Courses of Study : Mathematics

Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Mathematics (2015)
Grade(s): 7
All Resources: 1
Lesson Plans: 1
1 ) Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. [7-RP1]



NAEP Framework
NAEP Statement:
8NPO1b: Model or describe rational numbers or numerical relationships using number lines and diagrams.

NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.

NAEP Statement:
8NPO4a: Use ratios to describe problem situations.

NAEP Statement:
8NPO4b: Use fractions to represent and express ratios and proportions.

NAEP Statement:
8NPO4c: Use proportional reasoning to model and solve problems (including rates and scaling).


Mathematics (2015)
Grade(s): 7
All Resources: 1
Lesson Plans: 1
2 ) Recognize and represent proportional relationships between quantities. [7-RP2]

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-RP2a]

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. [7-RP2b]

c. Represent proportional relationships by equations. [7-RP2c]

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.

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-RP2d]


NAEP Framework
NAEP Statement:
4A1e: Recognize or describe a relationship in which quantities change proportionally.

NAEP Statement:
8A2c: Graph or interpret points represented by ordered pairs of numbers on a rectangular coordinate system.

NAEP Statement:
8M1i: Solve problems involving rates such as speed or population density.

NAEP Statement:
8NPO1b: Model or describe rational numbers or numerical relationships using number lines and diagrams.

NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.

NAEP Statement:
8NPO4a: Use ratios to describe problem situations.

NAEP Statement:
8NPO4b: Use fractions to represent and express ratios and proportions.

NAEP Statement:
8NPO4c: Use proportional reasoning to model and solve problems (including rates and scaling).


Mathematics (2015)
Grade(s): 7
All Resources: 1
Learning Activities: 1
3 ) Use proportional relationships to solve multistep ratio and percent problems. [7-RP3]

Examples: Sample problems may involve simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error.


NAEP Framework
NAEP Statement:
8NPO1b: Model or describe rational numbers or numerical relationships using number lines and diagrams.

NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.

NAEP Statement:
8NPO4a: Use ratios to describe problem situations.

NAEP Statement:
8NPO4b: Use fractions to represent and express ratios and proportions.

NAEP Statement:
8NPO4c: Use proportional reasoning to model and solve problems (including rates and scaling).

NAEP Statement:
8NPO4d: Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Mathematics (2015)
Grade(s): 7
All Resources: 0
4 ) 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-NS1]

a. Describe situations in which opposite quantities combine to make 0. [7-NS1a]

Example: A hydrogen atom has 0 charge because its two constituents are oppositely charged.

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-NS1b]

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-NS1c]

d. Apply properties of operations as strategies to add and subtract rational numbers. [7-NS1d]


NAEP Framework
NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO1g: Find or model absolute value or apply to problem situations.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO3e: Interpret rational number operations and the relationships between them.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Mathematics (2015)
Grade(s): 7
All Resources: 0
5 ) Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. [7-NS2]

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-NS2a]

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero 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-NS2b]

c. Apply properties of operations as strategies to multiply and divide rational numbers. [7-NS2c]

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-NS2d]


NAEP Framework
NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO3d: Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:
  • Zero, or
  • A number less than zero, or
  • A number between zero and one,
  • One, or
  • A number greater than one.


NAEP Statement:
8NPO3e: Interpret rational number operations and the relationships between them.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Mathematics (2015)
Grade(s): 7
All Resources: 0
6 ) Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) [7-NS3]


NAEP Framework
NAEP Statement:
8NPO1d: Write or rename rational numbers.

NAEP Statement:
8NPO1e: Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO3e: Interpret rational number operations and the relationships between them.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.

NAEP Statement:
8NPO5e: Apply basic properties of operations.


Expressions and Equations
Use properties of operations to generate equivalent expressions.
Mathematics (2015)
Grade(s): 7
All Resources: 0
7 ) Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. [7-EE1]


NAEP Framework
NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO4d: Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


Mathematics (2015)
Grade(s): 7
All Resources: 0
8 ) 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. [7-EE2]

Example: a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."


NAEP Framework
NAEP Statement:
8A3b: Write algebraic expressions, equations, or inequalities to represent a situation.

NAEP Statement:
8A3c: Perform basic operations, using appropriate tools, on linear algebraic expressions (including grouping and order of multiple operations involving basic operations, exponents, roots, simplifying, and expanding).

NAEP Statement:
8NPO3a: Perform computations with rational numbers.

NAEP Statement:
8NPO4d: Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).


Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Mathematics (2015)
Grade(s): 7
All Resources: 0
9 ) Solve multistep 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. [7-EE3]

Examples: 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.


NAEP Framework
NAEP Statement:
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement:
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.

NAEP Statement:
8NPO3f: Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.


Mathematics (2015)
Grade(s): 7
All Resources: 0
10 ) 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-EE4]

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. [7-EE4a]

Example: The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width'

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. [7-EE4b]

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.


NAEP Framework
NAEP Statement:
8A4a: Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).

NAEP Statement:
8A4b: Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.

NAEP Statement:
8A4c: Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).


Geometry
Draw, construct, and describe geometrical figures and describe the relationships between them.
Mathematics (2015)
Grade(s): 7
All Resources: 1
Lesson Plans: 1
11 ) 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. [7-G1]


NAEP Framework
NAEP Statement:
8G1d: Draw or sketch from a written description polygons, circles, or semicircles.

NAEP Statement:
8G1f: Demonstrate an understanding about the two-and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart.

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3c: Represent problem situations with simple geometric models to solve mathematical or real-world problems.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.

NAEP Statement:
8M3a: Solve problems involving indirect measurement such as finding the height of a building by comparing its shadow with the height and shadow of a known object.


Mathematics (2015)
Grade(s): 7
All Resources: 0
12 ) Draw (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. [7-G2]


NAEP Framework
NAEP Statement:
4G3c: Recognize two-dimensional faces of three-dimensional shapes.

NAEP Statement:
8G1d: Draw or sketch from a written description polygons, circles, or semicircles.

NAEP Statement:
8G1f: Demonstrate an understanding about the two-and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart.

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3c: Represent problem situations with simple geometric models to solve mathematical or real-world problems.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.


Mathematics (2015)
Grade(s): 7
All Resources: 0
13 ) Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. [7-G3]


NAEP Framework
NAEP Statement:
8G1e: Represent or describe a three-dimensional situation in a two-dimensional drawing from different views.

NAEP Statement:
8G2d: Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, cutting up and rearranging pieces).

NAEP Statement:
8G3b: Apply geometric properties and relationships in solving simple problems in two and three dimensions.

NAEP Statement:
8G3f: Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.

NAEP Statement:
8G4c: Visualize or describe the cross section of a solid.


Solve real-world and mathematical problems involving angle measure, area, surface area, and volume.
Mathematics (2015)
Grade(s): 7
All Resources: 0
14 ) Know 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. [7-G4]


NAEP Framework
NAEP Statement:
8A4e: Use and evaluate common formulas (e.g., relationship between a circle's circumference and diameter [C = pi d], distance and time under constant speed).

NAEP Statement:
8G2d: Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, cutting up and rearranging pieces).

NAEP Statement:
8M1f: Solve mathematical or real-world problems involving perimeter or area of plane figures such as triangles, rectangles, circles, or composite figures.


Mathematics (2015)
Grade(s): 7
All Resources: 1
Lesson Plans: 1
15 ) Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. [7-G5]

Mathematics (2015)
Grade(s): 7
All Resources: 1
Lesson Plans: 1
16 ) 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. [7-G6]


NAEP Framework
NAEP Statement:
8G2d: Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, cutting up and rearranging pieces).

NAEP Statement:
8M1h: Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

NAEP Statement:
8M3a: Solve problems involving indirect measurement such as finding the height of a building by comparing its shadow with the height and shadow of a known object.


Statistics and Probability
Use random sampling to draw inferences about a population.
Mathematics (2015)
Grade(s): 7
All Resources: 1
Learning Activities: 1
17 ) 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-SP1]


NAEP Framework
NAEP Statement:
8DASP1a: Read or interpret data, including interpolating or extrapolating from data.

NAEP Statement:
8DASP3b: Distinguish between a random and nonrandom sample.


Mathematics (2015)
Grade(s): 7
All Resources: 1
Learning Activities: 1
18 ) 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. [7-SP2]

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.


NAEP Framework
NAEP Statement:
8DASP1c: Solve problems by estimating and computing with data from a single set or across sets of data.

NAEP Statement:
8DASP2a: Calculate, use, or interpret mean, median, mode, or range.

NAEP Statement:
8DASP3a: Given a sample, identify possible sources of bias in sampling.

NAEP Statement:
8DASP4e: Determine the sample space for a given situation.


Draw informal comparative inferences about two populations.
Mathematics (2015)
Grade(s): 7
All Resources: 0
19 ) 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. [7-SP3]

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.


NAEP Framework
NAEP Statement:
4DASP2d: Compare two sets of related data.

NAEP Statement:
8DASP1c: Solve problems by estimating and computing with data from a single set or across sets of data.

NAEP Statement:
8DASP1e: Compare and contrast the effectiveness of different representations of the same data.

NAEP Statement:
8DASP2a: Calculate, use, or interpret mean, median, mode, or range.


Mathematics (2015)
Grade(s): 7
All Resources: 0
20 ) Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. [7-SP4]

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.


NAEP Framework
NAEP Statement:
4DASP2d: Compare two sets of related data.

NAEP Statement:
8DASP1a: Read or interpret data, including interpolating or extrapolating from data.

NAEP Statement:
8DASP1c: Solve problems by estimating and computing with data from a single set or across sets of data.

NAEP Statement:
8DASP2a: Calculate, use, or interpret mean, median, mode, or range.

NAEP Statement:
8DASP2b: Describe how mean, median, mode, range, or interquartile ranges relate to distribution shape.

NAEP Statement:
8DASP2d: Using appropriate statistical measures, compare two or more data sets describing the same characteristic for two different populations or subsets of the same population.


Investigate chance processes and develop, use, and evaluate probability models.
Mathematics (2015)
Grade(s): 7
All Resources: 0
21 ) 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-SP5]


NAEP Framework
NAEP Statement:
4DASP4a: Use informal probabilistic thinking to describe chance events (i.e., likely and unlikely, certain and impossible).

NAEP Statement:
8DASP4a: Analyze a situation that involves probability of an independent event.

NAEP Statement:
8DASP4g: Represent the probability of a given outcome using fractions, decimals, and percents.


Mathematics (2015)
Grade(s): 7
All Resources: 0
22 ) 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. [7-SP6]

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.


NAEP Framework
NAEP Statement:
4DASP4b: Determine a simple probability from a context that includes a picture.

NAEP Statement:
4DASP4g: Represent the probability of a given outcome using a picture or other graphic.

NAEP Statement:
8DASP4b: Determine the theoretical probability of simple and compound events in familiar contexts.

NAEP Statement:
8DASP4f: Use a sample space to determine the probability of possible outcomes for an event.

NAEP Statement:
8DASP4j: Interpret probabilities within a given context.


Mathematics (2015)
Grade(s): 7
All Resources: 1
Learning Activities: 1
23 ) 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-SP7]

a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. [7-SP7a]

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.

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. [7-SP7b]

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'


NAEP Framework
NAEP Statement:
4DASP4b: Determine a simple probability from a context that includes a picture.

NAEP Statement:
4DASP4e: List all possible outcomes of a given situation or event.

NAEP Statement:
8DASP4d: Use theoretical probability to evaluate or predict experimental outcomes


Mathematics (2015)
Grade(s): 7
All Resources: 0
24 ) Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7-SP8]

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-SP8a]

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-SP8b]

c. Design and use a simulation to generate frequencies for compound events. [7-SP8c]

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'


NAEP Framework
NAEP Statement:
8DASP4c: Estimate the probability of simple and compound events through experimentation or simulation.

NAEP Statement:
8DASP4f: Use a sample space to determine the probability of possible outcomes for an event.