Standards - Mathematics

2019 - Mathematics Course of Study Document
Mathematics (2019) Grade(s): 6

MA19.6.4

Interpret and compute quotients of fractions using visual models and equations to represent problems.

Unpacked Content

Knowledge

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.

Skills

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.

Understanding

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.

Vocabulary

  • Visual fraction models
  • Dividend
  • Divisor
  • Quotient
  • Equation
  • Numerator
  • Denominator
  • Mixed number
  • Improper fraction

Aligned Learning Resources

Varieties of Quadrilaterals
Mathematics (2019) Grade(s): 6

MA19.6.5

Fluently divide multi-digit whole numbers using a standard algorithm to solve real-world and mathematical problems.

Unpacked Content

Knowledge

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

Skills

Students are able to:
  • Strategically choose and apply appropriate strategies for dividing.
  • Accurately find quotients using the standard division algorithm.

Understanding

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.

Vocabulary

  • Standard algorithm
  • Dividend
  • Divisor
  • Quotient
Mathematics (2019) Grade(s): 6

MA19.6.6

Add, subtract, multiply, and divide decimals using a standard algorithm.

Unpacked Content

Knowledge

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.

Skills

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.

Understanding

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.

Vocabulary

  • Standard algorithms (addition, subtraction, multiplication, and division)
  • Quotient
  • Sum
  • Product
  • Difference
  • Tenths
  • Hundredths
  • Thousandths
  • Ten thousandths
  • Hundred thousandths

Aligned Learning Resources

Decimals: Carlos the Centipede Goes Shopping for Holiday Gifts!
Mathematics (2019) Grade(s): 6

MA19.6.7

Use 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.

Unpacked Content

Knowledge

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.

Skills

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.

Understanding

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.

Vocabulary

  • Greatest common factor
  • Distributive property
  • Parentheses
  • Decompose

Aligned Learning Resources

Dividing a Whole Number by a Unit Fraction
Graphing in Quadrant 1 in the Coordinate Plane
Mathematics (2019) Grade(s): 6

MA19.6.9

Use signed numbers to describe quantities that have opposite directions or values and to represent quantities in real-world contexts.

Unpacked Content

Knowledge

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

Skills

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

Understanding

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).

Vocabulary

  • Positive Numbers
  • Negative Numbers
  • Opposites
Mathematics (2019) Grade(s): 6

MA19.6.10

Locate integers and other rational numbers on a horizontal or vertical line diagram.

Unpacked Content

Knowledge

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.

Skills

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.

Understanding

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.

Vocabulary

  • Integers
  • Rational numbers
  • Horizontal line diagram
  • Vertical line diagram

Aligned Learning Resources

Wonder Number Line
Tell Temperature StudyJam
Lining Up With Integers
Jumping Around With Integers
Mathematics (2019) Grade(s): 6

MA19.6.10b

Use rational numbers in real-world and mathematical situations, explaining the meaning of 0 in each situation.

Mathematics (2019) Grade(s): 6

MA19.6.11

Find the position of pairs of integers and other rational numbers on the coordinate plane.

Unpacked Content

Knowledge

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

Skills

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).

Understanding

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).

Vocabulary

  • Coordinate plane
  • Quadrants
  • Coordinate values
  • ordered pairs
  • x axis
  • y axis
  • Reflection

Aligned Learning Resources

Michael Phelps.... or not?
Cyberchase Games: Pattern Recognition
Treasure Map Graphing
Graphville Shopping Excursion
Introducing the Four Quadrants of the Coordinate Plane
Getting to Know the Cartesian Plane
Finding the Four Quadrants
Graphing on the Coordinate Plane
Practice Graphing on the Coordinate Plane
A Basic Review of Plot Elements (Starring Spongebob Squarepants)
Mathematics (2019) Grade(s): 6

MA19.6.11a

Identify quadrant locations of ordered pairs on the coordinate plane based on the signs of the x and y coordinates.

Mathematics (2019) Grade(s): 6

MA19.6.11d

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane, including finding distances between points with the same first or second coordinate.

Mathematics (2019) Grade(s): 6

MA19.6.12

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

Unpacked Content

Knowledge

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

Skills

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

Understanding

Students understand that:
  • the absolute value of a number is its distance from zero.

Vocabulary

  • Absolute value
  • Inequality

Aligned Learning Resources

Integers StudyJam
Mathematics (2019) Grade(s): 6

MA19.6.13

Compare and order rational numbers and absolute value of rational numbers with and without a number line in order to solve real-world and mathematical problems.

Unpacked Content

Knowledge

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.

Skills

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.

Understanding

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.

Vocabulary

  • Absolute Value
  • Inequalities

Aligned Learning Resources

Integers StudyJam
Mathematics (2019) Grade(s): 7

MA19.7.4

Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

Unpacked Content

Knowledge

Students know:
  • a number and its opposite have a sum of 0.
  • A number and its opposite are called additive inverses.
  • Strategies for adding and subtracting two or more numbers.
  • Absolute value represents distance on a number line, therefore it is always non-negative.
  • Strategies for multiplying signed numbers.
  • Every quotient of integers (with non-zero divisor) is a rational number.
  • If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
  • The decimal form of a rational number terminates or eventually repeats.

Skills

Students are able to:
  • add rational numbers.
  • Subtract rational numbers.
  • Represent addition and subtraction on a number line diagram.
  • Describe situations in which opposite quantities combine to make 0.
  • Find the opposite of a number.
  • Interpret sums of rational numbers by describing real-world contexts.
  • Show that the distance between two rational numbers on the number line is the absolute value of their difference.
  • Use absolute value in real-world contexts involving distances.
  • Multiply and divide rational numbers.
  • Convert a rational number to a decimal using long division.

Understanding

Students understand that:
  • finding sums and differences of rational numbers (negative and positive) involves determining direction and distance on the number line.
  • Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
  • If a factor is multiplied by a number greater than one, the answer is larger than that factor.
  • If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
  • Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
  • Integers can be divided, provided that the divisor is not zero.

Vocabulary

  • Integers
  • Rational numbers
  • Additive inverses
  • opposite quantities
  • Absolute value
  • Terminating decimals
  • Repeating decimals

Aligned Learning Resources

Adding and Subtracting Rational Numbers Assessment
Cyberchase: Frog Hops, Part 1
Formulas: Volume
How Does it Change?
How to Avoid Inappropriate Shifts | No Nonsense Grammar
How to Use Commas for Introductory Elements | No Nonsense Grammar
How to Use Commas for Yes, No, Tag Questions, and Direct Address | No Nonsense Grammar
How to Use Coordinating Conjunctions | No Nonsense Grammar
Varieties of Quadrilaterals
Casey Runner: Adding and Subtracting Positive and Negative Numbers
A Trick for Adding and Subtracting Negative Numbers
Using Number Lines to Model Real-World Problems
Adding Integers Using Two-Color Counters
Locating Points on a Coordinate Plane
Music Concert: Multiplication and Division Part II
My New Bedroom
Mathematics (2019) Grade(s): 7

MA19.7.4a

Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.

Mathematics (2019) Grade(s): 7

MA19.7.4d

Use a number line to demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

Mathematics (2019) Grade(s): 7

MA19.7.4e

Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.

Mathematics (2019) Grade(s): 7

MA19.7.4f

Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non- zero divisor) as a rational number.

Mathematics (2019) Grade(s): 7

MA19.7.4g

Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats.

Mathematics (2019) Grade(s): 7

MA19.7.5

Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable.

Unpacked Content

Knowledge

Students know:
  • how to model real-world problems to include situations involving elevation, temperature changes, debits and credits, and proportional relationships with negative rates of change.
  • how to evaluate numerical expressions with greater fluency, using the properties of operations when necessary.

Skills

Students are able to:
  • Solve real-world and mathematical problems involving the four operations with rational numbers.

Understanding

Students understand that:
  • rational numbers can represent values in real-world situations.
  • properties of operations learned with whole numbers in elementary apply to rational numbers.

Vocabulary

  • Rational numbers
  • Complex fractions
  • properties of operations
Mathematics (2019) Grade(s): 8

MA19.8.1

Define the real number system as composed of rational and irrational numbers.

Unpacked Content

Knowledge

Students:know that any ratio a/b, where b is not equal to zero, has a quotient attained by dividing a by b.
  • know that the real number system contains natural numbers, whole numbers, integers, rational, and irrational numbers.
  • know that every real number has a decimal expansion that is repeating, terminating, or is non-repeating and non-terminating.

    • Skills

    Students are able to:
    • define the real number system by giving its components.
    • Explain the difference between rational and irrational numbers. specifically how their decimal expansions differ.
    • Convert a ratio into its decimal expansion and take a decimal expansion back to ratio form.

    Understanding

    Students understand that:
    • all real numbers are either rational or irrational and
    • Every real number has a decimal expansion that repeats, terminates, or is both non-repeating and non-terminating.

    Vocabulary

    • Real Number System
    • Ratio
    • Rational Number
    • Irrational Number

    Aligned Learning Resources

    Cyberchase Games: Rounding
    Mathematics (2019) Grade(s): 8

    MA19.8.1a

    Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates.

    Mathematics (2019) Grade(s): 8

    MA19.8.2

    Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.

    Unpacked Content

    Knowledge

    Students know:
    • the difference between a rational and an irrational number.
    • That real numbers and their decimal expansions can be approximated using a common place value to compare those expansions.

    Skills

    Students know:the difference between a rational and an irrational number.
  • That real numbers and their decimal expansions can be approximated using a common place value to compare those expansions.

    • Understanding

    Students understand that:
    • an estimation of the value of an irrational number can be used to compare an irrational number to other numbers and to place them on a number line.

    Vocabulary

    • Rational
    • Irrational

    Aligned Learning Resources

    Recursive and Explicit Equation Card Sort
    Recursive or Explicit, Why Not Both? Exit Ticket
    Same Area, Same Perimeter?
    Mathematics (2019) Grade(s): 7 - Grade 7 Accelerated

    MA19.7A.8

    Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

    Unpacked Content

    Knowledge

    Students know:
    • a number and its opposite have a sum of 0.
    • A number and its opposite are called additive inverses.
    • properties of operations.
    • Absolute value represents distance on a number line, therefore it is always non-negative.
    • Every quotient of integers (with non-zero divisor) is a rational number.
    • If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
    • The decimal form of a rational number terminates in 0s or eventually repeats.

    Skills

    Students are able to:
    • add rational numbers.
    • Subtract rational numbers.
    • Represent addition and subtraction on a number line diagram.
    • Describe situations in which opposite quantities combine to make 0.
    • Find the opposite of a number.
    • Interpret sums of rational numbers by describing real-world contexts.
    • Show that the distance between two rational numbers on the number line is the absolute value of their difference.
    • Use absolute value in real-world contexts involving distances.
    • Multiply and divide rational numbers.
    • Convert a rational number to a decimal using long division.

    Understanding

    Students understand that:
    • p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative.
    • Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
    • If a factor is multiplied by a number greater than one, the answer is larger than that factor.
    • If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
    • Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
    • Integers can be divided, provided that the divisor is not zero.

    Vocabulary

    • Integers
    • Rational numbers
    • Additive inverses
    • opposite quantities
    • Absolute value
    • Terminating decimals
    • Repeating decimals

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