value linear inequalities.

A2.6.GAnalyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d.

A2.6.HFormulate rational equations that model real-world situations.

A2.6.ISolve rational equations that have real solutions.

A2.6.JDetermine the reasonableness of a solution to a rational equation.

A2.6.KDetermine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation.

A2.6.LFormulate and solve equations involving inverse variation.

A2.7 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations.

A2.7.AAdd, subtract, and multiply complex numbers.

A2.7.BAdd, subtract, and multiply polynomials.

A2.7.CDetermine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two.

A2.7.DDetermine the linear factors of a polynomial function of degree three and of degree four using algebraic methods.

A2.7.EDetermine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping.

A2.7.FDetermine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two.

A2.7.GRewrite radical expressions that contain variables to equivalent forms.

A2.7.HSolve equations involving rational exponents.

A2.7.IWrite the domain and range of a function in interval notation, inequalities, and set notation.

A2.8 Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions.

A2.8.AAnalyze data to select the appropriate model from among linear, quadratic, and exponential models.

A2.8.BUse regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data.

A2.8.CPredict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.

Advanced Quantitative Reasoning

AQR.2 Numeric reasoning. The student applies the process standards in mathematics to generate new understandings by extending existing knowledge. The student generates new mathematical understandings through problems involving numerical data that arise in everyday life, society, and the workplace. The student extends existing knowledge and skills to analyze real-world situations.

AQR.2.AUse precision and accuracy in real-life situations related to measurement and significant figures.

AQR.2.BApply and analyze published ratings, weighted averages, and indices to make informed decisions.

AQR.2.CSolve problems involving quantities that are not easily measured using proportionality.

AQR.2.DSolve geometric problems involving indirect measurement, including similar triangles, the Pythagorean Theorem, Law of Sines, Law of Cosines, and the use of dynamic geometry software.

AQR.2.ESolve problems involving large quantities using combinatorics.

AQR.2.FUse arrays to efficiently manage large collections of data and add, subtract, and multiply matrices to solve applied problems, including geometric transformations.

AQR.2.GAnalyze various voting and selection processes to compare results in given situations.

AQR.2.HSelect and apply an algorithm of interest to solve real-life problems such as problems using recursion or iteration involving population growth or decline, fractals, and compound interest; the validity in recorded and transmitted data using checksums and hashing; sports rankings, weighted class rankings, and search engine rankings; and problems involving scheduling or routing situations using vertex-edge graphs, critical paths, Euler paths, and minimal spanning trees and communicate to peers the application of the algorithm in precise mathematical and nontechnical language.

AQR.3 Algebraic reasoning (expressions, equations, and generalized relationships). The student applies the process standards in mathematics to create and analyze mathematical models of everyday situations to make informed decisions related to earning, investing, spending, and borrowing money by appropriate, proficient, and efficient use of tools, including technology. The student uses mathematical relationships to make connections and predictions. The student judges the validity of a prediction and uses mathematical models to represent, analyze, and solve dynamic real-world problems.

AQR.3.ACollect numerical bivariate data to create a scatterplot, select a function to model the data, justify the model selection, and use the model to interpret results and make predictions.

AQR.3.BDescribe the degree to which uncorrelated variables may or may not be related and analyze situations where correlated variables do or do not indicate a cause-and-effect relationship.

AQR.3.CDetermine or analyze an appropriate growth or decay model for problem situations, including linear, exponential, and logistic functions.

AQR.3.DDetermine or analyze an appropriate cyclical model for problem situations that can be modeled with periodic functions.

AQR.3.EDetermine or analyze an appropriate piecewise model for problem situations.

AQR.3.FCreate, represent, and analyze mathematical models for various types of income calculations to determine the best option for a given situation.

AQR.3.GCreate, represent, and analyze mathematical models for expenditures, including those involving credit, to determine the best option for a given situation.

AQR.3.HCreate, represent, and analyze mathematical models and appropriate representations, including formulas and amortization tables, for various types of loans and investments to determine the best option for a given situation.

AQR.4 Probabilistic and statistical reasoning. The student uses the process standards in mathematics to generate new understandings of probability and statistics. The student analyzes statistical information and evaluates risk and return to connect mathematical ideas and make informed decisions. The student applies a problem-solving model and statistical methods to design and conduct a study that addresses one or more particular question(s). The student uses multiple representations to communicate effectively the results of student-generated statistical studies and the critical analysis of published statistical studies.

AQR.4.AUse a two-way frequency table as a sample space to identify whether two events are independent and to interpret the results.

AQR.4.BUse the Addition Rule, P(A or B)= P.A+ P.B- P(A and B), in mathematical and real-world problems.

AQR.4.CCalculate conditional probabilities and probabilities of compound events using tree diagrams, Venn diagrams, area models, and formulas.

AQR.4.DInterpret conditional probabilities and probabilities of compound events by analyzing representations to make decisions in problem situations.

AQR.4.EUse probabilities to make and justify decisions about risks in everyday life.

AQR.4.FCalculate expected value to analyze mathematical fairness, payoff, and risk.

AQR.4.GDetermine the validity of logical arguments that include compound conditional statements by constructing truth tables.

AQR.4.HIdentify limitations and lack of relevant information in studies reporting statistical information, especially when studies are reported in condensed form.

AQR.4.IInterpret and compare statistical results using appropriate technology given a margin of error.

AQR.4.JIdentify potential misuses of statistics to justify particular conclusions, including assertions of a cause-and-effect relationship rather than an association, and missteps or fallacies in logical reasoning.

AQR.4.KDescribe strengths and weaknesses of sampling techniques, data and graphical displays, and interpretations of summary statistics and other results appearing in a study, including reports published in the media.

AQR.4.LDetermine the need for and purpose of a statistical investigation and what type of statistical analysis can be used to answer a specific question or set of questions.

AQR.4.MIdentify the population of interest for a statistical investigation, select an appropriate sampling technique, and collect data.

AQR.4.NIdentify the variables to be used in a study.

AQR.4.ODetermine possible sources of statistical bias in a study and how bias may affect the validity of the results.

AQR.4.PCreate data displays for given data sets to investigate, compare, and estimate center, shape, spread, and unusual features of the data.

AQR.4.QAnalyze possible sources of data variability, including those that can be controlled and those that cannot be controlled.

AQR.4.RReport results of statistical studies to a particular audience, including selecting an appropriate presentation format, creating graphical data displays, and interpreting results in terms of the question studied.

AQR.4.SJustify the design and the conclusion(s) of statistical studies, including the methods used.

AQR.4.TCommunicate statistical results in oral and written formats using appropriate statistical and nontechnical language.

Algebraic Reasoning

AR.2 Patterns and structure. The student applies mathematical processes to connect finite differences or common ratios to attributes of functions.

AR.2.ADetermine the patterns that identify the relationship between a function and its common ratio or related finite differences as appropriate, including linear, quadratic, cubic, and exponential functions.

AR.2.BClassify a function as linear, quadratic, cubic, and exponential when a function is represented tabular using finite differences or common ratios as appropriate.

AR.2.CDetermine the function that models a given table of related values using finite differences and its restricted domain and range.

AR.2.DDetermine a function that models real-world data and mathematical contexts using finite differences such as the age of a tree and its circumference, figurative numbers, average velocity, and average acceleration.

AR.3 Patterns and structure. The student applies mathematical processes to understand the connections among representations of functions and combinations of functions, including the constant function, f(x) = x, f(x) = x^2, f(x) = √x, f(x) = 1/x, f(x) = x^3, f(x) = 3√x, f(x) = b^x, f(x) = |x|, and f(x) = logb (x) where b is 10 or e; functions and their inverses; and key attributes of these functions.

AR.3.ACompare and contrast the key attributes, including domain, range, maxima, minima, and intercepts, of a set of functions such as a set comprised of a linear, a quadratic, and an exponential function or a set comprised of an absolute value, a quadratic, and a square root function tabular, graphically, and symbolically.

AR.3.BCompare and contrast the key attributes of a function and its inverse when it exists, including domain, range, maxima, minima, and intercepts, tabular, graphically, and symbolically.

AR.3.CVerify that two functions are inverses of each other tabular and graphically such as situations involving compound interest and interest rate, velocity and braking distance, and Fahrenheit-Celsius conversions.

AR.3.DRepresent a resulting function tabular, graphically, and symbolically when functions are combined or separated using arithmetic operations such as combining a 20% discount and a 6% sales tax on a sale to determine h(x), the total sale, f(x) = 0.8x, g(x) = 0.06(0.8x), and h(x) = f(x) + g(x).

AR.3.EModel a situation using function notation when the output of one function is the input of a second function such as determining a function h(x) = g(f(x)) = 1.06(0.8x) for the final purchase price, h(x) of an item with price x dollars representing a 20% discount, f(x) = 0.8x followed by a 6% sales tax, g(x) = 1.06x.

AR.3.FCompare and contrast a function and possible functions that can be used to build it tabularly, graphically, and symbolically such as a quadratic function that results from multiplying two linear functions.

AR.4 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on functions represented in a variety of ways, including real-world situations.

AR.4.AConnect tabular representations to symbolic representations when adding, subtracting, and multiplying polynomial functions arising from mathematical and real-world situations such as applications involving surface area and volume.

AR.4.BCompare and contrast the results when adding two linear functions and multiplying two linear functions that are represented tabular, graphically, and symbolically.

AR.4.CDetermine the quotient of a polynomial function of degree three and of degree four when divided by a polynomial function of degree one and of degree two when represented tabular and symbolically.

AR.4.DDetermine the linear factors of a polynomial function of degree two and of degree three when represented symbolically and tabular and graphically where appropriate.

AR.5 Number and algebraic methods. The student applies mathematical processes to represent, simplify, and perform operations on matrices and to solve systems of equations using matrices.

AR.5.AAdd and subtract matrices.

AR.5.BMultiply matrices.

AR.5.CMultiply matrices by a scalar.

AR.5.DRepresent and solve systems of two linear equations arising from mathematical and real-world situations using matrices.

AR.5.ERepresent and solve systems of three linear equations arising from mathematical and real-world situations using matrices and technology.

AR.6 Number and algebraic methods. The student applies mathematical processes to estimate and determine solutions to equations resulting from functions and real-world applications with fluency.

AR.6.AEstimate a reasonable input value that results in a given output value for a given function, including quadratic, rational, and exponential functions.

AR.6.BSolve equations arising from questions asked about functions that model real-world applications, including linear and quadratic functions, tabular, graphically, and symbolically.

AR.6.CApproximate solutions to equations arising from questions asked about exponential, logarithmic, square root, and cubic functions that model real-world applications tabular and graphically.

AR.7 Modeling from data. The student applies mathematical processes to analyze and model data based on real-world situations with corresponding functions.

AR.7.ARepresent domain and range of a function using interval notation, inequalities, and set (builder) notation.

AR.7.BCompare and contrast between the mathematical and reasonable domain and range of functions modeling real-world situations, including linear, quadratic, exponential, and rational functions.

AR.7.CDetermine the accuracy of a prediction from a function that models a set of data compared to the actual data using comparisons between average rates of change and finite differences such as gathering data from an emptying tank and comparing the average rate of change of the volume or the second differences in the volume to key attributes of the given model.

AR.7.DDetermine an appropriate function model, including linear, quadratic, and exponential functions, for a set of data arising from real-world situations using finite differences and average rates of change.

AR.7.EDetermine if a given linear function is a reasonable model for a set of data arising from a real-world situation.

Discrete Mathematics for Problem Solving

DM.2 Graph theory. The student applies the concept of graphs to determine possible solutions to real-world problems.

DM.2.AExplain the concept of graphs.

DM.2.BUse graph models for simple problems in management science.

DM.2.CDetermine the valences of the vertices of a graph.

DM.2.DIdentify Euler circuits in a graph.

DM.2.ESolve route inspection problems by Eulerizing a graph.

DM.2.FDetermine solutions modeled by edge traversal in a graph.

DM.2.GCompare the results of solving the traveling salesman problem (TSP) using the nearest neighbor algorithm and using a greedy algorithm.

DM.2.HDistinguish between real-world problems modeled by Euler circuits and those modeled by Hamiltonian circuits.

DM.2.IDistinguish between algorithms that yield optimal solutions and those that give nearly optimal solutions.

DM.2.JFind minimum-cost spanning trees using Kruskal's algorithm.

DM.2.KUse the critical path method to determine the earliest possible completion time for a collection of tasks.

DM.2.LExplain the difference between a graph and a directed graph.

DM.3 Planning and scheduling. The student uses heuristic algorithms to solve real-world problems.

DM.3.AUse the list processing algorithm to schedule tasks on identical processors.

DM.3.BRecognize situations appropriate for modeling or scheduling problems.

DM.3.CDetermine whether a schedule is optimal using the critical path method together with the list processing algorithm.

DM.3.DIdentify situations appropriate for modeling by bin packing.

DM.3.EUse any of six heuristic algorithms to solve bin packing problems.

DM.3.FSolve independent task scheduling problems using the list processing algorithm.

DM.3.GExplain the relationship between scheduling problems and bin packing problems.

DM.4 Group decision making. The student uses mathematical processes to apply decision-making schemes. The student analyzes the effects of multiple types of weighted voting and applies multiple voting concepts to real-world situations.

DM.4.ADescribe the concept of a preference schedule and how to use it.

DM.4.BExplain how particular decision-making schemes work.

DM.4.CDetermine the outcome for various voting methods, given the voters' preferences.

DM.4.DExplain how different voting schemes or the order of voting can lead to different results.

DM.4.EDescribe the impact of various strategies on the results of the decision-making process.

DM.4.FExplain the impact of Arrow's Impossibility Theorem.

DM.4.GRelate the meaning of approval voting.

DM.4.HExplain the need for weighted voting and how it works.

DM.4.IIdentify voting concepts such as Borda count, Condorcet winner, dummy voter, and coalition.

DM.4.JCompute the Banzhaf power index and explain its significance.

DM.5 Fair division. The student applies the adjusted winner procedure and Knaster inheritance procedure to real-world situations.

DM.5.AUse the adjusted winner procedure to determine a fair allocation of property.

DM.5.BUse the adjusted winner procedure to resolve a dispute.

DM.5.CExplain how to reach a fair division using the Knaster inheritance procedure.

DM.5.DSolve fair division problems with three or more players using the Knaster inheritance procedure.

DM.5.EExplain the conditions under which the trimming procedure can be applied to indivisible goods.

DM.5.FIdentify situations appropriate for the techniques of fair division.

DM.5.GCompare the advantages of the divider and the chooser in the divider-chooser method.

DM.5.HDiscuss the rules and strategies of the divider-chooser method.

DM.5.IResolve cake-division problems for three players using the last-diminished method.

DM.5.JAnalyze the relative importance of the three desirable properties of fair division: equitability, envy-freeness, and Pareto optimality.

DM.5.KIdentify fair division procedures that exhibit envy-freeness.

DM.6 Game (or competition) theory. The student uses knowledge of basic game theory concepts to calculate optimal strategies. The student analyzes situations and identifies the use of gaming strategies.

DM.6.ARecognize competitive game situations.

DM.6.BRepresent a game with a matrix.

DM.6.CIdentify basic game theory concepts and vocabulary.

DM.6.DDetermine the optimal pure strategies and value of a game with a saddle point by means of the minimax technique.

DM.6.EExplain the concept of and need for a mixed strategy.

DM.6.FCompute the optimal mixed strategy and the expected value for a player in a game who has only two pure strategies.

DM.6.GModel simple two-by-two, bimatrix games of partial conflict.

DM.6.HIdentify the nature and implications of the game called "Prisoners' Dilemma".

DM.6.IExplain the game known as "chicken".

DM.6.JIdentify examples that illustrate the prevalence of Prisoners' Dilemma and chicken in our society.

DM.6.KDetermine when a pair of strategies for two players is in equilibrium.

DM.7 Theory of moves. The student analyzes the theory of moves (TOM). The student uses the TOM and game theory to analyze conflicts.

DM.7.ACompare and contrast TOM and game theory.

DM.7.BExplain the rules of TOM.

DM.7.CDescribe what is meant by a cyclic game.

DM.7.DUse a game tree to analyze a two-person game.

DM.7.EDetermine the effect of approaching Prisoners' Dilemma and chicken from the standpoint of TOM and contrast that to the effect of approaching them from the standpoint of game theory.

DM.7.FDescribe the use of TOM in a larger, more complicated game.

DM.7.GModel a conflict from literature or from a real-life situation as a two-by-two strict ordinal game and compare the results predicted by game theory and by TOM.

Geometry

G.2 Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures.

G.2.ADetermine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint.

G.2.BDerive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.

G.2.CDetermine an equation of a line parallel or perpendicular to a given line that passes through a given point.

G.3 Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and non-rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity).

G.3.ADescribe and perform transformations of figures in a plane using coordinate notation.

G.3.BDetermine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane.

G.3.CIdentify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane.

G.3.DIdentify and distinguish between reflectional and rotational symmetry in a plane figure.

G.4 Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships.

G.4.ADistinguish between undefined terms, definitions, postulates, conjectures, and theorems.

G.4.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.

G.4.CVerify that a conjecture is false using a counterexample.

G.4.DCompare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.

G.5 Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures.

G.5.AInvestigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.

G.5.BConstruct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge.

G.5.CUse the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships.

G.5.DVerify the Triangle Inequality theorem using constructions and apply the theorem to solve problems.

G.6 Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart.

G.6.AVerify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.

G.6.BProve two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

G.6.CApply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles.

G.6.DVerify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

G.6.EProve a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

G.7 Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems.

G.7.AApply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.

G.7.BApply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.

G.8 Similarity, proof, and trigonometry. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart.

G.8.AProve theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems.

G.8.BIdentify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.

G.9 Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles.

G.9.ADetermine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems.

G.9.BApply the relationships in special right triangles 30°-60°-90° and 45°-45°-90° and the Pythagorean theorem, including Pythagorean triples, to solve problems.

G.10 Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures.

G.10.AIdentify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations of two-dimensional shapes.

G.10.BDetermine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change.

G.11 Two-dimensional and three-dimensional figures. The student uses the process skills in the application of formulas to determine measures of two- and three-dimensional figures.

G.11.AApply the formula for the area of regular polygons to solve problems using appropriate units of measure.

G.11.BDetermine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure.

G.11.CApply the formulas for the total and lateral surface area of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.

G.11.DApply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.

G.12 Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles.

G.12.AApply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems.

G.12.BApply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems.

G.12.CApply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.

G.12.DDescribe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.

G.12.EShow that the equation of a circle with center at the origin and radius r is x^2 + y^2 = r^2 and determine the equation for the graph of a circle with radius r and center (h, k), (x - h)^2 + (y - k)^2 =r^2.

G.13 Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events.

G.13.ADevelop strategies to use permutations and combinations to solve contextual problems.

G.13.BDetermine probabilities based on area to solve contextual problems.

G.13.CIdentify whether two events are independent and compute the probability of the two events occurring together with or without replacement.

G.13.DApply conditional probability in contextual problems.

G.13.EApply independence in contextual problems.

Independent Study in Mathematics

ISM.1 Apply mathematics to problems arising in everyday life, society, and the workplace.

ISM.1.AApply mathematics to problems arising in everyday life, society, and the workplace.

ISM.2 Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

ISM.2.AUse a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

ISM.3 Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

ISM.3.ASelect tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

ISM.4 Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

ISM.4.ACommunicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

ISM.5 Create and use representations to organize, record, and communicate mathematical ideas.

ISM.5.ACreate and use representations to organize, record, and communicate mathematical ideas.

ISM.6 Analyze mathematical relationships to connect and communicate mathematical ideas.

ISM.6.AAnalyze mathematical relationships to connect and communicate mathematical ideas.

ISM.7 Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

ISM.7.ADisplay, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

ISM.ISMKnowledge and Skills: Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding.

Mathematical Models with Applications

MMA.2 Mathematical modeling in personal finance. The student uses mathematical processes with graphical and numerical techniques to study patterns and analyze data related to personal finance.

MMA.2.AUse rates and linear functions to solve problems involving personal finance and budgeting, including compensations and deductions.

MMA.2.BSolve problems involving personal taxes.

MMA.2.CAnalyze data to make decisions about banking, including options for online banking, checking accounts, overdraft protection, processing fees, and debit card/ATM fees.

MMA.3 Mathematical modeling in personal finance. The student uses mathematical processes with algebraic formulas, graphs, and amortization modeling to solve problems involving credit.

MMA.3.AUse formulas to generate tables to display series of payments for loan amortizations resulting from financed purchases.

MMA.3.BAnalyze personal credit options in retail purchasing and compare relative advantages and disadvantages of each option.

MMA.3.CUse technology to create amortization models to investigate home financing and compare buying a home to renting a home.

MMA.3.DUse technology to create amortization models to investigate automobile financing and compare buying a vehicle to leasing a vehicle.

MMA.4 Mathematical modeling in personal finance. The student uses mathematical processes with algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning.

MMA.4.AAnalyze and compare coverage options and rates in insurance.

MMA.4.BInvestigate and compare investment options, including stocks, bonds, annuities, certificates of deposit, and retirement plans.

MMA.4.CAnalyze types of savings options involving simple and compound interest and compare relative advantages of these options.

MMA.5 Mathematical modeling in science and engineering. The student applies mathematical processes with algebraic techniques to study patterns and analyze data as it applies to science.

MMA.5.AUse proportionality and inverse variation to describe physical laws such as Hook's Law, Newton's Second Law of Motion, and Boyle's Law.

MMA.5.BUse exponential models available through technology to model growth and decay in areas, including radioactive decay.

MMA.5.CUse quadratic functions to model motion.

MMA.6 Mathematical modeling in science and engineering. The student applies mathematical processes with algebra and geometry to study patterns and analyze data as it applies to architecture and engineering.

MMA.6.AUse similarity, geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in architecture.

MMA.6.BUse scale factors with two-dimensional and three-dimensional objects to demonstrate proportional and non-proportional changes in surface area and volume as applied to fields.

MMA.6.CUse the Pythagorean Theorem and special right-triangle relationships to calculate distances.

MMA.6.DUse trigonometric ratios to calculate distances and angle measures as applied to fields.

MMA.7 Mathematical modeling in fine arts. The student uses mathematical processes with algebra and geometry to study patterns and analyze data as it applies to fine arts.

MMA.7.AUse trigonometric ratios and functions available through technology to model periodic behavior in art and music.

MMA.7.BUse similarity, geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in art and photography.

MMA.7.CUse geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music.

MMA.7.DUse scale factors with two-dimensional and three-dimensional objects to demonstrate proportional and non-proportional changes in surface area and volume as applied to fields.

MMA.8 Mathematical modeling in social sciences. The student applies mathematical processes to determine the number of elements in a finite sample space and compute the probability of an event.

MMA.8.ADetermine the number of ways an event may occur using combinations, permutations, and the Fundamental Counting Principle.

MMA.8.BCompare theoretical to empirical probability.

MMA.8.CUse experiments to determine the reasonableness of a theoretical model such as binomial or geometric.

MMA.9 Mathematical modeling in social sciences. The student applies mathematical processes and mathematical models to analyze data as it applies to social sciences.

MMA.9.AInterpret information from various graphs, including line graphs, bar graphs, circle graphs, histograms, scatterplots, dot plots, stem-and-leaf plots, and box and whisker plots, to draw conclusions from the data and determine the strengths and weaknesses of conclusions.

MMA.9.BAnalyze numerical data using measures of central tendency (mean, median, and mode)and variability (range, interquartile range or IQR, and standard deviation) in order to make inferences with normal distributions.

MMA.9.CDistinguish the purposes and differences among types of research, including surveys, experiments, and observational studies.

MMA.9.DUse data from a sample to estimate population mean or population proportion.

MMA.9.EAnalyze marketing claims based on graphs and statistics from electronic and print media and justify the validity of stated or implied conclusions.

MMA.9.FUse regression methods available through technology to model linear and exponential functions, interpret correlations, and make predictions.

MMA.10 Mathematical modeling in social sciences. The student applies mathematical processes to design a study and use graphical, numerical, and analytical techniques to communicate the results of the study.

MMA.10.AFormulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions.

MMA.10.BCommunicate methods used, analyses conducted, and conclusions drawn for a data-analysis project through the use of one or more of the following: a written report, a visual display, an oral report, or a multi-media presentation.

Precalculus

PC.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems.

PC.2.AUse the composition of two functions to model and solve real-world problems.

PC.2.BDemonstrate that function composition is not always commutative.

PC.2.CRepresent a given function as a composite function of two or more functions.

PC.2.DDescribe symmetry of graphs of even and odd functions.

PC.2.EDetermine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations.

PC.2.FGraph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

PC.2.GGraph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems.

PC.2.HGraph arc sin x and arc cos x and describe the limitations on the domain.

PC.2.IDetermine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing.

PC.2.JAnalyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems.

PC.2.KAnalyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes.

PC.2.LDetermine various types of discontinuities in the interval (-∞, ∞) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities;

PC.2.MDescribe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities.

PC.2.NAnalyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems.

PC.2.ODevelop and use a sinusoidal function that models a situation in mathematical and real-world problems.

PC.2.PDetermine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems.

PC.3 Relations and geometric reasoning. The student uses the process standards in mathematics to model and make connections between algebraic and geometric relations.

PC.3.AGraph a set of parametric equations.

PC.3.BConvert parametric equations into rectangular relations and convert rectangular relations into parametric equations.

PC.3.CUse parametric equations to model and solve mathematical and real-world problems.

PC.3.DGraph points in the polar coordinate system and convert between rectangular coordinates and polar coordinates.

PC.3.EGraph polar equations by plotting points and using technology.

PC.3.FDetermine the conic section formed when a plane intersects a double-napped cone.

PC.3.GMake connections between the locus definition of conic sections and their equations in rectangular coordinates.

PC.3.HUse the characteristics of an ellipse to write the equation of an ellipse with center (h, k).

PC.3.IUse the characteristics of a hyperbola to write the equation of a hyperbola with center (h, k).

PC.4 Number and measure. The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and real-world problems.

PC.4.ADetermine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and real-world problems.

PC.4.BDescribe the relationship between degree and radian measure on the unit circle.

PC.4.CRepresent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position.

PC.4.DRepresent angles in radians or degrees based on the concept of rotation in mathematical and real-world problems, including linear and angular velocity.

PC.4.EDetermine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and real-world problems.

PC.4.FUse trigonometry in mathematical and real-world problems, including directional bearing.

PC.4.GUse the Law of Sines in mathematical and real-world problems.

PC.4.HUse the Law of Cosines in mathematical and real-world problems.

PC.4.IUse vectors to model situations involving magnitude and direction.

PC.4.JRepresent the addition of vectors and the multiplication of a vector by a scalar geometrically and symbolically.

PC.4.KApply vector addition and multiplication of a vector by a scalar in mathematical and real-world problems.

PC.5 Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms.

PC.5.AEvaluate finite sums and geometric series, when possible, written in sigma notation.

PC.5.BRepresent arithmetic sequences and geometric sequences using recursive formulas.

PC.5.CCalculate the nth term and the nth partial sum of an arithmetic series in mathematical and real-world problems.

PC.5.DRepresent arithmetic series and geometric series using sigma notation.

PC.5.ECalculate the nth term of a geometric series, the nth partial sum of a geometric series, and sum of an infinite geometric series when it exists.

PC.5.FApply the Binomial Theorem for the expansion of (a + b)^n in powers of a and b for a positive integer n, where a and b are any numbers.

PC.5.GUse the properties of logarithms to evaluate or transform logarithmic expressions.

PC.5.HGenerate and solve logarithmic equations in mathematical and real-world problems.

PC.5.IGenerate and solve exponential equations in mathematical and real-world problems.

PC.5.JSolve polynomial equations with real coefficients by applying a variety of techniques in mathematical and real-world problems.

PC.5.KSolve polynomial inequalities with real coefficients by applying a variety of techniques and write the solution set of the polynomial inequality in interval notation in mathematical and real-world problems.

PC.5.LSolve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems.

PC.5.MUse trigonometric identities such as reciprocal, quotient, Pythagorean, co-functions, even/odd, and sum and difference identities for cosine and sine to simplify trigonometric expressions.

PC.5.NGenerate and solve trigonometric equations in mathematical and real-world problems.

Statistics

S.2 Statistical process sampling and experimentation. The student applies mathematical processes to apply understandings about statistical studies, surveys, and experiments to design and conduct a study and use graphical, numerical, and analytical techniques to communicate the results of the study.

S.2.ACompare and contrast the benefits of different sampling techniques, including random sampling and convenience sampling methods.

S.2.BDistinguish among observational studies, surveys, and experiments.

S.2.CAnalyze generalizations made from observational studies, surveys, and experiments.

S.2.DDistinguish between sample statistics and population parameters.

S.2.EFormulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions.

S.2.FCommunicate methods used, analyses conducted, and conclusions drawn for a data-analysis project through the use of one or more of the following: a written report, a visual display, an oral report, or a multi-media presentation.

S.2.GCritically analyze published findings for appropriateness of study design implemented, sampling methods used, or the statistics applied.

S.3 Variability. The student applies the mathematical process standards when describing and modeling variability.

S.3.ADistinguish between mathematical models and statistical models.

S.3.BConstruct a statistical model to describe variability around the structure of a mathematical model for a given situation.

S.3.CDistinguish among different sources of variability, including measurement, natural, induced, and sampling variability.

S.3.DDescribe and model variability using population and sampling distributions.

S.4 Categorical and quantitative data. The student applies the mathematical process standards to represent and analyze both categorical and quantitative data.

S.4.ADistinguish between categorical and quantitative data.

S.4.BRepresent and summarize data and justify the representation.

S.4.CAnalyze the distribution characteristics of quantitative data, including determining the possible existence and impact of outliers.

S.4.DCompare and contrast different graphical or visual representations given the same data set.

S.4.ECompare and contrast meaningful information derived from summary statistics given a data set.

S.4.FAnalyze categorical data, including determining marginal and conditional distributions, using two-way tables.

S.5 Probability and random variables. The student applies the mathematical process standards to connect probability and statistics.

S.5.ADetermine probabilities, including the use of a two-way table.

S.5.BDescribe the relationship between theoretical and empirical probabilities using the Law of Large Numbers.

S.5.CConstruct a distribution based on a technology-generated simulation or collected samples for a discrete random variable.

S.5.DCompare statistical measures such as sample mean and standard deviation from a technology-simulated sampling distribution to the theoretical sampling distribution.

S.6 Inference. The student applies the mathematical process standards to make inferences and justify conclusions from statistical studies.

S.6.AExplain how a sample statistic and a confidence level are used in the construction of a confidence interval.

S.6.BExplain how changes in the sample size, confidence level, and standard deviation affect the margin of error of a confidence interval.

S.6.CCalculate a confidence interval for the mean of a normally distributed population with a known standard deviation.

S.6.DCalculate a confidence interval for a population proportion.

S.6.EInterpret confidence intervals for a population parameter, including confidence intervals from media or statistical reports.

S.6.FExplain how a sample statistic provides evidence against a claim about a population parameter when using a hypothesis test.

S.6.GConstruct null and alternative hypothesis statements about a population parameter.

S.6.HExplain the meaning of the p-value in relation to the significance level in providing evidence to reject or fail to reject the null hypothesis in the context of the situation.

S.6.IInterpret the results of a hypothesis test using technology-generated results such as large sample tests for proportion, mean, difference between two proportions, and difference between two independent means.

S.6.JDescribe the potential impact of Type I and Type II Errors.

S.7 Bivariate data. The student applies the mathematical process standards to analyze relationships among bivariate quantitative data.

S.7.AAnalyze scatterplots for patterns, linearity, outliers, and influential points.

S.7.BTransform a linear parent function to determine a line of best fit.

S.7.CCompare different linear models for the same set of data to determine best fit, including discussions about error.

S.7.DCompare different methods for determining best fit, including median-median and absolute value.

S.7.EDescribe the relationship between influential points and lines of best fit using dynamic graphing technology.

S.7.FIdentify and interpret the reasonableness of attributes of lines of best fit within the context, including slope and y-intercept.

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.