Common Core State Standards - Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. 1 Make sense of problems and persevere in solving them - Mathematically proficient students: v start by explaining to themselves the meaning of a problem and looking for entry points to its solution v analyze the problem and plan a solution pathway rather than simply jumping into a solution attempt v monitor and evaluate their progress and change course if necessary v check their answers to problems using a different method, and continually ask themselves, Does this make sense? v can understand the approaches of others 2 Reason abstractly and quantitatively - Mathematically proficient students: v make sense of quantities and their relationships in problem situations v create representations of the problem at hand, consider the units involved, and attend to the meaning of quantities, not just how to compute them 3 Construct viable arguments and critique the reasoning of others - Mathematically proficient students: v understand and use prior knowledge in constructing arguments v make conjectures and build a logical progression of statements to explore the truth of their conjectures v analyze situations, justify their conclusions, communicate their solutions to others, and respond to the arguments of others v construct arguments using concrete objects, drawings, diagrams, and actions; such arguments can make sense and be correct, even though they are not generalized or made formal until later grades v listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments 4 Model with mathematics - Mathematically proficient students: v apply the mathematics they already know to solve problems arising in everyday life v simplify situations and make necessary revisions as they move forward v identify important quantities in a practical situation v routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose 5 Use appropriate tools strategically - Mathematically proficient students: v consider the available tools when solving a mathematical problem (ex: pencil and paper, concrete models, a ruler, or a calculator) v are sufficiently familiar with tools appropriate for their grade to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations v detect possible errors by strategically using estimation and other mathematical knowledge 6 Attend to precision - Mathematically proficient students: v try to communicate precisely to others v try to use clear definitions in discussions with others and in their own reasoning v state the meaning of the symbols they choose, including using the equal sign consistently and appropriately v are careful about specifying units of measure v calculate accurately and efficiently v express numerical answers with a degree of precision appropriate for the problem context v give carefully formulated explanations to each other 7 Look for and make use of structure - Mathematically proficient students: v look closely to discern a pattern or structure 8 Look for and express regularity in repeated reasoning - Mathematically proficient students: v notice if calculations are repeated, and look for both general methods and shortcuts v maintain oversight of the process and attend to the details when working to solve a problem v continually evaluate the reasonableness of their intermediate results Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word understand are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Students require mathematical understanding in order to engage in the mathematical practices.
Mathematics Common Core State Standards Kindergarten Counting and Cardinality Know number names and the count sequence v Count to 100 by ones and tens v Count forward beginning from a given number within the known sequence (instead of having to begin at 1) v Write numbers from 0 to 20 v Represent a number of objects with a written numeral from 0-20 Count to tell the number of objects v When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object v The last number said tells the number of objects counted v The number of objects is the same regardless of the arrangement or order v Each successive number name refers to a quantity that is one larger v Count to answer how many? questions about as many as 20 things arranged in a line, rectangular array, or a circle, or as many as 10 things in a scattered configuration v Given a number from 1 20, count out that many objects Compare Numbers v Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group v Compare two numbers between 1 and 10 presented as written numerals Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from v Represent addition and subtraction with objects, fingers, etc. v Solve addition and subtraction word problems, and add and subtract within 10 v Decompose numbers less than or equal to 10 into pairs in more than one way v For any number from 1 to 9, find the number that makes 10 when added to the given number v Fluently add and subtract within 5 Work with numbers 11 19 to gain foundations for place value v Compose and decompose numbers from 11 to 19 into ten ones and some further ones by using objects or drawings, and record each composition or decomposition by a drawing or an equation Describe and compare measurable attributes v Describe measurable attributes of objects (ex: length or weight) v 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 Classify objects and count the number of objects in each category v Classify objects into given categories v Count the numbers of objects in each category and sort the categories by count Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres) v Describe objects in the 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 v Correctly name shapes regardless of their orientations or overall size v Identify objects as two-dimensional or three-dimensional Analyze, compare, create, and compose shapes v Analyze and compare 2-D and 3-D shapes using informal language to describe their similarities, differences, and parts v Build and draw shapes v Compose simple shapes to form larger shapes
Mathematics Common Core State Standards Grade One Represent and solve problems involving addition and subtraction v Use addition and subtraction within 20 to solve word problems with unknowns in all positions v Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 Understand and apply properties of operations and the relationship between addition and subtraction v Solve problems by applying properties of operations and using the relationship between addition and subtraction Add and subtract within 20 v Add and subtract within 20, demonstrating fluency for addition and subtraction within 10 using various strategies Work with addition and subtraction equations v Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false v Determine the unknown whole number in an addition or subtraction equation relating three whole numbers Extend the counting sequence v Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral Understand Place Value v Understand that the two digits of a two-digit number represent amounts of tens and ones v Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, <, and = Use place value understanding and properties of operations to add and subtract v 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 and strategies v Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count v Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 using concrete models or drawings and strategies based on place value Measure lengths indirectly and by iterating length units v Order three objects by length; compare the length of two objects indirectly by using a third object v Measure the length of an object by laying multiple copies of a shorter object end to end, with no gaps or overlaps Tell and Write Time v Tell and write time in hours and half-hours using analog and digital clocks v Organize, represent, and interpret data with up to three categories Reason with shapes and their attributes v Distinguish between defining (e.g., three-sided) versus non-defining (e.g., color) attributes of shapes v Build and draw shapes to possess defining attributes v Compose 2-D shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter circles) or 3-D shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape v 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
Mathematics Common Core State Standards Grade Two Represent and solve problems involving addition and subtraction v Use addition and subtraction within 100 to solve one- and two-step word problems with unknowns in all positions Add and subtract within 20 v Fluently add and subtract within 20 using mental strategies v By the end of grade 2, know from memory all sums of two one-digit numbers Work with equal groups of objects to gain foundations for multiplication v Determine whether a group of objects (up to 20) has an odd or even number of members; write an equation to express an even number as a sum of two equal addends v 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 Understand Place Value v Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones v Count within 1000; skip count by 5s, 10s, and 100s v Read and write numbers to 1000 using base-ten numerals, number names, and expanded form v Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, recording the results of comparisons with the symbols >, <, and = Use place value understanding and properties of operations to add and subtract v Fluently add and subtract within 100 v Add up to four two-digit numbers using strategies based on place value and properties of operations v Add and subtract within 1000, using concrete models, drawings, and strategies v Mentally add 10 or 100 to a given number 100 900, and mentally subtract 10 or 100 from a given number 100-900 Measure and estimate lengths in standard units v Measure the length of an object by selecting and using appropriate tools v Measure the length of an object twice, using length units of different lengths for the two measurements v Estimate lengths using units of inches, feet, centimeters, and meters v Measure to determine how much longer one object is than another Relate addition and subtraction to length v Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units; use drawings and equations with a symbol for the unknown number to represent the problem v 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 sums and differences within 100 on a number line Work with Time and Money v Tell and write time using analog and digital clocks to the nearest five minutes, using a.m. and p.m. v Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and symbols appropriately v Generate and represent measurement data using line plots, picture graphs, and bar graphs Reason with shapes and their attributes v Recognize and draw shapes having specified attributes; identify triangles, quadrilaterals, pentagons, hexagons, and cubes v Partition a rectangle into rows and columns of same-size squares and count to find the total number of them v 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
Mathematics Common Core State Standards Grade Three Represent and solve problems involving multiplication and division v Interpret whole-number products of multiplication problems and quotients of division problems v Use multiplication and division within 100 to solve word problems involving equal groups; use a symbol to represent the unknown number in an equation v Determine the unknown number in a multiplication or division equation relating three whole numbers Understand properties of multiplication and the relationship between multiplication and division v Apply properties of operations to multiply and divide v Understand division as an unknown-factor problem Multiply and divide within 100 v Fluently multiply and divide within 100 v By the end of grade 3, know from memory all products of two one-digit numbers Solve problems involving the four operations, and identify and explain patterns in arithmetic v Solve two-step word problems using the four operations v Represent problems using equations with a letter standing for the unknown quantity v Assess the reasonableness of answers using mental computation and estimation strategies including rounding v Identify and explain arithmetic patterns Use place value understanding and properties of operations to perform multi-digit arithmetic v Use place value understanding to round whole numbers to the nearest 10 or 100 v Fluently add and subtract within 1000 using strategies and algorithms v Multiply one-digit whole numbers by multiples of 10 in the range 10-90 Number and Operations Fractions Develop understanding of fractions as numbers v Understand part to whole relationships v Understand and represent fractions on a number line v Compare fractions and determine equivalence; use the symbols >, <, and = to record results; justify conclusions Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects v Tell and write time to the nearest minute and measure time intervals in minutes v Measure and estimate liquid volumes and masses of objects using standard units of grams, kilograms, and liters v Draw scaled picture graphs and bar graphs to represent a data set v Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch, and show the data by making a line plot marked off in appropriate units Geometric measurement: understand concepts of area and relate area to multiplication and to addition v Recognize area as an attribute of two-dimensional shapes v Measure area by counting unit squares v Find area by tiling and by multiplying the side lengths v Find total area by adding the area of parts of figures Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures v Solve real world and mathematical problems involving perimeters of polygons Reason with shapes and their attributes v Recognize attributes and categorize shapes based on these attributes v Partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole
Mathematics Common Core State Standards Grade Four Use the four operations with whole numbers to solve problems v Solve multistep word problems involving whole numbers v Represent multistep word problems using equations with a letter standing for the unknown quantity v Assess reasonableness of answers Gain familiarity with factors and multiples v Find factor pairs for a whole number in the 1 100 range v Determine if a number is a multiple of a one-digit number and if a number is prime or composite Generate and analyze patterns v Generate a number or shape pattern that follows a given rule Generalize place value understanding for multi-digit whole numbers v Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right v Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form v Round multi-digit whole numbers to any place; compare two multi-digit numbers using >, <, and = Use place value understanding and properties of operations to perform multi-digit arithmetic v Fluently add and subtract multi-digit whole numbers using the standard algorithm v Multiply a whole number of up to four digits by a one-digit whole number v Multiply two two-digit whole numbers v Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors Number and Operations Fractions Extend understanding of fraction equivalence and ordering v Recognize and generate equivalent fractions v Compare two fractions with different numerators and different denominators Build fractions from unit fractions by applying & extending previous understandings of operations on whole numbers v Add and subtract mixed numbers with like denominators v Solve word problems involving addition and subtraction of fractions with like denominators v Multiply a fraction by a whole number Understand decimal notation for fractions, and compare decimal fractions v Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with denominators 10 and 100 v Use decimal notation for fractions with denominators 10 or 100 v Compare two decimals to hundredths by reasoning about their size; record results with >, <, and = Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit v Know relative sizes of measurement units within one system of units v Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals v Apply the area and perimeter formulas for rectangles in real world and mathematical problems v Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, and 1/8) v Solve problems involving addition and subtraction of fractions by using information presented in line plots Geometric measurement: understand concepts of angle and measure angles v Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint v Understand how to measure angles as a fraction of a circle v Measure angles in whole-number degrees using a protractor; find unknown angles on a diagram Draw and identify lines and angles, and classify shapes by properties of their lines and angles v Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines v Classify 2-D shapes based on lines and angles and recognize a line of symmetry for a 2-D figure
Mathematics Common Core State Standards Grade Five Write and interpret numerical expressions v Use parentheses, brackets, or braces in numerical expressions v Evaluate expressions with these symbols v Write simple expressions that record calculations with numbers Analyze patterns and relationships v Generate numerical patterns using two given rules Understand the place value system v Recognize 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 v Explain patterns in the number of zeros when multiplying and dividing numbers by powers of 10 v Read, write, and compare decimals to thousandths v Use place value understanding to round decimals to any place Perform operations with multi-digit whole numbers and with decimals to hundredths v Fluently multiply multi-digit whole numbers using the standard algorithm v Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors v Add, subtract, multiply, and divide decimals to hundredths; record results and explain the reasoning used Number and Operations Fractions Use equivalent fractions as a strategy to add and subtract fractions v Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions v Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases with unlike denominators Apply and extend previous understandings of multiplication and division to multiply and divide fractions v Interpret a fraction as division of the numerator by the denominator, and solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers v Multiply fractions or whole numbers by a fraction v Explain the result of multiplying a given number by a fraction greater than one and a fraction less than one v Solve real world problems involving multiplication of fractions and mixed numbers v Solve real world problems involving division of fractions by non-zero whole numbers and division of whole numbers by unit fractions Convert like measurement units within a given measurement system v Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real world problems v Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8) Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition v Measure volume by counting unit cubes v Relate the counting of unit cubes to the formula for determining volume v Apply formulas for finding the volume of rectangular prisms v Find the volume of solid figures by adding the volumes of non-overlapping pieces Graph points on the coordinate plane to solve real-world and mathematical problems v Use axes to define a coordinate system v Understand the origin, x-axis, x-coordinates, y-axis, and y-coordinates v Represent real world and mathematical problems by graphing points in the first quadrant Classify two-dimensional figures into categories based on their properties v Classify two-dimensional figures in a hierarchy based on properties