Maryland Math Standards - Maryland College and Career Ready Standards for Mathematics

753 standards, Kindergarten through AP.

Maryland's College and Career Ready Standards for Mathematics were revised in July 2025 for kindergarten through Integrated Algebra 2. The revision organizes learning into Number and Operation Sense, Algebraic Thinking, Geometric Reasoning, and Data, Statistics, and Probability.

Kindergarten

K.AT Algebraic Thinking (AT)

K.AT.A Solve problems involving addition and subtraction.

K.AT.A.1Represent addition and subtraction situations (presented with numerals or mathematical symbols) within 10 with objects, fingers, drawings, sounds (e.g., claps), acting out situations, using verbal explanations, and/or writing expressions and equations.

K.AT.A.2Solve addition and subtraction problems in context (add to, take from, put together/take apart) within 10 and represent by using objects, drawings, and/or equations.

K.AT.B Understand repeating patterns.

K.AT.B.3Identify, extend, and create repeating patterns (AABAAB, ABCABC, or AABBCC) using concrete objects, drawings, sounds, or movements.

K.DS Reasoning with Data and Statistics (DS)

K.DS.A Classify objects into categories to represent data.

K.DS.A.1Organize data by classifying objects into given categories and counting the number of objects in each category (Limit the total in any one category to maximum of 20).

K.DS.A.2Analyze data sets by ordering the categories by count.

K.GR Geometric Reasoning and Measurement (GR)

K.GR.A Describe and compare measurable attributes.

K.GR.A.1Describe measurable attributes of an object such as length, height, weight, and capacity using appropriate vocabulary (e.g. long, short, tall, heavy, light, wide, narrow, full, empty).

K.GR.A.2Directly compare two objects with a measurable attribute in common, using words such as “greater/less,” “more/fewer,” “longer/shorter,” “lighter/heavier,” or “taller/shorter.”

K.GR.A.3Order up to 3 objects by a measurable attribute (e.g. greatest to least in height).

K.GR.B Work with two-dimensional and three-dimensional shapes.

K.GR.B.4Describe the relative positions of objects using terms such as above, below, beside, in front of, behind, and next to.

K.GR.B.5Identify and describe given shapes and shapes of objects in everyday situations including two-dimensional shapes (circle, triangle, rectangle, square, hexagon) and three-dimensional shapes (cone, cube, cylinder, and sphere) regardless of their orientation or overall size.

K.GR.B.6Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).

K.GR.B.7Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides, edges, and vertices/”corners”) and other attributes (e.g., having sides of equal length).

K.GR.B.8Draw two dimensional shapes and build or create models of three-dimensional shapes.

K.GR.B.9Compose simple shapes to form larger shapes (e.g. join two triangles with full sides touching to compose a rectangle).

K.NOS Number and Operation Sense (NOS)

K.NOS.A Know number names and the count sequence.

K.NOS.A.1Count to 100 by ones and by tens.

K.NOS.A.2Count forward from any given number within 100.

K.NOS.A.3Count backwards from any given number within 20.

K.NOS.A.4Write numbers from 0 to 20.

K.NOS.A.5Represent a number of objects with a written numeral within 20 (with 0 representing a count of no objects).

K.NOS.B Count to tell the number of objects.

K.NOS.B.6Use the relationship between numbers and quantities within 20 to count objects with one-to-one correspondence (arranged in a line, a rectangular array, a circle, or as many as 10 things in scatter configuration) and verbalize that the last number stated is the total when asked “How many?” (cardinality).

K.NOS.B.7When counting objects within 20, recognize that each successive number name refers to a quantity that is one larger.

K.NOS.B.8Recognize the number of objects in a set without counting (subitizing) with both unfamiliar patterns within 6 and familiar patterns within 10.

K.NOS.C Develop foundation for place value.

K.NOS.C.9Understand 10 as a group, collection, or bundle of ten ones called a “ten.” a. Compose and decompose numbers from 11 to 19 into ten ones and some further ones by using objects or drawings, and equations (e.g., 13 = 10 + 3). b. Describe a given number as one ten and the correct number of ones (e.g., 13 is one ten and three ones).

K.NOS.C.10Compare two quantities within 20 using greater than, equal to, or less than, with objects, location on number paths, and written numerals.

K.NOS.D Decompose numbers.

K.NOS.D.11Decompose numbers within 10 in more than one way, by using objects or drawings, and record each decomposition with a drawing or equation (e.g., 10 = 4 + 6 and 10 = 4+5 + 1).

Grade 1

1.AT Algebraic Thinking (AT)

1.AT.A Solve problems involving addition and subtraction.

1.AT.A.1Understand the meaning of the equal sign. Determine if equations involving addition and subtraction (on one or both sides of the equal sign) are true or false.

1.AT.A.2Add and subtract within 20 involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions (result unknown, change unknown, start unknown) by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem.

1.AT.A.3Determine the unknown whole number in an addition or subtraction equation relating three whole numbers (e.g., 8 + ? = 13; 6 = □ – 2; 8 + 8 = □).

1.DS Reasoning with Data and Statistics (DS)

1.DS.A Represent and interpret data.

1.DS.A.1Ask and answer questions by collecting, organizing and summarizing data. a. Craft a question that can be answered by collecting categorical data. b. Collect and organize categorical data (into up to three categories) using surveys or observations. c. Represent data by creating tally charts and picture graphs. d. Summarize the data presented in tally charts using “most,” “least,” “greater than,” “less than,” and “equal to”.

1.GR Geometric Reasoning and Measurement (GR)

1.GR.A Measure lengths indirectly and by iterative length units.

1.GR.A.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.

1.GR.A.2Use non-standard units (with no gaps or overlaps) to measure the length of an object to the nearest whole unit.

1.GR.B Reason with shapes and their attributes.

1.GR.B.3Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size).

1.GR.B.4Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

1.GR.C Work with time and money.

1.GR.C.5Tell time in hours, half hours, and quarter hours using a digital clock or analog clock. Estimate time intervals (1 hour, 30 minutes, 15 minutes, 1 minute) for activities (e.g., sharpen pencil, soccer game, etc).

1.GR.C.6Identify and know the value of coins (penny, nickel, dime, and quarter) and bills ($1, $5, $10, and $20).

1.NOS Number and Operation Sense (NOS)

1.NOS.A Extend the count sequence.

1.NOS.A.1Count forward and backward within 120. a. Count forward and backwards by ones starting with any number. b. Skip count forward and backwards by tens.

1.NOS.A.2Read, write numerals, and represent a number of objects with a written numeral within 120.

1.NOS.B Understand place value and representations of numbers.

1.NOS.B.3Extend understanding of 10 as a group, collection, or bundle of ten ones, called a “ten,” to compose and decompose two-digit numbers. a. Compose and decompose two-digit numbers into tens and ones using objects or drawings, and equations (e.g., 10 + 4 = 14; 38 = 30 + 8). b. Describe a given number as the correct number of tens and ones (e.g., 14 is one ten and four ones; 38 is three tens and eight ones). c. Compose and decompose two-digit numbers in more than one way (e.g. 46 = 41 + 5 or 46 = 23 + 23).

1.NOS.B.4Represent whole numbers as lengths from 0 on a number line (horizontal and vertical) with equally spaced points corresponding to whole numbers.

1.NOS.B.5Compare two numbers within 100 by reasoning about values of tens and ones digits and the location of the numbers on a number line. Record the results of comparisons with the symbols >, =, and <.

1.NOS.B.6Estimate the location of numbers on a number line by reasoning about their relationship to benchmark numbers (e.g. 10, 50, 100).

1.NOS.C Understand and apply properties of operations and the relationship between addition and subtraction.

1.NOS.C.7Apply the Commutative Property of Addition and Associative Property of Addition as a strategy to add.

1.NOS.C.8Use the inverse relationship between addition and subtraction to subtract.

1.NOS.D Add and subtract within 20.

1.NOS.D.9Recall or quickly derive addition and subtraction facts within 20. a. Use the count on and count back strategies to add and subtract (e.g., 4 + 1, 7 – 2, etc.). b. Use the make ten strategy to add and subtract with combinations of 10 (e.g., 7 + 3, 6 + 4, 10 – 3, 10 – 6, etc.). c. Use the ten more and ten less strategies to add and subtract (including those with a difference of 10). d. Use the doubles to add and subtract (e.g., 6 + 6 = 12, 14 – 7 = 7, etc.)

1.NOS.E Use place value understanding and properties to add and subtract.

1.NOS.E.10Add within 100 including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10. Understand that when adding two-digit numbers, tens are added to tens, ones are added to ones, and sometimes it is necessary to compose or decompose a ten. a. Use concrete models and drawings to add. b. Use counting strategies and strategies based on place value (e.g., partial sums, making tens, etc.) to add. c. Use properties of operations, and/or the inverse relationship between addition and subtraction to add. d. Represent and explain the calculation by connecting the strategy used to the meaning of addition.

1.NOS.E.11Given a two-digit number, identify 10 more or 10 less than the given number, without having to count, and explain the reasoning used.

1.NOS.E.12Subtract a multiple of 10 from another multiple of 10 within 90. a. Use concrete models or drawings to subtract. b. Use counting strategies, properties of operations, and/or the inverse relationship between addition and subtraction to subtract. c. Represent and explain the calculation by connecting the strategy used to the meaning of subtraction.

1.NOS.E.13Subtract a 1-digit number from a 2-digit number (without regrouping). a. Use concrete models or drawings to subtract. b. Use counting strategies, properties of operations, and/or the inverse relationship between addition and subtraction to subtract. c. Represent and explain the calculation by connecting the strategy used to the meaning of subtraction.

1.NOS.F Understand equal parts of a whole.

1.NOS.F.14Partition circles and rectangles into equal shares (halves and fourths) recognizing that decomposing into more equal shares creates smaller shares. Determine how many equal shares are needed to make a whole (e.g., two halves).

Grade 2

2.AT Algebraic Thinking (AT)

2.AT.A Solve problems involving addition and subtraction.

2.AT.A.1Use addition and subtraction within 100 to solve one-step problems in context involving an unknown in any situation or position by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

2.AT.A.2Use addition and/or subtraction within 100 to solve two-step problems in context involving any situation or position of an unknown by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

2.AT.B Recognize patterns in numbers.

2.AT.B.3Determine whether a group of objects within 20 has an odd or even number of members (e.g., by pairing objects or counting them by 2s). Write an equation to express an even number as a sum of two equal addends.

2.DS Reasoning with Data and Statistics (DS)

2.DS.A Represent and interpret data.

2.DS.A.1Ask and answer questions by collecting, organizing and summarizing data. a. Craft a question that can be answered by collecting categorical and numerical data. b. Collect and organize data using surveys, making observations, or measuring. c. Represent categorical data using picture graphs and bar graphs. Represent numerical data using line plots. d. Summarize the data presented in data visualizations using “most,” “least,” “greater than,” “less than,” and “equal to” and determine what questions can be answered with a given data set.

2.GR Geometric Reasoning and Measurement (GR)

2.GR.A Measure and estimate lengths in standard units.

2.GR.A.1Measure the length of an object to the nearest whole unit by using appropriate tools (e.g., rulers, yardsticks, meter sticks, and measuring tapes).

2.GR.A.2Measure the length of an object twice using different length units. Compare the two measurements relative to the size of the unit chosen.

2.GR.A.3Estimate lengths using units of inches, feet, centimeters, and meters.

2.GR.A.4Measure to determine how much longer one object is than another, expressing the difference in length in terms of a standard-length unit.

2.GR.B Reason with shapes and their attributes.

2.GR.B.5Identify and draw shapes (triangles, quadrilaterals, pentagons, hexagons, and cubes) having specific attributes, such as a given number of angles or a given number of equal faces.

2.GR.B.6Explain that a line of symmetry for a two-dimensional figure is a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

2.GR.C Work with time and money.

2.GR.C.7Tell and represent time to the nearest minute using multiple representations (e.g., digital clocks, analog clocks, number lines).

2.GR.C.8Estimate and calculate elapsed time to the hour and half hour (e.g. If the school day begins at 8am and ends at 3 pm, the school day lasts 7 hours).

2.GR.C.9Count mixed sets of coins and bills to solve problems in context (within $20.00) involving bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

2.NOS Number and Operation Sense (NOS)

2.NOS.A Understand place value and representations of numbers.

2.NOS.A.1Use understanding of 100 as a bundle of ten tens, called a “hundred,” and compose and decompose three-digit numbers. a. Compose and decompose three-digit numbers into hundreds, tens, and ones by using objects, drawings, and/or equations (e.g., 328 = 300 + 20 + 8 or 328 is thirty-two tens and 8 ones). b. Describe a given number as the correct number of hundreds, tens, and ones (e.g., 328 is three hundreds, two tens and eight ones). c. Compose and decompose three-digit numbers in more than one way (e.g., 439 = 400 + 39 or 439 = 435 + 4).

2.NOS.A.2Count forward and backward within 1000 starting with any number. a. Skip-count forward and backwards by 2s, 5s, 10s, and 100s. b. Use skip-counting to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns.

2.NOS.A.3Read and write numbers within 1000 using base-ten numerals, number names, and expanded form.

2.NOS.A.4Compare two numbers within 1000 by reasoning about the values of the hundreds, tens, and ones digits and/or the location of the numbers on a number line. Record the results of comparisons with the symbols >, =, and <.

2.NOS.A.5Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers (e.g. 10, 50, 100), and to assess reasonableness of sums and differences.

2.NOS.B Add and subtract within 20.

2.NOS.B.6Recall or quickly derive addition and subtraction facts within 20. a. Use counting, make ten, and ten more/less, doubling strategies to add and subtract. b. Use the use ten strategy (e.g., think of 9 + 1 + 6 to solve 9 + 7) and doubles plus one strategy (e.g., think of 4 + 4 + 1 to solve 4 + 5) to add and subtract. c. Use the inverse operation to add and subtract (e.g. think of 7 + ? = 15 to solve 15 – 7 = ?).

2.NOS.C Use representations, place value understanding, and properties of operations to add and subtract.

2.NOS.C.7Represent whole number sums and differences within 100 using lengths on a number line.

2.NOS.C.8Fluently add and subtract within 100. a. Use strategies based on counting (e.g., counting on, counting back) and place value (e.g., partial sums, making tens, etc.) to add and subtract. b. Use properties of operations, and the inverse relationship between addition and subtraction to add and subtract. c. Determine and explain when a strategy is most efficient.

2.NOS.C.9Add and subtract within 1000, recognizing that when adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. a. Use concrete models or drawings to add and subtract. b. Use strategies based on place value (counting on, partial sums, making tens, etc.) to add and subtract. c. Use properties of operations, and/or the relationship between addition and subtraction to add and subtract. d. Represent and explain the calculation by connecting the strategy used to the meaning of addition and subtraction.

2.NOS.C.10Mentally add and subtract 10 or 100 to a given number 100-900.

2.NOS.D Understand equal parts of a whole.

2.NOS.D.11Partition circles and rectangles into equal shares (halves, thirds, and fourths) recognizing that equal shares do not need to have the same shape. Determine how many equal shares are needed to make a whole (e.g., three thirds).

Grade 3

3.AT Algebraic Thinking (AT)

3.AT.A Solve problems involving four operations and identify and explain patterns in arthimetic.

3.AT.A.1Use multiplication and division within 100 to solve one-step problems in context involving equal groups, arrays, and area by using concrete materials, drawings, and/or equations.

3.AT.A.2Use the four operations to solve two-step problems in context (within number limits for each operation as stated in 3.NOS.B and 3.NOS.D). a. Represent these problems using equations with a letter standing for the unknown quantity. b. Assess the reasonableness of answers in terms of context, using mental computation and estimation strategies including, rounding, front-end estimation, and benchmark numbers.

3.AT.A.3Identify arithmetic patterns and explain them using properties of operations (e.g., observe the pattern that 4 times a number is always even and explain why 4 times a number can be decomposed into two equal addends).

3.DS Reasoning with Data and Statistics (DS)

3.DS.A Represent and interpret data.

3.DS.A.1Ask and answer questions by collecting, organizing and summarizing data, recognizing the importance of context when analyzing data. a. Create a scaled data visualization (e.g., picture graph, bar graph for categorial data; line plot with fraction units of halves and fourths for numerical data) to display collected or given data. b. Summarize data presented in scaled data visualizations (e.g., picture graph, bar graph for categorial data; line plot for numerical data) by describing the story the data is telling. c. Identify patterns in the data and evaluate whether these patterns might apply to different contexts or situations.

3.GR Geometric Reasoning and Measurement (GR)

3.GR.A Solve problems involving measurement and estimation of intervals of time, liquid, and masses of objects.

3.GR.A.1Measure the length of an object to the nearest half and quarter unit by using appropriate tools (e.g., rulers, yardsticks, meter sticks, and measuring tapes).

3.GR.A.2Solve problems in context that involve addition and subtraction of time intervals in minutes (e.g., represent and solve using a number line).

3.GR.A.3Measure and estimate liquid volumes and masses of objects using standard units of liters(l), grams (g), and kilograms (kg).

3.GR.A.4Add, subtract, multiply, or divide to solve one-step problems in measurement contexts involving masses or volumes that are given in the same units.

3.GR.B Reason with shapes and their attributes.

3.GR.B.5Explain that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category of shapes (e.g., quadrilaterals).

3.GR.C Understand concepts of area and perimeter.

3.GR.C.6Identify area as an attribute of a two-dimensional figure and measure area of rectangular space by counting square units (non-standard equal-sized units, square centimeters, square meters, square inches, square feet).

3.GR.C.7Relate area to multiplication and addition. a. Determine the area of a rectangle with whole-number side lengths by tiling and connecting the visual representation to multiplication of the side lengths to solve problems in context. b. Use tiling and area models to show that the area of a rectangle with whole number side lengths b and c + d is the sum of b × c and b × d, representing the Distributive Property of Multiplication. c. Determine the area of composite rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas.

3.GR.C.8Identify perimeter as an attribute of two-dimensional figures and measure the distance around the outside of a figure by counting units or adding lengths.

3.GR.C.9Solve problems in contexts involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and finding rectangles with the same perimeter and different areas or with the same area and different perimeters.

3.NOS Number and Operation Sense (NOS)

3.NOS.A Understand place value.

3.NOS.A.1Apply and extend place value understanding to whole numbers within 10,000. a. Read and write whole numbers using base-ten numerals, number names, and expanded form. b. Compose and decompose whole numbers. c. Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers, and rounding to nearest 10 or 100. d. Compare two whole numbers by reasoning about the values of the digits and the location of the numbers on a number line. Record the results of comparisons with the symbols >, =, and <.

3.NOS.B Represent and interpret problems involving multiplication and division.

3.NOS.B.2Represent and interpret multiplication of two factors (0-10) as equal groups or as a composed unit that can be repeated/iterated. Explain the relationship between the factors and products.

3.NOS.B.3Represent and interpret whole-number quotients (0-10) as dividing an amount into known number of groups (partitive) or as dividing an amount into groups of known size (quotative). Explain the relationship between the quotient, divisor, and dividend.

3.NOS.C Understand properties of multiplication and the relationship between multiplication and division.

3.NOS.C.4Identify and apply the Commutative Property of Multiplication, Associative Property of Multiplication, and Distributive Property of Multiplication as strategies to multiply.

3.NOS.C.5Explain and use the inverse relationship between multiplication and division to determine the unknown whole number in a multiplication or division equation involving three whole numbers (e.g., find 32 ÷ 8 by thinking of unknown-factor problem and finding the number that equals 32 when multiplied by 8).

3.NOS.D Use place value understanding and properties of operations to perform multi-digit arithmetic.

3.NOS.D.6Estimate sums and differences within 1,000 using benchmark numbers and front-end estimation. Determine when an estimate is appropriate and when an exact answer is needed.

3.NOS.D.7Fluently add and subtract within 1,000. a. Use computational strategies efficiently (e.g., decomposition, partials sums, compensation, make ten/hundred/thousand) to add and subtract. b. Use properties of operations, and/or the inverse relationship between addition and subtraction. c. Determine and explain when a strategy is most efficient.

3.NOS.D.8Multiply one-digit whole numbers by multiples of 10 in the range of 10-90 (e.g., 9 × 80;60 × 5) using strategies based on place value and properties of operations.

3.NOS.E Multiply and divide within 100.

3.NOS.E.9Recall or quickly derive multiplication and division facts within 100 (e.g., factors less than or equal to 10 and quotients less than or equal to 10). a. Skip count (2s, 5s, 10s) and apply properties of operations (0s, 1s) to derive foundational facts. b. Derive unknown facts from known facts using double facts (e.g., to solve 4 × 3, double 3 to get 6 and double 6 to get 12) to multiply and divide. c. Use properties of operations (e.g., to solve 8 × 7, think of (5 × 7) + (3 × 7)) to multiply. d. Use the inverse relationship between multiplication and division (e.g., think of 4 × ? = 28 to solve 28 ÷ 4 = ?) to multiply and divide.

3.NOS.F Develop understanding of fractions as numbers.

3.NOS.F.10Recognize and explain that a fraction represents a part of a whole. a. Partition shapes into parts with equal areas and express the area of each part using a unit fraction. b. Explain that a fraction 1/d represents one part of a whole that is divided into d equal parts, and that a fraction c/d represents c parts of size 1/d.

3.NOS.F.11Explain a fraction of a set as the quantity formed by c items as part of the given set of d objects.

3.NOS.F.12Represent a fraction as a number on the number line. a. Represent a fraction on a number line by defining the interval from 0 to 1 as the whole and partitioning it into d equal lengths. Recognize that each length measures and that the endpoints of the length starting at 0 represents the number on the number line. b. Represent a fraction on a number line by marking off c lengths of from 0. Recognize that the resulting interval has size and that the endpoint of the length locates the number on the number line.

3.NOS.F.13Explain equivalence of fractions and compare fractions by reasoning about their size, recognizing that comparisons are valid only when the two fractions refer to the same whole. a. Represent two fractions as equivalent if they are the same size, or they represent the same point on the number line. b. Identify and generate equivalent fractions, (e.g., 1/2 = 2/4, 4/6 = 2/3.) and explain why the fractions are equivalent using concrete materials, drawings, number lines, or equations. c. Express whole numbers as fractions, and identify fractions that are equivalent to whole numbers, (e.g., express 3 in the form 3 = 3/1; recognize that 6 = 6/1; locate 4/4 and 1 at the same point on a number line). d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by using concrete materials, drawings, number lines, and/or equations).

Grade 4

4.AT Algebraic Thinking (AT)

4.AT.A Represent and solve problems involving multiplication and division.

4.AT.A.1Interpret a multiplication situation as a comparison (e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5). Represent verbal statements of multiplicative comparisons as multiplication equations.

4.AT.A.2Multiply or divide to solve multiplicative comparison problems in context by using drawings and equations with a symbol for the unknown number to represent the problem

4.AT.A.3Distinguish multiplicative comparison from additive comparison.

4.AT.A.4Solve multistep problems in context involving whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. a. Represent these problems using equations with a letter standing for the unknown quantity. b. Assess the reasonableness of answers in terms of context, including interpreting remainders.

4.DS Reasoning with Data and Statistics (DS)

4.DS.A Represent and interpret data.

4.DS.A.1Ask and answer questions by collecting, organizing and summarizing data, recognizing the importance of context when analyzing data. a. Create scaled data visualization (e.g. bar graphs for categorical data; line plots with fraction units of halves, fourths, and eighths for numerical data) to display data to communicate an idea. b. Summarize data presented in scaled data visualizations (e.g. bar graphs, line plots) and draw conclusions about the data. c. Compare and contrast different data visualizations of the same data by varying attributes (e.g., reordering bars, changing the scale) and explain how changing the attributes affects the interpretation.

4.GR Geometric Reasoning and Measurement (GR)

4.GR.A Solve problems involving measurement conversion, area, and perimeter.

4.GR.A.1Apply the relationship between measurement units within a given measurement system (customary: in, ft, yd, oz, lb, sec, min, hr; metric: cm, m, km, g, kg, mL, L) to convert measurements from a larger unit to smaller unit.

4.GR.A.2Use the four operations to solve problems in context involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving common fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.

4.GR.A.3Apply the area and perimeter formulas for rectangles in context, including rectangles with missing dimensions.

4.GR.A.4Relate the area of a triangle to the area of a rectangle using decomposition and visual models. Derive and apply the formula to solve problems in contexts involving triangles positioned within or formed from the decomposition of rectangles

4.GR.B Undersatnd concepts of angle and measure angles.

4.GR.B.5Identify angles as geometric figures formed by two rays that share a common endpoint and describe angle size as the amount of rotation between the two rays measured in degrees with reference to a circle.

4.GR.B.6Measure angles in whole-number degrees using a protractor. Estimate and sketch angles of specified measures.

4.GR.B.7Explain that when an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in context (e.g., by using an equation with a symbol for the unknown angle measure).

4.GR.C Draw and identify lines and angles and classify shapes by properties of their lines and angles.

4.GR.C.8Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

4.GR.C.9Classify 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. Recognize right triangles as a category and identify right triangles.

4.NOS Number and Operation Sense (NOS)

4.NOS.A Generalize place value understanding for multi-digit whole numbers.

4.NOS.A.1Apply and extend place value understanding to multi-digit whole numbers within 1,000,000. a. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. b. Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers, and rounding to any place. c. Compare two multi-digit numbers by reasoning about the values of the digits and the location of the numbers on a number line. Record the results of comparisons with the symbols >, =, and <.

4.NOS.A.2Explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (e.g., recognize that 700 ÷ 70 = 10 by applying concepts of place value and division).

4.NOS.B Use place value understanding and properities of operations to perform multi-digit arithmetic.

4.NOS.B.3Fluently add and subtract multi-digit whole numbers within 1,000,000. a. Apply estimation strategies to estimate sums and differences. b. Use computational strategies (e.g., decomposition, partials sums) efficiently. c. Use a standard algorithm d. Determine and explain when a strategy or algorithm is most efficient.

4.NOS.B.4Extend place value understanding and basic facts to multiply a multi-digit whole number by a multiple of ten.

4.NOS.B.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers. a. Apply estimation strategies to estimate products. b. Use strategies based on place value (e.g., partial products) to multiply. c. Use properties of operations (e.g., use doubling and halving strategy to think of 12 × 32 as 6 × 64) to multiply. d. Represent and explain the computation by connecting rectangular arrays, area models, and/or equations to the meaning of multiplication.

4.NOS.B.6Divide whole numbers with up to four-digit dividends and one-digit divisors with and without remainders. a. Apply estimation strategies to estimate quotients. b. Use strategies based on place value (e.g., partial quotients) to divide. c. Use properties of operations and/or the inverse relationship between multiplication and division to divide. d. Represent and explain the computation by connecting rectangular arrays, area models, and/or equations to the meaning of division.

4.NOS.C Gain familiarity with factors and multiples.

4.NOS.C.7Recall or quickly derive multiplication and division facts within 100 (e.g., factors less than or equal to 10 and quotients less than are equal to 10). a. Skip count (2s, 5s, 10s) and apply properties of operations (0s, 1s) to derive foundational facts. b. Derive unknown facts from known facts using double facts (e.g., to solve 4 × 3, double 3 to get 6 and double 6 to get 12) to multiply and divide. c. Use properties of operations (e.g., to solve 8 × 7, think of (5 × 7 + (3 × 7)) to multiply. d. Use the inverse relationship between multiplication and division (e.g., think of 4 × ? = 28 to solve 28 ÷ 4 = ?) to multiply and divide.

4.NOS.C.8Explain and apply concepts of factors, multiples, and prime and composite numbers for whole numbers in the range of 1-100. a. Identify factor pairs for a whole number in the range of 1–100. b. Determine whether a given whole number in the range of 1–100 is a multiple of a one-digit number. c. Distinguish between factors and multiples and explain how they relate to a given number. d. Determine whether a given whole number in the range of 1–100 is prime, composite, or neither.

4.NOS.D Extend understanding of fraction equivalence and ordering.

4.NOS.D.9Understand and generate equivalent fractions. a. Identify and generate equivalent fractions using representations and the Identity Property of Multiplication. b. Explain why fractions are equivalent (c/d = (n × c)/(n × d)).

4.NOS.D.10Compare two fractions with different numerators and different denominators, understanding that comparisons are valid only when the two fractions refer to the same whole. a. Use benchmark numbers (e.g., 1/2, 1), common numerators, and common denominators to compare. b. Record comparisons with symbols >, =, <. c. Justify comparisons using representations and reasoning.

4.NOS.E Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NOS.E.11Apply understanding of a fraction c/d with d > 1 as a sum of unit fractions (c/d = 1/d + 1/d + ... + 1/d) to decompose a fraction (including fractions greater than 1) into a sum of fractions with the same denominator in more than one way, recording each decomposition as an equation. Justify decompositions using visual fraction models and equations (e.g., 3/8 = 1/8 + 1/8 + 1/8 or 3/8 = 1/8 + 2/8 or 2 1/8 = 1 + 1 + 1/8 or 2 1/8 = 8/8 + 8/8 + 1/8).

4.NOS.E.12Apply and extend previous understanding of addition and subtraction of whole numbers to add and subtract fractions with like denominators. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Estimate sums and differences by reasoning about benchmark numbers and assess reasonableness of answers (e.g., 3/4 + 3/4 will be greater than 1 because both fractions are greater than 1/2). c. Apply and extend whole number addition and subtraction strategies (e.g. counting on, making a whole, partial sums, compensation, properties of operations, the inverse relationship between addition and subtraction) to add and subtract fractions. d. Solve problems in context involving addition and subtraction of fractions using visual fraction models and equations to represent the problem.

4.NOS.E.13Apply and extend previous understandings of multiplication to multiply a fraction by a whole number (b × c/d). a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers (e.g., 4 × 1/2 will be less than 4 because I have 4 groups of something less than 1). b. Use understanding of multiplication as equal groups to multiply a unit fraction by whole number (e.g., interpret b × 1/d as b groups of 1/d). c. Use understanding of multiplication as equal groups to multiply a fraction by whole number (e.g., interpret b × c/d as b groups of c/d when c/d is greater than or less than 1). d. Solve problems in context involving multiplication of a fraction by a whole number, using visual fraction models and equations to represent the problem.

4.NOS.F Understand decimal notation for fractions and compare decimal fractions.

4.NOS.F.14Express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 to add two fractions with respective denominators of 10 and 100.

4.NOS.F.15Use decimal notation for fractions with denominators 10 and 100 (e.g., rewrite 0.62 as 62/100 or describe a length as 0.62 meters or locate 0.62 on a number line).

4.NOS.F.16Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, (e.g. using a visual model).

Grade 5

5.AT Algebraic Thinking (AT)

5.AT.A Write and interpret numerical expressions.

5.AT.A.1Interpret and evaluate numerical expressions with grouping symbols (e.g. parentheses, brackets, or braces).

5.AT.A.2Write expressions that record calculations with numbers and interpret numerical expressions without evaluating them.

5.AT.B Analyze patterns and relationships.

5.AT.B.3Generate and analyze number patterns. a. Generate a number pattern that follows a given rule and identify relationships between the numbers within the pattern. b. Generate two numerical patterns using two given rules and identify relationships between corresponding terms. c. Use tables, ordered pairs, and graphs to represent the relationship between quantities.

5.DS Reasoning with Data and Statistics (DS)

5.DS.A Represent and interpret data.

5.DS.A.1Ask and answer questions by collecting, organizing and summarizing data, recognizing the importance of context when analyzing data. a. Create scaled data visualization (e.g. bar graphs for categorical data; line plots with fraction units of halves, fourths, and eighths for numerical data) to display data and communicate relationships or support a claim. b. Summarize data presented in scaled visualizations (e.g. bar graphs, circle graphs, line plots, line graphs) by identifying the mode, the range, and any gaps in data and draw conclusions. c. Evaluate whether a data visualization accurately represents the data and allows for accurate interpretation given the context.

5.GR Geometric Reasoning and Measurement (GR)

5.GR.A Understand relationships between measurement units.

5.GR.A.1Apply the relationship between measurement units within a given measurement system to convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m) and use these conversions in solving multi-step problems.

5.GR.B Classify two-dimensional figures based on properties.

5.GR.B.2Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category (e.g., all rectangles have four right angles and squares are rectangles, so all squares have four right angles).

5.GR.B.3Classify two-dimensional figures based on properties. a. Classify triangles based on properties (e.g., angle measure, side lengths). b. Classify quadrilaterals in a hierarchy based on properties (e.g., side lengths, angle measures, presence of parallel or perpendicular sides).

5.GR.C Understand concepts of volume and relate volume to multiplication.

5.GR.C.4Identify volume as an attribute of three-dimensional figures and measure volume by counting cubic units (e.g., non-standard equal-sized units, cubic centimeters, cubic inches, and cubic feet) that fill a figure without gaps or overlaps.

5.GR.C.5Relate volume to the operations of multiplication and addition and solve problems in context. a. Find the volume of a right rectangular prism with whole-number side lengths by filling it with unit cubes in layers and show that the volume is the same as would be found by multiplying the edge lengths, or by multiplying the height by the area of the base. Represent products of three whole numbers as volumes, (e.g., to represent the associative property of multiplication). b. Apply the formulas V = (l)(w)(h) and V = (B)(h) for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve problems in the context. c. Determine the volumes of composite figures by decomposing them into non-overlapping right rectangular prisms and adding their volumes together to solve problems in context

5.GR.D Graph points on a coordinate plane to solve contextual and mathematical problems.

5.GR.D.6Define the coordinate system as a pair of perpendicular lines called axes that intersect at the origin. Use understanding that the coordinate values represent the distance from the origin on the x-axis and y-axis to graph and name coordinate points in the first quadrant using ordered pairs.

5.GR.D.7Represent problems by graphing points in the first quadrant of the coordinate plane and interpreting coordinate values of points in the context of the situation.

5.NOS Number and Operation Sense (NOS)

5.NOS.A Understand the place value system.

5.NOS.A.1Explain that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and of what it represents in the place to its left.

5.NOS.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10 and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of 10.

5.NOS.A.3Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form (e.g. 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000)). b. Estimate decimal quantities by reasoning about their location on a number line, their relationship to benchmark numbers (e.g. 0, 0.25, 0.5, 0.75, 1) and rounding to any place. c. Compare two decimals to thousandths by reasoning about the values of the digits and the location on the number line. Record the results of comparisons with the symbols >, =, <.

5.NOS.B Perform operations on multi-digit whole numbers and with decimals to thousandths.

5.NOS.B.4Fluently multiply multi-digit whole numbers. a. Apply estimation strategies to estimate products. b. Use computational strategies (e.g., partial products, doubling and halving) efficiently to multiply. c. Use a standard algorithm to multiply. d. Determine and explain when a strategy or algorithm is most efficient.

5.NOS.B.5Divide multi-digit whole numbers with up to four-digit dividends and two-digit divisors. a. Apply estimation strategies to estimate quotients. b. Use partial quotients efficiently to divide. c. Use properties of operations and/or the inverse relationship between multiplication and division to divide. d. Represent and explain the computation by connecting rectangular arrays, area models, and/or equations to the meaning of division.

5.NOS.B.6Apply and extend previous understanding of addition and subtraction of whole numbers to add and subtract decimals to hundredths. a. Apply estimation strategies to estimate sums and differences of decimals. b. Use concrete models, drawings, strategies based on place value (e.g., partial sums), properties of operations and/or the inverse relationship between addition and subtraction to add and subtract decimals. c. Use an algorithm to add and subtract decimals. d. Determine and explain when a strategy or algorithm is most efficient.

5.NOS.B.7Apply and extend previous understanding of multiplication and division of whole numbers to multiply and divide decimals to hundredths in context. a. Apply estimation strategies to estimate products and quotients. b. Use concrete models, drawings, and arrays to multiply and divide. c. Use strategies based on place value (e.g., partial products, partial quotients) to multiply and divide. d. Use properties of operations and/or the inverse relationship between multiplication and division to multiply and divide. e. Represent and explain the computation by connecting concrete models, drawings, and/or equations to the meaning of multiplication and division.

5.NOS.C Use equivalence fractions as a strategy to add and subtract fractions.

5.NOS.C.8Add and subtract fractions with unlike denominators. a. Estimate sums and differences by reasoning about benchmark numbers and assess reasonableness of answers (e.g., 2/3 + 3/4 will be more than 1 because both fractions are greater than 1/2). b. Apply and extend whole number addition and subtraction strategies (e.g. counting on, making a whole, partial sums, compensation, properties of operations, the inverse relationship between addition and subtraction) to add and subtract fractions. c. Use equivalent fractions to produce an equivalent sum or difference of fractions with like denominators (e.g. 2/3 + 5/4 = 8/12 + 15/12 = 23/12. In general, a/b + c/d = (ad + bc)/bd).

5.NOS.C.9Solve problems in context involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators using visual fraction models and/or equations to represent the problem.

5.NOS.D Apply and extend previous understanding of multiplication and division to multiply and divide fractions.

5.NOS.D.10Interpret a fraction as division of the numerator by the denominator (c/d = c ÷ d) in context using visual fraction models and equations to represent the problem (e.g., interpret 3/4 as the result of dividing 3 by 4 so when 3 wholes are shared equally among 4 people each person has a share of size 3/4).

5.NOS.D.11Apply and extend previous understandings of multiplication to multiply a whole number by a fraction (c/d × b). a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers (e.g., 3/4 × 2 will be less than 2 because I am creating a fractional part of 2). b. Multiply a whole number by a fraction (e.g., interpret c/d × b as c/d of b when c/d is greater than or less than 1).

5.NOS.D.12Apply and extend previous understandings of multiplication to multiply a fraction by a fraction. a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers (e.g., 1/2 × 3/4 will be less than 3/4 because I will have one half of 3/4). b. Apply and extend whole number multiplication strategies (e.g. area models). c. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths and show that the area is the same as would be found by multiplying the side lengths.

5.NOS.D.13Apply and extend previous understandings of multiplication to multiply a fraction by a fraction, with one or both factors greater than 1. a. Estimate products by reasoning about benchmark numbers and assess reasonableness of answers. b. Apply and extend whole number multiplication strategies (e.g. area model, partial products).

5.NOS.D.14Interpret multiplication as scaling (sizing) in context. a. Compare the size of a product to the size of one factor based on the size of the other factor, without performing the indicated multiplication. b. Explain why multiplying a given number by a fraction greater than a 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relate the principle of fraction equivalence to the Identity Property of Multiplication.

5.NOS.D.15Solve problems in contexts involving multiplication of fractions and mixed numbers, (e.g., by using visual fraction models or equations to represent the problem).

5.NOS.D.16Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions in context. a. Apply estimation strategies to estimate quotients and assess reasonableness of answers. b. Use visual fraction models and equations to interpret division of a unit fraction by a non-zero whole number and compute quotients. c. Use visual fraction models and equations to interpret division of a whole number by a unit fraction and compute quotients. d. Represent and explain the computation by connecting visual fraction models and/or equations to the meaning of division with unit fractions.

Grade 6

6.AT Algebraic Thinking (AT)

6.AT.A Understand ratio concepts and use ratio reasoning to solve problems.

6.AT.A.1Use ratio language in context (e.g., “__ to __,” “for every,” “per”) to describe a ratio relationship between two quantities, including part to part and part to whole (e.g., "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." )

6.AT.A.2Represent and use unit rates, written as fractions (a/b, b ≠ 0) with whole number numerators and denominators, in context. a. Use unit rate language in the context of a ratio relationship (e.g., “The recipe has a ratio of 3 cups of flour to 4 cups of sugar so there is 3/4 cup of flour for each cup of sugar.”) b. Solve unit rate problems, for example those involving unit pricing and constant speed (e.g., “If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?”).

6.AT.A.3Use ratio and rate reasoning to solve problems in context by reasoning about tables of equivalent ratios, tape diagrams, double number lines, or the Coordinate Plane. a. Generate, compare and find missing values of equivalent ratios and unit rates using multiple representations. b. Plot the pairs of values on the Coordinate Plane (in quadrant 1) to represent unit rates and make connections between representations. c. Use ratio reasoning to convert measurement units (e.g., money, time, length within the same system); manipulate and transform units appropriately when multiplying or dividing quantities.

6.AT.A.4Find a percent of a quantity in context as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity), including finding the whole given a part and the percent, by using tables, tape diagrams, and double number lines.

6.AT.B Apply and extend previous understandings of arithmetic to algebraic expressions.

6.AT.B.5Write numerical expressions involving whole number exponents and positive rational number bases. Use technology to evaluate numerical expressions involving whole number exponents and positive rational number bases.

6.AT.B.6Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers (e.g., express the calculation "Subtract y from 5" as 5 − y). b. Identify parts of an expression using mathematical language (e.g., sum, term, product, factor, quotient, coefficient) and view one or more parts of an expression as a single unit (e.g., describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single unit and a sum of two terms). c. Evaluate expressions given specific values of their variables in context. Include expressions that involve arithmetic operations and whole number exponents.

6.AT.B.7Apply the properties of operations (e.g., distributive, associative, commutative, etc.) to generate equivalent algebraic expressions and to identify when two expressions are equivalent (e.g., apply the distributive property to the expression 3(2 + 𝑥) to produce the equivalent expression 6 + 3𝑥; apply the distributive property to the expression 24𝑥 +18𝑦 to produce the equivalent expression 6(4𝑥 + 3𝑦); apply properties of operations to 𝑦 + 𝑦 + 𝑦 to produce the equivalent expression 3y.)

6.AT.C Reason about and solve one-variable equations and inequalities

6.AT.C.8Solve problems in context by writing and solving equations and interpreting the solutions in context. a. Explain that a variable can represent an unknown number and use substitution to determine whether a given number makes an equation true. b. Write and solve one-step equations of the form x + p = q, x − p = q, px = q and x/p = q for cases in which p, q and x are all nonnegative rational numbers. c. Represent solutions of such equations on a number line.

6.AT.C.9Write an inequality of the form 𝑥 > 𝑐 (using ≥, >, ≤, <) to represent a constraint or condition in context. Recognize that inequalities of this form have infinitely many solutions; represent solutions of such inequalities on number lines.

6.AT.C.10Solve problems in context by writing and solving inequalities and interpreting solution sets in context. a. Explain that a variable can represent an unknown number in a set and use substitution to determine whether a given number in a set makes an inequality true. b. Write and solve one-step inequalities of the form x + p > q, x − p > q, px > q and x/p > q (using ≥, >, ≤, <) where p and q are nonnegative rational numbers. c. Represent solutions of such inequalities on a number line.

6.AT.D Represent and analyze quantitative relationships between dependent and independent variables

6.AT.D.11Use variables to represent two quantities in context that change in relationship to one another. a. Write an equation (one-step) to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. b. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation (e.g., in a problem involving motion at constant speed, list and graph ordered pairs of distances and times and write the equation d = 65𝑡 to represent the relationship between distance and time).

6.DS Reasoning with Data and Statistics (DS)

6.DS.A Extend understanding of statistical variability.

6.DS.A.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers (e.g., "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages).

6.DS.A.2Use the language of probability (e.g., likely, unlikely, certain, impossible) to describe the likelihood of possible outcomes and to anticipate variability in data collected from statistical questions. Explain how the likelihood of outcomes helps predict which responses may be more or less common in the data and justify predictions using probability terms.

6.DS.A.3Explain how a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.DS.A.4Recognize that a measure of center (median and/or mean) for a numerical data set summarizes all the values with a single number, while a measure of variation describes how its values vary with a single number.

6.DS.B Summarize and describe distributions

6.DS.B.5Interpret numerical data in plots on a number line, including line plots, histograms, and box plots.

6.DS.B.6Summarize numerical data sets in relation to their context. a. Report the number of observations. b. Describe the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Give quantitative measures of center (median and/or mean) and variability (interquartile range), as well as describing outliers with reference to the context in which the data were gathered. d. Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

6.GR Geometric Reasoning and Measurement (GR)

6.GR.A Solve problems in context involving area, surface area, and volume.

6.GR.A.1Find the area of triangles, quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and quadrilaterals to solve problems.

6.GR.A.2Find the volume of a right rectangular prism with fractional edge lengths. a. By packing prisms with unit cubes and using estimation to show that the volume can be found by multiplying the edge lengths of the prism. b. Apply the formulas and (B is the area of the base) to find the volumes of right rectangular prisms with fractional edge lengths in context.

6.GR.A.3Draw polygons in the Coordinate Plane given coordinates for the vertices; use coordinates to find the length of a side joining endpoints with the same first coordinate or the same second coordinate. Apply these techniques in context.

6.GR.A.4Represent three-dimensional figures (triangular prism, rectangular prism, pyramid) using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures to solve problems.

6.NOS Number and Operation Sense (NOS)

6.NOS.A Apply and extend previous understanding of multiplication and division to divide fractions by fractions.

6.NOS.A.1Divide fractions by fractions in context. a. Extend estimation strategies to estimate and assess the reasonableness of quotients. b. Use and connect concrete and visual fraction models (e.g., linear, regional/area, and set models) and equations to divide. c. Represent and explain the calculation by connecting fraction models, and/or equations to the meaning of division.

6.NOS.B Compute fluently with multi-digit numbers and find common factors and multiples.

6.NOS.B.2Fluently divide multi-digit numbers. a. Extend estimation strategies to estimate and assess the reasonableness of quotients. b. Use strategies (e.g., partial quotients, inverse relationship between multiplication and division) to divide. c. Use a standard algorithm to divide. d. Determine and explain when a strategy or algorithm is most efficient.

6.NOS.B.3Fluently multiply and divide multi-digit decimals to the thousandths in context. a. Extend estimation strategies to estimate and assess the reasonableness of products and quotients. b. Generalize whole number strategies to multiply and divide decimals. c. Use a standard algorithm to multiply and divide decimals. d. Determine and explain when a strategy or algorithm is most efficient.

6.NOS.B.4Identify the common factors and multiples of two whole numbers. a. Identify the greatest common factor of two whole numbers within 100. b. Identify the least common multiple of two whole numbers within 12. c. Express the sum of two whole numbers within 100 using the distributive property to factor out the common factor of the addends (e.g., express 36 + 8 as 4(9 + 2)).

6.NOS.C Apply and extend previous understanding of numbers to the system of rational numbers.

6.NOS.C.5Use understanding of rational numbers as positive and negative numbers to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). Use rational numbers to represent quantities in context and explain the meaning of 0 in each situation.

6.NOS.C.6Represent a rational number as a point on vertical or horizontal number lines. Represent points on a number line with negative numbers. a. Estimate quantities by reasoning about their location on a number line, their relationship to benchmark numbers, and by rounding. b. Locate and interpret the positions of rational numbers on a number line. c. Recognize and locate opposite signs of rational numbers as indicating locations on opposite sides of 0 on the number line.

6.NOS.C.7Intersect perpendicular number lines at the point (0,0) as the x-axis and y-axis to introduce the four quadrants of the Coordinate Plane. Represent points on the plane with positive and negative coordinates. a. Use understanding of the relationship between signs and value of coordinates in ordered pairs to reason about the location of an ordered pair without plotting. b. Locate and interpret the positions of rational numbers on the Coordinate Plane, including how changes in the signs of ordered pairs reflect their location into different quadrants.

6.NOS.C.8Order and identify the absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line (e.g., interpret −3 > −7 as a statement that –3 is located to the right of –7 on a horizontal number line). b. Write, interpret, and explain statements of order for rational numbers in context (e.g., write −3℃ > −7℃ to express the fact that −3℃ is warmer than −7℃). c. Use understanding of the absolute value of a rational number as its distance from 0 on the number line to interpret absolute value as magnitude for a positive or negative quantity in context (e.g., for an account balance of –30 dollars, write to describe the size of the debt in dollars).

6.NOS.C.9Solve problems in context by graphing points (with the same first coordinate or the same second coordinate) in all four quadrants of the Coordinate Plane. a. Explain that points with the same x-coordinate or y-coordinate are located on same vertical or horizontal line respectively. b. Use absolute value to calculate length of vertical and horizontal lines on Coordinate Plane.

Grade 7

7.AT Algebraic Thinking (AT)

7.AT.A Represent and reason about proportional relationships to solve problems.

7.AT.A.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. Example: If a person walks mile in each hour, compute the unit rate as the complex fraction miles per hour, or equivalently 2 miles per hour.

7.AT.A.2Identify proportional relationships between quantities. a. Determine whether two quantities are in a proportional relationship, for example, by testing for equivalent ratios in a table or by graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and narrative descriptions of proportional relationships.

7.AT.A.3Represent proportional relationships between quantities. a. Represent proportional relationships using equations. b. Explain what a point (x, y) on the graph of a proportional relationship represents in terms of the context, with special attention to the points (0, 0) and (1, r), where r is the unit rate.

7.AT.A.4Solve ratio and percent problems in context (e.g., simple interest, tax, markups and markdowns, tips, commissions, fees, percent increase or decrease, and percent error) by applying proportional reasoning.

7.AT.B Look for and make use of structure to rewrite expressions in equivalent forms and reason about their properties.

7.AT.B.5Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.AT.B.6Rewrite an expression in different forms to shed light on the represented context and how the quantities in it are related. For example, a + 0.05a = 1.05a means that an "increase by 5%" is the same as "multiply by 1.05."

7.AT.C Model with and solve equations and inequalities

7.AT.C.7Analyze contextual situations to determine whether an equation or inequality best represents the relationship between quantities. Justify the choice based on the context.

7.AT.C.8Use variables to represent quantities and write and solve equations to model and reason about relationships in context. a. Write equations of the form px + q = r and p(x + q) = r where p, q, and r are specific rational numbers. b. Solve equations of these forms flexibly and efficiently. Explain the solution process and how it connects to the original statement of equality. c. Solve word problems leading to equations of these forms and interpret the solution in terms of the context.

7.AT.C.9Use variables to represent quantities and write and solve inequalities to model and reason about constraints and conditions in context. a. Write inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. b. Solve inequalities of these forms and graph the solution set on a number line. c. Interpret the solution set in terms of the context by describing what values are possible or reasonable in the given situation.

7.DS Reasoning with Data, Statistics, and Probability (DS)

7.DS.A Make sense of statistical inquiry.

7.DS.A.1Determine when generalizations about a population are appropriate based on how the sample was selected. a. Decide whether a sample is representative of a population by explaining how it was selected and how that selection affects the accuracy of conclusions. b. Explain how a random sample increases the likelihood that the sample represents the population.

7.DS.A.2Use data from random samples to make and evaluate estimates or predictions about a population. a. Use data from a single random sample to estimate an unknown characteristic of a population. b. Generate and compare multiple random samples of the same size and use the variability across samples to assess how reliable an estimate or prediction is.

7.DS.B Describe, analyze, and compare data using visual and numerical representations to model situations and draw inferences.

7.DS.B.3Compare two numerical data distributions using visual displays and numerical measures to draw inferences about differences between populations. a. Use histograms and other visualizations to describe similarities and differences in the range, shape, and clustering of the data sets. b. Use measures of center (e.g., mean or median) and measures of variability (e.g., range or interquartile range) to describe the size and significance of the difference between the distributions. c. Use both visual and numerical comparisons to describe how different the two groups are and whether the difference is meaningful in the context of the data.

7.DS.C Model and analyze probability to interpret chance events, compare models with observed data, and make predictions.

7.DS.C.4Explain that a probability of 0 means the event is impossible, 1 means it is certain, and values closer to 0 or 1 indicate how unlikely or likely the event is. Use this understanding to estimate the probability of chance events based on knowledge about the context, and compare the likelihood of different events.

7.DS.C.5Understand and apply long-run relative frequency to approximate probabilities and make predictions about chance events. a. Use data collection or simulation to approximate the probability of a chance event by observing its long-run relative frequency. b. Use a known or estimated probability to evaluate whether the number of outcomes in a given number of simulation trials is reasonable. c. Use visual representations (e.g., bar graphs, circle graphs) to compare and interpret probabilities.

7.DS.C.6Develop and use probability models to find probabilities of events and compare them to observed results. a. Create a uniform probability model for situations where outcomes are equally likely and use the model to calculate the probability of an event. b. Create a probability model using observed frequencies and use it to estimate the probability of an event. c. Compare probabilities from a model to observed data and explain possible reasons for any differences.

7.GR Geometric Reasoning and Measurement (GR)

7.GR.A Use structure and representation to analyze geometric figures.

7.GR.A.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

7.GR.A.2Describe the two-dimensional figures that result from slicing three-dimensional figures (e.g., right rectangular prisms, right cylinders) with a plane. Use the resulting shapes to explain how the structure of the solid connects to volume and surface area.

7.GR.B Reason with and about geometric formulas to solve problems and justify solutions in context.

7.GR.B.3Apply the formulas for the area and circumference of a circle to solve problems in context. Use informal reasoning and visual models to explain the relationship between the circumference and area of a circle.

7.GR.B.4Solve mathematical problems with and without context involving area, surface area, and volume of two- and three-dimensional objects. a. Solve problems involving area, surface area, and volume of polygonal figures and polyhedral solids, including triangles, quadrilaterals, other polygons, cubes, right prisms, and pyramids—figures defined by edges and flat faces. b. Solve problems involving volume of curved-surface solids, including cones, cylinders, and spheres—figures without edges. c. Interpret and justify solutions in context by comparing different geometric representations, estimating or checking for reasonableness, and explaining how the result applies to the context.

7.NOS Number and Operation Sense (NOS)

7.NOS.A Use structure and representations to reason with rational numbers.

7.NOS.A.1Add and subtract rational numbers using multiple strategies and representations. a. Describe situations in which opposite quantities combine to make 0. b. Model the addition of rational numbers as movement along a number line, where the direction and distance depend on the sign and magnitude of the addend. Show that a number and its opposite have a sum of 0 (are additive inverses). c. Rewrite subtraction as addition using the additive inverse p – q = p + (–q). Use strategies such as number lines and applying absolute value of the difference to determine the distance between two rational numbers and solve problems in context. d. Model and solve multistep contextual problems involving addition and subtraction of rational numbers, using efficient and flexible strategies.

7.NOS.A.2Multiply and divide rational numbers using multiple strategies and representations. a. Model multiplication of rational numbers using number lines, arrays, or area models to show how properties of operations (e.g., the distributive property) lead to rules for signed number multiplication, including (–1) × (–1) = 1. b. Interpret quotients of rational numbers in context and represent division of signed numbers in different forms, such as –p/q = (–p)/q = p/(–q) when the divisor is not zero. c. Use properties of operations to multiply and divide rational numbers flexibly and efficiently. d. Analyze the decimal representation of a quotient resulting from dividing rational numbers to determine whether it terminates or repeats and use this pattern to explain why all rational numbers have decimal forms that either terminate or repeat. e. Model and solve multistep contextual problems involving multiplication and division of rational numbers, using efficient and flexible strategies.

7.NOS.A.3Solve multistep problems involving positive and negative rational numbers in any form (i.e., fractions, decimals, percents) using all four operations. a. Apply properties of operations to efficiently compute with rational numbers and convert between forms when appropriate to support problem solving. b. Evaluate the reasonableness of solutions using estimation, mental math, and understanding of the context.

Grade 8

8.AT Algebraic Thinking (AT)

8.AT.A Use proportional relationships to represent and reason about linear equations.

8.AT.A.1Analyze and compare proportional relationships using slope. a. Graph proportional relationships and interpret the unit rate as the slope of the graph. b. Compare proportional relationships represented in different forms (e.g., graphs, equations, and tables) by analyzing their slopes.

8.AT.A.2Use proportional reasoning to identify slope and represent linear relationships. a. Use proportional reasoning to explain why the slope m is the same between any two points on a non-vertical line by showing that the ratios of vertical change to horizontal change between points are equivalent. b. Use this relationship to write equations of lines in the form for lines that pass through the origin, and in the form for lines that include a vertical shift from the origin, where b represents the y-intercept.

8.AT.B Make sense of and solve equations, inequalities, and systems of equations

8.AT.B.3Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solution. Justify each example by showing how it can be rewritten in an equivalent equation in the form x=a, a=a, or a=b (where a and b are different rational numbers) and explaining what this form reveals about the solution. b. Solve linear equations in one variable with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combing like terms.

8.AT.B.4Solve linear inequalities in one variable. a. Solve inequalities including those requiring the use of the distributive property and combining like terms. b. Represent the solution set on a number line and describe the set of possible values in the context of the problem.

8.AT.B.5Analyze and solve systems of two linear equations in two variables. a. Interpret the solution to a system of two linear equations as the point where their graphs intersect and verify that the coordinates of the intersection satisfy both equations. b. Solve systems of two linear equations by graphing and substitution. When using graphing, find an approximate or exact solution depending on the precision of the graph. Select a method strategically based on the structure of the system or the context of the problem and explain why the method is appropriate. c. Use structure to determine when a system has no solution, one solution, or infinitely many solutions and justify the conclusion. d. Model and solve contextual problems using systems of two linear equations.

8.AT.C Reason about functions

8.AT.C.6Determine whether a relationship is a function by analyzing whether each input is assigned to exactly one output. a. Given a clearly defined set of input-output pairs in a table, mapping, or real-world description, determine whether the relationship is a function and explain why. b. Given a graph, identify input-output pairs and determine whether the graph represents a function.

8.AT.C.7Compare the properties (i.e., rate of change, intercepts) of two linear functions each represented in a different way (algebraically, graphically, numerically in tables, or by narrative descriptions).

8.AT.C.8Identify whether a function is linear or non-linear given equations, graphs, or input-output pairs, and provide a justification using reasoning about the constant rate of change and its relationship to proportional reasoning.

8.AT.D Model with functions.

8.AT.D.9Interpret the graph of a linear function in the form y = mx + b. a. Given the graph of a linear function, identify the slope and y-intercept. b. Match a linear equation in the form y = mx + b to its graph.

8.AT.D.10Construct and interpret a linear function to model a relationship between two quantities. a. Given two input-output pairs, a graph, a table, or a description of a relationship, determine the rate of change and initial value and write the linear equation that models the situation. b. Interpret the rate of change and initial value in the context of the problem.

8.AT.D.11Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function based on a narrative description.

8.DS Reasoning with Data, Statistics, and Probability (DS)

8.DS.A Make sense of statistical inquiry.

8.DS.A.1Evaluate predictions and conclusions drawn from bivariate data. a. Determine whether observed associations in sample data justify generalizations about a broader population. b. Identify limitations or potential bias in data collection or interpretation when analyzing trends in scatter plots.

8.DS.B Describe, analyze, and compare data using visual and numerical representations to model situations and draw inferences.

8.DS.B.2Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.DS.B.3Compare the fit of different linear models shown on scatter plots of the same data by describing the overall pattern and the closeness of data points to each line.

8.DS.B.4Use a provided linear model to make predictions based on bivariate measurement data. Interpret the meaning of the slope and y-intercept in context.

8.DS.C Model and analyze probability to interpret chance events and make predictions

8.DS.C.5Construct a two-way table to organize data on two categorical variables collected from the same subjects. Calculate and interpret relative frequencies for rows and/or columns and use them to describe possible associations between the variables.

8.DS.C.6Find probabilities of compound events by representing or simulating the sample space or simulating events. a. Represent the sample space for a compound event using organized lists, tables, or tree diagrams. Given an event described in everyday language, identify the outcomes in the sample space that make up the event. b. Determine the probability of a compound event as a fraction of outcomes in the sample space that result in the event.

8.GR Geometric Reasoning and Measurement (GR)

8.GR.A Make conjectures about and verify geometric relationships.

8.GR.A.1Use the relationships between supplementary, complementary, vertical, and adjacent angles to write and solve equations for an unknown angle.

8.GR.A.2Draw or build triangles using tools (ruler, protractor, technology, or physical manipulatives) given three measures of angles or sides. Determine whether the given conditions result in a unique triangle, more than one triangle, or no triangle.

8.GR.A.3Use informal arguments to verify that the sum of the interior angles of a triangle is 180 degrees and why an exterior angle equals the sum of the two remote interior angles.

8.GR.B Apply the pythagorean theorem to reason about right triangles.

8.GR.B.4Understand and reason about the Pythagorean Theorem and its extensions. a. Use diagrams, examples, or informal reasoning to explain the Pythagorean Theorem and why it applies to all right triangles. b. Use the converse of the Pythagorean Theorem to determine whether a set of three side lengths forms a right triangle, and extend this reasoning to identify whether a triangle is acute or obtuse using Pythagorean inequalities.

8.GR.B.5Apply the Pythagorean Theorem to solve problems in context. a. Determine unknown side lengths in right triangles. b. Find the distance between two points in a coordinate system using the Pythagorean Theorem.

8.NOS Number and Operation Sense (NOS)

8.NOS.A Reason with irrational numbers.

8.NOS.A.1Identify a number as irrational when its decimal expansion neither terminates nor repeats, and explain that irrational numbers cannot be expressed as the ratio of two integers.

8.NOS.A.2Use rational approximations of irrational numbers to estimate their locations on a number line, compare their size, and apply this understanding to estimate the value of expressions (e.g., π^2) in context.

8.NOS.B Reason with exponents to express and interpret quantities.

8.NOS.B.3Know and apply the properties of integer exponents to generate equivalent numerical expressions.

8.NOS.B.4Use square and cube roots to represent solutions to equations and describe numbers. a. Use square and cube roots symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. b. Evaluate square roots of perfect squares from 1 to 100 and cube roots of perfect cubes from 1 to 125 by inspection.

8.NOS.B.5Apply scientific notation to represent and compare very large and very small quantities in context, and use technology to compute with numbers in scientific notation. a. Express numbers as a single digit times an integer power of 10 to estimate and compare magnitudes. b. Use scientific notation to model and compute with real-world quantities, choose appropriate units, and interpret their meaning based on the context.

Integrated Algebra 1

IA1.AT Algebraic Thinking (AT)

IA1.AT.A Look for and make use of structure to rewrite expressions in equivalent forms and reason about their properties.

IA1.AT.A.1Rewrite and interpret linear and exponential expressions to explain key properties of a relationship. a. Interpret components of linear and exponential expressions (e.g., coefficients, constants, bases, and exponents) as single entities and explain their meaning in context. b. Rewrite linear and exponential expressions in different equivalent forms to highlight and explain properties of the relationship (e.g., growth or decay rate, initial value, rate of change). c. Compare equivalent forms of linear and exponential expressions to determine which is most useful for understanding or solving a problem in context.

IA1.AT.B Make sense of and solve equations, inequalities, and systems of equations or inequalities.

IA1.AT.B.2Create and solve one-variable equations, justify the solution process, and interpret the solution(s) in context. a. Create linear and exponential equations in one variable to represent relationships between quantities. b. Solve linear equations in one variable using flexible and efficient strategies. Justify the solution process using logical steps and explanations. Interpret the meaning of the solution in context. c. Solve absolute value equations in one variable where the absolute value expression is isolated on one side of the equation. Justify the solution process and interpret the solution(s) in context. d. Solve exponential equations where both sides have comparable exponential expressions, requiring at most two steps to rewrite expressions to the same base, and explain the reasoning using properties of exponents.

IA1.AT.B.3Create and solve one-variable inequalities, justify the solution process, and interpret the solutions in context. a. Create inequalities in one variable to represent constraints or conditions. b. Solve linear inequalities in one variable, including those with variables on both sides, using flexible and efficient strategies. Represent the solution set using a number line and inequality notation. c. Solve absolute value inequalities in one variable where the absolute value expression is isolated on one side of the inequality. Represent the solution set using compound inequalities and a number line. d. Interpret the meaning of the solution set in the context of the problem, including what values are reasonable within the given situation.

IA1.AT.B.4Create linear and exponential equations in two variables, graph them, and use those graphs to represent and solve problems in context. a. Create linear and exponential equations in two variables to model relationships, and graph them on the coordinate plane. b. Use the understanding that the graph of an equation represents the set of all its solutions to identify solutions, describe patterns, and interpret the relationship between variables.

IA1.AT.B.5Create, solve, and analyze systems of two linear equations in two variables in context. a. Create systems of linear equations to represent relationships. b. Solve systems of linear equations using algebraic methods (i.e., substitution and linear combination). c. Approximate or verify solutions to systems of equations by graphing and identifying the point of intersection. d. Interpret the meaning of the solution in context.

IA1.AT.B.6Create linear inequalities in two variables to represent constraints in context and graph the solution set as a half-plane within the coordinate plane.

IA1.AT.B.7Represent and analyze systems of two linear inequalities in two variables in context. a. Create systems of linear inequalities in two variables to represent constraints and relationships. b. Graph the solution set to a system of linear inequalities as the intersection of the corresponding half-planes. c. Interpret the meaning of the solution set in context and determine whether given ordered pairs are solutions to the system.

IA1.AT.B.8Model and solve optimization problems using systems of linear inequalities. a. Represent contexts involving constraints and an objective to optimize (i.e., maximize or minimize) using a system of linear inequalities. b. Graph the feasible region and identify solutions that satisfy all constraints. c. Identify and justify the optimal solution(s) based on the objective, and interpret the meaning in context.

IA1.AT.B.9Use technology to solve systems of equations consisting of a linear equation and an exponential in two variables and interpret the solution within a given context.

IA1.AT.C Reason about functions.

IA1.AT.C.10Identify, represent, and analyze functions in terms of their domain and range. a. Determine whether a relationship is a function by verifying that each element of the domain is assigned to exactly one element of the range using tables, graphs, and equations. b. Determine whether a function is one-to-one by analyzing tables, graphs, or equations, and justify the reasoning based on the relationship between inputs and outputs. c. Use function notation to evaluate functions for specific input values, x, and describe how the output changes with respect to the input. Interpret the meaning of specific values and patterns in context. d. Represent the graph of a function as the set of ordered pairs (x, f(x)) and explain how it illustrates the relationship between domain and range.

IA1.AT.C.11Calculate or estimate and interpret the average rate of change over a specified interval of linear and exponential functions presented graphically, symbolically, or in a table.

IA1.AT.C.12Compare key properties (i.e., rates of change, intercepts, patterns of growth) of two functions, one linear and one exponential, when each is represented in a different way (i.e., algebraically, graphically, numerically in tables, or by narrative descriptions).

IA1.AT.D Model with functions.

IA1.AT.D.13Analyze and model arithmetic and geometric sequences using recursive and explicit representations. a. Identify whether a sequence is arithmetic or geometric and create an explicit rule to describe the pattern. b. Rewrite a recursive representation of an arithmetic or geometric sequence to an explicit rule. c. Model contexts involving arithmetic or geometric patterns using an explicit or recursive rule, and use the rule to solve problems.

IA1.AT.D.14Represent functions using tables and graphs, and interpret key features in context. Key features include: domain, intercepts, intervals of increase and decrease, positive and negative values, end behavior, asymptote, and points of transition between pieces. a. Analyze linear and exponential functions by identifying patterns in tables and graphs. Distinguish between constant and proportional growth. b. Extend the understanding of functions to include piecewise-defined and absolute value functions. Represent them in tables and graphs and explain how absolute value functions can be represented as a specific case of piecewise functions. c. Relate key features of a function’s graph and table to the characteristics of a context, and justify the appropriateness of a function model based on that context.

IA1.AT.D.15Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Given a relationship in context where the rate of change over equal intervals is constant, identify it as linear and create a linear function to model the relationship. b. Given a relationship in context where a quantity changes by a constant percent per unit interval relative to another, identify it as exponential and create an exponential function to model the relationship.

IA1.AT.D.16Analyze how changes to a function’s input or output affect the graph and its meaning in context. a. Identify how replacing f(x) with f(x + k) and f(x) + k changes the graph of a linear, exponential, or absolute value function. Describe the direction of the change and determine the value of k when given a graph. b. Explain how changing the input or output of a function affects the quantities represented in context. Use the structure and features of the parent function to justify how these changes impact the initial context.

IA1.AT.D.17Find and interpret the inverse of a linear function in context. a. Show that a linear function is one-to-one and therefore has an inverse that is also a function. b. Find the inverse of a linear function represented with an equation or a table. c. Interpret the meaning of the inverse in context as a relationship that maps outputs of the original function back to their corresponding inputs.

IA1.DS Reasoning with Data, Statistics, and Probability (DS)

IA1.DS.A Make sense of statistical inquiry.

IA1.DS.A.1Distinguish between correlation and causation.

IA1.DS.A.2Understand and apply statistics as a process for making inferences about population parameters using random samples. a. Explain the importance of randomization in selecting samples to ensure unbiased and representative data. b. Analyze the representativeness of a sample and its implications for making inferences about the population.

IA1.DS.A.3Analyze and interpret how data are represented and used in arguments or reports. a. Identify key components of data-based arguments, including claims, visualizations (e.g., graphs, tables), and the statistical language used to support conclusions. b. Evaluate the accuracy and effectiveness of visualizations in representing data and supporting a claim or conclusion. c. Describe how data were collected and whether the method (e.g., sampling, use of randomization) supports valid conclusions.

IA1.DS.A.4Plan statistical investigations. a. Formulate statistical questions that can be answered with data. b. Determine effective data collection methods that enhance accuracy, validity, and representativeness of the data.

IA1.DS.B Describe, analyze, and compare data using visual and numerical representations to model situations and draw inferences.

IA1.DS.B.5Analyze and compare one-variable data visualizations on the real number line to draw conclusions. a. Compare multiple data visualizations to identify the unique statistical information each provides and their usefulness in different contexts. b. Analyze and compare the shape, center, and spread of data sets using appropriate visualizations.

IA1.DS.B.6Construct and interpret two-way frequency tables to analyze associations between two categorical variables. a. Use the tables to decide if events are independent b. Use the tables to find joint and marginal probabilities.

IA1.DS.B.7Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a linear or exponential function to data; use functions fitted to data to solve problems in the context of the data. b. Informally assess the fit of a linear or exponential function visually by analyzing the graph and analytically by evaluating the appropriateness of the chosen model in representing the data and context.

IA1.DS.B.8Analyze and interpret the key components of linear and exponential models in the context of the data with and without technology. a. Interpret slope (rate of change) and the intercept (constant term) of a linear model to describe patterns and relationships in the data. b. Analyze the growth factor or average rate of change in exponential models to describe patterns, interpret trends, and make predictions in context.

IA1.GR Geometric Reasoning (GR)

IA1.GR.A Reason about transformations and congruence.

IA1.GR.A.1Apply definitions of rotations, reflections, and translations to transform figures and use these transformations to verify congruence. a. Describe rotations, reflections, and translations as functions that take points in the plane as inputs and give corresponding points as outputs. b. Verify experimentally that rigid transformations preserve distance, angle measure, and parallelism, and use this understanding to justify congruence.

IA1.GR.A.2Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

IA1.GR.B Conjecture about and verify geometric relationships.

IA1.GR.B.3Apply triangle congruence theorems (SSS, SAS, ASA, AAS, HL) to reason logically about geometric diagrams, including verifying congruence and solving for unknown measures.

IA1.GR.B.4Understand and apply theorems and criteria for parallel and perpendicular lines. a. Use theorems about parallel lines cut by a transversal to identify and apply angle pair relationships, including corresponding, alternate interior, alternate exterior, and consecutive interior angles. b. Verify the slope criteria for parallel and perpendicular lines and apply them to solve geometric problems (e.g., determine when lines are parallel or perpendicular and find equations of lines that satisfy given geometric conditions).

IA1.GR.B.5Use angle relationships and slope criteria to verify properties of parallelograms. a. Use angle relationships from parallel lines and transversals, along with triangle congruence theorems, to verify properties of parallelograms, including congruent opposite sides, congruent opposite angles, and diagonals that bisect each other. b. Apply the slope criteria for parallel and perpendicular lines to verify when a quadrilateral is a parallelogram in a coordinate plane.

Integrated Algebra 2

IA2.AT Algebraic Thinking (AT)

IA2.AT.A Look for and make use of structure to rewrite expressions in equivalent forms and reason about their properties.

IA2.AT.A.1Rewrite and interpret quadratic expressions to explain key properties of a relationship. a. Factor quadratic expressions to reveal the zeros of the function, and explain their significance in context. b. Complete the square to identify the maximum or minimum value of a relationship and interpret that value in context. c. Compare different forms of a quadratic expression (standard, factored, vertex) to determine which form is most useful for interpreting a given situation.

IA2.AT.A.2Given a polynomial of degree 3 or higher in factored form, use the Zero Product Property to identify its zeros and sketch a graph of the function.

IA2.AT.A.3Add, subtract, and multiply polynomials to rewrite expressions in equivalent forms for clarity, efficiency, or problem-solving applications. Use these operations to represent and analyze polynomial relationships in context. Multiplication of polynomials includes multiplying a binomial and a trinomial.

IA2.AT.B Make sense of and solve equations and systems of equations.

IA2.AT.B.4Create and solve quadratic equations in one variable. a. Create quadratic equations in one variable to represent relationships between quantities. b. Flexibly choose and apply methods to solve quadratic equations based on their initial form (i.e., solve equations by recognizing perfect squares (e.g., x² = 49), taking square roots, factoring when possible, completing the square, or applying the quadratic formula when necessary). Justify the selected method by reasoning about the structure of the equation and interpret the solution in context when applicable. c. Represent complex solutions as a ± bi for real numbers a and b.

IA2.AT.B.5Solve radical equations in one variable that contain a single radical term. Explain why some solutions may be extraneous and justify whether a given solution is valid based on a given context.

IA2.AT.B.6Create, solve, and analyze systems consisting of a linear and a quadratic equation in two variables in context. a. Create systems consisting of a linear and a quadratic equation to represent relationships. b. Solve these systems exactly and approximately using algebraic methods (i.e., substitution), graphical methods, or tables. c. Interpret the meaning of the solution(s) in context.

IA2.AT.B.7Use technology to solve systems of equations that include a combination of linear, exponential, quadratic, and/or radical equations. Interpret the solution(s) within a given context.

IA2.AT.C Reason about functions.

IA2.AT.C.8Calculate or estimate and interpret the average rate of change over a specified interval for exponential, quadratic, and radical functions presented graphically, symbolically or as a table.

IA2.AT.C.9Compare properties of two quadratic functions each represented in a different way (algebraically, graphically, numerically in tables, or by narrative descriptions), to analyze differences and similarities between the relationships the functions model.

IA2.AT.C.10Analyze and compare the behavior of functions across different families. a. Compare linear, quadratic, exponential, polynomial (degree of 3 or higher), and radical functions by analyzing differences in their patterns of change and key characteristics (e.g., symmetry, end behavior). Explain how these differences affect the modeling of various contexts. b. Analyze how the number and type of solutions to equations vary across function families and explain how those solutions are represented and interpreted in graphs and equations.

IA2.AT.D Model with functions.

IA2.AT.D.11Write a quadratic function to model relationships between quantities given a narrative description, table of values, or graph.

IA2.AT.D.12Represent and interpret quadratic functions using tables and graphs and apply them in context. a. Create tables and graphs to represent quadratic relationships and identify key features. Key features include: intercepts/zeros, symmetry, intervals of increase and decrease, relative extrema, end behavior, domain (if restricted in context), and range. b. Relate key features in the table and graph to characteristics of a context. c. Justify the appropriateness of a quadratic model by analyzing trends in the table or graph and explaining how the model’s features support predictions or contextual interpretations.

IA2.AT.D.13Find and interpret the inverse of quadratic and exponential functions using algebraic and graphical representations. a. Restrict the domain of a quadratic to make it one-to-one and express its inverse as a square root function. b. Express the inverse of an exponential function as a logarithmic function and evaluate expressions involving logarithms. c. Graph a function and its inverse and analyze features (e.g., domain and range, intercepts, points of intersection) and symmetry across the line y = x. d. Interpret the meaning of inverse functions in context.

IA2.AT.D.14Represent and interpret higher degree polynomial (degree 3 or higher), and radical functions in context using tables and graphs. a. Create tables and graphs to represent higher degree polynomial functions and identify key features. Key features include: intercepts/zeros, symmetry, intervals of increase and decrease, relative extrema, and end behavior. b. Create tables and graphs to represent radical functions and identify key features. Key features include: domain restrictions, intercepts, intervals of increase and decrease, and end behavior. c. Relate key features of higher-degree polynomial and radical functions to contexts.

IA2.AT.D.15Analyze how transformations affect the graph of a polynomial function given in vertex form and interpret their meaning in context. a. Identify how the graph of a quadratic function changes when f(x) is replaced by f(x + k), f(x) + k, and kf(x) and determine the value of k from the graph. Describe the type of transformation. b. Explain the meaning of each transformation by connecting the structure of the transformed function to the quantities it represents (e.g., how changes affect the vertex, direction of opening, and rate of change).

IA2.DS Reasoning with Data, Statistics, and Probability (DS)

IA2.DS.A Make sense of statistical inquiry.

IA2.DS.A.1Evaluate and refine statistical study designs and critically analyze data-based claims to improve the quality and validity of inferences. a. Assess the limitations of a sample or study design in terms of bias, randomness, and representativeness. b. Propose modifications to sampling methods or data collection strategies to enhance the validity and generalizability of conclusions. c. Identify potential sources of bias or misuse in data collection, representation, or modeling. d. Recognize when correlation is misused to imply causation.

IA2.DS.B Describe, analyze, and compare data using visual and numerical representations to model situations and draw inferences.

IA2.DS.B.2Represent data on two quantitative variables with a scatter plot, and describe how the variables are related. a. Fit a quadratic, higher degree polynomial, or radical function to data. Use functions fitted to data to solve problems in the context of the data. b. Informally assess the fit of a quadratic, higher degree polynomial, or radical function visually by analyzing the graph and analytically by evaluating the appropriateness of the chosen model in representing the data and context.

IA2.DS.B.3Construct and interpret two-way frequency tables to analyze associations between two categorical variables. a. Use the tables to decide if events are independent b. Approximate conditional probabilities and explain their meaning in context, including the relationship between conditional probability and independence.

IA2.GR Geometric Reasoning (GR)

IA2.GR.A Reason about transformations and similarity.

IA2.GR.A.1Describe the effect of dilations on two-dimensional figures using coordinates, including how scale factor and center of dilation impact side lengths and angle measures.

IA2.GR.A.2Use similarity transformations to determine and justify triangle similarity. a. Determine whether two triangles are similar by identifying a sequence of similarity transformations (rotations, reflections, translations, and dilations) that relates one triangle onto another and aligns corresponding parts. b. Justify triangle similarity by explaining how similarity transformations establish that corresponding angles are congruent and corresponding sides are proportional.

IA2.GR.A.3Apply properties of similar triangles to solve geometric problems in context.

IA2.GR.A.4Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

IA2.GR.B Reason about right triangle relationships.

IA2.GR.B.5Use similarity of right triangles to explain and define trigonometric ratios (sine, cosine, and tangent) for acute angles as properties of the angles, based on side ratios.

IA2.GR.B.6Justify the side length relationships in special right triangles (30°-60°-90° and 45°-45°-90°). a. Use geometric reasoning (e.g., dissecting an equilateral triangle or square) to explain why the side length relationship in special right triangles hold. b. Describe the consistent ratios of side lengths in these triangles (e.g., 1:1:√2 and 1:√3:2), and explain how these ratios relate to the angle measures.

IA2.GR.B.7Use trigonometric ratios, the Pythagorean Theorem, and special right triangle relationships to solve right triangles in applied contexts.

IA2.GR.C Make sense of circles and their properties.

IA2.GR.C.8Apply similarity to develop and use formulas involving arc length, radian measure, and sector area. a. Explain and demonstrate, using similarity, that arc length is proportional to the radius of a circle. b. Calculate the radian measure of a central angle as the ratio of arc length to radius. c. Apply the formula for the area of a sector using radian measure.

IA2.GR.C.9Create the equation of a circle given the center and radius; identify the center and radius of a circle given the equation.

IA2.GR.D Model with geometry.

IA2.GR.D.10Use geometric shapes, their measures, and their properties to represent real world objects, and solve related authentic modeling and design problems.

IA2.GR.D.11Rearrange geometric formulas to isolate and interpret a quantity of interest in context (e.g., solving for height in a volume formula), using reasoning aligned with solving equations and modeling with mathematics.

IA2.NOS Number and Operation Sense (NOS)

IA2.NOS.A Reason with rational exponents.

IA2.NOS.A.1Apply the properties of exponents to generate equivalent numerical expressions involving radicals and rational exponents.

IA2.NOS.B Reason about the complex number system.

IA2.NOS.B.2Use the knowledge that i² = –1 to write complex numbers in standard form, a + bi, and use them to represent solutions to quadratics that have no real solutions.

Number and Quantity

N.CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.

N.CN.A.1Know there is a complex number i such that i^2 = –1, and every complex number has the form a + bi with a and b real.

N.CN.A.2Use the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

N.CN.A.3Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

N.CN.B Represent complex numbers and their operations on the complex plane.

N.CN.B.4Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

N.CN.B.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

N.CN.B.6Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

N.CN.C Use complex numbers in polynomial identities and equations.

N.CN.C.7Solve quadratic equations with real coefficients that have complex solutions.

N.CN.C.8Extend polynomial identities to the complex numbers.

N.CN.C.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N.OA Operations and Algebraic Thinking

N.OA.A Represent and evaluate series.

N.OA.A.1Use the notation for the factorial of a non-negative integer, n!, to evaluate expressions.

N.Q.A Reason quantitatively and use units to solve problems.

N.Q.A.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N.Q.A.2Define appropriate quantities for the purpose of descriptive modeling.

N.Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N.RN The Real Number System

N.RN.A Extend the properties of exponents to rational exponents.

N.RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

N.RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.

N.RN.B Use properties of rational and irrational numbers.

N.RN.B.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

N.VM Vector and Matrix Quantities

N.VM.A Represent and model with vector quantities.

N.VM.A.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

N.VM.A.2Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N.VM.A.3Solve problems involving velocity and other quantities that can be represented by vectors.

N.VM.B Perform operations on vectors.

N.VM.B.4Add and subtract vectors.

N.VM.B.4.aAdd vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

N.VM.B.4.bGiven two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

N.VM.B.4.cUnderstand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

N.VM.B.5Multiply a vector by a scalar.

N.VM.B.5.aRepresent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).

N.VM.B.5.bCompute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

N.VM.B.5.cDetermine the dot product of two vectors.

N.VM.C Perform operations on matrices and use matrices in applications.

N.VM.C.6Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

N.VM.C.7Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

N.VM.C.8Add, subtract, and multiply matrices of appropriate dimensions.

N.VM.C.9Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

N.VM.C.10Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

N.VM.C.11Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

N.VM.C.12Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Algebra

A.APR Arithmetic with Polynomials and Rational Expressions

A.APR.A Perform arithmetic operations on polynomials.

A.APR.A.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A.APR.B Understand the relationship between zeros and factors of polynomials.

A.APR.B.2Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A.APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A.APR.B.3.aIdentify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A.APR.C Use polynomial identities to solve problems.

A.APR.C.4Prove polynomial identities and use them to describe numerical relationships.

A.APR.C.5Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

A.APR.D Rewrite rational expressions.

A.APR.D.6Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

A.APR.D.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

A.CED Creating Equations

A.CED.A Create equations that describe numbers or relationships.

A.CED.A.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A.CED.A.1.aCreate equations and inequalities in one variable and use them to solve problems.

A.CED.A.1.bCreate polynomial equations given roots.

A.CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.A.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

A.CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

A.REI Reasoning with Equations and Inequalities

A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.

A.REI.A.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A.REI.A.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A.REI.B Solve equations and inequalities in one variable.

A.REI.B.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.REI.B.4Solve quadratic equations in one variable.

A.REI.C Solve systems of equations.

A.REI.C.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A.REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A.REI.C.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

A.REI.C.7.aSolve systems of equations comprised of various combinations of all algebraic and transcendental functions in two variables.

A.REI.D Represent and solve equations and inequalities graphically.

A.REI.D.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A.REI.D.11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

A.REI.D.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

A.SSE Seeing Structure in Expressions

A.SSE.A Interpret the structure of expressions.

A.SSE.A.1Interpret expressions that represent a quantity in terms of its context.

A.SSE.A.2Use the structure of an expression to identify ways to rewrite it.

A.SSE.A.2.aAnalyze the structure of the general form of a second degree equation, Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, to identify the conic section represented by the equation.

A.SSE.B Write expressions in equivalent forms to solve problems.

A.SSE.B.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

A.SSE.B.3.cUse the properties of exponents to transform expressions for exponential functions.

A.SSE.B.3.dChoose and produce an equivalent form of a second degree equation, Ax^2 + Cy^2 + Dx + Ey + F = 0, to reveal and explain properties of the conic section represented by the equation.

A.SSE.B.3.eTranslate between the standard and general form, Ax^2 + Cy^2 + Dx + Ey + F = 0, of a second degree equation.

A.SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

A.SSE.B.4.aExpress the sums in a series using sigma notation.

A.SSE.B.5Determine the sum, if it exists, of an infinite geometric series.

Functions

F.BF Building Functions

F.BF.A Build a function that models a relationship between two quantities.

F.BF.A.1Write a function that describes a relationship between two quantities.

F.BF.A.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.

F.BF.A.1.cCompose functions.

F.BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

F.BF.B Build new functions from existing functions.

F.BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

F.BF.B.4Find inverse functions.

F.BF.B.4.bVerify by composition that one function is the inverse of another.

F.BF.B.4.cRead values of an inverse function from a graph or a table, given that the function has an inverse.

F.BF.B.4.dProduce an invertible function from a non-invertible function by restricting the domain.

F.BF.B.4.eBuild inverse trigonometric functions.

F.BF.B.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

F.IF Interpreting Functions

F.IF.A Understand the concept of a function and use function notation.

F.IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F.IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.A.2.aExtend evaluating functions to include operations with composite functions, e.g. f(x + 2) - f(x).

F.IF.A.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F.IF.B Interpret functions that arise in applications in terms of the context.

F.IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F.IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.C Analyze functions using different representations.

F.IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

F.IF.C.7.fGraph all functions including piecewise-defined functions, step functions and absolute value functions

F.IF.C.7.gDetermine the end behavior of the graph of a polynomial function using the degree and leading coefficient.

F.IF.C.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.C.8.cInterpret the behavior of the graph of a function using the concept of limits.

F.IF.C.8.dEstimate limits algebraically, numerically and graphically.

F.IF.C.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.IF.C.10Describe the behavior of a sequence.

F.LE Linear, Quadratic, and Exponential Models

F.LE.A Construct and compare linear, quadratic, and exponential models and solve problems.

F.LE.A.1Distinguish between situations that can be modeled with linear functions and with exponential functions.

F.LE.A.1.dDistinguish between situations that can be modeled with exponential functions and logistic functions.

F.LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

F.LE.A.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

F.LE.A.4For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

F.LE.A.4.aUse properties of logarithms, including both common and natural logarithms, to rewrite and solve exponential models.

F.LE.B Interpret expressions for functions in terms of the situation they model.

F.LE.B.5Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.B.5.aInterpret the parameters in a logistic function in terms of a context.

F.LE.B.6Build and interpret logistic functions to model real-world problems.

F.LE.B.6.aSketch and analyze the graphs of logistic functions.

F.LE.B.6.bCompare and contrast the exponential, logarithmic, and logistic models.

F.LE.B.6.cApply understanding of logarithmic and logistic functions to solve real-world problems.

F.TF Trigonometric Functions

F.TF.A Extend the domain of trigonometric functions using the unit circle.

F.TF.A.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F.TF.A.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

F.TF.A.3Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.

F.TF.A.4Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F.TF.B Model periodic phenomena with trigonometric functions.

F.TF.B.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

F.TF.B.6Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

F.TF.B.7Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

F.TF.C Prove and apply trigonometric identities.

F.TF.C.8Prove the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

F.TF.C.9.aUse trigonometric identities to rewrite expressions and as a tool when solving trigonometric equations.

Geometry

G.C.A Understand and apply theorems about circles

G.C.A.1Prove that all circles are similar.

G.C.A.2Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G.C.A.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G.C.B Find arc lengths and areas of sectors of circles

G.C.B.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G.CO Congruence

G.CO.A Experiment with transformations in the plane

G.CO.A.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.A.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G.CO.A.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G.CO.A.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G.CO.A.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G.CO.B Understand congruence in terms of rigid motions

G.CO.B.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G.CO.B.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.CO.B.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G.CO.C Prove geometric theorems

G.CO.C.9Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

G.CO.C.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G.CO.C.11Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

G.CO.D Make geometric constructions

G.CO.D.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.CO.D.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G.GMD Geometric Measurement and Dimension

G.GMD.A Explain volume formulas and use them to solve problems

G.GMD.A.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

G.GMD.A.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G.GMD.B Visualize relationships between two-dimensional and three-dimensional objects

G.GMD.B.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G.GMD.B.4.aIdentify the shapes of two-dimensional cross-sections of a right double cone.

G.GPE Expressing Geometric Properties with Equations

G.GPE.A Translate between the geometric description and the equation for a conic section

G.GPE.A.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

G.GPE.A.2Derive the equation of a parabola given a focus and directrix.

G.GPE.A.3Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

G.GPE.B Use coordinates to prove simple geometric theorems algebraically

G.GPE.B.4Use coordinates to prove simple geometric theorems algebraically.

G.GPE.B.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

G.GPE.B.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G.GPE.B.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

G.GPE.C Polar coordinates.

G.GPE.C.8Understand the relationship between polar coordinates and Cartesian coordinates.

G.GPE.C.9Convert between polar and rectangular coordinates.

G.GPE.C.10Plot points on a polar coordinate grid.

G.GPE.D Polar equations.

G.GPE.D.11Convert equations between polar and rectangular forms.

G.GPE.D.12Graph polar equations by hand and using technology.

G.GPE.D.13Solve systems of polar equations.

G.MG Modeling with Geometry

G.MG.A Apply geometric concepts in modeling situations

G.MG.A.1Use geometric shapes, their measures, and their properties to describe objects.

G.MG.A.2Apply concepts of density based on area and volume in modeling situations.

G.MG.A.3Apply geometric methods to solve design problems.

G.SRT Similarity, Right Triangles, and Trigonometry

G.SRT.A Understand similarity in terms of similarity transformations

G.SRT.A.1Verify experimentally the properties of dilations given by a center and a scale factor.

G.SRT.A.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G.SRT.A.3Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G.SRT.B Prove theorems involving similarity

G.SRT.B.4Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G.SRT.B.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G.SRT.C Define trigonometric ratios and solve problems involving right triangles

G.SRT.C.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G.SRT.C.7Explain and use the relationship between the sine and cosine of complementary angles.

G.SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

G.SRT.D Apply trigonometry to general triangles

G.SRT.D.10Prove the Laws of Sines and Cosines and use them to solve problems.

G.SRT.D.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Statistics and Probability

S.CP Conditional Probability & the Rules of Probability

S.CP.A Understand independence and conditional probability and use them to interpret data

S.CP.A.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

S.CP.A.2Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S.CP.A.3Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

S.CP.A.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

S.CP.A.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

S.CP.B Use the rules of probability to compute probabilities of compound events.

S.CP.B.6Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

S.CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

S.CP.B.8Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

S.CP.B.9Use permutations and combinations to compute probabilities of compound events and solve problems.

S.IC Making Inferences and Justifying Conclusions

S.IC.A Understand and evaluate random processes underlying statistical experiments

S.IC.A.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S.IC.A.1.aIntroduce sampling distributions as a means to explore variability in sample statistics and ultimately evaluate a claim about a population.

S.IC.A.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.?

S.IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies

S.IC.B.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S.IC.B.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

S.IC.B.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

S.IC.B.6Evaluate reports based on data.

S.IC.B.7Conduct statistical investigations.

S.IC.B.7.aConduct observational studies.

S.IC.B.7.bConduct statistical experiments.

S.ID Interpreting Categorical and Quantitative Data

S.ID.A Summarize, represent, and interpret data on a single count or measurement variable

S.ID.A.1Represent data with plots on the real number line (dot plots, histograms, and box plots).

S.ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S.ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S.ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

S.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables

S.ID.B.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

S.ID.B.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

S.ID.B.6.aFit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

S.ID.B.6.bInformally assess the fit of a function by plotting and analyzing residuals.

S.ID.B.6.cFit a linear function for a scatter plot that suggests a linear association.

S.ID.B.6.dFit a function to data represented by a scatterplot; use functions fitted to data to solve problems in the context of the data.

S.ID.C Interpret linear models

S.ID.C.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.C.8Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.C.9Distinguish between correlation and causation.

S.MD Using Probability to Make Decisions

S.MD.A Calculate expected values and use them to solve problems

S.MD.A.1Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

S.MD.A.2Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S.MD.A.3Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

S.MD.A.4Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

S.MD.B Use probability to evaluate outcomes of decisions

S.MD.B.5Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

S.MD.B.5.aFind the expected payoff for a game of chance.

S.MD.B.5.bEvaluate and compare strategies on the basis of expected values.

S.MD.B.6Use probabilities to make fair decisions.

S.MD.B.7Analyze decisions and strategies using probability concepts.

Precalculus (Maryland-specific standards)

P.CED Creating Equations

P.CED.A Create equations that describe numbers or relationships.

P.CED.A.1Create a single equation, using rectangular coordinates, that is equivalent to a pair of parametric equations.

P.CED.A.2Given a data set, create a parametric equation and a single equation using rectangular coordinates to fit the data.

P.IPE Interpreting Parametric Equations

P.IPE.A Interpret parametric equations.

P.IPE.A.1Sketch the curve defined by parametric equations.

P.IPE.A.2Use parametric equations to model and solve motion problems.

AP Calculus Learning Objectives

CB.Calc.LIM Limits

CB.Calc.LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

CB.Calc.LIM-1.ARepresent limits analytically using correct notation.

CB.Calc.LIM-1.BInterpret limits expressed in analytic notation.

CB.Calc.LIM-1.CEstimate limits of functions.

CB.Calc.LIM-1.DDetermine the limits of functions using limit theorems.

CB.Calc.LIM-1.EDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.

CB.Calc.LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

CB.Calc.LIM-2.AJustify conclusions about continuity at a point using the definition.

CB.Calc.LIM-2.BDetermine intervals over which a function is continuous.

CB.Calc.LIM-2.CDetermine values of x or solve for parameters that make discontinuous functions continuous, if possible.

CB.Calc.LIM-2.DInterpret the behavior of functions using limits involving infinity.

CB.Calc.LIM-3 The derivative of a function is defined as the limit of a difference quotient.

CB.Calc.LIM-3.AInterpret a limit as a definition of a derivative.

CB.Calc.LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.

CB.Calc.LIM-4.ADetermine limits of functions that result in indeterminate forms.

CB.Calc.LIM-5 Definite integrals can be approximated using geometric and numerical methods.

CB.Calc.LIM-5.AApproximate a definite integral using geometric and numerical methods.

CB.Calc.LIM-5.BInterpret the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-5.CRepresent the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-6 Improper integrals can be determined using limits of definite integrals.

CB.Calc.LIM-6.AEvaluate an improper integral or determine that the integral diverges.

CB.Calc.LIM-7 The behavior of a convergent or divergent series can be determined or estimated.

CB.Calc.LIM-7.ADetermine whether a series converges or diverges.

CB.Calc.LIM-7.BApproximate the sum of a series.

CB.Calc.LIM-8 Power series and Taylor series can represent functions.

CB.Calc.LIM-8.ARepresent a function at a point as a Taylor polynomial.

CB.Calc.LIM-8.BApproximate function values using a Taylor polynomial.

CB.Calc.LIM-8.CDetermine the error bound associated with a Taylor polynomial approximation.

CB.Calc.LIM-8.DDetermine the radius of convergence and interval of convergence for a power series.

CB.Calc.LIM-8.ERepresent a function as a Taylor series or a Maclaurin series.

CB.Calc.LIM-8.FInterpret Taylor series and Maclaurin series.

CB.Calc.LIM-8.GRepresent a given function as a power series.

CB.Calc.CHA Change

CB.Calc.CHA-1 The derivative of a function is the instantaneous rate of change of the function.

CB.Calc.CHA-1.AInterpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.

CB.Calc.CHA-2 Derivatives can be computed using definitions and rules.

CB.Calc.CHA-2.ADetermine average rates of change using difference quotients.

CB.Calc.CHA-2.BRepresent the derivative of a function as the limit of a difference quotient.

CB.Calc.CHA-2.CDetermine the equation of a line tangent to a curve at a given point.

CB.Calc.CHA-2.DEstimate derivatives.

CB.Calc.CHA-3 Derivatives can be used to model rates of change in applied contexts.

CB.Calc.CHA-3.AInterpret the meaning of a derivative in context.

CB.Calc.CHA-3.BCalculate rates of change in applied contexts.

CB.Calc.CHA-3.CInterpret rates of change in applied contexts.

CB.Calc.CHA-3.DCalculate related rates in applied contexts.

CB.Calc.CHA-3.EInterpret related rates in applied contexts.

CB.Calc.CHA-3.FApproximate a value on a curve using the equation of a tangent line.

CB.Calc.CHA-3.GCalculate derivatives of parametric functions.

CB.Calc.CHA-3.HCalculate derivatives of vector-valued functions.

CB.Calc.CHA-4 Integrals can be used to model accumulation and net change in applied contexts.

CB.Calc.CHA-4.AInterpret the meaning of areas associated with the graph of a rate of change in context.

CB.Calc.CHA-4.BDetermine the average value of a function using definite integrals.

CB.Calc.CHA-4.CDetermine values for positions and rates of change using definite integrals in problems involving rectilinear motion.

CB.Calc.CHA-4.DInterpret the meaning of a definite integral in accumulation problems.

CB.Calc.CHA-4.EDetermine net change using definite integrals in applied contexts.

CB.Calc.CHA-5 Definite integrals can be used to calculate areas, volumes, and lengths.

CB.Calc.CHA-5.ACalculate areas in the plane using the definite integral.

CB.Calc.CHA-5.BCalculate volumes of solids with known cross sections using definite integrals.

CB.Calc.CHA-5.CCalculate volumes of solids of revolution using definite integrals.

CB.Calc.CHA-5.DCalculate areas of regions defined by polar curves using definite integrals.

CB.Calc.CHA-6 Arc length can be calculated using definite integrals.

CB.Calc.CHA-6.ADetermine the length of a curve in the plane defined by a function, using a definite integral.

CB.Calc.CHA-6.BDetermine the length of a curve in the plane defined by parametric functions, using a definite integral.

CB.Calc.FUN Analysis of Functions

CB.Calc.FUN-1 Existence theorems can be used to draw conclusions about the behavior of functions.

CB.Calc.FUN-1.AExplain the behavior of a function on an interval using the Intermediate Value Theorem.

CB.Calc.FUN-1.BJustify conclusions about functions by applying the Mean Value Theorem over an interval.

CB.Calc.FUN-1.CJustify conclusions about functions by applying the Extreme Value Theorem.

CB.Calc.FUN-2 A function's derivative, when it exists, can be used to determine whether the function is increasing or decreasing.

CB.Calc.FUN-2.AExplain the relationship between differentiability and continuity.

CB.Calc.FUN-3 Derivatives can be calculated for various types of functions.

CB.Calc.FUN-3.ACalculate derivatives of familiar functions.

CB.Calc.FUN-3.BCalculate derivatives of products and quotients of differentiable functions.

CB.Calc.FUN-3.CCalculate derivatives of compositions of differentiable functions.

CB.Calc.FUN-3.DCalculate derivatives of implicitly defined functions.

CB.Calc.FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.

CB.Calc.FUN-3.FDetermine higher order derivatives of a function.

CB.Calc.FUN-3.GCalculate derivatives of functions written in polar coordinates.

CB.Calc.FUN-4 A function's derivative can be used to understand function behavior.

CB.Calc.FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.

CB.Calc.FUN-4.BCalculate minimum and maximum values in applied contexts or analysis of functions.

CB.Calc.FUN-4.CInterpret minimum and maximum values calculated in applied contexts.

CB.Calc.FUN-4.DDetermine critical points of implicit relations.

CB.Calc.FUN-4.EJustify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.

CB.Calc.FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.

CB.Calc.FUN-5.ARepresent accumulation functions using definite integrals.

CB.Calc.FUN-6 Recognizing and applying techniques for antidifferentiation.

CB.Calc.FUN-6.ACalculate a definite integral using areas and properties of definite integrals.

CB.Calc.FUN-6.BEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.

CB.Calc.FUN-6.CDetermine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

CB.Calc.FUN-6.DUse substitution and rearrangements into equivalent forms, such as long division and completing the square, for integrands requiring these techniques.

CB.Calc.FUN-6.EUse integration by parts for integrands requiring this technique.

CB.Calc.FUN-6.FUse integration by linear partial fractions for integrands requiring this technique.

CB.Calc.FUN-7 Solving differential equations allows us to model rates of change.

CB.Calc.FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.

CB.Calc.FUN-7.BVerify solutions to differential equations.

CB.Calc.FUN-7.CEstimate solutions to differential equations.

CB.Calc.FUN-7.DDetermine general solutions to differential equations.

CB.Calc.FUN-7.EDetermine particular solutions to differential equations.

CB.Calc.FUN-7.FInterpret the meaning of a differential equation and its variables in context.

CB.Calc.FUN-7.GDetermine general and particular solutions for problems involving differential equations in context.

CB.Calc.FUN-7.HInterpret the meaning of the logistic growth model in context.

CB.Calc.FUN-8 Solutions to differential equations can model motion in the plane.

CB.Calc.FUN-8.ADetermine a particular solution given a rate vector and initial conditions.

CB.Calc.FUN-8.BDetermine values for positions and rates of change in problems involving planar motion.

AP Statistics Learning Objectives

CB.Stats.VAR Variation and Distribution

CB.Stats.VAR-1 Exploring data through asking and answering statistical questions.

CB.Stats.VAR-1.AIdentify questions to be answered, based on variation in one-variable data.

CB.Stats.VAR-1.BIdentify variables in a set of data.

CB.Stats.VAR-1.CClassify types of variables.

CB.Stats.VAR-1.DIdentify questions to be answered about possible relationships in data.

CB.Stats.VAR-1.EIdentify questions to be answered about data collection methods.

CB.Stats.VAR-1.FIdentify questions suggested by patterns in data.

CB.Stats.VAR-1.GIdentify questions suggested by variation in statistics for samples collected from the same population.

CB.Stats.VAR-1.HIdentify questions suggested by variation in the shapes of distributions of samples taken from the same population.

CB.Stats.VAR-1.IIdentify questions suggested by probabilities of errors in statistical inference.

CB.Stats.VAR-1.JIdentify questions suggested by variation between observed and expected counts in categorical data.

CB.Stats.VAR-1.KIdentify questions suggested by variation in scatter plots.

CB.Stats.VAR-2 Normal distributions and comparing data.

CB.Stats.VAR-2.ACompare a data distribution to the normal distribution model.

CB.Stats.VAR-2.BDetermine proportions and percentiles from a normal distribution.

CB.Stats.VAR-2.CCompare measures of relative position in data sets.

CB.Stats.VAR-3 Designing and analyzing experiments.

CB.Stats.VAR-3.AIdentify the components of an experiment.

CB.Stats.VAR-3.BDescribe elements of a well-designed experiment.

CB.Stats.VAR-3.CCompare experimental designs and methods.

CB.Stats.VAR-3.DExplain why a particular experimental design is appropriate.

CB.Stats.VAR-3.EInterpret the results of a well-designed experiment.

CB.Stats.VAR-4 Probability rules and calculations.

CB.Stats.VAR-4.ACalculate probabilities for events and their complements.

CB.Stats.VAR-4.BInterpret probabilities for events.

CB.Stats.VAR-4.CExplain why two events are (or are not) mutually exclusive.

CB.Stats.VAR-4.DCalculate conditional probabilities.

CB.Stats.VAR-4.ECalculate probabilities for independent events and for the union of two events.

CB.Stats.VAR-5 Discrete random variables and probability distributions.

CB.Stats.VAR-5.ARepresent the probability distribution for a discrete random variable.

CB.Stats.VAR-5.BInterpret a probability distribution.

CB.Stats.VAR-5.CCalculate parameters for a discrete random variable.

CB.Stats.VAR-5.DInterpret parameters for a discrete random variable.

CB.Stats.VAR-5.ECalculate parameters for linear combinations of random variables.

CB.Stats.VAR-5.FDescribe the effects of linear transformations of parameters of random variables.

CB.Stats.VAR-6 Normal distributions and hypothesis testing for proportions.

CB.Stats.VAR-6.ACalculate the probability that a particular value lies in a given interval of a normal distribution.

CB.Stats.VAR-6.BDetermine the interval associated with a given area in a normal distribution.

CB.Stats.VAR-6.CDetermine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.

CB.Stats.VAR-6.DIdentify the null and alternative hypotheses for a population proportion.

CB.Stats.VAR-6.EIdentify an appropriate testing method for a population proportion.

CB.Stats.VAR-6.FVerify the conditions for making statistical inferences when testing a population proportion.

CB.Stats.VAR-6.GCalculate an appropriate test statistic and p-value for a population proportion.

CB.Stats.VAR-6.HIdentify the null and alternative hypotheses for a difference of two population proportions.

CB.Stats.VAR-6.IIdentify an appropriate testing method for the difference of two population proportions.

CB.Stats.VAR-6.JVerify the conditions for making statistical inferences when testing a difference of two population proportions.

CB.Stats.VAR-6.KCalculate an appropriate test statistic for the difference of two population proportions.

CB.Stats.VAR-7 t-distributions and hypothesis testing for means and regression.

CB.Stats.VAR-7.ADescribe t-distributions.

CB.Stats.VAR-7.BIdentify an appropriate testing method for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.CIdentify the null and alternative hypotheses for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.DVerify the conditions for the test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.ECalculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.FIdentify an appropriate selection of a testing method for a difference of two population means.

CB.Stats.VAR-7.GIdentify the null and alternative hypotheses for a difference of two population means.

CB.Stats.VAR-7.HVerify the conditions for the significance test for the difference of two population means.

CB.Stats.VAR-7.ICalculate an appropriate test statistic for a difference of two means.

CB.Stats.VAR-7.JIdentify the appropriate selection of a testing method for a slope of a regression model.

CB.Stats.VAR-7.KIdentify appropriate null and alternative hypotheses for a slope of a regression model.

CB.Stats.VAR-7.LVerify the conditions for the significance test for the slope of a regression model.

CB.Stats.VAR-7.MCalculate an appropriate test statistic for the slope of a regression model.

CB.Stats.VAR-8 Chi-square distributions and tests.

CB.Stats.VAR-8.ADescribe chi-square distributions.

CB.Stats.VAR-8.BIdentify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.CIdentify an appropriate testing method for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.DCalculate expected counts for the chi-square test for goodness of fit.

CB.Stats.VAR-8.EVerify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.

CB.Stats.VAR-8.FCalculate the appropriate statistic for the chi-square test for goodness of fit.

CB.Stats.VAR-8.GDetermine the p-value for the chi-square test for goodness of fit.

CB.Stats.VAR-8.HCalculate expected counts for two-way tables of categorical data.

CB.Stats.VAR-8.IIdentify the null and alternative hypotheses for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.JIdentify an appropriate testing method for comparing distributions in two-way tables of categorical data.

CB.Stats.VAR-8.KVerify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.

CB.Stats.VAR-8.LCalculate the appropriate statistic for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.MDetermine the p-value for a chi-square test for independence or homogeneity.

CB.Stats.UNC Uncertainty and Inference

CB.Stats.UNC-1 Representing and describing categorical and quantitative data.

CB.Stats.UNC-1.ARepresent categorical data using frequency or relative frequency tables.

CB.Stats.UNC-1.BDescribe categorical data represented in frequency or relative tables.

CB.Stats.UNC-1.CRepresent categorical data graphically.

CB.Stats.UNC-1.DDescribe categorical data represented graphically.

CB.Stats.UNC-1.ECompare multiple sets of categorical data.

CB.Stats.UNC-1.FClassify types of quantitative variables.

CB.Stats.UNC-1.GRepresent quantitative data graphically.

CB.Stats.UNC-1.HDescribe the characteristics of quantitative data distributions.

CB.Stats.UNC-1.ICalculate measures of center and position for quantitative data.

CB.Stats.UNC-1.JCalculate measures of variability for quantitative data.

CB.Stats.UNC-1.KExplain the selection of a particular measure of center and/or variability for describing a set of quantitative data.

CB.Stats.UNC-1.LRepresent summary statistics for quantitative data graphically.

CB.Stats.UNC-1.MDescribe summary statistics of quantitative data represented graphically.

CB.Stats.UNC-1.NCompare graphical representations for multiple sets of quantitative data.

CB.Stats.UNC-1.OCompare summary statistics for multiple sets of quantitative data.

CB.Stats.UNC-1.PCompare numerical and graphical representations for two categorical variables.

CB.Stats.UNC-1.QCalculate statistics for two categorical variables.

CB.Stats.UNC-1.RCompare statistics for two categorical variables.

CB.Stats.UNC-1.SRepresent bivariate quantitative data using scatterplots.

CB.Stats.UNC-2 Probability and simulation.

CB.Stats.UNC-2.AEstimate probabilities using simulation.

CB.Stats.UNC-3 Random variables and sampling distributions.

CB.Stats.UNC-3.AEstimate probabilities of binomial random variables using data from a simulation.

CB.Stats.UNC-3.BCalculate probabilities for a binomial distribution.

CB.Stats.UNC-3.CCalculate parameters for a binomial distribution.

CB.Stats.UNC-3.DInterpret probabilities and parameters for a binomial distribution.

CB.Stats.UNC-3.ECalculate probabilities for geometric random variables.

CB.Stats.UNC-3.FCalculate parameters of a geometric distribution.

CB.Stats.UNC-3.GInterpret probabilities and parameters for a geometric distribution.

CB.Stats.UNC-3.HEstimate sampling distributions using simulation.

CB.Stats.UNC-3.IExplain why an estimator is or is not unbiased.

CB.Stats.UNC-3.JCalculate estimates for a population parameter.

CB.Stats.UNC-3.KDetermine parameters of a sampling distribution for sample proportions.

CB.Stats.UNC-3.LDetermine whether a sampling distribution for a sample proportion can be described as approximately normal.

CB.Stats.UNC-3.MInterpret probabilities and parameters for a sampling distribution for a sample proportion.

CB.Stats.UNC-3.NDetermine parameters of a sampling distribution for a difference in sample proportions.

CB.Stats.UNC-3.ODetermine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.

CB.Stats.UNC-3.PInterpret probabilities and parameters for a sampling distribution for a difference in proportions.

CB.Stats.UNC-3.QDetermine parameters of a sampling distribution for a sample mean.

CB.Stats.UNC-3.RDetermine whether a sampling distribution of a sample mean can be described as approximately normal.

CB.Stats.UNC-3.SInterpret probabilities and parameters for a sampling distribution for a sample mean.

CB.Stats.UNC-3.TDetermine parameters of a sampling distribution for a difference in sample means.

CB.Stats.UNC-3.UDetermine whether a sampling distribution of a difference in sample means can be described as approximately normal.

CB.Stats.UNC-3.VInterpret probabilities and parameters for a sampling distribution for a difference in sample means.

CB.Stats.UNC-4 Confidence intervals for proportions and means.

CB.Stats.UNC-4.AIdentify an appropriate confidence interval procedure for a population proportion.

CB.Stats.UNC-4.BVerify the conditions for calculating confidence intervals for a population proportion.

CB.Stats.UNC-4.CDetermine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.

CB.Stats.UNC-4.DCalculate an appropriate confidence interval for a population proportion.

CB.Stats.UNC-4.ECalculate an interval estimate based on a confidence interval for a population proportion.

CB.Stats.UNC-4.FInterpret a confidence interval for a population proportion.

CB.Stats.UNC-4.GJustify a claim based on a confidence interval for a population proportion.

CB.Stats.UNC-4.HIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.

CB.Stats.UNC-4.IIdentify an appropriate confidence interval procedure for a comparison of population proportions.

CB.Stats.UNC-4.JVerify the conditions for calculating confidence intervals for a difference between population proportions.

CB.Stats.UNC-4.KCalculate an appropriate confidence interval for a comparison of population proportions.

CB.Stats.UNC-4.LCalculate an interval estimate based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.MInterpret a confidence interval for a difference of proportions.

CB.Stats.UNC-4.NJustify a claim based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.OIdentify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.PVerify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.QDetermine the margin of error for a given sample size for a one-sample t-interval.

CB.Stats.UNC-4.RCalculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.SInterpret a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.TJustify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.UIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.

CB.Stats.UNC-4.VIdentify an appropriate confidence interval procedure for a difference of two population means.

CB.Stats.UNC-4.WVerify the conditions for calculating confidence intervals for the difference of two population means.

CB.Stats.UNC-4.XDetermine the margin of error for the difference of two population means.

CB.Stats.UNC-4.YCalculate an appropriate confidence interval for a difference of two population means.

CB.Stats.UNC-4.ZInterpret a confidence interval for a difference of population means.

CB.Stats.UNC-5 Errors in statistical inference.

CB.Stats.UNC-5.AIdentify Type I and Type II errors.

CB.Stats.UNC-5.BCalculate the probability of a Type I and Type II errors.

CB.Stats.UNC-5.CIdentify factors that affect the probability of errors in significance testing.

CB.Stats.UNC-5.DInterpret Type I and Type II errors.

CB.Stats.DAT Data Analysis

CB.Stats.DAT-1 Bivariate data analysis and linear regression.

CB.Stats.DAT-1.ADescribe the characteristics of a scatter plot.

CB.Stats.DAT-1.BDetermine the correlation for a linear relationship.

CB.Stats.DAT-1.CInterpret the correlation for a linear relationship.

CB.Stats.DAT-1.DCalculate a predicted response value using a linear regression model.

CB.Stats.DAT-1.ERepresent differences between measured and predicted responses using residual plots.

CB.Stats.DAT-1.FDescribe the form of association of bivariate data using residual plots.

CB.Stats.DAT-1.GEstimate parameters for the least-squares regression line model.

CB.Stats.DAT-1.HInterpret coefficients for the least-squares regression line model.

CB.Stats.DAT-1.IIdentify influential points in regression.

CB.Stats.DAT-1.JCalculate a predicted response using a least-squares regression line for a transformed data set.

CB.Stats.DAT-2 Studies, sampling, and sources of bias.

CB.Stats.DAT-2.AIdentify the type of a study.

CB.Stats.DAT-2.BIdentify appropriate generalizations and determinations based on observational studies.

CB.Stats.DAT-2.CIdentify a sampling method, given a description of a study.

CB.Stats.DAT-2.DExplain why a particular sampling method is or is not appropriate for a given situation.

CB.Stats.DAT-2.EIdentify potential sources of bias in sampling methods.

CB.Stats.DAT-3 Interpreting and justifying conclusions from significance tests.

CB.Stats.DAT-3.AInterpret the p-value of a significance test for a population proportion.

CB.Stats.DAT-3.BJustify a claim about the population based on the results of a significance test for a population proportion.

CB.Stats.DAT-3.CInterpret the p-value of a significance test for a difference of population proportions.

CB.Stats.DAT-3.DJustify a claim about the population based on the results of a significance test for a difference of population proportions.

CB.Stats.DAT-3.EInterpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.DAT-3.FJustify a claim about the population based on the results of a significance test for a population mean.

CB.Stats.DAT-3.GInterpret the p-value of a significance test for a difference of population means.

CB.Stats.DAT-3.HJustify a claim about the population based on the results of a significance test for a difference of two population means in context.

CB.Stats.DAT-3.IInterpret the p-value for the chi-square test for goodness of fit.

CB.Stats.DAT-3.JJustify a claim about the population based on the results of a chi-square test for goodness of fit.

CB.Stats.DAT-3.KInterpret the p-value for the chi-square test for homogeneity or independence.

CB.Stats.DAT-3.LJustify a claim about the population based on the results of a chi-square test for homogeneity or independence.

CB.Stats.DAT-3.MInterpret the p-value of a significance test for the slope of a regression model.

CB.Stats.DAT-3.NJustify a claim about the population based on the results of a significance test for the slope of a regression model.

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