Courses of Study : Mathematics (Grade 6)

Students: Given contextual or mathematical situations involving multiplicative comparisons.

• Communicate the relationship of two or more quantities using ratio language.

Students know:

• Characteristics of additive situations.
• Characteristics of multiplicative situations

Students are able to:

• Compare and contrast additive vs. multiplicative contextual situations.
• Identify all ratios and describe them using "For every…, there are…"
• Identify a ratio as a part-to-part or a part-to whole comparison.
• Represent multiplicative comparisons in ratio notation and language (e.g., using words such as "out of" or "to" before using the symbolic notation of the colon and then the fraction bar. for example, 3 out of 7, 3 to 5, 6:7 and then 4/5).

Students understand that:

• In a multiplicative comparison situation one quantity changes at a constant rate with respect to a second related quantity. -Each ratio when expressed in forms: ie 10/5, 10:5 and/or 10 to 5 can be simplified to equivalent ratios, -Explain the relationships and differences between fractions and ratios.

Learning Objectives:
M.6.1.1: Define quantity, fraction, and ratio.
M.6.1.2: Identify the units or quantities being compared.
Example: Read 2/3 as 2 out of 3.
M.6.1.3: Write a ratio in appropriate notation;[a/b, a to b, a:b].
M.6.1.4: Draw a model of a given ratio or fraction.
M.6.1.5: Identify the numerator and denominator of a fraction.

Students:
Given contextual or mathematical situations involving multiplicative comparisons,

• Use unit rate to solve missing value problems (e.g., cost per item or distance per time unit).
• Use rate language to explain the relationships between ratio of two quantities as non-complex fractions and the associated unit rate of one of the quantities in terms of the other.

Students know:

• Characteristics of multiplicative comparison situations.
• Rate and ratio language.
• Techniques for determining unit rates.
• To use reasoning to find unit rates instead of a rule or using algorithms such as cross-products.

Students are able to:

• Explain relationships between ratios and the related unit rates.
• Use unit rates to name the amount of either quantity in terms of the other quantity flexibly.
• Represent contextual relationships as ratios.

Students understand that:

• A unit rate is a ratio (a:b) of two measurements in which b is one.
• A unit rate expresses a ratio as part-to-one or one unit of another quantity.

Learning Objectives:
M.6.2.1: Define unit rate, proportion, and rate.
M.6.2.2: Create a ratio or proportion from a given word problem.
M.6.2.3: Calculate unit rate by using ratios or proportions.
M.6.2.4: Write a ratio as a fraction.

Students:
Given contextual or mathematical situations involving ratio and rate (including those involving unit pricing, constant speed, and measurement conversions),

• Represent the situations using a variety of strategies (tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes) in order to solve problems, find missing values on tables and interpret relationships and results.
• Change given rates to unit rates in order to find and justify solutions to problems.

Given contextual or mathematical situations involving percents,

• Understand the relationship between ratios, fractions, decimals and percents.
• Interpret the percent as rate per 100.
• Solve problems and justify solutions when finding the whole, given a part and the percent.
• Solve problems and justify solutions when finding the part, given the whole and the percent.
• Solve problems and justify solutions when finding percent, given the whole and the part.

Students know:

• Strategies for representing contexts involving rates and ratios including. tables of equivalent ratios, changing to unit rate, tape diagrams, double number lines, equations, and plots on coordinate planes.
• Strategies for finding equivalent ratios,
• Strategies for using ratio reasoning to convert measurement units.
• Strategies to recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and denominator is the same.
• Strategies for converting between fractions, decimals and percents.
• Strategies for finding the whole when given the part and percent in a mathematical and contextual situation.
• Strategies for finding the part, given the whole and the percent in mathematical and contextual situation.
• Strategies for finding the percent, given the whole and the part in mathematical and contextual situation.

Students are able to:

• Represent ratio and rate situations using a variety of strategies (e.g., tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes).
• Use ratio, rates, and multiplicative reasoning to explain connections among representations and justify solutions in various contexts, including measurement, prices and geometry.
• Understand the multiplicative relationship between ratio comparisons in a table by writing an equation.
• Plot ratios as ordered pairs.
• Solve and justify solutions for rate problems including unit pricing, constant speed, measurement conversions, and situations involving percents.
• Solve problems and justify solutions when finding the whole given a part and the percent.
• Model using an equivalent fraction and decimal to percents.
• Use ratio reasoning, multiplication, and division to transform and interpret measurements.

Students understand that:

• A unit rate is a ratio (a:b) of two measurements in which b is one.
• A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.
• When computing with quantities the transformation and interpretation of the resulting unit is dependent on the particular operation performed.

Learning Objectives:
M.6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
M.6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation.
M.6.3.3: Calculate unit rate or rate by using ratios or proportions with or without a calculator.
M.6.3.4: Restate real-world problems or mathematical problems.
M.6.3.5: Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
M.6.3.6: Calculate missing input and/or output values in a table with or without a calculator.
M.6.3.7: Draw and label a table of equivalent ratios from given information.
M.6.3.8: Identify the parts of a table of equivalent ratios (input, output, etc.).
M.6.3.9: Compute the unit rate, unit price, and constant speed with or without a calculator.
M.6.3.10: Create a proportion or ratio from a given word problem.
M.6.3.11: Identify the two units being compared.
M.6.3.12: Define percent.
M.6.3.13: Calculate a proportion for missing information with or without a calculator.
M.6.3.14: Identify a proportion from given information.
M.6.3.15: Solve a proportion using part over whole equals percent over 100 with or without a calculator.
M.6.3.16: Form a ratio.
M.6.3.17: Convert like measurement units within a given system with or without a calculator. (Example: 120 min = 2 hrs).
M.6.3.18: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec.

Students:
Given a division problem involving a fraction divided by a fraction,

• Create an appropriate story context.
• Solve the problem using visual fraction models and an equation.
• Explain the relationship between the model and the problem.
• Interpret the solution.
• Use the inverse relationship between multiplication and division, or concept of division as repeated subtraction, to explain and justify the solution.

Students know:

• Strategies for representing fractions and operations on fractions using visual models,
• The inverse relationship between multiplication and division (a ÷ b = c implies that a = b x c).
• Strategies to solve mathematical and conceptual problems involving quotients of fractions.

Students are able to:

• Represent fractions and operations on fractions using visual models.
• Interpret quotients resulting from the division of a fraction by a fraction.
• Accurately determine quotients of fractions by fractions using visual models/equations.
• Justify solutions to division problems involving fractions using the inverse relationship between multiplication and division.

Students understand that:

• The operation of division is interpreted the same with fractions as with whole numbers.
• The inverse relationship between the operations of multiplication and division that was true for whole numbers continues to be true for fractions.
• The relationships between operations can be used to solve problems and justify solutions and solution paths.

Learning Objectives:
M.6.4.1: Define fraction (including numerator and denominator), reciprocal, equation, and quotient.
M.6.4.2: Construct an equation from a given word problem.
M.6.4.3: Discuss the process of multiplying by the reciprocal.
M.6.4.4: Interpret division of fractions by multiplying by the reciprocal.
M.6.4.5: Demonstrate division of fractions using a visual fraction model.
M.6.4.6: Demonstrate multiplication of fractions using a visual fraction model.

Students:
Given a context which calls for the division of two whole numbers,

• Choose the most appropriate strategy for computing the answer.
• Produce accurate results using a standard algorithm when appropriate.

Students know:

• strategies for computing answers to division mathematical and real-world problems using the standard division algorithm.

Students are able to:

• Strategically choose and apply appropriate strategies for dividing.
• Accurately find quotients using the standard division algorithm.

Students understand that:

• Mathematical problems can be solved using a variety of strategies, models, and representations.
• Efficient application of computation strategies is based on the numbers and operations in the problems,
• The steps used in the standard algorithms for division can be justified by using properties of operations and understanding of place value.
• Among all techniques and algorithms that may be chosen for accurately performing multi-digit computations, some procedures have been chosen with which all should be fluent for efficiency, communication, and use in other mathematics situations.

Learning Objectives:
M. 6.5.1: Define factors and multiples.
M. 6.5.2: Discuss the steps for solving a division problem.
M. 6.5.3: Recognize division and multiplication as inverse operations.
M. 6.5.4: Recall basic division and multiplication facts.
M. 6.5.5: Solve real-world division problems with and without models or a calculator.

Students:
Given a context which calls for complex computation involving multi-digit decimals,

• Choose the most appropriate strategy for computing the answer.
• Produce accurate results efficiently using a standard algorithm for each operation when appropriate.

Students know:

• Place value conventions (i.e., a digit in one place represents 10 times as much as it would represent in the place to its right and 1/10 of what it represents in the place to its left).
• Strategies for computing answers to complex addition, subtraction, multiplication, and division problems involving multi-digit decimals, including a standard algorithm for each operation.

Students are able to:

• Strategically choose and apply appropriate computation strategies.
• Accurately find sums, differences, products, and quotients using the standard algorithms for each operation.

Students understand that:

• Place value patterns and values continue to the right of the decimal point and allow the standard algorithm for addition and subtraction to be applied in the same manner as with whole numbers.
• Mathematical problems can be solved using a variety of strategies, models, and representations.
• Efficient application of computation strategies is based on the numbers and operations in the problem.
• The steps used in the standard algorithms for the four operations can be justified by using properties of operations and understanding of place value.
• Among all techniques and algorithms that may be chosen for accurately performing multi-digit computations, some procedures have been chosen with which all should be fluent for efficiency, communication, and use in other mathematics situations.

Learning Objectives:
M.6.6.1: Solve division problems involving multi-digit whole numbers and decimal numbers with or without a calculator.
M.6.6.2: Solve multiplication problems involving multi-digit whole numbers and decimal numbers with or without a calculator.
M.6.6.3: Recall basic multiplication and division facts.
M.6.6.4: Solve addition and subtraction of multi-digit decimal numbers (emphasis on alignment).
M.6.6.5: Solve addition and subtraction of multi-digit whole numbers.
M.6.6.6: Recognize place value of whole numbers and decimals.
M.6.6.7: Demonstrate addition, subtraction, multiplication, and division of whole numbers and decimals using manipulatives.

Students:

• Use the distributive property to write an equivalent expression for the sum of the two numbers as the product of the greatest common factor of the two numbers, and the sum of two whole numbers with no common factor. [if the two whole numbers are 36 and 8, 36+8 = 4(9+2)].

Students know:

• Distributive property of multiplication over addition.
• Strategies to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor by decomposing the numbers.

Students are able to:

• Use and model the distributive property to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor by decomposing the numbers.

Students understand that:

• Multiplication is distributive over addition.
• Composing and decomposing numbers provides insights into relationships among numbers.
• Quantities can be represented using a variety of equivalent expressions.

Learning Objectives:
M.6.7.1: Define greatest common factor, least common multiple, and the distributive property.
M.6.7.2: Design problems using greatest common factor and the distributive property.
M.6.7.3: Show an understanding of how to solve a problem using the distributive property, with or without the use of a calculator.

Students:
Given any two or more whole numbers,

• Strategically select and apply strategies for finding the greatest common factor of the two numbers and justify that the strategy used does produce the correct value for the greatest common factor.
• Strategically select and apply strategies for finding the least common multiple of the two numbers and justify that the strategy used does produce the correct value for the least common multiple.
• Use the relationship between factors and multiples to determine prime factorization.

Students know:

• Strategies for determining the greatest common factor of two or more numbers,
• Strategies for determining the least common multiple of two or more numbers,
• Strategies for determining the prime factorization of a number.

Students are able to:

• Apply strategies for determining greatest common factors and least common multiples.
• Apply strategies for determining the product of a number's prime factors in multiple forms which include exponential form and standard form.

Students understand that:

• Determining when two numbers have no common factors other than one means they are considered relatively prime.
• Composing and decomposing numbers provides insights into relationships among numbers.

Learning Objectives:
M.6.8.1: Identify the least common multiple of a given set of numbers, with or without the use of a calculator.
M.6.8.2: List multiples of any given whole number, with or without the use of a calculator.
M.6.8.3: Identify the greatest common factors of a given set of numbers, with or without the use of a calculator.
M.6.8.4: Define prime factorization.
M.6.8.5: List common factors of given whole numbers, with or without the use of a calculator.
M.6.8.6: Identify the prime factorization of a single digit number, with or without the use of a calculator.
M.6.8.7: Identify the prime factorization of any two digit whole number, with or without the use of a calculator.

Students:

• Given contextual or mathematical situations containing quantities that have opposite directions or values use positive, negative numbers, and their opposites to represent quantities in the contexts and explain the meaning of 0 in each situation.

Students know:

• notation for and meaning of positive and negative numbers, and their opposites in mathematical and real-world situations.

Students are able to:

• Use positive, negative numbers, and their opposites to represent quantities in real-world contexts.

Students understand that:

• Positive and negative numbers are used together to describe quantities having opposite directions or values (temperature above/below zero, elevation above/below sea level, credits/debits, or positive/negative electrical charges).

Learning Objectives:
M.6.9.1: Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
M.6.9.2: Discover that the opposite of the opposite of a number is the number itself.
M.6.9.3: Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.

Students:

• Create and interpret number line diagram.
• Given any rational number (positive or negative).
• Locate the number on a number line.
• Identify opposite signs of numbers as indicating the same distance from zero on the opposite side of zero, the opposite of the opposite, or a representation of its opposite as the point itself [-(-3) = 3], and zero as its own opposite.

Students know:

• Strategies for creating number line models of rational numbers (marking off equal lengths by estimation or recursive halving).
• Strategies for locating numbers on a number line.
• Notation for positive and negative numbers and zero.

Students are able to:

• Represent rational numbers and their opposites on a number line including both positive and negative quantities.
• Explain and justify the creation of number lines and placement of rational numbers on a number line.
• Explain the meaning of 0 in a variety of real-world contexts.

Students understand that:

• Representing rational numbers on number lines requires using both a distance and a direction,
• Locating numbers on a number line provides a representation of a mathematical context which aids in visualizing ideas and solving problems.

Learning Objectives:
M.6.10.1: Define integers, positive and negative numbers.
M.6.10.2: Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
M.6.10.3: Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
M.6.10.4: Discuss the measure of centering of 0 in relationship to positive and negative numbers.
M.6.10.5: Discover that the opposite of the opposite of a number is the number itself.
M.6.10.6: Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
M.6.10.7: Define rational number.
M.6.10.8: Plot pairs of integers and/or rational numbers on a coordinate plane.
M.6.10.9: Arrange integers and /or rational numbers on a horizontal or vertical number line.
M.6.10.10: Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
M.6.10.11: Identify a rational number as a point on the number line.
M.6.10.12: Name the pairs of integers and /or rational numbers of a point on a coordinate plane.

Students:

• Create and interpret coordinate axes with positive and negative coordinates.
• Given ordered pairs made up of rational numbers, locate and explain the placement of the ordered pair on a coordinate plane.
• Given two ordered pairs that differ only by signs, locate the points on a coordinate plane and explain the relationship of the locations of the points as reflections across one or both axes.
• Given real-world and mathematical problems where a coordinate graph will aid in the solution and given a graph of a real-world or mathematical situation, interpret the coordinate values of the points in the context of the situation including finding vertical and horizontal distances.

Students know:

• Strategies for creating coordinate graphs.
• Strategies for finding vertical and horizontal distance on coordinate graphs.

Students are able to:

• Graph points corresponding to ordered pairs,
• Represent real-world and mathematical problems on a coordinate plane.
• Interpret coordinate values of points in the context of real-world/mathematical situations.
• Determine lengths of line segments on a coordinate plane when the line segment joins points with the same first coordinate (vertical distance) or the same second coordinate (horizontal distance).

Students understand that:

• A graph can be used to illustrate mathematical situations and relationships. These representations help in conceptualizing ideas and in solving problems,
• Distances on lines parallel to the axes on a coordinate plane are the same as the related distance on the axis (number line).

Learning Objectives:
M.6.11.1: Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
M.6.11.2: Demonstrate an understanding of an extended coordinate plane.
M.6.11.3: Draw a four-quadrant coordinate plane.
M.6.11.4: Draw and extend vertical and horizontal number lines.
M.6.11.5: Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
M.6.11.6: Recall how to graph points in all four quadrants of the coordinate plane.
M.6.11.7: Define ordered pairs.
M.6.11.8: Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
M.6.11.9: Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
M.6.11.10: Identify which signs indicate the location of a point in a coordinate plane.
M.6.11.11: Recall how to plot ordered pairs on a coordinate plane.
M.6.11.12: Define reflections.
M.6.11.13: Calculate the distances between points having the same first or second coordinate using absolute value.

Students:

• Write, interpret, and explain the absolute values of the quantities.
• Distinguish comparisons of absolute value from statements about order (students will use logical reasoning to explain how an account balance less than $30 represents a debt greater than $30).

Students know:

• The meaning of absolute value and determine the absolute value of rational numbers in real-world contexts.

Students are able to:

• Understand that the absolute value of a number is the distance from zero in mathematical and real-world situations.

Students understand that:

• the absolute value of a number is its distance from zero.

Learning Objectives:
M.6.12.1: Define absolute value and rational numbers.
M.6.12.2: Recall how to order numbers.
M.6.12.3: Give examples of the magnitude for a positive or negative quantity in a real-world situations using absolute value.
M.6.12.4: Recognize the absolute value of a rational number is its' distance from 0 on a vertical and horizontal number line.

Students:

• Write, interpret, and explain inequalities that show order of the given numbers including absolute values of the quantities.
• Given contextual and mathematical situations involving quantities that can be represented as positive or negative rational numbers including absolute values of the quantities.

Students know:

• How to use and interpret inequality notation with rational numbers and absolute value.
• Strategies for comparing and ordering rational numbers and the absolute value of rational numbers with and without a number line in order to solve real-world and mathematical problems.

Students are able to:

• Use mathematical language to communicate the relationship between verbal representations of inequalities and the related number line and algebraic models.
• Distinguish comparisons of the absolute value of positive and negative rational numbers from statements about order.
• Use number line models to explain absolute value concepts in order to solve real-world and mathematical problems.

Students understand that:

• The absolute value of a number is its distance from zero on a number line regardless of direction,
• When using number lines to compare quantities those to the left are less than those to the right.

Learning Objectives:
M.6.13.1: Define rational number.
M.6.13.2: Plot pairs of integers and/or rational numbers on a coordinate plane.
M.6.13.3: Arrange integers and/or rational numbers on a horizontal or vertical number line.
M.6.13.4: Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
M.6.13.5: Evaluate a statement about order using comparisons of absolute value.
M.6.13.6: Recall how to order positive and negative numbers. (Use number line if needed.).

Students:

• Write whole numbers with indicated exponents and their equivalent form without exponents, and justify the equivalence.

Students know:

• Conventions of exponential notation.
• Factorization strategies for whole numbers.

Students are able to:

• Use factorization strategies to write equivalent expressions involving exponents.
• Accurately find products for repeated multiplication of the same factor in evaluating exponential expressions.

Students understand that:

• The use of exponents is an efficient way to write numbers as repeated multiplication of the same factor and this form reveals features of the number that may not be apparent in multiplied out form, (showing the prime factorization of two numbers with exponents helps determine how many of each factor).

Learning Objectives:
M.6.14.1: Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
M.6.14.2: Compute a numerical expression with exponents, with or without a calculator.
M.6.14.3: Restate exponential numbers as repeated multiplication.
M.6.14.4: Choose the correct value to replace each variable in the expression (Substitution).
M.6.14.5: Calculate the multiplication of single or multi-digit whole numbers, with or without a calculator.

Students:
Given contextual or mathematical problems both when known models exist (for example formulas) or algebraic models are unknown,

• Interpret the parts of the model in the original context.
• Create the algebraic model of the situation when appropriate.
• Use appropriate mathematical terminology to communicate the meaning of the expression.
• Evaluate the expressions for values of the variable including finding values following conventions of parentheses and order of operations.

Students know:

• Correct usage of mathematical symbolism to model the terms sum, term, product, factor, quotient, variable, difference, constant, and coefficient when they appear in verbally stated contexts.
• Conventions for order of operations.
• Convention of using juxtaposition (5A or xy) to indicate multiplication.

Students are able to:

• Translate fluently between verbally stated situations and algebraic models of the situation.
• Use operations (addition, subtraction, multiplication, division, and exponentiation) fluently with the conventions of parentheses and order of operations to evaluate expressions for specific values of variables in expressions.
• Use terminology related to algebraic expressions such as sum, term, product, factor, quotient, or coefficient, to communicate the meanings of the expression and the parts of the expression.

Students understand that:

• The structure of mathematics allows for terminology and techniques used with numerical expressions to be used in an analogous way with algebraic expressions, (the sum of 3 and 4 is written as 3 + 4, so the sum of 3 and y is written as 3 + y).
• When language is ambiguous about the meaning of a mathematical expression grouping, symbols and order of operations conventions are used to communicate the meaning clearly.
• Moving fluently among representations of mathematical situations (words, numbers, symbols, etc.), as needed for a given situation, allows a user of mathematics to make sense of the situation and choose appropriate and efficient paths to solutions.

Learning Objectives:
M.6.15.1: Define algebraic expression and variable.
M.6.15.2: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.3: Translate verbal and numerical expression using all operations.
M.6.15.4: Define coefficient, constant and term.
M.6.15.5: Match mathematical terms with correct mathematical symbols.
M.6.15.6: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.7: Calculate an expression in the correct order. with or without a calculator (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
M.6.15.8: Choose the correct value to replace each variable in the algebraic expression (Substitution).
M.6.15.9: Calculate a numerical expression, with or without a calculator (Ex. V=4x4x4).
M.6.15.10: Recognize the correct order to solve expressions with more than one operation.

Students:

• Given contextual or mathematical problems which may be modeled by algebraic expressions, use properties of the operations to produce combined and re-written forms of the expressions that are useful in resolving the problem.

Students know:

• the properties of operations, including inverse, identity, commutative, associative, and distributive and their appropriate application to be able to generate equivalent algebraic expressions.

Students are able to:

• Accurately use the properties of operations on algebraic expressions to produce equivalent expressions useful in a problem solving context.

Students understand that:

• The properties of operations used with numerical expressions are valid to use with algebraic expressions and allow for alternate but still equivalent forms of expressions for use in problem solving situations.

Learning Objectives:
M.6.16.1:Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.
M.6.16.2: Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
M.6.16.3: Combine terms that are alike of a given expression.
M.6.16.4: Recognize the property demonstrated in a given expression.
M.6.16.5: Simplify an expression by dividing by the greatest common factor.
Example: 18x + 6y = 6(3x + y).
M.6.16.6: Determine the greatest common factor in an algebraic expression.

Students:
Given a contextual or mathematical situation that could be represented algebraically,

• Explain by reasoning from the context why two expressions must be equivalent.
• Use properties of operations and equality to verify if two algebraic expressions are equivalent or not.

Students know:

• The properties of operations, including inverse, identity, commutative, associative, and distributive and their appropriate application to be able to determine whether two expressions are equivalent.
• Conventions of order of operations.

Students are able to:

• Accurately use the properties of operations to produce equivalent forms of an algebraic expression when interpreting mathematical and contextual situations.
• Use mathematical reasoning to communicate the relationships between equivalent algebraic expressions.

Students understand that:

• Manipulation of expressions via properties of the operations verifies mathematically that two expressions are equivalent.
• Reasoning about the context from which expressions arise allows for interpretation and meaning to be placed on each of the expressions and their equivalence.

Learning Objectives:
M.6.17.1: Define equivalent expressions.
M.6.17.2: Recognize equivalent expressions.
M.6.17.3: Substitute for the variable to find the value of a given expression.
M.6.17.4: Calculate a numerical expression.
M.6.17.5: Recognize that a variable without a written coefficient is understood to have a coefficient of one. (Ex. x = 1x).

Students:
Given situations that have been modeled with equations or inequalities:

• Substitute given specified values for the variables and the evaluate expressions.
• Determine if the resulting numerical sentence is true when the specified values are substituted for the variables.
• Explain with mathematical reasoning why a specified value is or is not a solution to a given equation or inequality.

Students know:

• Conventions of order of operations.
• The solution is the value of the variable that will make the equation or inequality true.
• That using various processes to identify the value(s) that when substituted for the variable will make the equation true.

Students are able to:

• Substitute specific values into algebraic equation or inequality and accurately perform operations of addition, subtraction, multiplication, division and exponentiation using order of operation.

Students understand that:

• Solving an equation or inequality means finding the value or values (if any) that make the mathematical sentence true.
• The solution to an inequality is often a range of values rather than a specific value.

Learning Objectives:
M.6.18.1: Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
M.6.18.2: Compute a numerical expression with exponents, with or without a calculator.
M.6.18.3: Restate exponential numbers as repeated multiplication.
M.6.18.4: Choose the correct value to replace each variable in the expression (Substitution).
M.6.18.5: Calculate the multiplication of single or multi-digit whole numbers, with or without a calculator.

Students:
Given contextual or mathematical situations which may be modeled by x + p = q or px = q (p,q, and x are rational and non-negative),

• Explain the role of the variable as a place holder where the variable stands for a particular number (y + 7 = 12) or a value in a formula (A = L × W) and where values are substituted for one or more variables another variable assumes different values.
• Write and solve equations modeling the situation, solve the resulting equations, and justify the solutions.

Students know:

• Correct translation between verbally stated situations and mathematical symbols and notation.
• How to write and solve a simple equation using non-negative rational numbers to solve mathematical and real-world problems.

Students are able to:

• Translate fluently between verbally stated situations and algebraic models of the situation.
• Use inverse operations and properties of equality to produce solutions to equations of the forms x + p = q or px = q.
• Use logical reasoning and properties of equality to justify solutions, reasonableness of solutions, and solution paths.

Students understand that:

• Variables may be unknown values that we wish to find.
• The solution to the equation is a value for the variable which, when substituted into the original equation, results in a true mathematical statement.
• A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.
• The structure of mathematics present in the properties of the operations and equality can be used to maintain equality while rearranging equations, as well as justify steps in the solutions of equations.

Learning Objectives:
M.6.19.1: Define equation and variable.
M.6.19.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
M.6.19.3: Solve the equation represented by the real-world situation.
M.6.19.4: Identify the unknown variable in a given situation.
M.6.19.5: List given information from the problem.
M.6.19.6: Explain the solution in the context of the problem.

Students:
Given contextual or mathematical situations which may be modeled by x > c or x < c,

• Write inequalities modeling the situation.
• Identify the set of values making the resulting inequalities true.
• Represent the solutions on a number line.

Students know:

• Correct translation between verbally stated situations and mathematical symbols and notation,
• Many real-world situations are represented by inequalities,
• The number line represents inequalities from various contextual and mathematical situations.

Students are able to:

• Translate fluently among verbally stated inequality situations, algebraic models of the situation ( x > c or x < c), and visual models on a number line.

Students understand that:

• Inequalities have infinitely many solutions.
• A symbolic or visual representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.

Learning Objectives:
M.6.20.1: Define inequality and solution set of an inequality.
M.6.20.2: Set up an inequality to represent the given situation, using correct mathematical operations and variable.
M.6.20.3: Identify solution set for the inequality used to represent the situation.
M.6.20.4: Recognize the inequality symbols; <, >, < , > , =, >, <, ?, and ?.
M.6.20.5: Construct and label a number line.
M.6.20.6: Graph the solution set on a number line for the inequality used to represent the situation.

Students:
Given a real-world problem involving two quantities that change in relationship to one another,

• Represent the context using graphs, tables, and equations.
• Explain the connections among the representations using mathematical vocabulary including dependent and independent variables.

Students know:

• Roles of dependent and independent variables.
• Correct translation between verbally stated situations and mathematical symbols and notation.

Students are able to:

• Represent real-world problems involving two quantities that change in relationship to one another using equations, graphs, and tables,
• Use mathematical vocabulary to explain connections among representations of function contexts.
• Analyze and interpret the relationship between the independent and the dependent variable in a given situation.

Students understand that:

• Equations with two variables represent mathematical relationships in which the value of the dependent variable varies with changes in the independent variable.
• A symbolic or visual representation of relevant features of a real-world problem can aid in interpretation of the situation.
• Translating between language, a table, an equation, or a graph represents the same relationship and provides a different perspective on the function.

Learning Objectives:
M.6.21.1: Define dependent variable, independent variable, ordered pairs, input, output, and coordinate plane.
M.6.21.2: Examine the graph and table to determine any relationship between the variables.
M.6.21.3: Recall how to draw a number line.
M.6.21.4: Draw and label a coordinate plane.
M.6.21.5: Analyze the pattern represented by the values in the table and develop an equation to express the relationship.
M.6.21.6: Relate the table and graph to the equation.
M.6.21.7: Plot independent (input) and dependent (output) values on a coordinate plane.
M.6.21.8: Create a table of independent and dependent values from the equation.

Students:
Given a variety of mathematical questions,

• Justify the classification of questions as either statistical or non-statistical.
• Write statistical and non-statistical questions.

Students know:

• Characteristics of statistical and non-statistical questions.

Students are able to:

• Justify the classification of mathematical questions as statistical or non-statistical questions.

Students understand that:

• Statistical questions have anticipated variability in the answers.
• Data are the numbers produced in response to a statistical question.

Learning Objectives:
M.6.22.1: Define statistical question.
M.6.22.2: Identify examples of statistical questions and non-statistical questions.
M.6.22.3: Compare and contrast statistical questions and non- statistical questions.

Students:
Given a set of numerical data, summarize the data by,

• Reporting the number of observations (n).
• Describing the nature of the attribute under investigation.
• Calculating, interpreting, and comparing the measures of center (median/mean/mode) in a real-world data set,
• Calculating, interpreting and comparing the measures of variability (interquartile range and range) in a real-world data set.
• Given a set of numerical data interpret the measures of center and variability in the context of a problem.
• Justify their choice of measures of center and variability to describe the data based on the data distribution and the context in which the data were gathered.

Students know:

• Measures of center and how they are affected by the data distribution and context.
• Measures of variability and how they are affected by the data distribution and context.
• Methods of determining mean, median, mode, interquartile range, and range.

Students are able to:

• Describe the nature of the attribute under investigation including how it was measured and its unit of measure using the context in which the data were collected.
• Determine measures of center and variability for a set of numerical data.
• Use characteristics of measures of center and variability to justify choices for summarizing and describing data.

Students understand that:

• Measures of center for a set of data summarize the values in the set in a single number and are affected by the distribution of the data.
• Measures of variability for a set of data describe how the values vary in a single number and are affected by the distribution of the data.

Learning Objectives:
M.6.23.1: Define numerical data set, measure of variation, and measure of center.
M.6.23.2: Relate the measure of variation, of a data set, with the concept of range.
M.6.23.3: Relate the measure of the center for a numerical data set with the concept of measure of center.
M.6.23.4: Define numerical data set, quantitative, measure of center, median, frequency distribution, and attribute.
M.6.23.5: Compare and contrast the center and variation.
M.6.23.6: Collect the data.
M.6.23.7: Organize the data.
M.6.23.8: Describe how attribute was measured including units of measurement.
M.6.23.9: Identify the attribute used to create the numerical set.

Students:
Given a set of numerical data,

• Analyze graphical representation of data by describing the center, spread, and shape including approx. symmetric or skewed.
• Reporting significant features in the shape of data including striking deviations, (e.g., extreme values, outliers, gaps, and clusters).
• Organize and display the data using plots on line plots, dot plots, stem and leaf plots, histograms, and box plots.

Students know:

• How to use graphical representations of real-world data to describe context, center, spread and shape from which they were collected.
• Techniques for constructing line plots, stem and leaf plots, dot plots, histograms, and box plots.

Students are able to:

• Organize and display data using dot plots, line plots, stem and leaf plots, histograms, and box plots.
• Describe the nature of the attribute under investigation including how it was measured and its unit of measure using the context in which the data were collected.
• Describe the shape of numerical data distribution including patterns and extreme values.
• Use graphical representations of real-world data to describe and summarize the context from which they were collected.

Students understand that:

• Sets of data can be organized and displayed in a variety of ways, each of which provides unique perspectives of the data set.
• Data displays help in conceptualizing ideas and in solving problems.
• The overall shape and other significant features of a set of data, (e.g., gaps, peaks, clusters and extreme values) are important in summarizing numerical data sets.

Learning Objectives:
M.6.24.1: Define dot plots, line plot, stem and leaf plots, upper quartile, lower quartile, median, histograms, and box plots.
M.6.24.2: Recall how to read a graph or table.
M.6.24.3: Calculate upper quartile median, lower quartile median, overall median, greatest value, and lowest value.
M.6.24.4: Create box plot using calculations.
M.6.24.5: Plot data on dot plots and histograms.
M.6.24.6: Construct and label the display.
M.6.24.7: Recognize the different types of displays.
M.6.24.8: Define distribution and skew.
M.6.24.9: Describe the shape of a set of data in a given distribution.
M.6.24.10: Describe the spread of a set of data in a given distribution.
M.6.24.11: Describe the center of a set of data in a given distribution.

Students:
Given real-world and mathematical problems involving the mapping of polygons onto a coordinate system,

• Determine the length of a side joining points with the same first coordinate or the same second coordinate.
• Determine missing vertices of a rectangle.
• Graph polygons in coordinate plane given vertices and solve real-world problems.

Students know:

• Terminology associated with coordinate systems.
• Correct construction of coordinate systems.

Students are able to:

• Graph points corresponding to ordered pairs.
• Represent real-world and mathematical problems on a coordinate plane.
• Interpret coordinate values of points in the context of real-world and mathematical situations.
• Determine lengths of line segments on a coordinate plane when the line segment joins points with the same first coordinate or the same second coordinate.

Students understand that:

• A variety of representations such as diagrams, number lines, charts, and graphs can be used to illustrate mathematical situations and relationships.
• These representations help in conceptualizing ideas and in solving problems.
• Distances on lines parallel to the axes on a coordinate plane are the same as the related distance on the axis (number line).

Learning Objectives:
M.6.25.1: Define vertices.
M.6.25.2: Apply absolute value to find the length of a side joining points with the same first coordinate or the same second coordinate.
M.6.25.3: Plot points on a coordinate plane., then connect points for the vertices to sketch a polygon.
M.6.25.4: Identify ordered pairs.
M.6.25.5: Recognize polygons.
M.6.25.6: Define perimeter and area.
M.6.25.7: Identify the length between vertices on a coordinate plane.
M.6.25.8: Calculate the perimeter and area using the distance between the vertices.

Students:
Given a variety of triangles and quadrilaterals:

• Find their area.
• Justify their solutions and solution paths by composing shapes into rectangles and decomposing into triangles or other shapes.

Given real-world and mathematical problems involving area of triangles and other polygons,

• Compose and decompose shapes to find solutions.
• Interpret solutions.

Students know:

• Appropriate units for measuring area: square inches, square units, square feet, etc..
• Strategies for composing and decomposing shapes to find area.

Students are able to:

• Communicate the relationship between models of area and the associated real-world mathematical problems.
• Use logical reasoning to choose and apply strategies for finding area by composing and decomposing shapes.
• Accurately compute area of rectangles using multiplication and the formula.

Students understand that:

• The area of a figure is measured by the number of same-size unit squares that exactly cover the interior space of the figure.
• Shapes can be composed and decomposed into shapes with related properties,
• Area is additive.

Learning Objectives:
M.6.26.1: Define area, special quadrilaterals, right triangles, and polygons.
M.6.26.2: Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
M.6.26.3: Apply area formulas to solve real-world mathematical problems.
M.6.26.4: Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
M.6.26.5: Explain how to find the area for rectangles.
M.6.26.6: Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
M.6.26.7: Recognize and demonstrate that two right triangles make a rectangle.

Students:
Given real-world and mathematical problems involving surface area,

• Use models of the relating net of the 3-D figure to explain and justify solutions and solution paths.

Students know:

• Measurable attributes of objects, specifically area and surface area.
• Strategies for representing the surface area of a 3-D shape as a 2-D net.

Students are able to:

• Communicate the relationships between rectangular models of area and multiplication problems.
• Model the surface area of 3-D shapes using 2-D nets.
• Accurately measure and compute area of triangles and rectangles.
• Strategically and fluently choose and apply strategies for finding surface areas of 3-D figures.

Students understand that:

• Area is additive.
• Surface area of a 3-D shape is represented by the sum of the areas of the faces of the object.
• Models represent measurable attributes of objects and help to solve problems.

Learning Objectives:
M.6.27.1: Define three-dimensional figures, surface area, and nets.
M.6.27.2: Identify three-dimensional figures.
M.6.27.3: Draw nets to find the surface area of a given three-dimensional figure.
M.6.27.4: Recall how to calculate the area of a rectangle.
M.6.27.5: Select and create a three-dimensional figure using manipulatives.

Students:
Given a right rectangular prism with fractional edge lengths within a real-world or mathematical problem context,

• Find and justify the volume of the prism as part or all of the problem's solution by relating a cube filled model to the corresponding multiplication problem(s).

Given cubes with appropriate unit fraction edge lengths,

• Create and explain rectangular prism models to show that the volume of a right rectangular prism with fractional edge lengths l, w, and h is represented by the formulas V = l w h and V = b h.

Students know:

• Measurable attributes of objects, specifically volume.
• Units of measurement, specifically unit cubes.
• Relationships between unit cubes and corresponding cubes with unit fraction edge lengths.
• Strategies for determining volume.
• Strategies for finding products of fractions.

Students are able to:

• Communicate the relationships between rectangular models of volume and multiplication problems.
• Model the volume of rectangles using manipulatives.
• Accurately measure volume using cubes with unit fraction edge lengths.
• Strategically and fluently choose and apply strategies for finding products of fractions.
• Accurately compute products of fractions.

Students understand that:

• The volume of a solid object is measured by the number of same-size cubes that exactly fill the interior space of the object.
• Generalized formulas for determining area and volume of shapes can be applied regardless of the level of accuracy of the shape's measurements (in this case, side lengths).

Learning Objectives:
M.6.28.1: Define volume, rectangular prism, edge, and formula.
M.6.28.2: Recall how to multiply fractional numbers.
M.6.28.3: Evaluate the volumes of rectangular prisms in the context of solving real-world and mathematical problems.
M.6.28.4: Use models and volume formulas (V=lwh and V=Bh) to find volumes in the context of solving real-world and mathematical problems.
M.6.28.5: Calculate the volume of a rectangular prism using fractional lengths.
M.6.28.6: Test the formula V= lwh and V=Bh with the experimental findings.
M.6.28.7: Experiment with finding the volume using a variety of sizes of rectangular prisms manipulatives.