In this unit, students learn about negative numbers and ways to represent them on a number line and the coordinate plane. They write and graph simple inequalities in one variable and determine the greatest common factor and least common multiple of two whole numbers. 

• Interpret a rational number and the absolute value of a number in context.

• Plot rational numbers and their opposites on a number line; know that a number and its opposite have the same absolute value.

• Use words and symbols to compare rational numbers, where a rational number could also be the absolute value of a number.

• Determine whether a given value is a solution to a given inequality.

• Draw and label a number line diagram to represent the solutions to an inequality.

• Write an inequality statement to represent a constraint.

• 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.

• Use coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

• Working in all four quadrants, plot a point given its coordinates or identify the coordinates of a given point in the coordinate plane.

• List the factors of a number and identify common factors for two numbers in a real-world situation.

• List the multiples of a number and identify common multiples for two numbers in a real-world situation.

• Generate a list of ordered pairs to create an image in the coordinate plane, and explain (orally) the reasoning.

A: Negative Numbers and Absolute Value

• I can explain what 0, positive numbers, and negative numbers mean in the context of temperature and elevation.

• I can use positive and negative numbers to describe temperature and elevation.

• I know what positive and negative numbers are.

• I can represent negative numbers on a number line.

• I can tell or approximate the value of any point on a number line.

• I understand what it means for numbers to be opposites.

• I can explain how to use the positions of numbers on a number line to compare them.

• I can use inequalities to compare positive and negative numbers.

• I can compare and order rational numbers.

• I can use phrases like “greater than,” “less than,” and “opposite” to compare rational numbers.

• I can explain and use negative numbers in situations involving money.

• I can interpret and use negative numbers in different contexts.

• I can explain what the absolute value of a number is.

• I can find the absolute values of rational numbers.

• I can recognize and use the notation for absolute value.

• I can explain what absolute value means in situations involving elevation.

• I can use absolute values to describe elevations.

• I can use inequalities to compare rational numbers and the absolute values of rational numbers.

B: Inequalities

• I can graph inequalities on a number line.

• I can write an inequality to represent a situation.

• I can explain what it means for a number to be a solution to an inequality.

• I can graph the solutions to an inequality on a number line.

• I can tell if a particular number is a solution to an inequality.

• I can explain what the solution to an inequality means in a situation.

• I can write inequalities that involve more than one variable.

C: The Coordinate Plane

• I can plot points with negative coordinates in the coordinate plane.

• I know what a coordinate plane is and can describe the four quadrants.

• I know what negative numbers in coordinates tell us.

• When given points to plot, I can construct a coordinate plane with an appropriate scale and pair of axes.

• I can explain how rational numbers represent balances in a money context.

• I can explain what points in a four-quadrant coordinate plane represent in a situation.

• I can plot points in a four-quadrant coordinate plane to represent situations and solve problems.

• I can find horizontal and vertical distances between points on the coordinate plane.

• I can find the lengths of horizontal and vertical segments in the coordinate plane.

• I can plot polygons on the coordinate plane when I have the coordinates for the vertices.

D: Common Factors and Common Multiples

• I can explain what a common factor is.

• I can explain what the greatest common factor is.

• I can find the greatest common factor of two whole numbers.

• I can explain what a common multiple is.

• I can explain what the least common multiple is.

• I can find the least common multiple of two whole numbers.

• I can solve problems using common factors and multiples.

In this unit, students learn to write and solve expressions 

• Solve equations of the form x+p=q or px=q and explain the solution method.

• Understand that solving an equation with a variable means finding a value for the variable that makes the equation true, and use substitution to determine whether a number is a solution to the equation.

• Justify whether two expressions are “equivalent,” or equal to each other for every value of their variable.

• Use the distributive property to write equivalent algebraic expressions.

• Evaluate expressions with whole-number exponents at specific values of their variables.

• Justify whether numerical expressions involving whole-number exponents are equal by evaluating the expressions and performing operations in the conventional order.

• Create a table, graph, and equation to represent the relationship between two quantities.

• Identify the independent and dependent variable in a situation.

A: Equations in one Variable

• I can tell whether or not an equation could represent a tape diagram.

• I can use a tape diagram to represent an equation.

• I can replace a variable in an equation with a number that makes the equation true, and know that this number is called a “solution” to the equation.

• I can compare the process of removing or grouping weights to keep a hanger diagram balanced and the process of subtracting or dividing numbers to solve an equation.

• I can explain what a balanced hanger diagram and a true equation have in common.

• I can write equations that could represent the weights on a balanced hanger diagram.

• I can solve addition and multiplication equations with one variable.

• I can explain how an equation with a variable represents a real-world problem.

• I can use equations with variables to solve real-world problems.

• I can solve percent problems by writing and solving equations.

B: Equal and Equivalent

• I can use an expression that represents a situation to find an amount in a story.

• I can write an expression with a variable to represent a calculation where I do not know one of the numbers.

• I can explain what it means for two expressions to be equivalent.

• I can use what I know about operations to decide whether two expressions are equivalent.

• I can use a diagram of a rectangle split into two smaller rectangles to write different expressions representing its area.

• I can use the distributive property to explain how two expressions with numbers are equivalent.

• I can use a diagram of a split rectangle to write different expressions with variables representing its area.

• I can use the distributive property to write equivalent expressions with variables.

C: Expressions with Exponents

• I can find the value of expressions with exponents and write expressions with exponents that are equal to a given number.

• I understand the meaning of an expression with an exponent 

• I can decide if expressions with exponents are equal by finding the values of the expressions or by understanding what exponents mean.

• I know how to find the value of expressions that have both an exponent and addition or subtraction.

• I know how to find the value of expressions that have both an exponent and multiplication or division.

• I can find solutions to equations with exponents in a list of numbers.

• I can replace a variable with a number in an expression with exponents and use the correct order of operations to find the value of the expression.

D: Relationships Between Quantities

• I can create tables and graphs that show the relationship between two amounts.

• I can write an equation with two variables that shows the relationship between two amounts.

• I can create tables and graphs to represent the relationship between distance and time for something moving at a constant speed.

• I can write an equation with variables to represent the relationship between distance and time for something moving at a constant speed.

• I can create tables and graphs that show different kinds of relationships between amounts.

• I can write equations that describe relationships with area and volume. 

In this unit, students learn adding, subtracting, multiplying, and dividing numbers with decimals

• Explain the sum or difference of two decimals in terms of combining like base-ten units and composing (or decomposing) a larger unit from (or into 10) units of a lower value.

• Use the standard algorithm to add or subtract decimals with multiple non-zero digits.

• Calculate the product of two decimals and explain the solution method.

• Use an algorithm to calculate the product of whole numbers, and justify how to use that value to find the product of two decimals with the same significant digits.

• Calculate the quotient of a whole-number or a decimal dividend and a decimal divisor, and explain the solution method.

• Use long division to divide a multi-digit whole number or a decimal by a whole number.

A: Exploring, Adding, and Subtracting Decimals

• I can use decimals to make estimates and calculations about money.

• I can use diagrams to represent and reason about addition and subtraction of decimals.

• I can use place value to explain addition and subtraction of decimals.

• I can use vertical calculations to represent and reason about addition and subtraction of decimals.

• I can tell whether writing or removing a zero in a decimal will change its value.

• I know how to solve subtraction problems with decimals that require decomposing.

• I can solve problems that involve addition and subtraction of decimals.

B: Multiplying Decimals

• I can use place value and fractions to reason about the multiplication of decimals.

• I can use area diagrams to represent and reason about multiplication of decimals.

• I know and can explain more than one way to multiply decimals using fractions and place value.

• I can use area diagrams and partial products to represent and find products of decimals.

• I can describe and apply a method for multiplying decimals.

• I know how to use a product of whole numbers to find a product of decimals.

C: Dividing Decimals

• I can use base-ten diagrams to represent division of whole numbers and division of a decimal by a whole number.

• I can use partial quotients to find a quotient of two whole numbers.

• I can use long division to find a quotient of two whole numbers when the quotient is a whole number.

• I can use long division to divide two whole numbers when the quotient is not a whole number, or to divide a decimal by a whole number.

• I can explain how multiplying the dividend and the divisor by the same power of 10 can help me find a quotient of two decimals.

• I can find the quotient of two decimals.

In this unit, students learn multiply and divide with fractional forms

• Create a diagram and a multiplication or division equation to represent the relationship in situations involving equal-size groups, and coordinate these representations.

• Interpret a division expression in two ways: as an answer to a "How many groups?" question or to a "How many in each group?" question.

• Create a diagram, a multiplication equation, and a division equation to represent a problem that involves a fractional divisor and that asks "How many groups?" or "How much in one group?"

• Solve problems involving division of fractions by using diagrams, writing equations, or reasoning about the relationship between multiplication and division. Explain the solution methods.

• Generalize a process for dividing a number n by a fraction 

• Use the fraction division algorithm to calculate quotients.

• Determine the volume of a rectangular prism by packing it with unit cubes with appropriate unit-fraction edge lengths and by multiplying the edge lengths.

• Solve problems about length comparison, the side lengths and area of a rectangle, and the edge lengths and volume of a rectangular prism using fraction division.

A: Making Sense of Division

• When dividing, I know how the size of a divisor affects the quotient.

• I can explain how multiplication and division are related.

• I can explain two ways of interpreting a division expression such as 27÷3

• When given a division equation, I can write a multiplication equation that represents the same situation.

• I can create a diagram or write an equation that represents division and multiplication questions.

• I can decide whether a division question is asking “How many groups?” or “How many in each group?”.

B: Meanings of Fraction Division

• I can find how many groups there are when the amount in each group is not a whole number.

• I can use diagrams and multiplication and division equations to represent “How many groups?” questions.

• I can find how many groups there are when the number of groups and the amount in each group are not whole numbers.

• I can use a tape diagram to represent equal-size groups and to find the number of groups.

• I can tell when a question is asking for the number of groups and that number is less than 1.

• I can use diagrams and multiplication and division equations to represent and answer “What fraction of a group?” questions.

• I can tell when a question is asking for the amount in one group.

• I can use diagrams and multiplication and division equations to represent and answer “How much in each group?” questions.

• I can find the amount in one group in different real-world situations.

C: Algorithm for Fraction Division

• I can divide a number by a non-unit fraction by reasoning with the numerator and denominator, which are whole numbers.

• I can divide a number by a unit fraction 

•  by reasoning with the denominator, which is a whole number.

• I can describe and apply a rule to divide numbers by any fraction.

D: Fractions in Lengths, Areas, and Volumes

• I can use division and multiplication to solve problems involving fractional lengths.

• I can use division and multiplication to solve problems involving areas of rectangles with fractional side lengths.

• I can explain how to find the volume of a rectangular prism using cubes that have a unit fraction as their edge length.

• I can use division and multiplication to solve problems involving areas of triangles with fractional bases and heights.

• I know how to find the volume of a rectangular prism even when the edge lengths are not whole numbers.

• I can solve volume problems that involve fractions.

In this unit, students learn unit rates and percentages

• Recognize that when we measure things in two different units, the pairs of measurements are equivalent ratios.

• Apply this understanding to convert a measurement from one unit to another unit.

• Use a “rate per 1” to solve problems involving unit conversion.

• Apply reasoning about ratios and unit rates to solve problems and explain the solution methods.

• Calculate the two unit rates associated with a ratio and interpret them in the context of a situation.

• Choose a strategy to solve problems involving percentages and explain the solution method.

• Create tape diagrams, double number line diagrams, or tables to represent situations involving percentages.

A: Units of Measurement and Unit Conversion

• I can name common objects that are about as long as 1 inch, foot, yard, mile, millimeter, centimeter, meter, or kilometer.

• I can name common objects that weigh about 1 ounce, pound, ton, gram, or kilogram, or that hold about 1 cup, quart, gallon, milliliter, or liter.

• When I read or hear a unit of measurement, I know whether it is used to measure length, weight, or volume.

• When I know a measurement in one unit, I can decide whether it takes more or less of a different unit to measure the same quantity.

• I can convert measurements from one unit to another, using double number lines, tables, or by thinking about “how much for 1.”

• I know that when we measure things in two different units, the pairs of measurements are equivalent ratios.

B: Rates

• I understand that if two ratios have the same rate per 1, they are equivalent ratios.

• When measurements are expressed in different units, I can decide who is traveling faster or which item is the better deal by comparing “how much for 1” of the same unit.

• I can choose which unit rate to use based on how I plan to solve the problem.

• When I have a ratio, I can calculate its two unit rates and explain what each of them means in the situation.

• I can give an example of two equivalent ratios and show that they have the same unit rates.

• I can multiply or divide by the unit rate to calculate missing values in a table of equivalent ratios.

• I can choose how to use unit rates to solve problems.

• I can see that thinking about “how much for 1” is useful for solving different types of problems.

• I can solve more complicated problems about constant speed situations.

C: Percentages

• I can create a double number line diagram with percentages on one line and dollar amounts on the other line.

• I can explain the meaning of percentages using dollars and cents as an example.

• I can use double number line diagrams to solve different problems like “What is 40% of 60?” or “60 is 40% of what number?”

• I can use tape diagrams to solve different problems like “What is 40% of 60?” or “60 is 40% of what number?”

• When I read or hear that something is 10%, 25%, 50%, or 75% of an amount, I know what fraction of that amount they are referring to.

• I can choose and create diagrams to help me solve problems about percentages.

• I can solve different problems like “What is 40% of 60?” by dividing and multiplying.

• I can solve different problems like “60 is what percentage of 40?” by dividing and multiplying.

In this unit, students learn ratios and equivalent ratios.

• Create and interpret discrete diagrams that represent situations involving ratios.

• Create and interpret sentences that describe ratios.

• Explain equivalent ratios in terms of the quantities in a recipe being multiplied by the same number to create a different size batch of something with the same taste or same shade.

• Generate equivalent ratios and justify that they are equivalent.

• Given prices and quantities or distances and times, calculate the price for 1 object or the distance traveled in 1 unit of time. Express each using the word “per.”

• Justify whether two situations do or do not happen at the same rate, by determining if the ratios in the situations are equivalent.

• Use a double number line diagram to represent and find equivalent ratios.

• Choose multipliers strategically when solving multi-step problems involving equivalent ratios.

• Use a table of equivalent ratios to solve problems.

• Choose representations and solution methods to reason about ratios and sums of quantities.

• Use diagrams or other strategies to solve problems involving ratios and the total amount.

A: What are Ratios?

• I can write or say a sentence that describes a ratio.

• I know how to say words and numbers in the correct order to accurately describe the ratio.

• I can draw a diagram that represents a ratio and explain what the diagram means.

• I include labels when I draw a diagram that represents a ratio, so that the meaning of the diagram is clear.

B: Equivalent Ratios

• I can explain what it means for two ratios to be equivalent using a recipe as an example.

• I can use a diagram to represent a recipe and to represent a double batch and a triple batch of the recipe.

• I know what it means to double or triple a recipe.

• I can explain the meaning of equivalent ratios using a color mixture as an example.

• I can use a diagram to represent a single batch, a double batch, and a triple batch of a color mixture.

• I know what it means to double or triple a color mixture.

• If I have a ratio, I can create a new ratio that is equivalent to it.

• If I have two ratios, I can decide whether they are equivalent to each other.

C: Representing Equivalent Ratios

• I can label a double number line diagram to represent batches of a recipe or color mixture.

• When I have a double number line that represents a situation, I can explain what it means.

• I can create a double number line diagram and correctly place and label tick marks to represent equivalent ratios.

• I can explain what the word "per" means.

• I can choose and create diagrams to help me reason about prices.

• I can explain what the phrase “at this rate” means, using prices as an example.

• If I know the price of multiple things, I can find the price per thing.

• I can choose and create diagrams to help me reason about constant speed.

• If I know that an object is moving at a constant speed, and I know two of these things: the distance it travels, the amount of time it takes, and its speed, I can find the other thing.

• I can decide whether or not two situations are happening at the same rate.

• I can explain what it means when two situations happen at the same rate.

• I know some examples of situations where things can happen at the same rate.

D: Solving Ratio and Rate Problems

• If I am looking at a table of values, I know where the rows are and where the columns are.

• When I see a table representing a set of equivalent ratios, I can come up with numbers to make a new row.

• When I see a table representing a set of equivalent ratios, I can explain what the numbers mean.

• I can solve problems about situations happening at the same rate by using a table and finding a “1” row.

• I can use a table of equivalent ratios to solve problems about unit price.

• I can create a table that represents a set of equivalent ratios.

• I can explain why sometimes a table is easier to use than a double number line to solve problems involving equivalent ratios.

• I include column labels when I create a table, so that the meaning of the numbers is clear.

• I can decide what information I need to know to be able to solve problems about situations happening at the same rate.

• I can explain my reasoning using diagrams that I choose.

E: Part-part-whole Ratios

• I can create tape diagrams to help me reason about problems that involve both a ratio and a total amount.

• I can solve problems when I know a ratio and a total amount.

• I can choose and create diagrams to help think through my solution.

• I can solve all kinds of problems about equivalent ratios.

• I can use diagrams to help someone else understand why my solution makes sense.

In this unit, students learn to find areas of polygons by decomposing, rearranging, and composing shapes.

• Find the area of a two-dimensional region with straight boundaries by decomposing, rearranging, subtracting, or enclosing shapes, and explain the solution method.

• Apply the formula for area of a parallelogram to find the area, the length of a base, or the height.

• Understand bases and heights in a parallelogram and recognize base-height pairs to use to calculate its area.

• Find the area of a polygon by decomposing it into parallelograms and triangles.

• Understand bases and heights in a triangle and recognize base-height pairs to use to find the area of a triangle.

• Understand why the process of finding the area of a triangle can be abstracted as 12⋅b⋅h (or equivalent) and apply the formula to find the area of a triangle.

• Identify or create a net that represents a prism or pyramid.

• Use a net to calculate the surface area of a prism or pyramid and explain the solution method.

• Interpret and write expressions with exponents 2 and 3 to represent the area of a square or the volume of a cube. Write expressions, with or without exponents, to represent the surface area of a given cube.

A: Reasoning to find Area

• I can explain the meaning of "area."

• I can explain how to find the area of a figure that is composed of other shapes.

• I know how to find the area of a figure by decomposing it and rearranging the parts.

• I know what it means for two figures to have the same area.

• I can use different reasoning strategies to find the area of shapes.

B: Parallelograms

• I can use reasoning strategies and what I know about the area of a rectangle to find the area of a parallelogram.

• I know how to describe the characteristics of a parallelogram using mathematical vocabulary.

• I can identify pairs of base and height of a parallelogram.

• I can write and explain the formula for the area of a parallelogram.

• I know what the terms "base" and "height" refer to in a parallelogram.

• I can use the area formula to find the area of any parallelogram.

C: Triangles and Other Polygons

• I can explain the special relationship between a pair of identical triangles and a parallelogram.

• I can use what I know about parallelograms to reason about the area of triangles.

• I can use the area formula to find the area of any triangle.

• I can write and explain the formula for the area of a triangle.

• I know what the terms “base” and “height” refer to in a triangle.

• I can identify pairs of base and corresponding height of any triangle.

• When given information about a base of a triangle, I can identify and draw a corresponding height.

• I can describe the characteristics of a polygon using mathematical vocabulary.

• I can reason about the area of any polygon by decomposing and rearranging it, and by using what I know about rectangles and triangles.

D: Surface Area

• I know what the surface area of a three-dimensional object means.

• I can describe the features of a polyhedron using mathematical vocabulary.

• I can explain the difference between prisms and pyramids.

• I understand the relationship between a polyhedron and its net.

• I can match polyhedra to their nets and explain how I know.

• When given a net of a prism or a pyramid, I can calculate its surface area.

• I can calculate the surface area of prisms and pyramids.

• I can draw the nets of prisms and pyramids.

• I can explain how it is possible for two polyhedra to have the same surface area but different volumes, or to have different surface areas but the same volume.

• I know how one-, two-, and three-dimensional measurements and units are different.

E: Squares and Cubes

• I can write and explain the formula for the volume of a cube, including the meaning of the exponent.

• When I know the edge length of a cube, I can find the volume and express it using appropriate units.

• I can write and explain the formula for the surface area of a cube.

• When I know the edge length of a cube, I can find its surface area and express it using appropriate units.