Focus Standards
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1 Write and evaluate numeric expressions involving whole-number exponents.
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract from ” as .
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 as a product of two factors; view as both a single entity and a sum of two terms.
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 and to find the volume and surface area of a cube with sides of length .
6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression to produce the equivalent expression ; apply the distributive property to the expression to produce the equivalent expression ; apply properties of operations to to produce the equivalent expression .
6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions
and are equivalent because they name the same number regardless of which number stands for.
Reason about and solve one-variable equations and inequalities.
6.EE.B.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.B.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.B.7 Solve real-world and mathematical problems by writing and solving equations in the form and for cases in which , and are all nonnegative rational numbers.
6.EE.B.8 Write an inequality of the form or to represent a constraint or condition in a real-world mathematical problem. Recognize that inequalities of the form or have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.C.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 to represent the relationship between distance and time.
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1 Write and evaluate numeric expressions involving whole-number exponents.
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract from ” as .
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 as a product of two factors; view as both a single entity and a sum of two terms.
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 and to find the volume and surface area of a cube with sides of length .
6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression to produce the equivalent expression ; apply the distributive property to the expression to produce the equivalent expression ; apply properties of operations to to produce the equivalent expression .
6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions
and are equivalent because they name the same number regardless of which number stands for.
Reason about and solve one-variable equations and inequalities.
6.EE.B.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.B.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.B.7 Solve real-world and mathematical problems by writing and solving equations in the form and for cases in which , and are all nonnegative rational numbers.
6.EE.B.8 Write an inequality of the form or to represent a constraint or condition in a real-world mathematical problem. Recognize that inequalities of the form or have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.C.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 to represent the relationship between distance and time.
Foundational Standards
Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.B.3 Apply properties of operations as strategies to add and subtract.[1] Examples: If is known, then is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so . (Associative property of addition.)
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide.2 Examples: If is known, then is also known. (Commutative property of multiplication.) can be found by , then , or by , then . (Associative property of multiplication.) Knowing that and , one can find as (Distributive property.)
Gain familiarity with factors and multiples.
4.OA.B.4 Find all factors for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
Geometric measurement: understand concepts of angle and measure angles.
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through of a circle is called a “one-degree angle,” and can be used to measure angles.
b. An angle that turns through one-degree angles is said to have an angle measure of degrees.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
Write and interpret numerical expressions.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add and , then multiply by ” as . Recognize that is three times as large as , without having to calculate the indicated sum or product.
Analyze patterns and relationships.
5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add ” and the starting number , and given the rule “Add ” and the starting number , generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Understand the place value system.
5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of , and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of . Use whole-number exponents to denote powers of .
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the on each line and given point the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., -axis and -coordinate, -axis and -coordinate).
5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.3 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.
a. 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.
b. Solve unit rate problems including those involving unit pricing and constant speed. For 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?
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.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 as .
[1] Students need not use formal terms for these properties.
Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.B.3 Apply properties of operations as strategies to add and subtract.[1] Examples: If is known, then is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so . (Associative property of addition.)
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide.2 Examples: If is known, then is also known. (Commutative property of multiplication.) can be found by , then , or by , then . (Associative property of multiplication.) Knowing that and , one can find as (Distributive property.)
Gain familiarity with factors and multiples.
4.OA.B.4 Find all factors for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
Geometric measurement: understand concepts of angle and measure angles.
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through of a circle is called a “one-degree angle,” and can be used to measure angles.
b. An angle that turns through one-degree angles is said to have an angle measure of degrees.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
Write and interpret numerical expressions.
5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add and , then multiply by ” as . Recognize that is three times as large as , without having to calculate the indicated sum or product.
Analyze patterns and relationships.
5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add ” and the starting number , and given the rule “Add ” and the starting number , generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Understand the place value system.
5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of , and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of . Use whole-number exponents to denote powers of .
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the on each line and given point the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., -axis and -coordinate, -axis and -coordinate).
5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.3 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.
a. 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.
b. Solve unit rate problems including those involving unit pricing and constant speed. For 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?
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.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 as .
[1] Students need not use formal terms for these properties.
Focus Standards for Mathematical Practice
MP.2 Reason abstractly and quantitatively. Students connect symbols to their numerical referents. They understand exponential notation as repeated multiplication of the base number. Students realize that is represented as , with a product of , and explain how differs from , where the product is . Students determine the meaning of a variable within a real-life context. They write equations and inequalities to represent mathematical situations. Students manipulate equations using the properties so that the meaning of the symbols and variables can be more closely related to the real-world context. For example, given the expression represents how many beads are available to make necklaces, students rewrite as when trying to show the portion each person gets if there are three people, or rewrite as if there are two people sharing. Students recognize that these expressions are equivalent. Students can also use equivalent expressions to express the area of rectangles and to calculate the dimensions of a rectangle when the area is given. Also, students make connections between a table of ordered pairs of numbers and the graph of those data.
MP.6 Attend to Precision. Students are precise in defining variables. They understand that a variable represents one number. For example, students understand that in the equation , the variable can only represent one number to make the equation true. That number is , so . When variables are represented in a real-world problem, students precisely define the variables. In the equation , students define the variable as weight in pounds (or some other unit) rather than just weight. Students are precise in using operation symbols and can connect between previously learned symbols and new symbols ( can be represented with parentheses or with the multiplication dot ; similarly is also represented with the fraction bar ). In addition, students use appropriate vocabulary and terminology when communicating about expressions, equations, and inequalities. For example, students write expressions, equations, and inequalities from verbal or written descriptions. “A number increased by is equal to ” can be written as . Students refer to as a term or expression, whereas is referred to as an equation.
MP.7 Look for and make use of structure. Students look for structure in expressions by deconstructing them into a sequence of operations. They make use of structure to interpret an expression’s meaning in terms of the quantities represented by the variables. In addition, students make use of structure by creating equivalent expressions using properties. For example, students write as , , , or other equivalent expressions. Students also make sense of algebraic solutions when solving an equation for the value of the variable through connections to bar diagrams and properties. For example, when there are two copies of , this can be expressed as either , , or as . Students use tables and graphs to compare different expressions or equations to make decisions in real-world scenarios. These models also create structure as students gain knowledge in writing expressions and equations.
MP.8 Look for and express regularity in repeated reasoning. Students look for regularity in a repeated calculation and express it with a general formula. Students work with variable expressions while focusing more on the patterns that develop than the actual numbers that the variable represents. For example, students move from an expression such as to the general form , or . Similarly, student move from expressions such as to the general form . These are especially important when moving from the general form back to a specific value for the variable.
MP.2 Reason abstractly and quantitatively. Students connect symbols to their numerical referents. They understand exponential notation as repeated multiplication of the base number. Students realize that is represented as , with a product of , and explain how differs from , where the product is . Students determine the meaning of a variable within a real-life context. They write equations and inequalities to represent mathematical situations. Students manipulate equations using the properties so that the meaning of the symbols and variables can be more closely related to the real-world context. For example, given the expression represents how many beads are available to make necklaces, students rewrite as when trying to show the portion each person gets if there are three people, or rewrite as if there are two people sharing. Students recognize that these expressions are equivalent. Students can also use equivalent expressions to express the area of rectangles and to calculate the dimensions of a rectangle when the area is given. Also, students make connections between a table of ordered pairs of numbers and the graph of those data.
MP.6 Attend to Precision. Students are precise in defining variables. They understand that a variable represents one number. For example, students understand that in the equation , the variable can only represent one number to make the equation true. That number is , so . When variables are represented in a real-world problem, students precisely define the variables. In the equation , students define the variable as weight in pounds (or some other unit) rather than just weight. Students are precise in using operation symbols and can connect between previously learned symbols and new symbols ( can be represented with parentheses or with the multiplication dot ; similarly is also represented with the fraction bar ). In addition, students use appropriate vocabulary and terminology when communicating about expressions, equations, and inequalities. For example, students write expressions, equations, and inequalities from verbal or written descriptions. “A number increased by is equal to ” can be written as . Students refer to as a term or expression, whereas is referred to as an equation.
MP.7 Look for and make use of structure. Students look for structure in expressions by deconstructing them into a sequence of operations. They make use of structure to interpret an expression’s meaning in terms of the quantities represented by the variables. In addition, students make use of structure by creating equivalent expressions using properties. For example, students write as , , , or other equivalent expressions. Students also make sense of algebraic solutions when solving an equation for the value of the variable through connections to bar diagrams and properties. For example, when there are two copies of , this can be expressed as either , , or as . Students use tables and graphs to compare different expressions or equations to make decisions in real-world scenarios. These models also create structure as students gain knowledge in writing expressions and equations.
MP.8 Look for and express regularity in repeated reasoning. Students look for regularity in a repeated calculation and express it with a general formula. Students work with variable expressions while focusing more on the patterns that develop than the actual numbers that the variable represents. For example, students move from an expression such as to the general form , or . Similarly, student move from expressions such as to the general form . These are especially important when moving from the general form back to a specific value for the variable.