Standards
Pose statistical investigative questions about a broader population using samples taken from the population.
Generate resourceUnderstand information from a sample is valid only if the sample is representative of that population. Understand data can be used to make comparisons between different groups at one point in time and the same group over time.
Generate resourceIdentify, determine, and interpret measures of center (mean and median) and measures of variability (range, interquartile range) to answer a statistically investigative question, summarizing the distribution of data using the measures of center and variability. Use reasoning about distributions to compare two groups based on the variables.
Generate resourceAcknowledge that looking beyond the data is feasible and recognize the uncertainty caused by sample-to-sample variability when making comparisons and/or conclusions from data to answer the investigative question.
Generate resourceFlexibly, efficiently, and accurately use properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Generate resourceUnderstand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
Generate resourceFlexibly, efficiently, and accurately 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. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
Generate resourceUse variables to represent quantities in a real-world or mathematical problem and write simple equations and inequalities to flexibly, efficiently, and accurately solve problems by reasoning about the quantities. Compare solving the same problem algebraically vs. with arithmetic, explaining the steps involved in each approach. Graph the solutions of these inequalities and interpret them in context of the problem.
Generate resourceFlexibly, efficiently, and accurately solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Generate resourceDraw geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Generate resourceDescribe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Generate resourceKnow the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Generate resourceUse facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Generate resourceSolve 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.
Generate resourceFlexibly, efficiently, and accurately 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 showing the distance between two numbers is the absolute value of their difference, understand the concept of opposite quantities combining to zero (additive inverse), representing operations on number lines, and interpreting real-world scenarios in context.
Generate resourceFlexibly, efficiently, and accurately apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers including the distributive property and properties of operations. Understand integers can be divided as long as the divisor isn't zero, resulting in rational numbers and convert rational numbers into decimals using long division, recognizing that the decimal form either ends in 0s or repeats eventually, and interpreting realworld contexts.
Generate resourceFlexibly, efficiently, and accurately compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
Generate resourceRecognize and represent proportional relationships between quantities, including using equivalent ratios in a table, graphing on the coordinate plane to see of the graph is a straight line through origin, identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions, write equations for proportional relationships, and analyze graphs to understand what the data points tell them about the real-world situation, focusing on points like (0, 0) which represents no change and (1, r) where r is the unit rate.
Generate resourceFlexibly, efficiently, and accurately use proportional relationships to solve multistep ratio and percent problems.
Generate resourceUnderstand 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.
Generate resourceUse data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
Generate resourceInformally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
Generate resourceUse measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
Generate resourceUnderstand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Generate resourceApproximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
Generate resourceDevelop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Generate resourceFind probabilities of compound events using organized lists, tables, tree diagrams, and simulation, understanding the probability of a compound event is a fraction of the outcomes of the sample space. Design and use a simulation to generate frequencies for compound events.
Generate resourceUse properties of operations to generate equivalent expressions.
Generate resourceApply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Generate resourceUnderstand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
Generate resourceSolve real-life and mathematical problems using numerical and algebraic expressions and equations.
Generate resourceSolve 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. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
Generate resourceUse 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.
Generate resourceSolve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
Generate resourceSolve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
Generate resourceDraw, construct, and describe geometrical figures and describe the relationships between them.
Generate resourceSolve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Generate resourceDraw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Generate resourceDescribe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Generate resourceSolve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Generate resourceKnow the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Generate resourceUse facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Generate resourceSolve 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.
Generate resourceApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Generate resourceApply 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.
Generate resourceDescribe situations in which opposite quantities combine to make 0.
Generate resourceUnderstand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
Generate resourceUnderstand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Generate resourceApply properties of operations as strategies to add and subtract rational numbers.
Generate resourceApply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Generate resourceUnderstand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
Generate resourceUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
Generate resourceApply properties of operations as strategies to multiply and divide rational numbers.
Generate resourceConvert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Generate resourceSolve real-world and mathematical problems involving the four operations with rational numbers.
Generate resourceAnalyze proportional relationships and use them to solve real-world and mathematical problems.
Generate resourceCompute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
Generate resourceRecognize and represent proportional relationships between quantities.
Generate resourceDecide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Generate resourceIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Generate resourceExplain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Generate resourceUse proportional relationships to solve multistep ratio and percent problems.
Generate resourceUnderstand 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.
Generate resourceUse data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
Generate resourceDraw informal comparative inferences about two populations.
Generate resourceInformally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
Generate resourceUse measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
Generate resourceInvestigate chance processes and develop, use, and evaluate probability models.
Generate resourceUnderstand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Generate resourceApproximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
Generate resourceDevelop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Generate resourceDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
Generate resourceDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
Generate resourceFind probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Generate resourceUnderstand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
Generate resourceRepresent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
Generate resourceDesign and use a simulation to generate frequencies for compound events.
Generate resourceConstruct viable arguments and critique the reasoning of others.
Generate resourceAnalyze proportional relationships and use them to solve real-world and mathematical problems.
Generate resourceDraw, construct, and describe geometrical figures and describe the relationships between them.
Generate resourceSolve real-world and mathematical problems involving area, surface area, and volume.
Generate resourceInvestigate chance processes and develop, use, and evaluate probability models.
Generate resourceFlexibly, efficiently, and accurately solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Generate resource