Mathematics

Ratios and Proportional Relationships

• 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
• 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.

The Number System

• interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions
• fluently divide multi-‐digit numbers using the standard algorithm
• fluently add, subtract, multiply, and divide multi-‐digit decimals using the standard algorithm for each operation
• 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
• apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems
• understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-‐ world contexts, explaining the meaning of 0 in each situation
• 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
• understand ordering and absolute value of rational numbers.

Expressions and Equations

• write and evaluate numerical expressions involving whole-‐number exponents write, read, and evaluate expressions in which letters stand for numbers
• 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
• 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
• 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.

Geometry, Statistics and Probability

• solve real-‐world and mathematical problems involving area, surface area, and volume develop understanding of statistical variability
• summarize and describe distributions.

Ratios and Proportional Relationships

• compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units
• recognize and represent proportional relationships between quantities
• use proportional relationships to solve multi-‐step ratio and percent problems.

The Number System

• apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram
• apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers
• solve real-‐world and mathematical problems involving the four operations with  rational numbers
• compare, order, estimate, and translate among integers, fractions and mixed numbers (i.e., rational numbers), decimals, and percents.

Expressions and Equations

• apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients
• solve multi-‐step real-‐life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically
• use variables to represent quantities in a real-‐world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Geometry

• solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale
• describe the two-‐dimensional figures that result from slicing three-‐dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids
• know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle
• use facts about supplementary, complementary, vertical, and adjacent angles in a multi-‐step problem to write and use them to solve simple equations for an unknown angle in a figure
• solve real-‐world and mathematical problems involving area, volume, and surface area of two-‐ and three-‐dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Statistics and Probability

• understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population; understand that random sampling tends to produce representative samples and support valid inferences
• use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations
• understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring
• find probabilities of compound events using organized lists, tables, tree diagrams, and simulation

The Number System

• know that numbers that are not rational are called irrational; understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number
• use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.

Expressions and Equations

• know and apply the properties of integer exponents to generate equivalent numerical expressions use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number; evaluate square roots of small perfect squares and cube roots of small perfect cubes; know that √2 is irrational.
• use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used; use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading); interpret scientific notation that has been generated by technology
• graph proportional relationships, interpreting the unit rate as the slope of the graph; compare two different proportional relationships represented in different ways
• solve linear equations in one variable
• analyze and solve pairs of simultaneous linear equations.

Functions

• understand that a function is a rule that assigns to each input exactly one output; the graph of a function is the set of ordered pairs consisting of an input and the corresponding output
• compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)
• interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear
• construct a function to model a linear relationship between two quantities; determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph; interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values
• describe qualitatively the functional relationship between two quantities by analyzing a graph; sketch a graph that exhibits the qualitative features of a function that has been described verbally

Geometry

• verify experimentally the properties of rotations, reflections, and translations:
  • lines are taken to lines, and line segments to line segments of the same length
  • angles are taken to angles of the same measure
  • parallel lines are taken to parallel lines
• understand that a two-‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
• describe the effects of dilations, translations, rotations, and reflections on two-‐dimensional figures using coordinates
• understand that a two-‐dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-‐ dimensional figures, describe a sequence that exhibits the similarity between them
• apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-‐world and mathematical problems in two and three dimensions
• apply the Pythagorean Theorem to find the distance between two points in a coordinate system know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-‐world and mathematical problems.

Statistics

• construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities; describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association
• know that straight lines are widely used to model relationships between two quantitative variables; for scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line
• use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.