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CCRS StandardStandard IDEvidence of Student AttainmentTeacher VocabularyKnowledgeSkillsUnderstandingResources1. Understand the concept of a ratio, and use ratio language to describe a ratio relationship between two quantities. Examples: The ratio of wings to beaks in the bird house at the zoo was 2:1 because for every 2 wings there was 1 beak. For every vote candidate A received, candidate C received nearly three votes.Ratios & Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.6.RP.1Students: Given contextual or mathematical situations involving multiplicative comparisons,
l Communicate the relationship of two quantities using ratio language.l Ratio
l Ratio languageStudents know:
l Characteristics of additive situations (Table 1),
l Characteristics of multiplicative situations (Table 2).Students are able to:
l Compare and contrast additive vs. multiplicative contextual situations,
l Represent multiplicative comparisons in ratio notation and language.Students understand that:
l In a multiplicative comparison situation one quantity changes at a constant rate with respect to a second related quantity.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53830" \t "_blank" ALEX Resources2. Understand the concept of a unit rate a/b associated with a ratio a:b with b `" 0, and use rate language in the context of a ratio relationship. Examples: This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.1 (1 Expectations for unit rates in this grade are limited to non-complex fractions.)Ratios & Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.6.RP.2Students:Given contextual or mathematical situations involving multiplicative comparisons,
l Use rate language to explain the relationships between ratio of two quantities and the associated unit rate of one of the quantities in terms of the other.l Unit rate
l Ratio
l Rate languageStudents know:
l Characteristics of multiplicative comparison situations,
l Rate and ratio language,
l Techniques for determining unit rates.Students are able to:
l Explain relationships between ratios and the related unit rates,
l Represent contextual relationships as ratios.Students understand that:
l A unit rate is a ratio (a:b) of two measurements in which b is one. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53832" \t "_blank" ALEX Resources3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Solve unit rate problems including those involving unit pricing and constant speed. Example: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.Ratios & Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.6.RP.3Students:Given contextual or mathematical situations involving ratio and rate (including those involving unit pricing, constant speed, and measurement conversions),
l Represent the 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) in order to solve problems, find missing values on tables and interpret relationships and results,
l Change given rates to unit rates in order to find and justify solutions to problems.
Given contextual or mathematical situations involving percents,
l Interpret the percent as rate per 100,
l Solve problems and justify solutions when finding the whole given a part and the percent. (e.g., Brian has made 3 batches of bagels so far this morning. His boss told him that he has only completed 30% of the work she expects done during the shift. How many batches of bagels did Brian's boss expect him to make during each shift?). l Rate
l Ratio
l Rate reasoning
l Ratio reasoning
l Transform units
l QuantitiesStudents know:
l 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,
l Strategies for finding equivalent ratios,
l Strategies for using ratio reasoning to convert measurement units.Students are able to:
l 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),
l Use ratio and rate reasoning to explain connections among representations and justify solutions,
l Solve and justify solutions for rate problems including unit pricing, constant speed, measurement conversions, and situations involving percents,
l Solve problems and justify solutions when finding the whole given a part and the percent,
l Use ratio reasoning, multiplication, and division to transform and interpret measurements. Students understand that:
l A unit rate is a ratio (a:b) of two measurements in which b is one,
l A symbolic representation of relevant features of a real world problem can provide for resolution of the problem and interpretation of the situation,
l When computing with quantities the transformation and interpretation of the resulting unit is dependent on the particular operation performed (e.g.. 4cm + 5cm = 9cm, 4cm x 5cm = 20cm2, 100mi. 4hr. = 25mi./hr.). Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53834" \t "_blank" ALEX Resources4. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Example: Create a story context for (2/3) (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Compute fluently with multi-digit numbers and find common factors and multiples.The Number SystemApply and extend previous understandings of multiplication and division to divide fractions by fractions.6.NS.1Students:Given a division problem involving a fraction divided by a fraction,
l Create an appropriate story context,
l Solve the problem using visual fraction models or an equation,
l Explain the relationship between the model and the problem,
l Interpret the solution,
l Use the inverse relationship between multiplication and division, or concept of division as repeated subtraction, to explain and justify the solution.l Visual fraction modelsStudents know:
l Strategies for representing fractions and operations on fractions using visual models,
l The inverse relationship between multiplication and division (a b = c implies that a = b x c).Students are able to:
l Represent fractions and operations on fractions using visual models,
l Interpret quotients resulting from the division of a fraction by a fraction,
l Accurately determine quotients of fractions by fractions using visual models/equations,
l Justify solutions to division problems involving fractions using the inverse relationship between multiplication and division.Students understand that:
l The operation of division is interpreted the same with fractions as with whole numbers,
l The inverse relationship between the operations of multiplication and division that was true for whole numbers continues to be true for fractions,
l The relationships between operations can be used to solve problems and justify solutions and solution paths.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53841" \t "_blank" ALEX Resources5. Fluently divide multi-digit numbers using the standard algorithm.The Number SystemApply and extend previous understandings of multiplication and division to divide fractions by fractions.6.NS.2Students:Given a context which calls for the division of two whole numbers,
l Choose the most appropriate strategy for computing the answer,
l Produce accurate results using the standard algorithm when appropriate. l Standard algorithm (long division)Students know:
l Strategies for computing answers to division problems, including the standard division algorithm.Students are able to:
l Strategically choose and apply appropriate strategies for dividing,
l Accurately find quotients using the standard division algorithm. Students understand that:
l Mathematical problems can be solved using a variety of strategies, models, and representations,
l Efficient application of computation strategies is based on the numbers and operations in the problems,
l The steps used in the standard algorithms for division can be justified by using properties of operations and understanding of place value,
l 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.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53844" \t "_blank" ALEX Resources6. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.The Number SystemApply and extend previous understandings of multiplication and division to divide fractions by fractions.6.NS.3Students:Given a context which calls for complex computation involving multi-digit decimals,
l Choose the most appropriate strategy for computing the answer,
l Produce accurate results efficiently using the standard algorithm for each operation when appropriate. l Standard algorithms (addition, subtraction, multiplication, and division)Students know:
l 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),
l Strategies for computing answers to complex addition, subtraction, multiplication, and division problems involving multi-digit decimals, including the standard algorithm for each operation.Students are able to:
l Strategically choose and apply appropriate computation strategies,
l Accurately find sums, differences, products, and quotients using the standard algorithms for each operation.Students uderstand that:
l 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,
l Mathematical problems can be solved using a variety of strategies, models, and representations,
l Efficient application of computation strategies is based on the numbers and operations in the problem,
l The steps used in the standard algorithms for the four operatons can be justified by using properties of operations and understanding of place value,
l 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.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53845" \t "_blank" ALEX Resources7. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: Express 36 + 8 as 4 (9 + 2).The Number SystemApply and extend previous understandings of multiplication and division to divide fractions by fractions.6.NS.4Students:Given any two whole numbers less than or equal to 100,
l 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,
l 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. [i.e., if the two whole numbers are 36 and 8, 36+8 = 4(9+2)].
Given two whole numbers less than or equal to 12,
l 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.l Greatest common factor
l Least common multiple
l Distributive propertyStudents know:
l Strategies for determining the greatest common factor of two numbers,
l Strategies for determining the least common multiple of two numbers,
l Distributive Property of Multiplication over addition.Students are able to:
l Apply strategies for determining greatest common factors and least common multiples,
l Use the distributive property to express the sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor.Students understand that:
l Multiplication is distributive over addition,
l Composing and decomposing numbers provides insights into relationships among numbers,
l Quantities can be represented using a variety of equivalent expressions.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53846" \t "_blank" ALEX Resources8. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts explaining the meaning of 0 in each situationThe Number SystemApply and extend previous understandings of numbers to the system of rational numbers.6.NS.5Students:Given contextual or mathematical situations containing quantities that have opposite directions or values,
l Use positive and negative numbers to represent quantities in the contexts and explain the meaning of 0 in each situation.l Positive and negative numbersStudents know:
l Notation for and meaning of positive and negative numbers and zero.Students are able to:
l Use positive and negative numbers to represent quantities in real-world contexts,
l Explain the meaning of zero in a variety of real-world contexts.Students understand that:
l Positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, or positive/negative electrical charges.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53849" \t "_blank" ALEX Resources9. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.The Number SystemApply and extend previous understandings of numbers to the system of rational numbers.6.NS.6Students:
l Create and interpret number line diagrams and coordinate axes with positive and negative coordinates.
Given any rational number (positive or negative),
l Locate the number on a number line,
l 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.
Given ordered pairs made up of rational numbers,
l Locate and explain the placement of the ordered pair on a coordinate plane.
Given two ordered pairs that differ only by signs,
l Locate the points on a coordinate plane and explain the relationship of the locations of the points as reflections across one or both axes.l Coordinate axes
l Ordered pairs
l Coordinate planeStudents know:
l Strategies for creating number line models of rational numbers (e.g., marking off equal lengths by estimation or recursive halving) and coordinate axes for plotting points with rational coordinates,
l Strategies for locating numbers on a number line or ordered pairs of numbers on a coordinate system.Students are able to:
l Represent rational numbers and their opposites on a number line including both positive and negative quantities,
l Explain and justify the creation of number lines and placement of rational numbers on a number line,
l Explain how to identify the coordinates of a point on a coordinate system in any of the four quadrants,
l Graph points corresponding to ordered pairs made up of two rational numbers.Students understand that:
l Representing rational numbers on number lines requires using both a distance and a direction,
l Locating numbers on a number line or graphing points on a coordinate plane provides a representation of a mathematical context which aids in visualizing ideas and solving problems.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53850" \t "_blank" ALEX Resources10. Understand ordering and absolute value of rational numbers.
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. Example: Interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right.
Write, interpret, and explain statements of order for rational numbers in real-world contexts. Example: Write 3 oC > 7 oC to express the fact that 3 oC is warmer than 7 oC.
Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. Example: For an account balance of 30 dollars, write |30| = 30 to describe the size of the debt in dollars.
Distinguish comparisons of absolute value from statements about order. Example: Recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars.The Number SystemApply and extend previous understandings of numbers to the system of rational numbers.6.NS.7Students:Given contextual or mathematical situations involving quantities that can be represented as positive or negative rational numbers,
l Write, interpret,and explain inequalities that show order of the given numbers,
l Write, interpret, and explain the absolute values of the quantities,
l Distinguish comparisons of absolute value from statements about order, (i.e., students will use logical reasoning to explain how an account balance less than -30 dollars represents a debt greater than 30 dollars).l Absolute value
l InequalityStudents know:
l Use and interpretation of absolute value and inequality notation.Students are able to:
l Use mathematical language to communicate the relationship between verbal representations of inequalities and the related number line and algebraic models,
l Distinguish comparisons of absolute value of positive and negative rational numbers from statements about order,
l Use number line models to explain absolute value concepts.Students understand that:
l The absolute value of a number is its distance from zero on a number line regardless of direction,
l When using number lines to compare quantities those to the left are less than those to the right.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53854" \t "_blank" ALEX Resources11. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.The Number SystemApply and extend previous understandings of numbers to the system of rational numbers.6.NS.8Students:Given real world and mathematical problems where a coordinate graph will aid in the solution,
l Create the corresponding graph on a coordinate plane and explain its relationship to the context of the problem, (e.g., Tickets cost $0.75 each. If 20 tickets are sold, the cost is $15, if 21 tickets are sold the cost is $15.75. Create a graph that shows the cost of any number of tickets).
Given a graph of a real world or mathematical situation,
l Interpret the coordinate values of the points in the context of the situation including finding vertical and horizontal distances.l Coordinate plane
l Quadrants
l Coordinate valuesStudents know:
l Strategies for creating coordinate graphs,
l Strategies for finding vertical and horizontal distance on coordinate graphs including using absolute value. Students are able to:
l Graph points corresponding to ordered pairs,
l Represent real world and mathematical problems on a coordinate plane,
l Interpret coordinate values of points in the context of real world/mathematical situations,
l 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:
l 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,
l Distances on lines parallel to the axes on a coordinate plane are the same as the related distance on the axis (number line),
l When finding vertical and horizontal distance on a coordinate grid only positive results are used therefore absolute value must be considered. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53863" \t "_blank" ALEX Resources12. Write and evaluate numerical expressions involving whole-number exponents.Expressions & EquationsApply and extend previous understandings of arithmetic to algebraic expressions.6.EE.1Students: - Write whole numbers with indicated exponents and their equivalent form without exponents, (e.g., given 108 the student will write 22 x 33 or given 22 x 33 will write 108) and justify the equivalence.l Numerical expression
l ExponentStudents know:
l Conventions of exponential notation,
l Factorization strategies for whole numbers.Students are able to:
l Use factorization strategies to write equivalent expressions involving exponents,
l Accurately find products for repeated multiplication of the same factor in evaluating exponential expressions.Students understand that:
l 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, (i.e., showing the prime factorization of two numbers with exponents helps determine how many of each factor should be used in finding the least common multiple).Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53865" \t "_blank" ALEX Resources13. Write, read, and evaluate expressions in which letters stand for numbers.
Write expressions that record operations with numbers and with letters standing for numbers. Example: Express the calculation, Subtract y from 5, as 5 y.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. Example: Describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Example: Use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2Expressions & EquationsApply and extend previous understandings of arithmetic to algebraic expressions.6.EE.2Students:Given contextual or mathematical problems both when known models exist (for example formulas) or algebraic models are unknown
l Interpret the parts of the model in the original context,
l Create the algebraic model of the situation when apprpopriate,
l Use appropriate mathematical terminology to communicate the meaning of the expression,
l Evaluate the expressions for values of the variable including finding values following conventions of parentheses and order of operations.l Expressions
l Term
l CoefficientStudents know:
l Correct usage of mathematical symbolism to model the terms sum, term, product, factor, quotient, and coefficient when they appear in verbally stated contexts,
l Conventions for order of operations,
l Convention of using juxtaposition (e.g., 5A or xy) to indicate multiplication.Students are able to:
l Translate fluently between verbally stated situations and algebraic models of the situation,
l 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,
l Use terminology related to algebraic expressions (sum, term, product, factor, quotient, or coefficient) to communicate the meanings of the expression and the parts of the expression.Students understand that:
l The structure of mathematics allows for terminology and techniques used with numerical expressions to be used in an anlaogous way with algebraic expressions, (i.e., the sum of 3 and 4 is written as 3 + 4, so the sum of 3 and y is written as 3 + y),
l 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,
l 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.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53866" \t "_blank" ALEX Resources14. Apply the properties of operations to generate equivalent expressions. Example: Apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.Expressions & EquationsApply and extend previous understandings of arithmetic to algebraic expressions.6.EE.3Students:Given contextual or mathematical problems which may be modeled by algebraic expressions,
l Use properties of the operations to produce combined and re-written forms of the expressions that are useful in resolving the problem.l Properties of operations
l Distributive propertyStudents know:
l The properties of operations listed in Table 3 and their appropriate application.Students are able to:
l Accurately use the properties of operations on algebraic expressions to produce equivalent expressions useful in a problem solving context.Students understand that:
lThe 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.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53873" \t "_blank" ALEX Resources15. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). Example: The expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities.Expressions & EquationsApply and extend previous understandings of arithmetic to algebraic expressions.6.EE.4Students:Given a contextual or mathematical situation that could be represented algebraically,
l Explain by reasoning from the context why two expressions must be equivalent, (e.g., the student will demonstrate that each of these three ways of thinking about the following problem will always result in the same answer when n is known. When determining the number of tiles needed for a border around a square pool of side n, the expression 4n+4 shows counting 4 sides and then 4 corners. The expression 4(n+1) shows counting four sides which each include one corner. The expression 4(n+2) - 4 shows counting the outer border then subtracting the corners as they have been counted twice).
l Use properties of operations and equality to verify if two algebraic expressions are equivalent or not.l Equivalent
l ExpressionsStudents know:
l The properties of operations listed in Table 3 and their appropriate application,
l Conventions of order of operations.Students are able to:
l Accurately use the properties of operations to produce equivalent forms of an algebraic expression when interpreting mathematical and contextual situations,
l Use mathematical reasoning to communicate the relationships between equivalent algebraic expressions.Students understand that:
l Manipulation of expressions via properties of the operations verifies mathematically that two expressions are equivalent,
l Reasoning about the context from which expressions arise allows for interpretaion and meaning to be placed on each of the expressions and their equivalence.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53875" \t "_blank" ALEX Resources16. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Expressions & EquationsReason about and solve one-variable equations and inequalities.6.EE.5Students:Given situations that have been modeled with equations or inequalities,
l Substitute given specified values for the variables and the evaluate expressions,
l Determine if the resulting numerical sentence is true when the specified values are substituted for the variables,
l Explain with mathematical reasoning why a specified value is or is not a solution to a given equation or inequality.l Substitution
l Equation
l InequalityStudents know:
l Conventions of order of operations.Students are able to:
l Substitute specific values into algebraic expressions and accurately perform operations of addition, subtraction, multiplication, division and exponentiation on numerical expressions,
l Use conventions of order of operations to evaluate expressions.Students understand that:
l Solving an equation or inequality means finding the value or values (if any) that make the mathematical sentence true,
l The solution to an inequality is often a range of values rather than a specific value.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53878" \t "_blank" ALEX Resources17. Use variables to represent numbers, and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or, depending on the purpose at hand, any number in a specified set.Expressions & EquationsReason about and solve one-variable equations and inequalities.6.EE.6Students:Given a contextual problem,
l Model the problem with mathematical symbols, variables, and expressions that aid in solving the problem,
l 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 xmvwx VZ
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W) where as values are substituted for one or more variables another variable assumes different values. l Variable
l ExpressionStudents know:
l Correct translation between verbally stated situations and mathematical symbols and notation.Students are able to:
l Translate fluently between verbally stated situations and algebraic models of the situation.Students understand that:
l Variables may be unknown values that we wish to find,
l Variables may be used in generalized statements to represent the truth of a statement for all values that satisfy the relationship modeled (i.e., formulas).Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53879" \t "_blank" ALEX Resources18. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.Expressions & EquationsReason about and solve one-variable equations and inequalities.6.EE.7Students: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),
l Write equations modeling the situation, solve the resulting equations, and justify the solutions.l Equations
l Nonnegative rational numbersStudents know:
l Correct translation between verbally stated situations and mathematical symbols and notation.Students are able to:
l Translate fluently between verbally stated equality situations to algebraic models of the situation,
l Use inverse operations and properties of equality to produce solutions to equations of the forms x + p = q or px = q,
l Use logical reasoning and properties of equality to justify solutions, reasonableness of solutions, and solution paths.Students understand that:
l Variables may be unknown values that we wish to find,
l The solution to the equation is a value for the variable which, when substituted into the original equation, results in a true mathematical statement,
l A symbolic representation of relevant features of a real world problem can provide for resolution of the problem and interpretation of the situation,
l 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.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53880" \t "_blank" ALEX Resources19. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Expressions & EquationsReason about and solve one-variable equations and inequalities.6.EE.8Students:Given contextual or mathematical situations which may be modeled by x > c or x < c,
l Write inequalities modeling the situation,
l Identify the set of values making the resulting inequalities true. (e.g., the temperature of the meat must stay below 28 degrees F; the store owner must make at least $1000 in order to pay all his employees),
l Represent the solutions on a number line.l Inequalities
l ConstraintStudents know:
l Correct translation between verbally stated situations and mathematical symbols and notation.Students are able to:
l 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:
l Inequalities have infinitely many solutions,
l 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.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53881" \t "_blank" ALEX Resources20. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Example: In a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.Expressions & EquationsRepresent and analyze quantitative relationships between dependent and independent variables.6.EE.9Students:Given a real world problem involving two quantities that change in relationship to one another,
l Represent the context using graphs, tables, and equations,
l Explain the connections among the representations using mathematical vocabulary including dependent and independent variables.l Dependent variables
l Independent variablesStudents know:
l Roles of dependent and independent variables,
l Correct translation between verbally stated situations and mathematical symbols and notation.Students are able to:
l Represent real world problems involving two quantities that change in relationship to one another using equations, graphs, and tables,
l Use mathematical vocabulary to explain connections among representations of function contexts,
l Analyze and interpret the relationship between the independent and the dependent variable in a given situation.Students understand that:
l Equations with two variables represent mathematical relationships in which the value of the dependent variable varies with changes in the independent variable,
l A symbolic or visual representation of relevant features of a real world problem can aid in interpretation of the situation.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53883" \t "_blank" ALEX Resources21. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.GeometrySolve real-world and mathematical problems involving area, surface area, and volume.6.G.1Students:Given a variety of triangles and quadrilaterals,
l Find their area,
l Justify their solutions and solution paths by composing shapes into rectangles and decomposing into triangles or other shapes, (e.g., combining two congruent right triangles into a rectangle to show that the area of one triangle is half of the area of the corresponding rectangle or decomposing a non-rectangular parallelogram into two right triangles and a rectangle to show that the area is the same as sum of the areas of the two right triangles and the rectangle).
Given real world and mathematical problems involving area of triangles and other polygons,
l Compose and decompose shapes to find solutions,
l Interpret solutions.l Right triangles
l Special quadrilaterals
l PolygonsStudents know:
l Appropriate units for measuring area: square inches, square units, square feet, etc.,
l Strategies for composing and decomposing shapes to find area.Students are able to:
l Communicate the relationship between models of area and the associated real world mathematical problems,
l Use logical reasoning to choose and apply strategies for finding area by composing and decomposing shapes,
l Accurately compute area of rectangles using multiplication and the formula A = l x w. Students understand that:
l The area of a figure is measured by the number of same-size unit squares that exactly cover the interior space of the figure,
l Shapes can be composed and decomposed into shapes with related properties,
l Area is additive.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53886" \t "_blank" ALEX Resources22. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.GeometrySolve real-world and mathematical problems involving area, surface area, and volume.6.G.2Students:Given a right rectangular prism with fractional edge lengths within a real world or mathematical problem context,
l 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,
l 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.l Right rectangular prism
l V = b h (Volume of a right rectangular prism = the area of the base x the height)Students know:
l Measurable attributes of objects, specifically volume,
l Units of measurement, specifically unit cubes,
l Relationships between unit cubes and corresponding cubes with unit fraction edge lengths,
l Strategies for determining volume,
l Strategies for finding products of fractions.Students are able to:
l Communicate the relationships between rectangular models of volume and multiplication problems,
l Model the volume of rectangles using manipulatives,
l Accurately measure volume using cubes with unit fraction edge lengths,
l Strategically and fluently choose and apply strategies for finding products of fractions,
l Accurately compute products of fractions.Students understand that:
l The volume of a solid object is measured by the number of same-size cubes that exactly fill the interior space of the object,
l 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). Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53887" \t "_blank" ALEX Resources23. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.GeometrySolve real-world and mathematical problems involving area, surface area, and volume.6.G.3Students:Given real world and mathematical problems involving the mapping of polygons onto a coordinate system,
l Determine the length of a side joining points with the same first coordinate or the same second coordinate.l Polygon
l Coordinate planeStudents know:
l Terminology associated with coordinate systems,
l Correct construction of coordinate systems.Students are able to:
l Graph points corresponding to ordered pairs,
l Represent real world and mathematical problems on a coordinate plane,
l Interpret coordinate values of points in the context of real world and mathematical situations,
l 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:
l A variety of representations such as diagrams, number lines, charts, and graphs can be used to illustrate mathematical situations and relationships,
l These representations help in conceptualizing ideas and in solving problems,
l Distances on lines parallel to the axes on a coordinate plane are the same as the related distance on the axis (numberline).Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53888" \t "_blank" ALEX Resources24. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.GeometrySolve real-world and mathematical problems involving area, surface area, and volume.6.G.4Students:Given real world and mathematical problems involving surface area,
l Use models of the relating net of the 3-D figure to explain and justify solutions and solution paths.l Nets
l Surface areaStudents know:
l Measureable attributes of objects, specifically area and surface area,
l Strategies for representing the surface area of a 3-D shape as a 2-D net.Students are able to:
l Communicate the relationships between rectangular models of area and multiplication problems,
l Model the surface area of 3-D shapes using 2-D nets,
l Accurately measure and compute area of triangles and rectangles,
l Strategically and fluently choose and apply strategies for finding surface areas of 3-D figures.Students understand that:
l Area is additive,
l Surface area of a 3-D shape is represented by the sum of the areas of the faces of the object,
l Models represent measurable attributes of objects and help to solve problems. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53889" \t "_blank" ALEX Resources25. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Example: How old am I? is not a statistical question, but How old are the students in my school? is a statistical question because one anticipates variability in students ages.Statistics & ProbabilityDevelop understanding of statistical variability.6.SP.1Students:Given a variety of mathematical questions,
l Justify the classification of questions as either statistical or non-statistical l Statistical questions
l VariabilityStudents know:
l Characteristics of statistical and non-statistical questions.Students are able to:
l Justify the classification of mathematical questions as statistical or non-statistical questions.Students understand that:
l Statistical questions have anticipated variability in the answers.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53891" \t "_blank" ALEX Resources26. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.Statistics & ProbabilityDevelop understanding of statistical variability.6.SP.2Students:
l Generate statistical questions,
l Collect and organize the data to address the questions,
l Describe the distributions of the data using measures of center (e.g., mean and median), spread, and overall shape including outliers. l Statistical question
l Distribution
l Measure of center
l Spread
l ShapeStudents Know:
l How to identify the significant features of a data set, [e.g., measures of center (mean and median), spread, and overall shape].Students are able to:
l Calculate the mean, median, and range,
l Organize data in ways that aid in identifying significant features of the data (e.g. putting data in order to find median, displaying data in a graph to see overall shape),
l Describe the distribution of a set of data by referring to measures of center, spread, and shape.Students understand that:
l A set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape,
l Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53893" \t "_blank" ALEX Resources27. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.Statistics & ProbabilityDevelop understanding of statistical variability.6.SP.3Students:Given a set of numerical data,
l Determine and interpret measures of center (mean and median) and variability (interquartile range and mean absolute deviation). l Measures of center
l Measures of variation Students know:
l Measures of center (mean and median) and how they are affected by the data distribution,
l Measures of variability (interquartile range and mean absolute deviation) and how they are affected by data distribution.Students are able to:
l Determine measures of center and variability for a set of numerical data,
l Interpret measures of center and variability for a set of numerical data.Students understand that:
l Measures of center for a set of data summarize the values in the set in a single number,
l Measures of variability for a set of data describe how the values vary in a single number. Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53894" \t "_blank" ALEX Resources28. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.Statistics & ProbabilitySummarize and describe distributions.6.SP.4Students: Given a set of numerical data,
l Organize and display the data using plots on a number line, including dot plots, histograms, and box plots.l Dot plot
l Histograms
l Box plotsStudents know:
l Techniques for constructing dot plots, histograms, and box plots.Students are able to:
l Organize and display data using dot plots, histograms, and box plots.Students understnd that:
l Sets of data can be organized and displayed in a variety of ways, each of which provides unique perspectives of the data set,
l Data displays help in conceptualizing ideas and in solving problems.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53896" \t "_blank" ALEX Resources29. Summarize numerical data sets in relation to their context, such as by:
Reporting the number of observations.
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation) as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.Statistics & ProbabilitySummarize and describe distributions.6.SP.5Students:Given a set of numerical data, summarize the data by,
l Reporting the number of observations (n),
l Describing the nature of the attribute under investigation (how it was measured and its units of measure),
l Determining the measures of center (median/mean),
l Determining the measures of variability (interquartile range and mean absolute deviation),
l Reporting significant features in the shape of data including striking deviations, (e.g., outliers, gaps, and clusters).
Given a set of numerical data,
l 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.l Data distribution
l Measures of center
l Measures of variability
l Mean
l Median
l Interquartile range
l Mean absolute deviations
l Striking deviationsStudents know:
l Measures of center and how they are affected by the data distribution and context,
l Measures of variability and how they are affected by the data distribution and context,
l Methods of determining mean, median, interquartile range, and mean absolute deviation.Students are able to:
l 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,
l Describe the shape of numerical data distribution including patterns and outliers,
l Determine measures of center and variability for a set of numerical data,
l Use characteristics of measures of center and variability to justify choices for summarizing and describing data.Students understand that:
l 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,
l 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,
l The overall shape and other significant features of a set of data, (e.g., outliers, gaps, and clusters) are important in summarizing numerical data sets.Click below to access all ALEX resources aligned to this standard.
HYPERLINK "http://alex.state.al.us/all.php?std_id=53897" \t "_blank" ALEX Resources
HYPERLINK "http://alex.state.al.us/insight/lauderdaleco/standardprint/getstandards" \l "#" print
HYPERLINK "http://alex.state.al.us/insight/lauderdaleco/standardprint/getstandards" \l "#" print
Grade 6 Mathematics CCRS Standards and Alabama COS
Franklin County Schools
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