Massachusetts State Standards for Mathematics: Grade 7

MA.7.N. Number Sense and Operations: Students engage in problem solving, communicating, reasoning, connecting, and representing.

7.N.1. Compare, order, estimate, and translate among integers, fractions and mixed numbers (i.e., rational numbers), decimals, and percents. This standard is intentionally the same as standard 8.N.1.

7.N.2. Use ratios and proportions in the solution of problems involving unit rates, scale drawings, and reading of maps.

7.N.3. Represent numbers in scientific notation (positive powers of ten only) and use in problem situations.

7.N.4. Demonstrate an understanding of absolute value, e.g., |-3| = |3| = 3. This standard is intentionally the same as standard 8.N.6.

7.N.5. Apply the rules of positive integer exponents to the solution of problems. Extend the Order of Operations to include positive integer exponents.

7.N.6. Use the inverse relationships of addition and subtraction, and of multiplication and division, to simplify computations and solve problems, e.g. multiplying by 1/2 or 0.5 is the same as dividing by 2.

7.N.7. Estimate and compute with fractions (including simplification of fractions), integers, decimals, and percents (including those greater than 100 and less than 1). This standard is intentionally the same as standard 8.N.10.

7.N.8. Determine when an estimate rather than an exact answer is appropriate and apply in problem situations. This standard is intentionally the same as standard 8.N.11.

7.N.9. Select and use appropriate operations - addition, subtraction, multiplication, division, and positive integer exponents - to solve problems with rational numbers (including negatives). This standard is intentionally the same as standard 8.N.12.

MA.7.P. Patterns, Relations, and Algebra: Students engage in problem solving, communicating, reasoning, connecting, and representing.

7.P.1. Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic expressions. Include arithmetic and geometric progressions, e.g., compounding. This standard is intentionally the same as standard 8.P.1.

7.P.2. Evaluate simple algebraic expressions for given variable values, e.g., 3a squared - b for a = 3 and b = 7. This standard is intentionally the same as standard 8.P.2.

7.P.3. Create and use symbolic expressions for linear relationships and relate them to verbal, tabular, and graphical representations.

7.P.4. Solve linear equations using tables, graphs, models, and algebraic methods.

7.P.5. Identify, describe, and analyze linear relationships between two variables. Compare positive rate of change, e.g., y = 3x + 1, to negative rate of change, e.g., y = -3x + 1.

7.P.6. Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate. This standard is intentionally the same as standard 8.P.9.

MA.7.G. Geometry: Students engage in problem solving, communicating, reasoning, connecting, and representing.

7.G.1. Analyze, apply, and explain the relationship between the number of sides and the sums of the interior angle measures of polygons.

7.G.2. Classify figures in terms of congruence and similarity, and apply these relationships to the solution of problems. This standard is intentionally the same as standard 8.G.2.

7.G.3. Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines cut by a transversal. This standard is intentionally the same as standard 8.G.3.

7.G.4. Graph points and identify coordinates of points on the Cartesian coordinate plane (all four quadrants).

7.G.5. Use a ruler, protractor, and compass to draw polygons and circles.

7.G.6. Predict the results of translations and reflections of figures on unmarked or coordinate planes and draw the transformed figure.

7.G.7. Identify three-dimensional figures (e.g., prisms, pyramids) by their physical appearance, distinguishing attributes, and spatial relationships such as parallel faces. This standard is intentionally the same as standard 8.G.7.

MA.7.M. Measurement: Students engage in problem solving, communicating, reasoning, connecting, and representing.

7.M.1. Select, convert (within the same system of measurement), and use appropriate units of measurement or scale. This standard is intentionally the same as standard 8.M.1.

7.M.2. Given the formulas, convert from one system of measurement to another. Use technology as appropriate. This standard is intentionally the same as standard 8.M.2.

7.M.3. Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, and cylinders. Use technology as appropriate.

MA.7.D. Data Analysis, Statistics, and Probability: Students engage in problem solving, communicating, reasoning, connecting, and representing.

7.D.1. Select, create, interpret, and utilize the following tabular and graphical representations of data: circle graphs, Venn diagrams, stem-and-leaf plots, tables, and charts.

7.D.2. Find, describe, and interpret appropriate measures of central tendency (mean, median, and mode) and spread (range) that represent a set of data. Use these notions to compare different sets of data. This standard is intentionally the same as standard 8.D.3.

7.D.3. Use tree diagrams, tables, organized lists, and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.

MA.CC.7.RP. Ratios and Proportional Relationships

7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

7.RP.2. Recognize and represent proportional relationships between quantities.

7.RP.2.a. Decide 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.

7.RP.2.b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.2.c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

7.RP.2.d. Explain 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.

7.RP.3. Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

MA.CC.7.NS. The Number System

7.NS.1. 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.

7.NS.1.a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

7.NS.1.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.

7.NS.1.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.

7.NS.1.d. Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

7.NS.2.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.

7.NS.2.b. Understand 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.

7.NS.2.c. Apply properties of operations as strategies to multiply and divide rational numbers.

7.NS.2.d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.

MA.CC.7.EE. Expressions and Equations

7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.2. Understand 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. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

7.EE.3. Solve multi-step real-world 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. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

7.EE.4. 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.

7.EE.4.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. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

7.EE.4.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 solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

7.EE.MA.4c. Extend analysis of patterns to include analyzing, extending, and determining an expression for simple arithmetic and geometric sequences (e.g., compounding, increasing area), using tables, graphs, words, and expressions.

MA.CC.7.G. Geometry

7.G.MA.7. Solve real-world and mathematical problems involving the surface area of spheres.

MA.CC.7.SP. Statistics and Probability

7.SP.1. 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.

7.SP.2. 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. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

7.SP.3. Informally 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. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

7.SP.4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

7.SP.5. 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.

7.SP.6. 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. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

7.SP.7. 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.

7.SP.7.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.

7.SP.7.b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

7.SP.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

7.SP.8.a. Understand 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.

7.SP.8.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.

7.SP.8.c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?