### Seventh Grade North Carolina Standards

Ratios and Proportional Relationships

Analyze proportional relationships and use them to solve real-world and mathematical problems.

• 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.

• o  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.

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

o  Representproportional relationships by equations. For example, if total cost t isproportional to the number n of items purchased at a constant price p, therelationship between the total cost and the number of items can be expressed ast = pn.

o  Explainwhat a point (xy) on the graph of a proportional relationshipmeans 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 proportionalrelationships to solve multistep ratio and percent problems. Examples: simpleinterest, tax, markups and markdowns, gratuities and commissions, fees, percentincrease and decrease, percent error.

• The Number System

Apply and extend previousunderstandings of operations with fractions to add, subtract, multiply, anddivide rational numbers.

• 7.NS.1. Apply and extendprevious understandings of addition and subtraction to add and subtractrational numbers; represent addition and subtraction on a horizontal orvertical number line diagram.

• o  Describesituations in which opposite quantities combine to make 0. For example, ahydrogen atom has 0 charge because its two constituents are oppositely charged.

o  Understandp + q as the number located a distance |q| from p,in the positive or negative direction depending on whether q is positiveor negative. Show that a number and its opposite have a sum of 0 (are additiveinverses). Interpret sums of rational numbers by describing real-worldcontexts.

o  Understandsubtraction of rational numbers as adding the additive inverse, p – qp + (–q). Show that the distance between two rational numberson the number line is the absolute value of their difference, and apply thisprinciple in real-world contexts.

o  Applyproperties of operations as strategies to add and subtract rational numbers.

• 7.NS.2. Apply and extendprevious understandings of multiplication and division and of fractions tomultiply and divide rational numbers.

• o  Understandthat multiplication is extended from fractions to rational numbers by requiringthat operations continue to satisfy the properties of operations, particularlythe distributive property, leading to products such as (–1)(–1) = 1 and therules for multiplying signed numbers. Interpret products of rational numbers bydescribing real-world contexts.

o  Understandthat integers can be divided, provided that the divisor is not zero, and everyquotient of integers (with non-zero divisor) is a rational number. If pand q are integers, then –(p/q) = (–p)/q = p/(–q).Interpret quotients of rational numbers by describing real-world contexts.

o  Applyproperties of operations as strategies to multiply and divide rational numbers.

o  Converta rational number to a decimal using long division; know that the decimal formof a rational number terminates in 0s or eventually repeats.

• 7.NS.3. Solve real-world andmathematical problems involving the four operations with rational numbers.1

• Expression and Equations

Use properties of operations togenerate equivalent expressions.

• 7.EE.1. Apply properties ofoperations as strategies to add, subtract, factor, and expand linearexpressions with rational coefficients.

• 7.EE.2. Understand thatrewriting an expression in different forms in a problem context can shed lighton 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.”

• Solve real-life and mathematicalproblems using numerical and algebraic expressions and equations.

• 7.EE.3. Solve multi-stepreal-life and mathematical problems posed with positive and negative rationalnumbers in any form (whole numbers, fractions, and decimals), using toolsstrategically. Apply properties of operations to calculate with numbers in anyform; convert between forms as appropriate; and assess the reasonableness ofanswers using mental computation and estimation strategies. For example: Ifa woman making \$25 an hour gets a 10% raise, she will make an additional 1/10of her salary an hour, or \$2.50, for a new salary of \$27.50. If you want toplace a towel bar 9 3/4 inches long in the center of a door that is 27 1/2inches wide, you will need to place the bar about 9 inches from each edge; thisestimate can be used as a check on the exact computation.

• 7.EE.4. Use variables torepresent quantities in a real-world or mathematical problem, and constructsimple equations and inequalities to solve problems by reasoning about thequantities.

• o  Solveword problems leading to equations of the form px + q = rand p(x + q) = r, where pq, and rare specific rational numbers. Solve equations of these forms fluently. Comparean algebraic solution to an arithmetic solution, identifying the sequence ofthe operations used in each approach. For example, the perimeter of arectangle is 54 cm. Its length is 6 cm. What is its width?

o  Solveword problems leading to inequalities of the form px + q > ror px + q < r, where pq, and r arespecific rational numbers. Graph the solution set of the inequality andinterpret 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 beat least \$100. Write an inequality for the number of sales you need to make,and describe the solutions.

Geometry

Draw construct, and describegeometrical figures and describe the relationships between them.

• 7.G.1. Solve problemsinvolving scale drawings of geometric figures, including computing actuallengths and areas from a scale drawing and reproducing a scale drawing at adifferent scale.

• 7.G.2. Draw (freehand, withruler and protractor, and with technology) geometric shapes with givenconditions. Focus on constructing triangles from three measures of angles orsides, noticing when the conditions determine a unique triangle, more than onetriangle, or no triangle.

• 7.G.3. Describe thetwo-dimensional figures that result from slicing three-dimensional figures, asin plane sections of right rectangular prisms and right rectangular pyramids.

• Solve real-life and mathematicalproblems involving angle measure, area, surface area, and volume.

• 7.G.4. Know the formulasfor the area and circumference of a circle and use them to solve problems; givean informal derivation of the relationship between the circumference and areaof a circle.

• 7.G.5. Use facts aboutsupplementary, complementary, vertical, and adjacent angles in a multi-stepproblem to write and solve simple equations for an unknown angle in a figure.

• 7.G.6. Solve real-world andmathematical problems involving area, volume and surface area of two- andthree-dimensional objects composed of triangles, quadrilaterals, polygons,cubes, and right prisms.

•  Statistics and Probability

Use random sampling to draw inferencesabout a population.

• 7.SP.1. Understand thatstatistics can be used to gain information about a population by examining asample of the population; generalizations about a population from a sample arevalid only if the sample is representative of that population. Understand thatrandom sampling tends to produce representative samples and support validinferences.

• 7.SP.2. Use data from arandom sample to draw inferences about a population with an unknowncharacteristic of interest. Generate multiple samples (or simulated samples) ofthe same size to gauge the variation in estimates or predictions. Forexample, estimate the mean word length in a book by randomly sampling wordsfrom the book; predict the winner of a school election based on randomlysampled survey data. Gauge how far off the estimate or prediction might be.

• Draw informal comparative inferencesabout two populations.

• 7.SP.3. Informally assessthe degree of visual overlap of two numerical data distributions with similarvariabilities, measuring the difference between the centers by expressing it asa multiple of a measure of variability. For example, the mean height ofplayers on the basketball team is 10 cm greater than the mean height of playerson the soccer team, about twice the variability (mean absolute deviation) oneither team; on a dot plot, the separation between the two distributions ofheights is noticeable.

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

• Investigate chance processes anddevelop, use, and evaluate probability models.

• 7.SP.5.7.SP.5. Understand that the probability of a chance event is anumber between 0 and 1 that expresses the likelihood of the event occurring.Larger numbers indicate greater likelihood. A probability near 0 indicates anunlikely event, a probability around 1/2 indicates an event that is neitherunlikely nor likely, and a probability near 1 indicates a likely event.

• 7.SP.6. Approximate theprobability of a chance event by collecting data on the chance process thatproduces it and observing its long-run relative frequency, and predict theapproximate relative frequency given the probability. For example, whenrolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly200 times, but probably not exactly 200 times.

• 7.SP.7. Develop aprobability model and use it to find probabilities of events. Compareprobabilities from a model to observed frequencies; if the agreement is notgood, explain possible sources of the discrepancy.

• o  Developa uniform probability model by assigning equal probability to all outcomes, anduse the model to determine probabilities of events. For example, if astudent is selected at random from a class, find the probability that Jane willbe selected and the probability that a girl will be selected.

o  Developa probability model (which may not be uniform) by observing frequencies in datagenerated from a chance process. For example, find the approximateprobability that a spinning penny will land heads up or that a tossed paper cupwill land open-end down. Do the outcomes for the spinning penny appear to beequally likely based on the observed frequencies?

• 7.SP.8. Find probabilitiesof compound events using organized lists, tables, tree diagrams, andsimulation.

• o  Understandthat, just as with simple events, the probability of a compound event is thefraction of outcomes in the sample space for which the compound event occurs.

o  Representsample spaces for compound events using methods such as organized lists, tablesand tree diagrams. For an event described in everyday language (e.g., “rollingdouble sixes”), identify the outcomes in the sample space which compose theevent.

o  Designand use a simulation to generate frequencies for compound events. Forexample, use random digits as a simulation tool to approximate the answer tothe question: If 40% of donors have type A blood, what is the probability thatit will take at least 4 donors to find one with type A blood?