Gateway Regional School District VERTICAL ARTICULATION OF MATHEMATICS STANDARDS Grades K-4

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1 NUMBER SENSE & OPERATIONS K.N.1 Count by ones to at least 20. When you count, the last number word you say tells the number of items in the set. Counting a set of objects in a different order does not change the total. There is a number word and a matching symbol that tell exactly how many items are in a group. Adding 1-10 counters to a filled ten-frame results in the teen numbers or the numbers Zero is a number that tells how many objects there are when there are none. Use objects to represent and count the quantities 1 through 20. Recognize and write the numeral that describe the quantity 0. 2.N.1 Name and write (in numerals) whole numbers to 1000, identify the place values of the digits, and order the numbers. There is a specific order to the set of whole numbers. Numbers 11 through 19 can be shown as a group of 10 with fewer than 10 left over. The counting sequence to 100 is built on the repetition of 0 to 9 in the ones place of each new decade. In a standard numeral, the tens are written to the left of the ones. You can add the values of the digits of a number together to get the actual number. If a number is represented with 10 or more ones, 10 ones may be grouped to form another 10 and the value remains the same. Except for decade or century changes, a number is increased or decreased by 1 when the ones digit is changed by 1 and a number is increased or decreased by 10 when the tens digit is changed by 1. A number ending with 5 ones is halfway between two multiples of 10 and numbers ending in 1, 2, 8, or 9 are closest to the tens. 2.N.1 Name and write (in numerals) whole numbers to 1000, identify the place values of the digits, and order the numbers. The decade numbers to 100 are built on groups of ten with oral names that are similar to, but not the same as, the number of tens counted. In a two-digit number, the tens digit tells how many groups of ten and the ones digit tells the number of ones. Numbers 21 through 99 are each written by joining two number words with a hyphen. Numbers 1 through 20 are each represented by a unique number word. For any two-digit numbers, the one with more tens is the greater number; if the 2 numbers have the same number of tens, then the number with more ones is greater. Except for decade changes, a number is increased or decreased by 1 when the ones digit is changed by tens make 100, and 10 hundreds make 1,000 Each digit in a three-digit number tell how many hundreds, 3.N.1 Exhibit an understanding of the values of the digits in the base ten number system by reading, modeling, writing, comparing, and ordering whole numbers through 9,999. Our number system is based on groups of ten. Whenever we get 10 in one place value, we move to the next greater place value. Place value can be used to write numbers in different, but equivalent forms. You can regroup whole numbers by breaking numbers apart using place value. Read and write numbers up to 9,999. Generate equivalent representation for a number by composing and decomposing numbers. Regroup a two- or three-digit number in preparation for subtraction. 4.N.1 Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at least 100,000; demonstrating an understanding of the values of the digits; and comparing and ordering the numbers. Place value can be used to write numbers in different but equivalent forms. We use place-value periods to help us understand, read, and write larger numbers. Place value can help us compare and order numbers. Place value relationships can help estimate how much. Use place value ideas to write multiples of 100, 1,000, and 10,000 in different ways. Read, write, compare, and order numbers through 999,999. Estimate totals made up of large numbers. Page 1 of 51

2 Ordering three numbers is similar to comparing two numbers because each number must be compared to both of the others. 10 tens make 1 hundred. The number that comes before or after a given number will change the ones place by 1 except at the decade changes. Write the numbers just before, after, or between two given numbers. Order numbers through 12. Read and write the teen numbers as a group of 10 and some left over. Count and write numbers to 100 on the hundred chart. Given a quantity shown with tens and ones, tell how many tens and ones there are and write the number. Exchange a ten for 10 ones or 10 ones for a ten and write the new representation in expanded form. Given a two-digit number, write the numbers that are 10 more/10 less and 1 more/1 less. Estimate the positions of numbers on a number line marked only in multiples of 10. Given 3 two-digit numbers, order them from least to greatest or from greatest to least. tens, or ones are in that number. Adding together the values of the three-digit number produces the total value of the number. Count groups of ten, up to 10 tens, and write how many. Use groups of tens and ones to show a given two-digit number. Read and write number words for given numbers. Use place value to ompare numbers using the greater-than, less-than. Identify and write numbers that are one before, one after, or between given numbers. Count by hundreds to 1,000. Count sets grouped in hundreds, tens, and ones. Read and write three-digit numbers using expanded form, standard form, and number words. Page 2 of 51

3 K.N.2 Match quantities up to at least 10 with numerals and words. There is a number word and a matching symbol that tell exactly how many items are in a group. Numbers can be compared by counting or by matching corresponding sets. There is a specific order to the set of whole numbers. A graph is a tool that can be used to organize information in order to solve problems. When the objects in two sets are matched one-to-one, the set which leftover objects at the end has more. Recognize and write the numerals that describe the quantities 0 to 20. Recognize number words from 0 to 10. Use objects to order numbers from 0 to 5 in sequence. Solve problems by making and reading a real graph and a picture graph. 2.N.2 Identify and distinguish among multiple uses of numbers, including cardinal (to tell how many) and ordinal (to tell which one in an ordered list), and numbers as labels and as measurements. Ordinal positions in a row or list can be determined by counting; the ordinal words are similar to the counting words. Use ordinals through twentieth to identify position. 2.N.2 Identify and distinguish among multiple uses of numbers, including cardinal (to tell how many) and ordinal (to tell which one in an ordered list), and numbers as labels and as measurements. Ordinal positions in a row or list can be determined by counting and the ordinal words are similar to the counting words. Use ordinals through twentieth to identify position. 3.N.2 Represent, order, and compare numbers through 9,999. Represent numbers using expanded notation (e.g., 853 = 8 x x ), and written out in words (e.g., eight hundred fifty-three). Our number system is based on groups of ten. Whenever we get 10 in one place value, we move to the next greater place value. Place value can be used to write numbers in different, but equivalent forms. Place value can help us compare and order numbers. Read and write numbers through 9,999. Generate equivalent representation for a number by composing and decomposing numbers. Regroup a two- or three-digit number in preparation for subtraction. Compare whole numbers through 10, N.2 Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g., 853 = 8 x x Place value can be used to write numbers in different but equivalent forms. We use place-value periods to help us understand, read, and write larger numbers. Use place value ideas to write multiples of 100, 1,000, and 10,000 in different ways. Read, write, compare, and order numbers through 999,999. Page 3 of 51

4 K.N.3 Identify positions of objects in sequences (e.g., first, second) up to fifth. Ordinal position in a row or list can be determined by counting and the ordinal words are similar to the counting words. Use the words first through fifth to identify ordinal positions. 2.N.3 Identify and represent common fractions (1/2, 1/3, 1/4) as parts of wholes, parts of groups, and numbers on the number line. A shape can be divided into any number of equal parts in a variety of ways. A half of a region is one of two equally sized parts. A unit fraction of a region names one of a number of equally sized parts. Groups can be divided into equal parts in the same way that shapes can be divided into equal parts. Determine whether a shape has been divided into equal or unequal parts and count the number of equal parts into which it has been divided. Identify and show half of a region. Identify and show half of a group objects. 2.N.3 Identify and represent common fractions (1/2, 1/3, 1/4) as parts of wholes, parts of groups, and numbers on the number line. Equal parts means that each part is the same. A unit fraction names one of a number of equal parts into which a shape or region has been divided. The bottom number of a fraction tells the number of equal parts. The top number tells how many equal parts are being named. Fractions can be used to name part of a set as well as part of a region. Determine whether a shape has been divided into equal or unequal parts; identify halves, thirds, and fourths. Identify and show a unit fraction of a region. Identify and show any fraction of a region. Identify and show fractions of a set of objects. 3.N.3 Identify and represent fractions (between 0 and 1 with denominators through 10) as parts of unit wholes and parts of groups. Model and represent a mixed number (with denominator 2, 3, or 4) as a whole number and a fraction, e.g., 1 2/3, 3 1/2. A region can be divided into equal parts in different ways and parts that are equal in size can have different shapes. A fraction is relative to the size of the whole. The denominator of a fraction gives the number of equal parts in all, and the numerator tells how many equal parts are described. A fraction is relative to the size of the whole. Different fractions used to name the same amount are equivalent. Finding the number of objects in a fractional part of a set involves division. Fractions in which the numerator is greater than the denominator may be expressed as mixed numbers or as improper fractions. Identify regions that have been divided into equal-sized parts and divide regions into equal-sized parts. Identify and draw fractional parts of regions. 4.N.4 Select, use, and explain models to relate common fractions and mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, and 1 1 /2), find equivalent fractions, mixed numbers, and decimals, and order fractions. The same fractional part can have different names that are equivalent. Equivalent fractions are found by multiplying or dividing the numerator and denominator of a fraction by the same non-zero number. Fractions can be expressed in their simplest form by dividing the numerator and denominator by their greatest common factor When two fractions have the same denominator, the greater fraction has the greater numerator, and when two fractions have the same numerator, the fraction with the greater denominator is less. Fractions with a common denominator or a common numerator are easy to compare and order. Identify fractions that are equivalent and find fractions equivalent to a given fraction using models and/or a computational procedure. Page 4 of 51

5 K.N.4 Compare sets of up to at least 10 concrete objects using appropriate language (e.g., none, more than, fewer than, same number of, one more than) and order numbers. When the objects in two sets are matched one-to-one, the set with leftover objects at the end has more. Numbers can be compared by counting or by matching corresponding sets. There is a specific order to the set of whole numbers. A graph is a tool that can be used to organize information in order to solve problems. You can use 5 and 10 as benchmarks and compare other numbers to these benchmarks. On a calendar, the number to the right or left of another number is either one more or one less than the other. One-to-one correspondence can be used to compare groups. 2.N.4 Compare whole numbers using terms and symbols, e.g., less than, equal to, greater than (<, =, >). 10 is a foundational number in our number system and can be made up to two 5s; 5 and 10 provide points of reference to which other numbers can be related. There is a specific order to the set of whole numbers. For any 2 two-digit numbers, the one with more tens is the greater number; if they have the same number of tens, the one with more ones is greater. Ordering three numbers is similar to comparing two numbers because each number must be compared to both of the others. Compare a given number to 5 and N.4 Compare whole numbers using terms and symbols, e.g., less than, equal to, greater than (<, =, >). For any two-digit numbers, the one with more tens is the greater number; if the 2 numbers have the same number of tens, then the number with more ones is greater. Numbers are compared by beginning with the place of greatest value, the place farthest to the left, and then moving to the right as far as is needed. Compare numbers using the greater-than, less-than, and equal to symbols. Compare three-digit numbers using the symbols <, >, and =. Find equivalent fractions using models such as fraction strips. Compare and order fractions. Find the number of objects in a fractional part of a set where the numerator is 1. Read and write mixed numbers and use objects or pictures to show mixed numbers. Write fractions. 3.N.4 Locate on the number line and compare fractions (between 0 and 1 with denominators 2, 3, or 4, e.g., 2/3). Different fractions used to name the same amount are equivalent. Fractions with a common denominator or a common numerator can be compared and ordered. A set can be considered a whole, and fractional parts are parts of the set. The denominator of the fraction tells the total number of things in the set and the numerator tells the number of parts being described. Find equivalent fractions using models such as fraction strips. Compare and order fractions. Identify fractional parts of sets or groups and divide sets to show fractional parts. Express fractions in simplest form. Determine which of two fractions is greater (or less). Compare fractions using >, <, and =, and order fractions. 4.N.3 Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on the number line. The denominator of a fraction, set, or group gives the number of equal parts in all, and the numerator tells how many equal parts are described. When the numerator and denominator are equal, the fraction is equal to 1 or the entire region. The distance between 0 and 1 on a number line can be divided into fractional parts, and the points can be named with fractions. Identify and draw fractional parts of a region. Identify fractional parts of sets or groups and divide sets to show fractional parts. Locate and name fractions on a number line. Page 5 of 51

6 Use one-to-one correspondence and counting to compare groups and determine which has more, which has fewer, or whether the groups are the same. Use objects to order numbers from 0 to 5 in sequence. Solve problems by making and reading a real graph and a picture graph. Given a number from 1 to 10, tell whether it is more or less than 10. Use a number line to order numbers from 0 through 10. Compare two numbers to decide which number is greater and which is less. Find, identify, and record numbers through 31 on a calendar. K.N.5 Understand the concepts of whole and half Parts of a whole are equal when they are the same size. When objects or shapes are cut into 2 equal parts, the parts are called halves. Identify equal parts of a whole Order numbers through 12. Given 2 two-digit numbers, determine if the first is greater than, less than, or equal to the second. Given 3 two-digit numbers, order them from least to greatest or from greatest to least. 2.N.5 Identify odd and even numbers and determine whether a set of objects has an odd or even number of elements. Even numbers can be broken into two equal parts; odd numbers cannot. Given a number less than 60, determine if it is odd or even. 2.N.5 Identify odd and even numbers and determine whether a set of objects has an odd or even number of elements. Even numbers can be broken into two equal parts; odd numbers cannot. Any number that ends in 2, 4, 6, 8, or 0 is even. Any number that ends in 1, 3, 5, 7, 9 is odd. 3.N.5 Recognize classes to which a number may belong (odd numbers, even numbers, and multiples of numbers through 10). Identify the numbers in those classes, e.g., the class of multiples of 7 between 1 and 29 consists of 7, 14, 21, 28. Continue number patterns and use place-value patterns to find sums and differences. Addition doubles facts and multiplying by 2 give the same result. Patterns and properties can help you remember multiplication 4.N.7 Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may belong, and identify the numbers in those classes. Use these in the solution of problems. Fractions can be expressed in their simplest form by dividing the numerator and denominator by their greatest common factor. Express fractions in simplest form. Page 6 of 51

7 and identify halves. Identify any number as odd or even. facts. Multiplication facts help you find the products for other facts. Count on or count back easily using place values, Find products of one-digit numbers times 0 through 10. K.N.6 Identify U.S. coins by name. The name of a penny, nickel, dime, quarter, and dollar bill. Identify a penny, nickel, dime, quarter, and dollar bill. 2.N.6 Identify the value of all U.S. coins, and $1, $5, $10, and $20 bills. Find the value of a collection of coins and dollar bills and different ways to represent an amount of money up to $5. Use appropriate notation, e.g., 69, $1.35. A penny has a value of 1 cent, nickel has a value of 5 cents, and a dime has a value of 10 cents; pennies can be counted by 1s, nickels can be counted by 5s, and dimes can be counted by 10s. To count a mixed group of coins, skip count by the largest coin present and then shift to skip count by the next largest coin present and keep going until no coins remain. A quarter is worth 25 cents. A half-dollar has a value of 50 cents and a dollar has a value of 100 cents. Identify the value of a group of nickels and pennies through 25 2.N.6 Identify the value of all U.S. coins, and $1, $5, $10, and $20 bills. Find the value of a collection of coins and dollar bills and different ways to represent an amount of money up to $5. Use appropriate notation, e.g., 69, $1.35. To count a mixed group of coins, skip count by the greatest coin value present then shift to skip counting by the next greatest coin value present and then keep going until no coins remain. When counting a set of coins, begin with the coin or coins that have the greatest value and then count on to the coin or coin of least value. The number of coins in a set does not necessarily indicate which of two sets has the greater value. There are many combinations of coins that show the same amount of money. One dollar is worth 100 cents. Both a dollar bill and a dollar 3.N.8 Select and use appropriate operations (addition, subtraction, division, and multiplication) to solve problems, including those involving money. When counting money it is often easiest to start with the bills or coins that have the greatest value. Counting up rather than subtraction is a common way to make change. Making change can be thought of as part (price) plus part (change) equals whole (amount paid) Writing a number sentence is one way of representing what we know and what we need to find out in a word problem. The algorithm for adding and subtracting whole numbers can be extended to adding and subtracting money. 4.N.10 Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those involving money. Word problems tell what is known and what needs to be figured out. When counting money, it is often easiest to start with the bills or coins that have the greatest value. The kinds of numbers in a calculation and the ease with which one can apply different calculation methods together determine an appropriate computation method. Writing a number expression is one way of representing what we know in a word problem. Word phrases that express mathematical situations can be translated into specific expressions using numbers and Page 7 of 51

8 cents. Identify the value of a group of dimes and pennies through 99 cents. Identify the value of a group of dimes and nickels through 95 cents. Identify the value of a group of dimes, nickels, and pennies through 99 cents. Identify a quarter and find groups of coins that have the same value as a quarter. Count a collection of coins that includes half-dollars, quarters, dimes, nickels, and pennies. Identify a dollar bill, a dollar coin, a half-dollar coin, and combinations of coins worth amounts up to $1.00. coin stand for 100 cents. Identify the value of a group of dimes, nickels, and pennies through 99 cents. Count a collection of coins that includes half-dollars, quarters, dimes, nickels, and pennies. Compare the values of two sets of coins. Show the same amount of money using different sets of coins. Identify the value of a dollar bill and a dollar coin. Make change by counting on. Find the value of money ($5 and $1 bills, half-dollars, quarters, dimes, nickels, and pennies) Write number sentences for word problems and use complete sentences to write answers to word problems. Add and subtract money up to $100. Use multiplication facts along with addition and subtraction to solve problems. operations. Solving problems using the Try, Check, and Revise strategy is based on initial estimates. After the difference between the estimate and the desired total is known it is easier to determine the exact answer. The algorithm for multiplying whole numbers can be extended to multiplying money. To find the product of three factors you can start by finding the product of any two. Choosing certain pairs of factors may enable you to compute the product of the three factors using mental math. Properties of whole numbers explain why you can choose which numbers to multiply first. Remainders expressed as whole numbers give a specific type of information that is sometimes more useful than expressing remainders as fractions or as the decimal part of the quotient. Algorithms for dividing whole numbers can be extended to dividing money. Writing a number sentence is one way you can represent what you know and what you need to find out in a word problem. Tell in words what is known and what needs to be determined in given word problems. Page 8 of 51

9 Find the value of a given assortment of bills and coins and tell how to make a given money amount with the fewest bills and/or coins. For a variety of problems, state the computation method to be used and add or subtract using that method. Write number expressions for phrases. Choose and evaluate the number expression that matches a word phrase. Solve problems using the Try, Check and Revise strategy. Reading for the main idea in a problem helps in identifying the operation or operations needed to solve it. Phrases like how many, how many more, and how many times as many are clues to the correct operation, but these phrases also must be read in context. Decide how to use the quotient and remainder to answer the question in a division problem. Compute and estimate quotients involving money amounts. Write number sentences for word problems and use complete sentences to write answers to K.N. 8 Estimate the number of objects in a group and verify results 2.N.7 Demonstrate an understanding of various meanings of addition and subtraction, e.g., addition as combination (plus, combined 2.N.7 Demonstrate an understanding of various meanings of addition and subtraction, e.g., addition as combination (plus, combined 3.N.10 Add and subtract (up to four-digit numbers) and multiply (up to two-digit numbers by a one-digit number) accurately and efficiently. word problems. 4.N.12 Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently. Page 9 of 51

10 Objects can be used to determine the approximate quantities of similar groups. Use objects to estimate the quantities of groups. with, more); subtraction as comparison (how much less, how much more), equalizing (how many more are needed to make these equal), and separation (how much remaining). Most numbers can be described in terms of two parts in a variety of ways. 1 more than expresses the relationship between two numbers 1 less than expresses the relationship between two numbers Addition can be used to represent joining situations. Given two parts, addition can be used to name the whole. Joining groups can be shown in an addition sentence that uses the plus symbol and the equals sign. The sum of zero and a number, or a number and zero, is that number. Subtraction can be used to represent separating situations. Given the whole and one part, subtraction can be used to name the other part. Separating groups can be shown in a subtraction sentence that uses the minus symbol and the equals sign. The difference between a number and zero is that number; the difference between a number and itself is zero. with, more); subtraction as comparison (how much less, how much more), equalizing (how many more are needed to make these equal), and separation (how much remaining). Two groups can be combined and then counted to see how many objects there are in all. When one part is removed from a whole, the remaining part can be counted to see how many remain. Two groups can be compared by counting the leftovers after doing one-to-one matching. Changing the order of the addends does not change the sum. Counting on is a strategy that can be used for solving missingaddend problems. Join two groups together and write an addition sentence to tell how many in all. Take away a number of objects from a group and count to find how many are left. Compare two groups to find out how many more or how many fewer. Use the commutative property to find sums. Given a quantity and one of its parts, find the missing part by counting on or counting back. The algorithm for adding, subtracting, and multiplying whole numbers. Add and subtract numbers to four digits. Multiply up to a two-digit number by a one-digit number. There are different ways to calculate mentally. Most involve breaking numbers apart or replacing them with numbers that are easy to compute with. Three or more numbers can be added in any order. The process for adding two whole numbers is just repeated when adding more than two numbers. The algorithm for adding and subtracting whole numbers can be extended to adding and subtracting money. The kinds of numbers in a calculation and the ease with which one can apply different calculation methods together determine an appropriate computation method. Making an array with placevalue blocks enables you to model and visualize the partial products used in the expanded algorithm for multiplying. The expanded algorithm involves multiplying the parts of numbers based on their place value. Both the expanded and the traditional or standard multiplication algorithms involve breaking the overall calculation Page 10 of 51

11 One-to-one correspondence can be used to compare groups. Subtraction can be used to represent both separating and comparing situations. Show ways the numbers 6, 7, 8, 9, and 10 can be divided into two parts. Find the numbers that are 1 and 2 more than a given number. Tell and act out joining stories to find how many in all. Find the sum of two addends. Write an addition sentence to find the sum in a joining situation. Write an addition sentence using zero. Tell and act out separating stories to find out how many are left. Find the difference between two numbers. Write a subtraction sentence to find the difference in a separating situation. Write subtraction sentences using zero. Compare two groups to find out how nany more or how many fewer. Write subtraction sentences to compare and tell how many more or how many fewer. Use counting on to find missing parts of 1,000. into simpler calculations. The traditional or standard algorithm is a shortcut for the expanded algorithm. Both the expanded and the traditional or standard multiplication algorithms can be extended to multiply greater numbers. Compute sums of numbers mentally. Compute differences of numbers mentally. Find the sums of three or more whole numbers or money amounts. Use the standard algorithm to find differences using whole number amounts and money amounts. For a variety of problems, state the computation method to be used and add or subtract using that method. Make arrays with place-value blocks to find products. Use the standard algorithm to multiply two-digit numbers and/or three-digit numbers by one-digit numbers. Use arrays to find products involving two-digit factors. 2.N.8 Understand and use the 2.N.8 Understand and use the 3.N.6 Select, use, and explain 4.N.8 Select, use, and explain Page 11 of 51

12 inverse relationship between addition and subtraction (e.g., = 14 is equivalent to 14 6 = 8 and is also equivalent to 14 8 = 6) to solve problems and check solutions. Doubles facts can be used to find differences for their related subtraction facts. Every addition fact has at least one related subtraction fact. Fact families use the same three numbers and can be used to show how addition and subtraction are related. An addition fact can be used to find the difference in a related subtraction fact. Find differences by using doubles facts. Write the addition and subtraction sentences that make up a fact family. Find differences by using known addition facts. Write related addition and subtraction facts with sums through 18. inverse relationship between addition and subtraction (e.g., = 14 is equivalent to 14 6 = 8 and is also equivalent to 14 8 = 6) to solve problems and check solutions. Fact families use the same three numbers and can be used to show how addition and subtraction are related. Doubles facts can be used to find differences for their related subtraction facts. An addition fact can be used to find the difference in a related subtraction fact. Pictures frequently contain important information that helps to solve problems. Subtraction and addition are inversely related so one can undo the other. Write the addition and subtraction sentences that make up a fact family. Find differences by using doubles facts. Find differences by using addition facts. Use pictures to help find missing addends in number sentences. Relate addition to subtraction by using one operation to check the other. various meanings and models of multiplication (through 10 x 10). Relate multiplication problems to corresponding division problems, e.g., draw a model to represent 5 x 6 and Combining equal groups is one meaning of multiplication. Arrays are a special kind of arrangement of equal groups and multiplication can be used to find the total. Two ways of thinking of division are sharing equally and repeated subtraction. Multiplication and division are inverse operations. Write multiplication number sentences for given situations using the X symbol. Write multiplication sentences for arrays and use arrays to find multiplication facts. Use repeated subtraction to find answers. Give all the facts in a multiplication/division fact family. various meanings and models of multiplication and division of whole numbers. Understand and use the inverse relationship between the two operations. Use patterns to find products with factors of 0, 1, 2, 5, and 9. Fractions can be expressed in their simplest form by dividing the numerator and denominator by their greatest common factor. Patterns can help you remember multiplication facts. Express fractions in simplest form. Page 12 of 51

13 2.N.9 Know addition facts (addends to ten) and related subtraction facts, and use them to solve problems. Given two parts, addition can be used to name the whole. Joining groups can be shown in an addition sentence that uses the plus (+) symbol and the equals (=) sign. The sum of zero and a number or a number and zero is that number. Writing a number sentence is one strategy that can be used to solve a problem. Given the whole and one part, subtraction can be used to name the part. Separating groups can be shown in a subtraction sentence that uses the minus (-) symbol and the equals (=) sign. The difference between a number and zero is that number; the difference between a number and itself is zero. Subtraction can be written in a horizontal or vertical form, both representing the same situation. Addition and subtraction can both be useful in answering joining and separating questions. When counting on, the last number word said tells how 2.N.9 Know addition facts (addends to ten) and related subtraction facts, and use them to solve problems. Sums to twenty and related subtraction facts. The difference between combining and separating problems. Write an addition sentence to find the sum of two addends up to a sum of twenty in a joining situation. Tell and act out separating stories to find out how many are left. Write a subtraction sentence to find the difference in a separating situation with numbers of twenty or less. After reading a story problem, identify combining or separating situations and choose the appropriate strategy for solving the problem. 3.N.9 Know multiplication facts through 10 x 10 and related division facts, e.g., 9 x 8 = 72 and 72 9 = 8. Use these facts to solve related problems, e.g., 3 x 5 is related to 3 x 50. Addition doubles facts and multiplying by 2 give the same result. Patterns and properties can help you remember multiplication facts. Word problems tell what is known and what needs to be figured out. Multiplication and division are inverse operations. You can use the inverse relationship between multiplication and division to find division facts. Patterns can help you when dividing with 0 and 1. Patterns can help you divide with divisors of 10. Place value, multiplication, and division facts and patterns can help you multiply by multiples of 10 and 100. You can use multiplication facts you know to help you find the products for other facts. You can use multiplication to compare the size of two groups. (For example, twice as many) 4.N.11 Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication problems and compute related problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x 500. Basic facts and place-value patterns can help you multiply a one-digit number by multiples of 10, 100, and 1,000. Basic facts and place-value patterns can help you multiply a two-digit number by multiples of 10, 100, and 1,000. Multiply any number by 10, 100, or 1,000. Mentally multiply any number by 10, 100, or 1,000. Page 13 of 51

14 many there are all together. Changing the order of the addends does not change the sum. It saves time when you count on if you begin with the greater number. Moving to the right on a number line is one way to represent addition. Some of the information given in a problem may not be needed in order to solve the problem. If you know a doubles fact, it can help you figure out a doubles-plus 1 fact. 10 is an important benchmark in our numeration system so other numbers can be thought of in relation to what we know about the number 10. Knowing how to draw pictures to solve problems is helpful in checking to see that answers make sense. Moving to the left on a number line is one way to show subtraction. When counting back, the last number said tells how many are left. Doubles facts can be used to find differences for their related subtraction facts. An addition fact can be used to find the difference in a related subtraction fact. If you know a doubles fact, it can help you figure out a Find products of one-digit numbers from 0 to 10. Give all the facts in a multiplication/division fact family. Give quotients for division facts with divisors from 0 to 10. Recognize which numbers are divisible by 10. Mentally multiply any number by 10 and 100. Memorize multiplication facts. Use multiplication and comparison to find the size of a group. Recognize patterns on a multiplication fact table. Page 14 of 51

15 doubles-plus 1 fact or a doublesminus 1 fact. There is a place-value pattern that can be used to add 10 to a single-digit number mentally. Making a group of 10 can change a difficult addition fact to one that is easier to add mentally. There are a variety of strategies to use to find addition fact sums; which one is best depends on the addends. Addition fact strategies can be applied to finding sums of three numbers. Every addition fact has at least one related subtraction fact. Fact families use the same three numbers and can be used to show how addition and subtraction are related. An addition fact can be used to find the difference in a related subtraction fact. Making a 10 before subtracting can change a difficult subtraction fact to one that is easier to subtract mentally. There are a variety of strategies to use to find subtraction fact differences; which one is best depends on the numbers involved. The answer to one problem can be used as information needed to solve another problem. Find the sum of two addends. Page 15 of 51

16 Write an addition sentence to find the sum in a joining situation. Write an addition sentence using zero. Solve problems by writing addition sentences. Find the difference between two numbers. Write a subtraction sentence to find the difference in a separating situation. Write subtraction sentences using zero. Write the differences for horizontal and vertical forms of subtraction. Solve problems by choosing addition or subtraction. Find sums by counting on 1, 2, or 3 using counters. Use turn around facts ( the commutative property) to find sums. Count on 1, 2, or 3 to add starting with the greater number. Use a number line to count on 1, 2, or 3. Solve problems by identifying unnecessary information and writing number sentences. Recognize doubles as a strategy for remembering sums. Use doubles facts to learn doubles-plus 1 facts. Recognize facts that have sums of 10. Solve problems by drawing pictures. Page 16 of 51

17 Solve problems by writing subtraction sentences. Use a number line to count back 1 or 2. Find differences by counting back 1 or 2. Find differences by using doubles facts. Write related addition and subtraction facts and sentences that make up a fact family. Find differences by using known addition facts. Recognize doubles as a strategy for remembering sums to 18. Use doubles facts to learn doubles-plus-1 facts and doublesminus-1 facts. Use a pattern to add numbers 1 to 8 to the number 10. Find sums by making a 10 when adding to 8 or 9. Select and apply addition fact strategies. Use the associative property to find sums of three numbers. Write related addition and subtraction facts with sums through 18. Find differences using a tenframe. Select and apply subtraction strategies to use to find subtraction fact differences; which one is best depends on the numbers involved. Solve multiple-step problems by using the answer to the first Page 17 of 51

18 questions to answer the second question. 2.N.10 Demonstrate the ability to add and subtract three-digit numbers accurately and efficiently. Standard is not developmentally appropriate for 1. 2.N.10 Demonstrate the ability to add and subtract three-digit numbers accurately and efficiently. In the standard algorithm, ones, tens, and hundreds are added separately beginning with the ones. Whenever there are 10 or more in a column the regrouping is recorded in the next column to the left. Difference between two numbers can be estimated by subtracting close-but-easier numbers. Use paper and pencil to add 2 three-digit numbers with one regrouping. Use estimation to select two numbers that have a given difference. 3.N.7 Use the commutative (order) and identity properties of addition and multiplication on whole numbers in computations and problem situations, e.g., = = There are certain relationships for whole numbers and addition that always hold true. Arrays are a special kind of arrangement of equal groups and multiplication can be used to find the total. Patterns and properties can help you remember multiplication facts. When multiplying three numbers you can multiply the product of any two of the numbers by the third number. Apply commutative and identity properties in addition and multiplication. Use mental math to add numbers by breaking them apart using place value. Add mentally by rounding with multiples of ten. Write multiplication sentences for arrays and use arrays to find multiplication facts. Multiply three numbers (for example, 3 X 4 X 8) 4.N.9 Select, use, and explain the commutative, associative, and identity properties of operations on whole numbers in problem situations, e.g., 37 x 46 = 46 x 37, (5 x 7) x 2 = 5 x (7 x 2). Use patterns to find products with factors of 0, 1, 2, 5, and 9. Dollars, dimes, and pennies represent whole numbers, tenths, and hundredths in our decimal number system. When counting money, it is often easiest to start with the bills or coins that have the greatest value. There are different ways to calculate mentally. Most involve breaking numbers apart or replacing them with numbers that are easy to compute with. Making a table can help to represent what you know in solving a problem. Give money amounts in dollars, dimes, and pennies and in ones, tenths, and hundredths. Find the value of a given assortment of bills and coins and tell how to make a given money amount with the fewest bills and/or coins. Compute sums of money Page 18 of 51

19 mentally. Make tables and use them to solve word problems. Patterns can help you remember multiplication facts. K.N.7 Use objects and drawings to model and solve related addition and subtraction problems to ten. 1 more than expresses the relationship between two numbers: 4 is more than 3. 1 less than expresses the relationship between two numbers: 3 is less than 4. Addition can be used to represent joining numbers and/or objects. Subtraction can be used to represent separating numbers and/or objects. Given two parts, addition can be used to name the whole. Knowing how to draw pictures to solve problems is helpful in checking to see that answers make sense. Joining groups can be shown in an addition phrase that use the plus (+) sign and the equal (=) sign. 2.N.11 Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition (two 3-digit numbers and three 2-digit numbers) and subtraction (two 3-digit numbers). Standard is not developmentally appropriate for 1. 2.N.11 Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition (two 3-digit numbers and three 2-digit numbers) and subtraction (two 3-digit numbers). Adding or subtracting hundreds or tens is similar to adding or subtracting single-digit numbers. Numbers can be mentally added by combining any parts in any order, but it is easiest to add the greatest place values first. When adding like place values are combined. If there are 10 or more of one value, then 10 of them can be exchanged for 1 of the next-higher value. In the standard algorithm, ones, tens, and hundreds are added separately beginning with the ones. Whenever there are 10 or more in a column the regrouping is recorded in the next column to the left. Horizontal addition or subtraction problems can be written vertically so that the 4.N.14 Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up to five-digit numbers), and multiplication (up to three digits by two digits). Three or more numbers can be added in any order. The process for adding two whole numbers is just repeated when adding more than two numbers. The algorithm for adding and subtracting whole numbers can be extended to adding and subtracting money. Both the expanded and the traditional or standard multiplication algorithms involve breaking the overall calculation into simpler calculations. The traditional or standard algorithm is a shortcut for the expanded algorithm. Both the expanded and the traditional or standard multiplication algorithms can be extended to multiply greater numbers. Page 19 of 51

20 Separating groups can be shown with a subtraction phrase the uses the minus (-) symbol. Use counters to show 2 to 10 in two parts. Use a ten-frame to show 10 in different ways. Act out number stories that involve joining or separating two groups. Interpret illustrations that show joining groups and write the corresponding numbers. Solve problems by drawing pictures about joining two groups. Use the plus (+) sign to represent joining groups when recording addition. Use the minus sign (-) to represent take way situations when recording subtraction. Identify and use the equal sign; add and write the sum or subtract and write the difference. standard algorithm can be used. Missing parts can be found by counting on to the known part or by counting back from the whole. A hundred can be regrouped for 10 tens when you need to subtract more tens than are present in the tens place of the top number. When regrouping, the numbers written above the columns do not change the original amount; they simply indicate a change in the way that amount is being represented. When adding 2 two-digit numbers, you may or may not have enough ones altogether to make another ten. Adding or subtracting hundreds or tens is similar to adding or subtracting single-digit numbers. Add and subtract multiples of 10 or 100 to and from a threedigit number without regrouping. Add three-digit numbers mentally without regrouping. Use place-value models to add 2 three-digit numbers without regrouping. Use paper and pencil to add 2 three-digit numbers with one regrouping. Add 2 three-digit numbers in vertical form when they are given in horizontal form. Use models to subtract three- Add and subtract whole numbers and money amounts (to five digits). Find the sums of three or more whole numbers or money amounts. Use the standard algorithm to find differences using whole number amounts and money amounts. Use the standard algorithm to multiply two-digit numbers and/or three-digit numbers by one-digit numbers. Page 20 of 51

21 digit numbers with regrouping. Use the standard algorithm to subtract three-digit numbers with regrouping. Subtract three-digit numbers written in horizontal form. Add a two-digit number to a two-digit number using models 2.N.12 Estimate, calculate, and solve problems involving addition and subtraction of twodigit numbers. Describe differences between estimates and actual calculations. or mental math. 2.N.12 Estimate, calculate, and solve problems involving addition and subtraction of twodigit numbers. Describe differences between estimates and actual calculations. 3.N.12 Understand and use the strategies of rounding and regrouping to estimate quantities, measures, and the results of whole-number computations (addition, subtraction, and multiplication) up to two-digit whole numbers and amounts of money to $100, and to judge the reasonableness of the answer. 4.N.17 Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate quantities, measures, and the results of whole-number computations up to three-digit whole numbers and amounts of money to $1000, and to judge the reasonableness of the answer. Standard is not developmentally appropriate for 1. When adding 2 two-digit numbers, you may or may not have enough ones altogether to make another ten. An estimate of a two-digit sum can be made by adding the tens and then deciding whether the ones will increase the sum beyond the next multiple of ten. An estimate of a two-digit difference can be made by subtracting tens then considering the ones. Add a two-digit number to a two-digit number using models or mental math. Strategies for rounding and regrouping. Use rounding and regrouping strategies to estimate quantities, measures, and whole-number computations. There are different ways to estimate sums and differences. Most involve replacing numbers with other numbers that are close and easy to compute. The numbers used determine whether an estimate is reasonable. The specific numbers used to make an estimate determine whether an estimate is reasonable. There are different ways to estimate products and quotients. Most involve replacing numbers with other numbers that are close and easy to compute. Page 21 of 51

22 Estimate the sum of two twodigit numbers. Estimate the difference between two two-digit numbers. Use rounding and front-end estimation to estimate sums and differences. Indicate whether an estimate is reasonable. Use rounding and compatible numbers to estimate products. Use rounding and place value to estimate products of larger numbers. Estimate quotients 3.N.11 Round whole numbers through 1,000 to the nearest 10, 100, and 1, N.16 Round whole numbers through 100,000 to the nearest 10, 100, 1000, 10,000, and 100,000. Strategies for rounding whole numbers. Round whole numbers through 1,000 to the nearest 10, 100, and 1,000. Rounding is a process for finding the multiple of 10, 100, etc, closest to a given number. Round whole numbers through one hundred thousand. 4.N.13 Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and efficiently. Interpret any remainders. When you divide whole numbers, sometimes there is a remainder. The remainder must be less than the divisor. Page 22 of 51