NUMBER SENSE & OPERATIONS 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. Order and compare fractions, decimals and percents correctly To convert fractions into equal decimals and percents so that they can be ordered and compared. Given a set of numbers, mixed numbers and positive fractions, decimals and percents, students will be able to put them in order of value. 5.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 1000 + 7 x 100 + 2 x 10 + 4. 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 6.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 1000 + 7 x 100 + 2 x 10 + 4. 7.N.1 Compare, order, estimate, and translate among integers, fractions and mixed numbers i.e., rational numbers), decimals, and percents. Review content in standard 6.N.5, 6.N.7 Definition of rational number Integers, fractions, mixed numbers, percents, and some decimals are rational numbers. Compare, order, estimate, define, identify and translate (convert) ratios, fractions, mixed numbers, decimals, integers, and percents. Express a given value/data in ratio, fraction, decimal and percent form. Calculate percent scores given the number correct out of the total number 7.N.3 Represent numbers in scientific notation (positive powers of ten only) and use that notation in problem situations. 8.N.1 Compare, order, estimate, and translate among integers, fractions and mixed numbers (i.e., rational numbers), decimals, and percents. Equivalent fractions, decimals, and percents Definitions of integers and rational numbers Translate/convert equivalent fractions, decimals, and percents into common set to be able to compare, order, and estimate Write equivalent fractions with common denominators 8.N.4 Represent numbers in scientific notation, and use them in calculations and problem situations. How to represent numbers from thousandths place through millions in various forms such as standard form, standard notation and expanded form. How to represent numbers from thousandths place to billions in various forms How to represent numbers in scientific notation, using positive powers of ten How to translate into scientific notation Place value As we read to the right numbers become smaller and bigger as we read to the left. Page 1 of 31
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.1 Demonstrate an understanding of (positive integer) powers of ten, e.g., 10 2, 10 5. 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.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. Concept of fractions 1/2, 1/3, 1/4, 1/5, 1/10 as ratios Expanded notation without exponents Standard and written form 6.N.1 Demonstrate an understanding of positive integer exponents, in particular, when used in powers of ten, e.g., 10 2, 10 5 Concept of exponents Compute powers of ten Compute exponents other than ten 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. Concept of fractions through 1/12 as ratios Use scientific notation to solve problems Translate a number written in standard form to scientific notation with or without a calculator. Translate a number written in scientific notation to standard form with or without a calculator. Use scientific notification to solve problems including comparison of numbers written in scientific notation. 7.N.5 Apply the rules of positive integer exponents to the solution of problems. Extend the Order of Operations to include positive integer exponents. Order of operations including exponents How to apply positive exponents to solve problems Solve problems using order of operations including positive exponents Simplify numerical expressions using the order of operations 7.N.2 Use ratios and proportions in the solution of problems involving unit rates, scale drawings, and reading of maps. Definition of ratio, proportion, scale, Rates can be represented as ratios. Represent numbers in scientific notation using positive and negative powers of ten Translate from standard form into scientific notation 8.N.7 Apply the rules of powers and roots to the solution of problems. Extend the Order of Operations to include positive integer exponents and square roots. Order of operations including exponents How to apply positive exponents to solve problems Square roots as an inverse concept of exponents Solve problems using positive exponents and square roots Evaluate numeric expressions containing exponents and square roots 8.N.3 Use ratios and proportions in the solution of problems, in particular, problems involving unit rates, scale factors, and rate of change. Ratios can be written with colons, as fractions, or as written phrases Page 2 of 31
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. 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. Express and order above fractions as ratios Set up and solve proportions Create, interpret scale drawings Solve for missing values given appropriate rates or unit rates (ex - distance) Interpret distances on map given scale Set up proportions to convert from one unit of measurement to another Convert from one measurement to another to obtain an accurate ratio Given a rate, calculate the corresponding unit rate. Cross multiply to solve for an unknown value Set up proportions using variable representation Translate verbal ratio models to fraction or colon notation 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 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, Page 3 of 31
.60). How to reduce fractions using the concept of common factors. connection between common denominator and multiples. 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) percents 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 6.N.2 Demonstrate an understanding of place value to billions and thousandths. Place value from thousandths through billions Order numbers 7.N.4 Demonstrate an understanding of absolute value, e.g., -3 = 3 = 3. Concept of absolute value Define and determine absolute value Compare and order absolute values given a set of absolute values Evaluate expressions involving absolute values 8.N.6 Demonstrate an understanding of absolute value, e.g., -3 = 3 = 3. Concept of absolute value Application process for comparison symbols Operations for addition and subtraction of integers Define and determine absolute value Compare greater absolute value of any two given numbers using comparison symbols Page 4 of 31
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. Evaluate expressions that contain absolute value using order of operations 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 5.N.6 Find and position whole numbers, positive fractions and mixed numbers (with denominators 2, 4, 5, 10) 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. 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. Use number line to add/subtract integers 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 8.N.2 Define, compare, order, and apply frequently used irrational numbers, such as 2 and π. Concept of irrational numbers Frequently used irrational numbers What perfect squares are Square root is the inverse operation of squaring Define, compare and order frequently used irrational numbers Page 5 of 31
5.N.9 Solve problems involving multiplication and division of whole numbers, and multiplication of positive fractions with whole numbers. 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) 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. 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. 6.N.15 Add and subtract integers, with the exception of subtracting negative integers. 7.N.9 Select and use appropriate operations addition, subtraction, multiplication, division, and positive integer exponents to solve problems with rational numbers (including negatives). Steps for addition, subtraction, multiplication, and division for rational numbers (including negative numbers and exponents) Recognize when to add, subtract, multiply and divide rational numbers, including negatives and exponents to solve problems. Given an original value, calculate a new value as a fraction of the original Select and apply a problem solving strategy to accurately solve a multi-step problem Explain, verbally and in writing, the strategy used to solve a problem 8.N.12 Select and use appropriate operations addition, subtraction, multiplication, division, and positive integer exponents to solve problems with rational numbers (including negatives). Steps for addition, subtraction, multiplication, and division for rational numbers (including negative numbers and exponents) Recognize when to add, subtract, multiply and divide rational numbers, including negatives and exponents to solve problems. Calculate problems of increasing complexity using exponents. Analyze questions then choose the appropriate operation to solve a problem Page 6 of 31
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. 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. 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 6.N.16 Estimate results of computations with whole numbers, and with positive fractions, mixed numbers, decimals, and percents. Describe reasonableness of estimates. 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 7.N.7 Estimate and compute with fractions (including simplification of fractions), integers, decimals, and percents (including those greater than 100 and less than 1). How to round numbers and simplify fractions before computing Concept of percents less than 1 and greater than 100 How to decide if the answer is reasonable A proportion that can be used to solve percent problems is is/of = %/100 Translate percent problems into the proportion is/of = %/100 then crossmultiply and divide to solve for the missing value. Perform addition, subtraction, multiplication, and division with rational numbers Estimate fractions, integers and decimals, and percents greater than 100 8.N.10 Estimate and compute with fractions (including simplification of fractions), integers, decimals, and percents (including those greater than 100 and less than 1). How to estimate and compute with fractions, integers, decimals and percents (greater than 100 and less than 1) Determine when an estimate rather than an exact answer is appropriate and apply in problem situations Compute with fractions and percents Page 7 of 31
and less than 1. Compute with fractions, integers and decimals, and percents greater than 100 and less than accurately and proficiently using standard algorithms and other methods. 7.N.8 Determine when an estimate rather than an exact answer is appropriate and apply in problem situations. How to round prior to calculating How to estimate calculations There are many times when an estimate is appropriate rather than an exact answer. 8.N.11 Determine when an estimate rather than an exact answer is appropriate and apply in problem situations. Review rounding from grade 7 Sometimes estimation is appropriate and how sometimes exact calculation is more appropriate 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 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 Determine whether an estimate rather than an exact answer is appropriate Recognize convenient numbers Given a formula and values, substitute rounded values to determine an appropriate estimated answer 7.N.6 Use the inverse relationships of addition and subtraction, and of multiplication and division, to simplify computations and solve problems, e.g., multiplying by 1/2 or 0.5 is the same as dividing by 2. The inverse operations of addition, subtraction, multiplication and division Multiply by 1/2 or.5 and/or divide by 2 to solve the same problem Use inverse operations to write Determine whether an estimate rather than an exact answer is appropriate and apply to the solution of the problem 8.N.9 Use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems, e.g. multiplying by 1/2 or 0.5 is the same as dividing by 2. The inverse operations of addition, subtraction, multiplication, division, squares and square roots Strong grasp of standard 8.N.1 Apply inverse relationships to simplify Page 8 of 31
difference between 9 and 17. how to use inverse relationship between addition and subtraction to help prove or check answers. inverse operations Solve problems using the inverse relationships of addition and subtraction equivalent expressions Use opposite operations to check answers for accuracy computations and solve problems (including squaring and square roots) Use inverse operations to solve simple addition and subtraction equations Solve problems that involve the use of inverse operations 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 5.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 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 6.N.13 Accurately and efficiently add, subtract, multiply, and divide (with double-digit divisors) whole numbers and positive decimals. 8.N.8 Demonstrate an understanding of the properties of arithmetic operations on rational numbers. Use the associative, commutative, and distributive properties; properties of the identity and inverse elements (e.g., -7 + 7 = 0; 3/4 x 4/3 = 1); and the notion of closure of a subset of the rational numbers under an operation (e.g., the set of odd integers is closed under multiplication but not under addition). The associative, commutative and distributive properties in numeric and algebraic notation The identity and inverse properties affect on evaluation of expressions and equation solutions Multiplication of odd numbers yields an odd number Apply arithmetic properties to the solution of problems. Recognize applications of properties Page 9 of 31
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 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. 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 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 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. 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 8.N.5 Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems. Prime factorization Relatively prime numbers Use factor tree to state the prime factorization of a given number Break down numbers into their component factors Use factor tree or lists to find the greatest common factor Represent prime factorization using exponents Page 10 of 31
with divisibility rules. 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. 6.N.14 Accurately and efficiently add, subtract, multiply, and divide positive fractions and mixed numbers. Simplify fractions. 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 How to simplify fractions in cases when both the numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor What simplifying fractions means and how the two fractions are still of equal value. connection between common denominators and common multiples. 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) Simplify fractions with denominator having 2, 3, 4, 5, or 10 as a common factor. Accurately and efficiently add and subtract positive fractions and mixed numbers with like denominators and with unlike denominators (2, 4, 5, 10 only) Page 11 of 31
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, 27... 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 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 PATTERNS, RELATIONS, & ALGEBRA 6.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,. 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. 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 7.P.1 Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic expressions. Include arithmetic and geometric progressions, e.g., compounding. Geometric and numerical patterns and compounding. How to use a table of values (e.g., to plot coordinates, find patterns, write expressions, etc.) Write the rule for a progression as a verbal, numeric, or algebraic expression when provided with a table, graph, pictoral model. Given a model, table, or graph, will be able to continue geometric pattern to a specified step. Given a rule represented as verbal, numeric, or algebraic expression, select the corresponding pattern. Given a pattern select the verbal, numeric, or algebraic rule. 7.P.2 Evaluate simple algebraic expressions for given variable values, e.g., 3a 2 b for a = 3 and b = 7. Substitute values for multiple variables and perform calculations according to the order of operations 8.P.1 Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic expressions. Include arithmetic and geometric progressions, e.g., compounding. How to find the missing numbers in a progression by using a variety of methods. Geometric and numerical patterns and compounding. How to use a table of values (e.g., to plot coordinates, find patterns, write equations, etc.) Write an equation from a table of values Calculate the missing numbers in a progression by using a variety of methods. 8.P.2 Evaluate simple algebraic expressions for given variable values, e.g., 3a 2 b for a = 3 and b = 7. Concept of what satisfies an expression Decimal, fraction, and integer operations Page 12 of 31
Problems Like: b + 2 when b = 4 b - 2 when b = 7 2b when b = 3 b/2 when b = 6 Replace variables with given values Evaluate and simplify equations Substitute given values in multi-step algebraic expressions and apply the order of operations to determine correct value Select the appropriate answer to an algebraic equation with a substituted value Substitute given numerical values for variables Perform the order of operations correctly to simplify the expression. 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 5.P.5 Solve problems involving proportional relationships using concrete models, tables, graphs, and paper-pencil methods. How to solve problems using proportional relationships using a variety of media What a proportional relationship is with relation to fractions How to set up proportions Solve problems using proportional 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 7.P.6 Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate. How to substitute values in given formulas How to apply formulas in problem solving How to set up and solve proportions using cross products How to calculate unit rates 8.P.3 Demonstrate an understanding of the identity (-x)(-y) = xy. Use this identity to simplify algebraic expressions, e.g., (-2)(-x+2) = 2x - 4. How to apply rules for integer operations How to recognize and apply variable notation Apply rules of identity to simplify algebraic expressions Apply the distributive property to algebraic expressions 8.P.9 Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate. How to substitute values in given formulas How to apply formulas in problem solving How to set up and solve proportions Write linear equations to model a set of Page 13 of 31
relationships using a variety of media 6.P.5 Solve linear equations using concrete models, tables, graphs, and paper-pencil methods. What a linear equation is How to read and interpret tables and graphs Set up and solve linear equations Given proportional information in a table format, match corresponding equation with charted information Given written or tabular information, write and solve proportions Select the appropriate proportion to describe a mathematic situation Given a table depicting a relationship between two variables, select the appropriate algebraic equation 7.P.4 Solve linear equations using tables, graphs, models, and algebraic methods. How to read and interpret tables and graphs Use opposite operations to solve oneand two-step equations for an unknown variable Given a model, will be able to select the appropriate algebraic equation data Determine if the relationship between data points is linear Apply linear equations and proportionality to solve indirect measurement problems 8.P.7 Set up and solve linear equations and inequalities with one or two variables, using algebraic methods, models, and/or graphs. A linear equation a line on a Cartesian plane How to set up as well as solve multistep equations for unknown variables Recognize a given linear equation as a solution to a problem How to create tables and graphs using a given set of data to solve linear equations Translate word problems to linear equations or inequalities Solve linear equations or inequalities for an unknown value Graph linear equations 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 7.P.5 Identify, describe, and analyze linear relationships between two variables. Compare positive rate of change, e.g., y = 3x + 1, to negative rate of change, e.g., y = 3x + 1. How to recognize positive, negative and 8.P.5 Identify the slope of a line as a measure of its steepness and as a constant rate of change from its table of values, equation, or graph. Apply the concept of slope to the solution of problems. Slope shows the rate of change Page 14 of 31
5.P.6 Interpret graphs that represent the relationship between two variables in everyday situations. 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 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. 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 6.P.6 Produce and interpret graphs that represent the relationship between two variables in everyday situations. How to read, interpret and produce graphs with two variables such as temperatures in a given time period Read, interpret and produce graphs with two variables such as temperatures in a given time period 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. undefined slopes How to use ratio to describe rate of change How to analyze a positive and negative rate of change Identify a positive, negative or undefined rate of change (slope) for y on a graph Given a linear equation, identify the positive or negative point of change. Given graph showing relationship between x and y, will be able to describe the relationship Given a table showing the relationship between two values, write an algebraic equation describing the relationship between the two values Formula for rate of change How to relate slopes to grades and other real-life concepts Slope intercept form of linear equations Visually & mathematically determine rates of change 8.P.4 Create and use symbolic expressions and relate them to verbal, tabular, and graphical representations. Content from 8.D.2 Written math problems have key words that are clues to how to solve the problem Write an algebraic expression from a word problem Translate data from tables and graphs into algebraic expressions Page 15 of 31
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 Represent real situations using words, symbols, models, tables, and graphs Create basic algebraic expressions to represent a rule 7.P.3 Create and use symbolic expressions for linear relationships and relate them to verbal, tabular, and graphical representations. How to identify or interpret word problems mathematically How to translate word problems to algebraic expressions How to translate data from tables to algebraic equations How to translate data from graphs to algebraic equations Connect algebraic expression or equation to a given pattern or situation Write an algebraic expression or equation from a word problem Given a blank table, students will fill in values according to a given rate or rule Given a table following a mathematical pattern, write an equation showing the 8.P.10 Use tables and graphs to represent and compare linear growth patterns. In particular, compare rates of change and x- and y-intercepts of different linear patterns. How to compare slopes to define relationships between linear equations, such as parallel and perpendicular How to complete a table and recognize patterns of relationships in the columns and write an equation Draw lines on a Cartesian plane given specific slope, y-intercept or point Page 16 of 31
correspondence between two variables 8.P.6 Identify the roles of variables within an equation, e.g., y = mx + b, expressing y as a function of x with parameters m and b. How to verbally define each variable within an expression How to identify slope and y-intercept as variables Describe the relationship between variables within an equation 8.P.8 Explain and analyze both quantitatively and qualitatively, using pictures, graphs, charts, or equations how a change in one variable results in a change in another variable in functional relationships, e.g., C = πd, A = πr 2 (A as a function of r), A rectangle = lw (A rectangle as a function of l and w). How to recognize independent and dependent variables in a function in a picture, graph, chart or equation How to explain the relationship between parts of an equation and the ending result MCAS 2004 question 30 5.G.1 Identify, describe, and compare special types of triangles (isosceles, equilateral, right) and quadrilaterals (square, rectangle, parallelogram, 6.G.1 Identify polygons based on their properties, including types of interior angles, perpendicular or parallel sides, and congruence of sides, e.g., squares, GEOMETRY 7.G.1 Analyze, apply, and explain the relationship between the number of sides and the sums of the interior angle measures of polygons. 8.G.1 Analyze, apply, and explain the relationship between the number of sides and the sums of the interior and exterior angle measures of polygons. Page 17 of 31
rhombus, trapezoid), e.g., recognize that all equilateral triangles are isosceles, but not all isosceles triangles are equilateral. How to identify, describe, and compare special types of triangles (isosceles, equilateral, right) and quadrilaterals (square, rectangle, parallelogram, rhombus, trapezoid) Recognize and identify differences and similarities among special types of triangles Recognize and identify differences and similarities among various types of quadrilaterals 5.G.2 Identify, describe, and compare special types of three-dimensional shapes (cubes, prisms, spheres, pyramids) based on their properties, such as edges and faces. Definitions of faces and edges How to identify cubes, prisms, spheres and pyramids rectangles, rhombuses, parallelograms, trapezoids, and isosceles, equilateral, and right triangles How to identify polygons based on their properties, Types of interior angles Definitions of perpendicular, parallel, congruency, acute, obtuse, right angles Identify polygons based upon their properties, e.g., types of angles, congruency, and sides, including perpendicular and parallel 6.G.2 Identify three-dimensional shapes (e.g., cubes, prisms, spheres, cones, and pyramids) based on their properties, such as edges and faces. Definitions of faces, edges and vertices How to identify cubes, prisms, spheres, cones and pyramids The sum of interior angles in any triangle is equal to 180 How to subdivide a polygon using one vertex into a bunch of triangles Discover a pattern between the number of sides and the sum of interior angles Calculate the sum of interior angles of a polygon given the number of sides Given measures of some interior angles in a polygon, will be able to calculate missing angle Given two angles in a quadrilateral, with the unknowns being congruent, calculate the value of the unknown angles Prequisite knowledge is how to classify quadrilaterals and triangles Identify congruent figures Vocabulary congruent Calculate the unknown angle of a polygon given other angles Calculate sum of interior angles Recognize, compare and contrast cubes, prisms, sphere and pyramids Identify the number of faces and edges on a 3D figure Recognize, compare and contrast cubes, prisms, spheres, cones and pyramids Identify the number of faces, edges and vertices on a 3D figure 5.G.3 Identify relationships among points and lines, e.g., intersecting, parallel, perpendicular. 6.G.3 Identify relationships among points, lines, and planes, e.g., intersecting, parallel, perpendicular. 7.G.3 Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines 8.G.3 Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines Page 18 of 31
cut by a transversal. cut by a transversal. Definitions of points and lines, including intersecting, parallel, and perpendicular Identify how points came to make perpendicular, parallel, or intersecting lines. Identify specific lines based on a definition 5.G.4 Using ordered pairs of whole numbers (including zero), graph, locate, and identify points, and describe paths on the Cartesian coordinate plane. The x-axis is horizontal and the y-axis is vertical First number in ordered pair goes with x- axis To graph coordinate pairs Identify coordinates of given points within a Cartesian plane Definitions of points, planes, lines, including the terms intersecting, parallel, and perpendicular Identify relationships among points, lines, and planes Use terms such as intersecting, parallel and perpendicular to describe points, lines and planes 6.G.4 Graph points and identify coordinates of points on the Cartesian coordinate plane (all four quadrants). The location of and the labels (I, II, III, IV) for the four quadrants of the Cartesian plane The x-axis is horizontal and the y-axis is vertical To graph coordinate pairs Draw a Cartesian plane Identify coordinates of given points within a Cartesian plane How to identify adjacent, complimentary, supplementary, corresponding, vertical, alternate interior and exterior angles Which angle relationships result in congruency Calculate the complementary and supplementary angles Given intersecting lines and the measure of one angle, calculate the missing angle 7.G.4 Graph points and identify coordinates of points on the Cartesian coordinate plane (all four quadrants). Review the location of and the labels (I, II, III, IV) for the four quadrants of the Cartesian plane The x-axis is horizontal and the y-axis is vertical To graph coordinate pairs Draw a Cartesian plane Identify coordinates of given points within a Cartesian plane Copy an x and y axis onto a grid; plot and label specific points, determine missing points given length and width of polygon; write coordinates of missing points; connect points to form specified polygon; draw diagonals, write coordinates of intersecting diagonals Given lines of symmetry identify How to identify adjacent, complimentary, supplementary, corresponding, vertical, alternate interior and exterior angles Find angle measures using the relationship between angles Identify parallel lines and angles formed by parallel lines Page 19 of 31
6.G.5 Find the distance between two points on horizontal or vertical number lines. Identify value of points on a number line How to find the difference between two points Calculate the difference between two points on a number line corresponding points 7.G.5 Use a ruler, protractor, and compass to draw polygons and circles. The differences among and uses of a ruler, protractor and compass Make a one-dimensional measurement using a ruler Measure angles using a protractor Draw circles using a compass Create polygons and circles utilizing rulers, protractors and compasses 8.G.5 Use a straightedge, compass, or other tools to formulate and test conjectures, and to draw geometric figures. Properties of geometric figures, including similar triangles and congruent figures Construct two-dimensional enclosed figures using a straightedge, compass, etc. 8.G.4 Demonstrate an understanding of the Pythagorean theorem. Apply the theorem to the solution of problems. Properties of a right triangle The definition of hypotenuse and legs Concept of squares and square roots Supplementary angles sum to 180 is background knowledge Pythagorean theorem Apply the Pythagorean theorem to calculate missing length on a right triangle 5.G.6 Identify and describe line symmetry in two-dimensional shapes, including shapes that have multiple lines of symmetry. The definition of symmetry How to identify lines in two-dimensional 6.G.7 Identify types of symmetry, including line and rotational. Definition of congruency How to measure sides and angles 7.G.7 Identify three-dimensional figures (e.g., prisms, pyramids) by their physical appearance, distinguishing attributes, and spatial relationships such as parallel faces. A prism has two parallel polygonal bases and rectangular lateral faces 8.G.7 Identify three-dimensional figures (e.g., prisms, pyramids) by their physical appearance, distinguishing attributes, and spatial relationships such as parallel faces. A prism has two parallel polygonal bases and rectangular lateral faces Page 20 of 31
shapes Identify mirror images Draw lines of symmetry through squares, rectangles, hexagons, triangles and other shapes with multiple lines of symmetry Describe lines of symmetry in squares, rectangles, hexagons, triangles and other shapes Series of motions including translations, rotations and reflections Determine if two polygons are congruent Prove that two polygons are congruent by using motion concepts such as translations, rotations and reflections A pyramid has a polygonal base and triangular lateral faces A prism or pyramid is named by the shape of it s base The definition of a polyhedron Construct pyramids and prisms from nets or three-dimensional software Identify given three-dimensional figures from given attributes (prisms and pyramids) A pyramid has a polygonal base and triangular lateral faces A prism or pyramid is named by the shape of it s base The definition of a polyhedron, cone and cylinder Count faces and edges accurately Identify given three-dimensional figures by comparing various attributes 5.G.5 Describe and perform transformations on two-dimensional shapes, e.g., translations, rotations, and reflections. How to slide, flip and rotate images Definitions of translation, rotation, and reflection Manipulate two-dimensional shapes in a translation, rotation and/or reflection Describe resulting translations, rotations, and reflections 6.G.6 Predict, describe, and perform transformations on two-dimensional shapes, e.g., translations, rotations, and reflections. How to slide, flip and rotate images Definitions of translation, rotation, and reflection Manipulate two-dimensional shapes in a translation, rotation and/or reflection Describe resulting translations, rotations, and reflections Predict resulting coordinates when a two-dimensional figure has been translated, rotated and/or reflected 7.G.6 Predict the results of translations and reflections of figures on unmarked or coordinate planes and draw the transformed figure. Definition of translation and reflection Definition of line of symmetry Definition of tessellation Draw translated or reflected figures across the line of symmetry Create tessellations using reflections or translations Choose the appropriate reflection of a figure across a specified axis 8.G.6 Predict the results of transformations on unmarked or coordinate planes and draw the transformed figure, e.g., predict how tessellations transform under translations, reflections, and rotations. Definition of translation, reflection, rotations, and tessellation. Draw transformations including translations, reflections and rotations 5.G.7 Determine if two triangles or two quadrilaterals are congruent by measuring sides or a combination of sides and angles, as necessary; or by motions or series of motions, e.g., translations, 6.G.8 Determine if two shapes are congruent by measuring sides or a combination of sides and angles, as necessary; or by motions or series of motions, e.g., translations, rotations, and 7.G.2 Classify figures in terms of congruence and similarity, and apply these relationships to the solution of problems. 8.G.2 Classify figures in terms of congruence and similarity, and apply these relationships to the solution of problems. Page 21 of 31
rotations, and reflections. Definition of congruency How to measure sides and angles Series of motions including translations, rotations and reflections Determine if two triangles or quadrilaterals are congruent Prove that two figures are congruent by using motion concepts such as translations, rotations and reflections reflections. Definition of congruency How to measure sides and angles Series of motions including translations, rotations and reflections Determine if two polygons are congruent Prove that two polygons are congruent by using motion concepts such as translations, rotations and reflections Definitions of congruence and similarity Concepts of ratios and proportions Recognize the difference between congruence and similarity Apply knowledge of similarity and congruence in solving real-life problems Use ratios to solve unknown dimension given similar polygons Difference between congruence and similarity Similar triangles have similar angles Classify triangles and quadrilaterals in terms of congruence and similarity Given a figure, determine congruence and similarity to calculate missing angles or measurements in a triangle or quadrilateral 6.G.9 Match three-dimensional objects and their two-dimensional representations, e.g., nets, projections, and perspective drawings. The three dimensional equivalent of twodimensional layout What nets, projections, and perspective drawings are 8.G.8 Recognize and draw twodimensional representations of threedimensional objects, e.g., nets, projections, and perspective drawings. The two dimensional equivalent of threedimensional figure What nets, projections, and perspective drawings are Match 3-D objects to their 2-D representations Recognize, match and draw 2-D layouts of 3-D figures i.e perspective drawings 5.M.3 Solve problems involving simple unit conversions within a system of measurement. Equivalent units within customary units of measurement, e.g., inches, feet, yards, 6.M.3 Solve problems involving proportional relationships and units of measurement, e.g., same system unit conversions, scale models, maps, and speed. How to set up proportions How to solve a problem using proportions MEASUREMENT 7.M.1 Select, convert (within the same system of measurement), and use appropriate units of measurement or scale. How to choose the appropriate unit of measurement to answer a question 8.M.1 Select, convert (within the same system of measurement), and use appropriate units of measurement or scale. How to choose the appropriate unit of measurement to answer a question Page 22 of 31