Grade 7

Pose statistical investigative questions about a broader population using samples taken from the population.

Understand 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.

Identify, 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.

Acknowledge 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.

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 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.

Informally assess the degree of 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.

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. 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.

Approximate 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.

Develop 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.

1. a.Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
2. b.Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

1. a.Understand the probability of a compound event is a fraction of the outcomes of the sample space.
2. b.Represent 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.
3. c.Design and use a simulation to generate frequencies for compound events.

Efficiently, flexibly, 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.

1. a.Describe situations in which opposite quantities combine to make 0.
2. b.Understand 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.
3. c.Understand 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.
4. d.Apply properties of operations as strategies to add and subtract rational numbers.

Efficiently, flexibly, and accurately apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

1. a.Understand 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.
2. b.Understand integers can be divided as long as the divisor isn't zero, resulting in rational numbers.
3. c.Apply properties of operations as strategies to multiply and divide rational numbers.
4. d.Convert rational numbers into decimals using flexible, efficient, and accurate strategies, recognizing that the decimal form either ends in 0s or repeats eventually.

Efficiently, flexibly, 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.

Efficiently, flexibly, and accurately 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.

1. a.Decide whether the relationship between two quantities is proportional using equivalent ratios in a table, graphing on the coordinate plane to see of the graph is a straight line through origin.
2. b.Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
3. c.Write equations for proportional relationships.
4. d.Analyze graphs to understand what the data points indicate about the real-world situation, focusing on points like (0, 0) and (1, r) where r is the unit rate.

Efficiently, flexibly, and accurately use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Efficiently, flexibly, and accurately use properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

Understand that rewriting an expression in different forms in a problem context can bring awareness to parts of the problem and how the quantities in it are related.

Efficiently, flexibly, 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.

Use variables to represent quantities in a real-world or mathematical problem and write equations and inequalities to efficiently, flexibly, and accurately to solve problems by reasoning about the quantities.

1. a.Solve 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. Compare solving the same problem algebraically vs. with arithmetic, explaining the steps involved in each approach.
2. b.Solve 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 solutions of these inequalities and interpret them in context of the problem.

Draw 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.

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 use them to solve problems; informally show 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 solve simple equations for an unknown angle in a figure.

Efficiently, flexibly, and accurately 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.