Gateway Regional School District VERTICAL ALIGNMENT OF MATHEMATICS STANDARDS Grades 3-6

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1 NUMBER SENSE & OPERATIONS 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. 5.N.2 Demonstrate an understanding of place value through millions and thousandths. Place value from thousandths through millions The value of each number place. Understand how increasing and decreasing its individual place values change the overall value of a number. Order numbers through millions and thousandths. Identify the place value of a given number within a larger number. (What place is the 4 in, in the number 645,783?) Give the value of any number within a larger number. (What is the value of the 4 in that number?) Increase and decrease any number by increasing or decreasing one of its place values. 6.N.2 Demonstrate an understanding of place value to billions and thousandths. Place value from thousandths through billions Order numbers 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 fiftythree). 4.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 N.3 Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded notation without exponents, e.g., 9724 = 9 x x x N.3 Represent and compare very large (billions) and very small (thousandths) positive numbers in various forms such as expanded notation without exponents, e.g., 9724 = 9 x x x Page 1 of 38

2 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.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. 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, 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. How to represent numbers from thousandths place through millions in various forms such as standard form, standard notation and expanded form. Expanded notation without exponents Standard and written form read large numbers when it is presented in various forms. work from one form to another (reading and writing large numbers) 5.N.5 Identify and determine common equivalent fractions (with denominators 2, 4, 5, 10) and mixed numbers (with denominators 2, 4, 5, 10), decimals, and percents (through one hundred percent), e.g., 3/4 = 0.75 = 75%. Concept of equivalent fractions (with denominators 2, 3, 4, 5 and 10), decimals and percents How to work from decimal, fraction and percent given 1 value (3/5 equals, 60% or.60). How to reduce fractions using the concept of common factors. Connection between common denominator and multiples. How to represent numbers from thousandths place to billions in various forms Expanded notation without exponents Standard and written form 6.N.5 Identify and determine common equivalent fractions, mixed numbers, decimals, and percents. Common equivalent fractions, mixed numbers, decimals and percents Convert among fractions, decimals, percents Page 2 of 38

3 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. 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 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. 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 Create equivalent fractions by reducing or multiplying the numerator and the denominator by a common number. Convert among fractions, decimals and percents. Given a single fraction, decimal or percent, produce it equivalent in the other two forms. List remainders in division as decimals. (Denominators 2, 3,4, 5, 10) Identify remainders in division first as a fraction, then as its equivalent decimal. Organize (least to greatest) fractions, decimals and percents when given in different forms. Students will be able to convert all the items to a common form (convert all the fractions to decimals) 5.N.6 Find and position whole numbers, positive fractions, positive mixed numbers, and positive decimals on a number line. Concept of positive numbers, fractions, mixed numbers and decimals. Understand that fractions and decimals are located between whole numbers and how they are organized on a number line. 6.N.6 Find and position integers, fractions, mixed numbers, and decimals (both positive and negative) on the number line Concept of positive and negative integers, fractions, mixed numbers and decimals Locate and place positive and negative whole integers, fractions, mixed numbers and decimals on a number line Page 3 of 38

4 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. 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. Locate and place positive whole numbers, fractions, mixed numbers and decimals on a number line. Given a blank number line with two consecutive whole numbers at each end, students will be able place fractions in their absolute location. Identify fractions that are equivalent and find fractions equivalent to a given fraction using models and/or a computational procedure. Express fractions in simplest form. Determine which of two fractions is greater (or less). Compare fractions using >, <, and =, and order fractions. 4.N.5 Identify and generate equivalent forms of common decimals and fractions less than one whole (halves, quarters, fifths, and tenths). 5.N.4 Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection, and as locations on the number line. 6.N.4 Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection, and as locations on the number line. Page 4 of 38

5 Decimals show fractional parts of a whole. Relate decimals to common fraction benchmarks in 10ths and 100ths. Concept of fractions 1/2, 1/3, 1/4, 1/5, 1/10 as ratios Fractions, (without whole numbers) are part of a whole and have a value less than 1. How the denominator of each fractions determines the amount of parts that make up a whole. How the numerator of a fraction determines how many parts we are looking at. How to place a given fraction correctly in its place on a number line. Concept of fractions through 1/12 as ratios Express and order above fractions as ratios Express and order above fractions as ratios Place fractions correctly on a number line that contains both fractions and whole numbers. assign a fractional representation when given such things as: 3 out 12 or at a ration of 1 for every 4. Given a number line that has only whole numbers represented, students will be able to identify how the lines between the wholes numbers are divided (fourths, fifths, thirds). Students will then be able to place given fraction on the line correctly. 4.N.6 Exhibit an understanding of the base ten number system by reading, naming, and writing decimals between 0 and 1 up to the hundredths 5.N.1 Demonstrate an understanding of (positive integer) powers of ten, e.g., 10 2, N.1 Demonstrate an understanding of positive integer exponents, in particular, when used in powers of ten, e.g., 10 2, 10 5 Page 5 of 38

6 3.N.13 Use concrete objects and visual models to add and subtract (only when the answer is greater than or equal to zero) common fractions (halves, thirds, fourths, sixths, and eighths) with like denominators. To add or subtract fractions with like denominators add or subtract the numerators and use the same denominator. 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. Decimals show fractional parts of a whole. Decimal place value is an extension of a whole number place value based on groups of ten. 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 Read, write, and shade grids to show tenths and hundredths expressed as decimals. Write decimals in tenths and hundredths. 4.N.18 Use concrete objects and visual models to add and subtract common fractions. When adding or subtracting fractions with like denominators, you are adding or subtracting portions of the same size, so you can add the numerators the numbers of pieces or portions without changing the denominator. Concept of exponents How to write the exponent out in long form (105 = 10x10x10x10x10) to show understanding Compute powers of ten Given a larger number (10,000), Students will give its equivalent using exponents. Show understanding of how the number of zeros in a number will determine its exponent. Express a given number in standard and exponent form. 5.N.6Find and position whole numbers, positive fractions, positive mixed numbers, and positive decimals on a number line. Concept of positive numbers, fractions, mixed numbers and decimals Locate and place positive whole numbers, fractions, mixed numbers and Concept of exponents Compute powers of ten Compute exponents other than ten 6.N.6 Find and position integers, fractions, mixed numbers, and decimals (both positive and negative) on the number line Concept of positive and negative integers, fractions, mixed numbers and decimals Locate and place positive and negative Page 6 of 38

7 Add and subtract fractions with like denominators. To add or subtract fractions or mixed numbers with unlike denominators, change the number sentence to a simpler one with like denominators. For example, eights, fourths, and halves or thirds and sixths where at least one denominator stays the same - not thirds and fourths to equal twelfths. decimals on a number line whole integers, fractions, mixed numbers and decimals on a number line 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 placevalue 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 facts. Multiplication facts help you find the products for other facts. Add fractions with like denominators, using models and paper and pencil. Add fractions with unlike denominators, using models and paper and pencil. 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. 5.N.7 Compare and order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents. Fractions 1/2, 1/3, 1/4, 1/5, 1/10 and equivalent decimals and percents How to order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents given in different forms. How to convert a number into each of the three forms listed so that they can be compared in the same form. 6.N.7 Compare and order integers (including negative integers), and positive fractions, mixed numbers, decimals, and percents. Fractions through 1/12 and equivalent decimals and percents Compare, order and place appropriately on a number line Count on or count back easily using place values, Find products of one-digit numbers times 0 through 10. Order and compare fractions, decimals and percents correctly To convert fractions into equal decimals and percents so that they can be ordered and compared. Page 7 of 38

8 Given a set of numbers, mixed numbers and positive fractions, decimals and percents, students will be able to put them 3.N.6 Select, use, and explain 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. 4.N.8 Select, use, and explain 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. in order of value. 5.N.11 Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to simplify computation and solve problems. The inverse operations of addition and subtraction. Students will understand that subtraction means the difference between two numbers. Using this information to understand that this can be used to solve addition problems. (9 +? = 17 What is the difference between 9 and 17. Understand how to use inverse relationship between addition and subtraction to help prove or check answers. Use inverse operations to solve simple addition and subtraction equations Solve problems that involve the use of inverse operations 6.N.12 Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to simplify computation and solve problems. The inverse operations of addition, subtraction, multiplication and division Use inverse operations to solve simple equations Solve problems that involve the use of inverse operations Solve problems using the inverse relationships of addition and subtraction 3.N.8 Select and use appropriate operations (addition, subtraction, division, and multiplication) to solve problems, including those involving money. This standard is intentionally the same as standard 4.N N.10 Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those involving money. 5.N.9 Solve problems involving multiplication and division of whole numbers, and multiplication of positive fractions with whole numbers. 6.N.9 Select and use appropriate operations to solve problems involving addition, subtraction, multiplication, division, and positive integer exponents with whole numbers, and with positive fractions, mixed numbers, decimals, and percents. Page 8 of 38

9 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. 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. 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 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 Steps for multiplying and dividing whole numbers (through millions) and fractions (with denominators 2, 4, 5, 10). All whole numbers have a numerator of 1. Mathematical reasoning of why numbers decrease when multiplying by a fraction. (Multiplying by less than 1) Multiply/divide whole numbers (through millions) and fractions (with denominators 2, 4, 5, 10). When given a word problem or problem solving activity, students will be able to rationalize which operation would be appropriate. (division or multiplication) Steps for addition, subtraction, multiplication, and division of whole numbers through billions (with positive exponents), positive fractions (with denominators up to 12), mixed numbers, decimals and percents Positive integer exponents 3^2 + 3^2 = 18 Add, subtract, multiply and divide whole numbers through billions (with positive exponents), positive fractions (denominators to12), mixed numbers, decimals and percents. Add, subtract, multiply and divide whole numbers (with positive exponents), positive fractions, mixed numbers, decimals and percents to solve problems. Page 9 of 38

10 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. 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 Page 10 of 38

11 problems and use complete sentences to 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) write answers to word problems. 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 mentally. Make tables and use them to solve word problems. Patterns can help you remember multiplication facts. 5.N.10 Demonstrate an understanding of how parentheses affect expressions involving addition, subtraction, and multiplication, and use that understanding to solve problems, e.g., 3 x (4 + 2) = 3 x 6. The order of operations involving parentheses Solve problems involving order of operations Solve problems accurately involving parentheses in addition, subtraction and multiplication 6.N.11 Apply the Order of Operations for expressions involving addition, subtraction, multiplication, and division with grouping symbols (+,, x, ). The order of operations involving parentheses Solve problems that involve order of operations Solve problems that involve parentheses Apply the order of operations for expressions involving addition, subtraction, multiplication and division with grouping symbols Page 11 of 38

12 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, N.12 Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-digit divisors) whole numbers. Multiply positive decimals with whole numbers. How to add and subtract whole numbers and positive decimals How to multiply and divide whole numbers (up to two digits) How to multiply positive decimals with whole numbers Add, subtract, multiply and divide whole numbers accurately using traditional algorithms Multiply a decimal by a whole number MCAS 2006 question 11 6.N.13 Accurately and efficiently add, subtract, multiply, and divide (with double-digit divisors) whole numbers and positive decimals. How to add, subtract, multiply and divide (with double-digit divisors) whole numbers and decimals Add, subtract, multiply and divide whole numbers (with double digit divisors) and decimals accurately using traditional algorithms Find products of one-digit numbers from 0 to 10. Give all the facts in a multiplication/division fact family. Page 12 of 38

13 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. 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. 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. 4.N.12 Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently. 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 place-value 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. 6.N.10 Use the number line to model addition and subtraction of integers, with the exception of subtracting negative integers. Concept of positive/negative integers Use number line to add/subtract integers Page 13 of 38

14 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. 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. 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). 6.N.15 Add and subtract integers, with the exception of subtracting negative integers. Page 14 of 38

15 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. Steps for adding positive and negative integers up to the billions place. Steps for subtracting positive integers up to the billions place. Add and subtract positive integers through the billions place. Subtract positive integers through the billions place. 3.N.11 Round whole numbers through 1,000 to the nearest 10, 100, and 1,000. Strategies for rounding whole numbers. Round whole numbers through 1,000 to 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. 4.N.16 Round whole numbers through 100,000 to the nearest 10, 100, 1000, 10,000, and 100,000. Rounding is a process for finding the multiple of 10, 100, etc, closest to a given number. Page 15 of 38

16 the nearest 10, 100, and 1, 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 twodigit whole numbers and amounts of money to $100, and to judge the reasonableness of the answer. Round whole numbers through one hundred thousand. 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 wholenumber computations up to three-digit whole numbers and amounts of money to $1000, and to judge the reasonableness of the answer. 5.N.14 Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer. 6.N.16 Estimate results of computations with whole numbers, and with positive fractions, mixed numbers, decimals, and percents. Describe reasonableness of estimates. 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. 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 Steps for rounding numbers prior to adding, subtracting or multiplying, in order to help estimate sums, differences and products. Steps for adding, subtracting whole numbers, positive fractions and positive decimals. Steps for multiplying positive decimals and whole numbers. If an answer to an addition, subtraction or multiplication problem is reasonable or not. Estimate sums, differences and products of whole numbers positive fractions and decimals using rounding techniques prior to adding, subtracting or multiplying How to round numbers before computing How to decide if the answer is reasonable Compute whole numbers by whole numbers Convert whole numbers to fractions Compute whole numbers with fractions Compute fractions with fractions Compute mixed numbers with whole numbers and fractions Compute decimals with decimals Compute simple percents Estimate results of computations Given a multiple choice word problem, select the reasonable answer see question 33 from 2006 MCAS Page 16 of 38

17 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. Division algorithms involve breaking the calculation into smaller, simpler calculations using place value. Algorithms for dividing can be extended to greater numbers. Remainders expressed as whole numbers give a specific type of information. The steps for dividing do not change when there are zeros in the quotient. Estimation is an effective strategy for checking the reasonableness of a quotient. 5.N.13 Accurately and efficiently add and subtract positive fractions and mixed numbers with like denominators and with unlike denominators (2, 4, 5, 10 only); multiply positive fractions with whole numbers. Simplify fractions in cases when both the numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor. Add and subtract positive fractions and mixed numbers with like denominators and with unlike denominators (2, 4, 5, 10 only) Multiply positive fractions with whole numbers Simplify fractions in cases when both the numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor MCAS 2006 question 13 (open response) 6.N.14 Accurately and efficiently add, subtract, multiply, and divide positive fractions and mixed numbers. Simplify fractions. How to add, subtract, multiply and divide positive fractions and mixed numbers. How to simplify fractions Use basic operations with fractions and mixed numbers. (ex. 1/2 + 2/3; 1 2/5 3/4; 1 1/2 * 2 3/8; 3/4 2/3) Use models to find quotients and remainders. Use models and the standard algorithm to divide 2-digit and 3-digit numbers by 1- digit numbers. Divide with zeros in the quotient. 4.N.15 Demonstrate in the classroom an understanding of and the ability to use the conventional algorithm for division of up to a three-digit whole number with a single-digit divisor (with or without remainders). 5.N.8 Apply the number theory concepts of common factor, common multiple, and divisibility rules for 2, 3, 5, and 10 to the solution of problems. Demonstrate an understanding of the concepts of prime and composite numbers. 6.N.8 Apply number theory concepts including prime and composite numbers, prime factorization, greatest common factor, least common multiple, and divisibility rules for 2, 3, 4, 5, 6, 9, and 10 to the solution of problems. Page 17 of 38

18 Division algorithms involve breaking the calculation into smaller, simpler calculations using place value. Algorithms for dividing can be extended to greater numbers. Use a standard algorithm to divide a two-digit number by a one-digit number. Use a standard algorithm to divide a three-digit number by a one-digit number. Factoring Common Multiples Divisibility rules for 2, 3, 5, 6 and 10 Prime and composite numbers Connection between multiplication and factors and multiples. To tell, using only divisibility rules, if a given number (2, 3, 5, 6, and 10) will go in evenly in to a larger number. Understand the connection between factors and divisibility rules. Understand how divisibility rules connect with division and later on reducing fractions. Determine if a numbers is prime or composite based on divisibility rules. How to use the concept of prime and composite numbers to solve problems How to use the concept of prime factorization to solve problems How to use the concept of greatest common factor to help solve problems involving fractions How to use the concept of lowest common multiple to help solve problems involving fractions How to use divisibilty rules for 2, 3, 4, 5, 6, 9, and 10 to solve problems Apply number theory concepts which include prime and composite numbers, prime factorization, GCF and LCM along with divisibility rules. 3.P.1 Create, describe, extend, and explain symbolic (geometric) patterns and addition and subtraction patterns, e.g., 2, 6, 10, ; and 50, 45, 40. PATTERNS, RELATIONS, & ALGEBRA 4.P.1 Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including multiplication patterns like 3, 30, 300, 3000, 5.P.1 Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC; 1, 5, 9, 13 ; 3, 9, P.1 Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC; 1, 5, 9, 13 ; 3, 9, 27,. We can count on or count back using place values. Applying an addition or subtraction rule to quantities results in a certain type of numerical pattern. Addition doubles facts and multiplying by 2 give the same result. Patterns can help you remember multiplication facts. Patterns are recognized when numerical quantities are organized in a table. The Place value can be used to write numbers in different but equivalent forms. Patterns can help you remember multiplication facts. Patterns and breaking apart can help you find products with factors of 10, 11, and 12. The pattern or rule that relates two quantities can be represented using words, tables, and symbols. Recognizing the pattern that relates two quantities helps to Find and apply rules for extending patterns and progressions Recognize the difference between addition and multiplication sequences Find the next number when given a multiplication or addition sequence MCAS 2006 question 24 Find and apply rules for extending patterns and progressions Vocabulary: arithmetic sequence, geometric sequence, and symbolic sequence Find patterns in addition, subtraction, multiplication, and division facts. Use patterns to find solutions to problems. Page 18 of 38

19 pattern or rule that relates two quantities can be represented using words, tables, and symbols. An explanation of the solution to a problem includes information that is known and how you have used this information. Place value and multiplication facts and patterns can help you multiply by multiples of 10 and 100. Continue number patterns and use placevalue patterns to find sums and differences. Complete tables representing patterns and give the rules for the patterns. Find products of one-digit numbers from 0 to 10 Write to explain a pattern. Use mental math to multiply by multiples of 10, and P.3 Determine the value of a variable (through 10) in simple equations involving addition, subtraction, or multiplication,(e.g., 2 + = 9) Relationships between numbers, between numbers and expressions, and between expressions can be expressed using variables. Compare numbers and expressions using numbers and supply numbers that make given inequalities/equalities true. extend the table. 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. Use place value ideas to write multiples of 100, 1,000, and 10,000 in different ways. Use patterns to find products with factors of 0, 1, 2, 5, 9, 10,11, and 12. Find the rule for a pattern presented in a table and use the rule to add inputs and outputs to the table. Multiply any number by 10, 100, or 1,000. Mentally multiply any number by 10, 100, or 1, P.3 Determine values of variables in simple equations, e.g., 4106 = 37, 5 = + 3, and = 3. Equations can be solved by trying out specific values, checking the result, and trying other values until a true statement results. Find the solution to an equation by testing a set of values for the variable. Find the solution to an equation informally by substituting values for the variable. 5.P.2 Replace variables with given values and evaluate/simplify, e.g., 2( ) + 3 when = 4. How to substitute numbers for variables and perform necessary calculations Problems Like: b + 2 when b = 4 b - 2 when b = 7 2b when b = 3 b/2 when b = 6 Use a variable to describe general patterns Find the n th number when given a multiplication or addition sequence Extend symbolic arithmetic and geometric patterns and progressions 6.P.2 Replace variables with given values and evaluate/simplify, e.g., 2( ) + 3 when = 4. Constants and variables Expressions Replace variables with given values Evaluate and simplify equations Page 19 of 38

20 3.P.2 Determine which symbol (<, >, or =) is appropriate for a given number sentence, e.g., 7 x 8.? Relationships between numbers, between numbers and expressions, and between expressions can be expressed using the relational symbols = (equals), < (less than), and >(greater than). Compare numbers and expressions using relational symbols that make given inequalities/equalities true. 4.P.2 Use symbol and letter variables (e.g.,, x) to represent unknowns or quantities that vary in expressions and in equations or inequalities (mathematical sentences that use =, <, >). Real situations can be represented by writing variable expressions, and these expressions can be evaluated by substituting values for the variable Word sentences that express mathematical situations can be translated into equations and equations can help you solve problems. Write and evaluate variable expressions that involve a single operation. Write equations for word sentences. 5.P.3 Use the properties of equality to solve problems with whole numbers, e.g., if + 7 = 13, then = 13 7, therefore = 6 ; if 3 x = 15, then = 15 3, therefore = 5. The property of equality as it applies to whole numbers and the four basic operations Inverse operations with whole numbers Solve one step equations with whole numbers Solve for the unknown 6.P.3 Use the properties of equality to solve problems, e.g., if + 7 = 13, then = 13 7, therefore = 6; if 3 x = 15, then 1 / 3 x 3 x = 1 / 3 x 15, therefore = 5. The property of equality as it applies to whole numbers and fractions and the four basic operations Inverse operations with whole numbers and fractions How to calculate fractions using the four basic operations Solve one step equations with whole numbers and fractions Solve for the unknown 4.P.5 Solve problems involving proportional relationships, including unit pricing (e.g., four apples cost 80, so one apple costs 20 ) and map interpretation (e.g., one inch represents five miles, so two inches represent ten miles). 5.P.5 Solve problems involving proportional relationships using concrete models, tables, graphs, and paper-pencil methods. 6.P.5 Solve linear equations using concrete models, tables, graphs, and paper-pencil methods. Making a table can help to represent what you know in solving a problem. Make tables and use them to solve word problems. How to read and interpret graphs with two variables such as gas used and miles traveled Read and interpret graphs with two variables such as gas used and miles traveled What a linear equation is How to read and interpret tables and graphs Set up and solve linear equations Page 20 of 38

21 3.P.4 Write number sentences using +,, x,, <, =, and/or > to represent mathematical relationships in everyday situations. Apply number operations and symbols to represent mathematical relationships in everyday situations. Write number expressions for phrases. 4.P.6 Determine how change in one variable relates to a change in a second variable, e.g., input-output tables Find the rule for a pattern presented in a table and use the rule to add inputs and outputs to the table. The pattern or rule that relates two quantities can be represented using words, tables, and symbols. Recognizing the pattern that relates two quantities helps to extend the table. 6.P.7 Identify and describe relationships between two variables with a constant rate of change. Contrast these with relationships where the rate of change is not constant. Definition of constant rate of change Difference between constant rate of change and one that is not constant Identify a constant rate of change using tables Identify a rate of change that is not constant Find the rule for a pattern presented in a table and use the rule to add inputs and outputs to the table. 4.P.4 Use pictures, models, tables, charts, graphs, words, number sentences, and mathematical notations to interpret mathematical relationships. 5.P.4 Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and with symbols, e.g., input-output tables. 6.P.4 Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and with symbols, e.g., input-output tables. Find the rule for a pattern presented in a table and use the rule to add inputs and outputs to the table. Making a table can help to represent what you know in solving a problem. Word sentences that express mathematical situations can be translated into equations and equations can help you solve problems. Writing a number expression is one way of representing what we know in a word How to use models, tables, graphs and rules to represent real situations How to use words and symbols to represent real situations How to read, interpret and produce input/output tables Represent real situations using words, symbols, models, tables, and graphs How to use models, tables, graphs and rules to represent real situations How to use words and symbols, including algebraic symbols, to represent real situations How to read, interpret and produce input/output tables How to identify rules and put into standard algebraic form Page 21 of 38

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