compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line

develop meaning for percents greater than 100 and less than 1

understand and use ratios and proportions to represent quantitative relationships

develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation

use factors, multiples, prime factorization, and relatively prime numbers to solve problems

develop meaning for integers and represent and compare quantities with them

work flexibly with fractions, decimals, and percents to solve problems

select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods

develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use

develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results

develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios

work flexibly with fractions, decimals, and percents to solve problems

compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line

develop meaning for percents greater than 100 and less than 1

understand and use ratios and proportions to represent quantitative relationships

develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation

use factors, multiples, prime factorization, and relatively prime numbers to solve problems

develop meaning for integers and represent and compare quantities with them

represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules

relate and compare different forms of representation for a relationship

identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations

develop an initial conceptual understanding of different uses of variables

explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope

use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships

recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

model and solve contextualized problems using various representations, such as graphs, tables, and equations

use graphs to analyze the nature of changes in quantities in linear relationships

precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties

understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects

create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship

use coordinate geometry to represent and examine the properties of geometric shapes

use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides

describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling

examine the congruence, similarity, and line or rotational symmetry of objects using transformations

draw geometric objects with specified properties, such as side lengths or angle measures

use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume

use visual tools such as networks to represent and solve problems

use geometric models to represent and explain numerical and algebraic relationships

recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life

understand both metric and customary systems of measurement

understand relationships among units and convert from one unit to another within the same system

understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume

use common benchmarks to select appropriate methods for estimating measurements

select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision

develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes

develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders

solve problems involving scale factors, using ratio and proportion

solve simple problems involving rates and derived measurements for such attributes as velocity and density

formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population

select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots

find, use, and interpret measures of center and spread, including mean and interquartile range

discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots

use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken

make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit

use conjectures to formulate new questions and plan new studies to answer them

understand and use appropriate terminology to describe complementary and mutually exclusive events

use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations

compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models