All Lessons

(A) Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values

(B) Determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one

(E) Multiply and divide positive rational numbers fluently.

(Y7) Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)

6.NS.1 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. For 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?

(D) Determine the distance between two points on a coordinate plane using the Pythagorean Theorem.

(Y9) Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)

6.NS.8 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.

(C) Solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm

(Y5) Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101)

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

(E) Multiply and divide positive rational numbers fluently.

(Y7) Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)

6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

(F) Analyze mathematical relationships to connect and communicate mathematical ideas

(Y5) Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098)

6.NS.4 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 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

(B) Represent solutions for one-variable, one-step equations and inequalities on number lines

(Y10) Solve linear inequalities and graph their solutions on a number line (ACMNA236)

6.NS.7.A Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For 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.

6.NS.7.B Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3 oC > -7 oC to express the fact that -3 oC is warmer than -7 oC.

(A) Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

(Y7) Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149)

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

(Y8) Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution (ACMNA194)

6.EE.5 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.

6.EE.6 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.

6.EE.7 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.

(6) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships.

(Y7) Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)

(Y7) Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176)

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

(D) Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

(Y7) Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177)

6.EE.3 Apply the properties of operations to generate equivalent expressions. For 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.

(I) Write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

(Y11-12) Identify the dependent and independent variable (ACMEM140)

6.EE.9 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. For 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.

(A) Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

(Y7) Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149)

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

(A) Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

(Y6) Explore the use of brackets and order of operations to write number sentences (ACMNA134)

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

(C) Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.

(A) Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems

(Y7) Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)

6.EE.2.A Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 - y.

(A) Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

(Y6) Explore the use of brackets and order of operations to write number sentences (ACMNA134)

6.EE.2.C 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). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

(D) Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

6.EE.3 Apply the properties of operations to generate equivalent expressions. For 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.

(6) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships.

(Y7) Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

6.EE.2.B 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. For 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.

(D) Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

(Y6) Explore the use of brackets and order of operations to write number sentences (ACMNA134)

(Y11-12) Apply arithmetic operations according to their correct order (ACMEM002)

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

(6) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships.

(Y7) Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

(7) Expressions, equations, and relationships. The student applies mathematical process standards to develop concepts of expressions and equations.

(Y7) Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176)

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

6.EE.2.C 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). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.