K-12

Generate questions to investigate situations within the classroom.M.K.DA.DS.1

Collect or consider data through organizing objects or drawing pictures to represent and communicate observations.M.K.DA.DS.2

Analyze data sets by noticing and describing patterns in data-rich situations.M.K.DA.DS.3

Interpret and communicate results through structured answers with teacher guidance.M.K.DA.DS.4

Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.M.K.DA.MD.1

Directly compare two objects with a measurable attribute in common to see which object has “more of”/“less of” the attribute and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.M.K.DA.MD.2

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.M.K.DA.MD.3

Count to 100 by ones and by tens.M.K.Q.CC.1

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).M.K.Q.CC.2

Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).M.K.Q.CC.3

Understand the relationship between numbers and quantities; connect counting to the number of items in the set.M.K.Q.CC.4

Count to answer “how many?” questions about as many as 20 objects arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.M.K.Q.CC.5

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.M.K.Q.CC.6

Compare two numbers between 1 and 10 presented as written numerals.M.K.Q.CC.7

Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.M.K.R.OA.1

Efficiently, flexibly, and accurately solve addition and subtraction word problems, and add and subtract within 10, e.g., using objects or drawings to represent the problem.M.K.R.OA.2

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition with a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).M.K.R.OA.3

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.M.K.R.OA.4

Efficiently, flexibly, and accurately add and subtract within 5.M.K.R.OA.5

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition with a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.M.K.R.NBT.1

Describe objects in the student environment using names of shapes and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.M.K.SR.G.1

Correctly name shapes regardless of their orientations or overall size.M.K.SR.G.2

Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).M.K.SR.G.3

Analyze 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 and vertices/"corners") and other attributes (e.g., having sides of equal length).M.K.SR.G.4

Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and/or drawing shapes.M.K.SR.G.5

Use 2-dimensional shapes to compose a variety of larger shapes. For example, "Can you join these two triangles with full sides touching to make a rectangle?"M.K.SR.G.6

Generate questions to investigate situations within the classroom.M.1.DA.DS.1

Collect and use data to consider and decide what data will answer the investigative question. Organize data with drawings, tally marks, or other visual representations.M.1.DA.DS.2

Analyze data sets with up to three categories by making comparisons and/or identifying patterns in the data.M.1.DA.DS.3

Interpret and communicate results through structured answers with teacher guidance.M.1.DA.DS.4

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less in each category.M.1.DA.MD.4

Use addition and subtraction within 20 to efficiently, flexibly, and accurately solve real world word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem.M.1.Q.OA.1

Efficiently, flexibly, and accurately solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem.M.1.Q.OA.2

Apply and extend properties of operations by selecting and demonstrating strategies to add and subtract.M.1.Q.OA.3

Demonstrate understanding of subtraction as an unknown-addend problem. For example, subtract 10-8 by finding the number that makes 10 when added to 8.M.1.Q.OA.4

Apply and extend counting strategies to addition and subtraction (e.g., by counting on 2 to add 2).M.1.Q.OA.5

Efficiently, flexibly, and accurately add within 20, and subtract within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14), decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).M.1.Q.OA.6

Demonstrate understanding of the meaning of the equal sign and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.M.1.Q.OA.7

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?.M.1.Q.OA.8

Order three objects according to their length; compare the lengths of two objects indirectly by using a third object.M.1.Q.MD.1

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlap.M.1.Q.MD.2

Tell and write time to the hour and half-hour using analog and digital clocks.M.1.Q.MD.3

Count to 120, starting at any number less than 120. In this range, recognize and represent numerals and represent a number of objects with a written numeral.M.1.R.NBT.1

Understand that the digits of a two-digit number represent amounts of tens and ones.M.1.R.NBT.2

Compare two two-digit numbers based on the values of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.M.1.R.NBT.3

Efficiently, flexibly, and accurately add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.M.1.R.NBT.4

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.M.1.R.NBT.5

Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.M.1.R.NBT.6

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus nondefining attributes (e.g., color, orientation, overall size) and represent shapes (build and/or draw) to possess defining attributes.M.1.SR.G.1

Compose 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 create new shapes from the composite shape.M.1.SR.G.2

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.M.1.SR.G.3

Generate questions to investigate topics of interest to students within the classroom, school, or community.M.2.DA.DS.1

Collect and use data to consider and decide what data will answer the investigative question. Organize data with pictographs, line plots and bar graphs with single-unit scales. Recognize that data can vary for a variety of reasons.M.2.DA.DS.2

Analyze data sets with up to four categories by making comparisons, looking for patterns and/or making predictions.M.2.DA.DS.3

Interpret and communicate results through structured answers with teacher guidance. Make a statement(s) about the data collected to support the answer to the investigative question.M.2.DA.DS.4

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.M.2.DA.MD.9

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve one step put-together, take-apart, and compare problems using information presented in a bar graph.M.2.DA.MD.10

Count within 1000; skip-count by 5s, 10s, and 100s.M.2.Q.NBT.2

Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.M.2.Q.NBT.3

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.M.2.Q.NBT.4

Efficiently, flexibly, and accurately add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.M.2.Q.NBT.5

Add up to four two-digit numbers using strategies based on place value and properties of operations.M.2.Q.NBT.6

Efficiently, flexibly, and accurately add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Demonstrate understanding that in 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.M.2.Q.NBT.7

Use flexible and efficient strategies to mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.M.2.Q.NBT.8

Explain with words, numbers, and /or pictures why addition and subtraction strategies work, using place value and the properties of operations.M.2.Q.NBT.9

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.M.2.Q.MD.1

Measure the length of an object twice, using length units of different lengths for the two measurements; explain how the two measurements relate to the size of the unit chosen.M.2.Q.MD.2

Estimate lengths using units of inches, feet, centimeters, and meters.M.2.Q.MD.3

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

Efficiently, flexibly, and accurately use addition and subtraction within 100 to solve real world word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.M.2.Q.MD.5

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, . . . and represent whole-number sum s and differences within 100 on a number line diagram.M.2.Q.MD.6

Efficiently, flexibly, and accurately solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?M.2.Q.MD.8

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:M.2.R.NBT.1

Use addition and subtraction within 100 to efficiently, flexibly, and accurately solve one- and two-step real world word problems involving situations of adding to, taking from , putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.M.2.R.OA.1

Efficiently, flexibly, and accurately add and subtract within 20 using flexible and efficient strategies.M.2.R.OA.2

Determine whether a group of objects (up to 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.M.2.R.OA.3

Identify and draw shapes based on specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.M.2.SR.G.1

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them .M.2.SR.G.2

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Demonstrate that equal shares of identical wholes need not have the same shape.M.2.SR.G.3

Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.M.2.SR.MD.7

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.M.2.SR.OA.4

Generate questions to investigate topics of interest to students that can be answered with a variety of data or data sets.M.3.DA.DS.1

Collect and consider data in a variety of ways including surveys, groupings, measurement, etc., and ask in what ways the data can be collected to capture as much information as necessary to inform the investigative question.M.3.DA.DS.2

Represent data in a variety of ways including technology. Critically analyze data visualizations, including bar graphs, line plots, and scaled picture graphs with various scales. Analyze data sets with several categories by making comparisons, looking for patterns and/or making predictions and recognize the source and amount of data collected may impact the accuracy.M.3.DA.DS.3

Interpret and communicate results, describing difference between groups, with teacher guidance. Make a statement(s) about the data collected to support the answer to the investigative question.M.3.DA.DS.4

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to efficiently, flexibly, and accurately solve one-step real world word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.M.3.DA.MD.2

Create a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs.M.3.DA.MD.3

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units according to the data— whole numbers, halves, or quarters.M.3.DA.MD.4

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.M.3.Q.OA.1

Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.M.3.Q.OA.2

Use multiplication and division within 100 to efficiently, flexibly, and accurately solve real world word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.M.3.Q.OA.3

Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ? ÷ 3, 6 × 6 = ?.M.3.Q.OA.4

Use strategies to multiply and divide by applying and extending understanding of the properties of operations.M.3.Q.OA.5

Demonstrate understanding of division as an unknown-factor problem.M.3.Q.OA.6

Efficiently, flexibly, and accurately multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations.M.3.Q.OA.7

Efficiently, flexibly, and accurately solve two-step real world word problems using the four operations. Represent these problems using visual models and equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental and estimation strategies.M.3.Q.OA.8

Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations.For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.M.3.Q.OA.9

Use place value understanding to round whole numbers to the nearest 10 or 100.M.3.Q.NBT.1

Efficiently, flexibly, and accurately add and subtract within 1000 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.M.3.Q.NBT.2

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

Understand a unit fraction as the quantity formed when a whole is partitioned into equal parts and explain that a unit fraction is one of those parts; understand fractions are quantities composed of unit fractions.M.3.R.NF.1

Understand a fraction as a number; and represent fractions on a number line diagram.M.3.R.NF.2

Explain equivalence of fractions and compare fractions by reasoning about their size.M.3.R.NF.3

Recognize area as an attribute of plane figures and understand concepts of area measurement.M.3.R.MD.5

Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).M.3.R.MD.6

Relate area to the operations of multiplication and addition.M.3.R.MD.7

Demonstrate understanding 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 (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.M.3.SR.G.1

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.M.3.SR.G.2

Tell and write time to the nearest minute and measure time intervals in minutes. Efficiently, flexibly, and accurately solve real world word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.M.3.SR.MD.1

Efficiently, flexibly, and accurately solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.M.3.SR.MD.8

Generate questions of interest to the students that elicit data, generate ideas based on the questions, and refine the question as necessary.M.4.DA.DS.1

Determine strategies for collecting and considering data in a variety of ways including with the use of technology, evaluate whether additional data that should be collected to completely address the investigative question.M.4.DA.DS.2

Critically analyze data visualizations, including tables, bar graphs, line plots, or spreadsheets to support a claim related to the investigative question. Ask whether the data collected sufficiently addresses the investigative question.M.4.DA.DS.3

Interpret and communicate results, describing difference between groups, with teacher guidance. Make a statement(s) about the data collected to support the answer to the investigative question.M.4.DA.DS.4

Make a line plot to display a data set of measurements in fractions of a unit. Efficiently, flexibly, and accurately solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.M.4.DA.MD.4

Multiply or divide to efficiently, flexibly, and accurately solve real world word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.M.4.Q.OA.2

Efficiently, flexibly, and accurately solve multistep real world word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using visual models and equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using estimation strategies.M.4.Q.OA.3

Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.M.4.Q.OA.4

Generate a number or shape pattern that follows a given rule. Identify and explain apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain why the numbers will continue to alternate in this way.M.4.Q.OA.5

Read, write, and compare multi-digit whole numbers using base-ten numerals, number names, and expanded form using the meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.M.4.Q.NBT.2

Use place value understanding of multi-digit whole numbers to generate estimates to any place less than or equal to 1,000,000 using a variety of estimation strategies.M.4.Q.NBT.3

Efficiently, flexibly, and accurately add and subtract multi-digit whole numbers using strategies or algorithms.M.4.Q.NBT.4

Efficiently, flexibly, and accurately multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.M.4.Q.NBT.5

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using multiple strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.M.4.Q.NBT.6

Interpret a multiplication equation 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 these comparison statements as multiplication equations.M.4.R.OA.1

Understand that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.M.4.R.NBT.1

Demonstrate understanding of why a fraction is equivalent to another fraction by using visual fraction models (e.g., tape diagrams and number lines), with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Understand and use general principles to recognize and generate equivalent fractions.M.4.R.NF.1

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing them to a benchmark fraction. Understand that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model.M.4.R.NF.2

Efficiently, flexibly, and accurately compose and decompose fractions with a numerator greater than 1 into unit fractions, including fractions greater than one or mixed numbers, to solve situations in context with addition and subtraction of fractions with like denominators.M.4.R.NF.3

Flexibly apply and extend previous understandings of multiplication to multiply a fraction by a whole number using visual models in the context of word problems.M.4.R.NF.4

Explore and explain using models, words, and/or numbers that a fraction with a denominator of 10 is an equivalent fraction with denominator of 100 and use this technique to add two fractions with respective denominators of 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.M.4.R.NF.5

Explore and explain decimal notation for fractions with denominators of 10 and 100 using models, words, and/or numbers. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.M.4.R.NF.6

Compare two decimals to hundredths by reasoning about their size. Understand 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 by using multiple strategies and/or visual models.M.4.R.NF.7

Use the four operations to efficiently, flexibly, and accurately solve real world word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using multiple visual models.M.4.R.MD.2

Demonstrate understanding of angles as geometric figures that are formed wherever two rays share a common endpoint and accurately measure an angle.M.4.R.MD.5

Demonstrate understanding 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. Efficiently, flexibly, and accurately solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.M.4.R.MD.7

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these features in two-dimensional figures.M.4.SR.G.1

Classify 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.M.4.SR.G.2

Recognize a line of symmetry for a two-dimensional figure as a line separating the figure such that the figure can be folded along the line into identical parts. Identify line-symmetric figures and draw lines of symmetry.M.4.SR.G.3

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec., and express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...M.4.SR.MD.1

Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.M.4.SR.MD.3

Measure angles in whole-number degrees using a protractor. Represent angles of specified measure.M.4.SR.MD.6

Generate questions of interest to the students that elicit data, generate ideas based on the questions, and refine the question as necessary. Pose statistical questions that can compare two variables withing a group, setting, or situation.M.5.DA.DS.1

Determine strategies for collecting and considering data in a variety of ways including with the use of technology. Understand that data may contain errors (missing values, etc.) and decisions have to be made on how to account for or resolve these issues.M.5.DA.DS.2

Critically analyze data visualizations, including tables, bar graphs, line plots, or spreadsheets to support a claim related to the investigative question. Compare and contrast different data visualizations to determine which transparently communicate results and interpretations.M.5.DA.DS.3

Interpret and communicate results, describing difference between groups, with teacher guidance. Make a statement(s) about the data collected to support the answer to the investigative question. Describe the difference between two groups with different conditions.M.5.DA.DS.4

Make a line plot to display a data set of measurements in fractions of a unit. Use operations on fractions to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.M.5.DA.MD.2

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.M.5.DA.OA.3

Add and subtract fractions with unlike denominators (including mixed numbers) using flexible and efficient strategies, including replacing given fractions with equivalent fractions with like denominators. Justify using visual models (e.g., tape diagrams or number lines).M.5.Q.NF.1

Efficiently, flexibly, and accurately solve real world word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.M.5.Q.NF.2

Interpret a fraction as division, where a quantity (the numerator) is divided into equal parts (the denominator). Flexibly and efficiently solve real world word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Assess the reasonableness of answers using estimation strategies. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?M.5.Q.NF.3

Apply and extend previous understandings of multiplication to efficiently, flexibly, and accurately multiply a fraction or whole number by a fraction.M.5.Q.NF.4

Interpret multiplication as scaling (resizing) by estimating whether a product will be larger or smaller than a given factor based on the size of the other factor, without performing the indicated multiplication.M.5.Q.NF.5

Efficiently and flexibly solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using fraction models or equations to represent the problem. Assess the reasonableness of answers using mental and estimation strategies.M.5.Q.NF.6

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions using fraction models and equations to represent the problem.M.5.Q.NF.7

Read, write, and compare decimals to thousandths.M.5.Q.NBT.3

Efficiently, flexibly, and accurately multiply multi-digit whole numbers using strategies or algorithms.M.5.Q.NBT.5

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.M.5.R.OA.1

Write expressions that record calculations with numbers and interpret numerical expressions without evaluating them . For example, express the calculation “add 8 and 7, then multiply by 2” as 2 x (8+7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.M.5.R.OA.2

Understand 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 1/10 of what it represents in the place to its left.M.5.R.NBT.1

Explain 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.M.5.R.NBT.2

Use place value understanding of decimals to generate estimates to any place using a variety of estimation strategies.M.5.R.NBT.4

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors using strategies based on place value, properties of operations, and connected to the relationship between multiplication and division including rectangular arrays, partial quotients, and/or area models.M.5.R.NBT.6

Efficiently, flexibly, and accurately add, subtract, multiply, and divide decimals to hundredths, using models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.M.5.R.NBT.7

Use a pair of perpendicular number lines, called axes, to define a coordinate system , with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).M.5.SR.G.1

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation.M.5.SR.G.2

Demonstrate understanding that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.M.5.SR.G.3

Classify two-dimensional figures into categories based on properties.M.5.SR.G.4

Convert among different-sized standard measurement units within a given measurement system and use these conversions in solving multi-step, real world problems. Assess the reasonableness of answers using mental and estimation strategies.M.5.SR.MD.1

Identify volume as an attribute of solid figures and understand concepts of volume measurement.M.5.SR.MD.3

Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.M.5.SR.MD.4

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.M.5.SR.MD.5

Formulate and recognize statistical investigative questions that are of interest to students to collect data from online sources and websites, smartphones, sensors, publicly available government agencies, and other modern devices.M.6.DA.DS.1

Collect and record data with technology to identify and describe the characteristics of data sets. Understand that data can be collected (primary data) or existing data can be obtained from other sources (secondary data).M.6.DA.DS.2

Analyze data visualizations and describe measures of center and variability of quantitative data using appropriate displays (dot plots, boxplots). Describe key features of distributions for the variables including center, variability, and shape.M.6.DA.DS.3

Use statistical evidence from analyses to answer the statistical investigative question and communicate results with comprehensive answers with some teacher guidance.M.6.DA.DS.4

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "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.M.6.DA.SP.1

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.M.6.DA.SP.2

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.M.6.DA.SP.3

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.M.6.DA.SP.4

Summarize numerical data sets in relation to their context.M.6.DA.SP.5

Efficiently, flexibly, and accurately divide multi-digit numbers using strategies or algorithms.M.6.Q.NS.2

Efficiently, flexibly, and accurately add, subtract, multiply, and divide multi-digit decimals using strategies or algorithms for each operation.M.6.Q.NS.3

Explain how positive and negative numbers are used together 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 positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.M.6.Q.NS.5

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to place any number (integer or rational, positive or negative) on the line (horizontal or vertical), and in the plane.M.6.Q.NS.6

Understand ordering and absolute value of positive and negative rational numbers.M.6.Q.NS.7

Explain the concept of a ratio and efficiently, flexibly, and accurately use ratio language to describe a ratio relationship between two quantities.M.6.R.RP.1

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.M.6.R.RP.2

Efficiently, flexibly, and accurately demonstrate understanding of ratio and rate reasoning by solving real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations to find different ways to express the same ratio.M.6.R.RP.3

Interpret and efficiently, flexibly, and accurately determine quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.M.6.R.NS.1

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.M.6.R.NS.4

Efficiently, flexibly, and accurately write and evaluate numerical expressions involving whole-number exponents.M.6.R.EE.1

Read, and evaluate expressions efficiently, flexibly, and accurately.M.6.R.EE.2

Apply the properties of operations (including the distributive property) efficiently, flexibly, and accurately to generate equivalent expressions.M.6.R.EE.3

Identify whether two expressions are equivalent (including expressions with one or more variables) as both expressions will always yield the same outcome for any value of a given variable.M.6.R.EE.4

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.M.6.R.EE.5

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or depending on the purpose at hand, any number in a specified set.M.6.R.EE.6

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p,q and x are all nonnegative rational numbers.M.6.R.EE.7

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.M.6.R.EE.8

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables and relate these to the equation.M.6.R.EE.9

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by efficiently, flexibly, and accurately composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.M.6.SR.G.1

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.M.6.SR.G.2

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.M.6.SR.G.3

Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.M.6.SR.G.4

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.M.6.SR.NS.8

Pose statistical investigative questions about a broader population using samples taken from the population.M.7.DA.DS.1

Understand information from a sample is valid only if the sample is representative of that population. Understand data can be used to make comparisons between different groups at one point in time and the same group over time.M.7.DA.DS.2

Identify, determine, and interpret measures of center (mean and median) and measures of variability (range, interquartile range) to answer a statistically investigative question, summarizing the distribution of data using the measures of center and variability. Use reasoning about distributions to compare two groups based on the variables.M.7.DA.DS.3

Acknowledge that looking beyond the data is feasible and recognize the uncertainty caused by sample-to-sample variability when making comparisons and/or conclusions from data to answer the investigative question.M.7.DA.DS.4

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.M.7.DA.SP.1

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.M.7.DA.SP.2

Informally assess the degree of overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.M.7.DA.SP.3

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.M.7.DA.SP.4

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.M.7.DA.SP.5

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency and predict the approximate relative frequency given the probability.M.7.DA.SP.6

Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.M.7.DA.SP.7

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.M.7.DA.SP.8

Efficiently, flexibly, and accurately apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.M.7.Q.NS.1

Efficiently, flexibly, and accurately apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.M.7.Q.NS.2

Efficiently, flexibly, and accurately solve 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.M.7.Q.G.1

Efficiently, flexibly, and accurately compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.M.7.R.RP.1

Recognize and represent proportional relationships between quantities.M.7.R.RP.2

Efficiently, flexibly, and accurately use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.M.7.R.RP.3

Efficiently, flexibly, and accurately use properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.M.7.R.EE.1

Understand that rewriting an expression in different forms in a problem context can bring awareness to parts of the problem and how the quantities in it are related.M.7.R.EE.2

Efficiently, flexibly, and accurately solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.M.7.R.EE.3

Use variables to represent quantities in a real-world or mathematical problem and write equations and inequalities to efficiently, flexibly, and accurately to solve problems by reasoning about the quantities.M.7.R.EE.4

Draw geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.M.7.SR.G.2

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.M.7.SR.G.3

Know the formulas for the area and circumference of a circle and use them to solve problems; informally show the relationship between the circumference and area of a circle.M.7.SR.G.4

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve equations for an unknown angle in a figure.M.7.SR.G.5

Efficiently, flexibly, and accurately solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.M.7.SR.G.6

Formulate statistical investigative questions to articulate research topics and uncover patterns of association seen in bivariate categorical data, that multiple investigative questions may exist for a research topic and must take into account context.M.8.DA.DS.1

Understand how to interrogate the data to determine how the data were collected, from whom they were collected, what types of variables are in the data, how the variables were measured, and possible outcomes for the variables.M.8.DA.DS.2

Create data visualizations about a data set. Organize and present the data in appropriate ways, including in tables and scatter plots, and incorporate other relevant information that helps to tell a story and support a claim about the data.M.8.DA.DS.3

Generalize beyond the sample providing statistical evidence for the conclusion, being sure to address limitations of the sample, evidenced in the data. Consider the reasonableness of the results.M.8.DA.DS.4

Construct 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.M.8.DA.SP.1

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.M.8.DA.SP.2

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.M.8.DA.SP.3

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.M.8.DA.SP.4

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers efficiently, flexibly, and accurately show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.M.8.Q.NS.1

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).M.8.Q.NS.2

Know and apply the properties of integer exponents to generate equivalent numerical expressions.M.8.Q.EE.1

Use square roots and cube roots where p is a positive rational number. Use square root symbols to represent solutions to equations of the form x² = p. Evaluate square roots of perfect squares. Use cube root symbols to represent solutions to equations of the form x³ = p and evaluate cube roots of perfect cubes. Know that √2 is irrational.M.8.Q.EE.2

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.M.8.Q.EE.3

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.M.8.Q.EE.4

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.M.8.R.EE.5

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.M.8.R.EE.6

Efficiently, flexibly, and accurately solve linear equations in one variable.M.8.R.EE.7

Analyze and efficiently, flexibly, and accurately solve pairs of simultaneous linear equations.M.8.R.EE.8

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.M.8.R.F.1

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).M.8.R.F.2

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.M.8.R.F.3

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.M.8.R.F.4

Describe 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 that has been described informally.M.8.R.F.5

Verify experimentally the properties of rotations, reflections, and translations.M.8.SR.G.1

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.M.8.SR.G.2

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.M.8.SR.G.3

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.M.8.SR.G.4

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.M.8.SR.G.5

Efficiently, flexibly, and accurately explain a proof of the Pythagorean Theorem and its converse.M.8.SR.G.6

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.M.8.SR.G.7

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.M.8.SR.G.8

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.M.8.SR.G.9

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.M.HS.DA.DS.1

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.M.HS.DA.DS.2

Create and analyze data sets using technology to clean, sort, or filter data. Summarize, and describe relationships between quantitative variables using data displays including but not limited to scatter plots, regressions, histograms, and boxplots.M.HS.DA.DS.3

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.M.HS.DA.DS.4

Understand that a function has a domain (input, independent elements) and range (output, dependent elements), and assigns to each domain element exactly one range element. If f is a function and x is a value of its domain, then the output of f corresponds to the input x. The graph of f is the graph of the equation y=f(x).M.HS.DA.FIF.1

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.M.HS.DA.FIF.2

Represent data with plots on the real number line (dot plots, histograms, and box plots).M.HS.DA.SID.1

Use 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.M.HS.DA.SID.2

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).M.HS.DA.SID.3

Use 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.M.HS.DA.SID.4

Summarize 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.M.HS.DA.SID.5

Represent data on two quantitative variables on a scatter plot and describe how the variables are related. Solve problems in context by fitting functions to the data and explaining trends and relationships within the data.M.HS.DA.SID.6

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.M.HS.DA.SID.7

Compute (using technology) and interpret the correlation coefficient of a linear fit.M.HS.DA.SID.8

Distinguish between correlation and causation.M.HS.DA.SID.9

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.M.HS.DA.SIC.1

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.M.HS.DA.SIC.2

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.M.HS.DA.SIC.3

Use 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.M.HS.DA.SIC.4

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.M.HS.DA.SIC.5

Evaluate reports based on data.M.HS.DA.SIC.6

Describe 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").M.HS.DA.SCP.1

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

Understand 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.M.HS.DA.SCP.3

Construct 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.M.HS.DA.SCP.4

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.M.HS.DA.SCP.5

Find 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.M.HS.DA.SCP.6

Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.M.HS.DA.SCP.7

(+) Apply 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.M.HS.DA.SCP.8

(+) Use permutations and combinations to compute probabilities of compound events and solve problems.M.HS.DA.SCP.9

(+) Define 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.M.HS.DA.SMD.1

(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.M.HS.DA.SMD.2

(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.M.HS.DA.SMD.3

(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.M.HS.DA.SMD.4

(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.M.HS.DA.SMD.5

(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).M.HS.DA.SMD.6

(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).M.HS.DA.SMD.7

Use 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.M.HS.Q.NQ.1

Define appropriate quantities for the purpose of descriptive modeling.M.HS.Q.NQ.2

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.M.HS.Q.NQ.3

Interpret expressions that represent a quantity in terms of its context in any family of functions.M.HS.Q.ASSE.1

Use the structure of an expression to identify ways to rewrite it.M.HS.Q.ASSE.2

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and use the formula to solve problems.M.HS.Q.ASSE.4

Graph families of functions from their symbolic forms by hand or with technology, and identify key features including intercepts, intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries, and end behavior.M.HS.Q.FIF.7

Efficiently, flexibly, and accurately, write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.M.HS.Q.FIF.8

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions can include polynomial, radical, rational, logarithms, absolute value, piecewise, trigonometric, and increasingly complex linear, exponential, and quadratic relationships.M.HS.Q.FIF.9

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.M.HS.Q.FIF.3

Represent 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. May include any families of functions.M.HS.Q.ACED.3

Efficiently, flexibly, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. May include any families of functions.M.HS.Q.ACED.4

Efficiently, flexibly, and accurately write a function that describes a relationship between two quantities, including linear and exponential relationships, and arithmetic and geometric sequences in context.M.HS.Q.FBF.1

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model linear and exponential situations, and translate between the two forms.M.HS.Q.FBF.2

Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Using a variety of strategies, experiment with cases and illustrate an explanation of the effects on the graph using technology.M.HS.Q.FBF.3

Efficiently, flexibly, and accurately explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values using a variety of strategies, allowing for a notation for radicals in terms of rational exponents.M.HS.Q.NRN.1

Rewrite expressions involving radicals and rational exponents using the properties of exponents.M.HS.Q.NRN.2

Explain 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.M.HS.Q.NRN.3

Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.M.HS.Q.NCN.1

Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.M.HS.Q.NCN.2

(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.M.HS.Q.NCN.3

(+) Represent 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.M.HS.Q.NCN.4

(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.M.HS.Q.NCN.5

(+) Calculate 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.M.HS.Q.NCN.6

Solve quadratic equations with real coefficients that have complex solutions.M.HS.Q.NCN.7

(+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).M.HS.Q.NCN.8

(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.M.HS.Q.NCN.9

(+) Recognize 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).M.HS.Q.NVM.1

(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.M.HS.Q.NVM.2