Courses of Study : Mathematics

The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
Mathematics (2016)
Grade(s): 8
All Resources: 0
1 ) Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. [8-NS1]


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.


Mathematics (2016)
Grade(s): 8
All Resources: 0
2 ) Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). [8-NS2]

Example: By truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Expressions and Equations
Work with radicals and integer exponents.
Mathematics (2016)
Grade(s): 8
All Resources: 0
3 ) Know and apply the properties of integer exponents to generate equivalent numerical expressions. [8-EE1]

Example: 32 x 3-5 = 3-3 = 1/33 = 1/27.

Mathematics (2016)
Grade(s): 8
All Resources: 1
Lesson Plans: 1
4 ) Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.


NAEP Framework
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:
8NPO2a: Establish or apply benchmarks for rational numbers and common irrational numbers (e.g., ?) in contexts.

NAEP Statement:
8NPO2d: Estimate square or cube roots of numbers less than 1,000 between two whole numbers.


Mathematics (2016)
Grade(s): 8
All Resources: 0
5 ) Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. [8-EE3]

Example: Estimate the population of the United States as 3 x 108 and the population of the world as 7 x 109, and determine that the world population is more than 20 times larger.


NAEP Framework
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:
8NPO1f: Express or interpret numbers using scientific notation from real-life contexts.


Mathematics (2016)
Grade(s): 8
All Resources: 0
6 ) Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. [8-EE4]


NAEP Framework
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:
8NPO1f: Express or interpret numbers using scientific notation from real-life contexts.

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


Understand the connections among proportional relationships, lines, and linear equations.
Mathematics (2016)
Grade(s): 8
All Resources: 0
7 ) Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [8-EE5]

Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.


NAEP Framework
NAEP Statement:
8A1f: Interpret the meaning of slope or intercepts in linear functions.

NAEP Statement:
8A2f: Identify or represent functional relationships in meaningful contexts including proportional, linear, and common nonlinear (e.g., compound interest, bacterial growth) in tables, graphs, words, or symbols.

NAEP Statement:
8A4d: Interpret relationships between symbolic linear expressions and graphs of lines by identifying and computing slope and intercepts (e.g., know in y = ax + b, that a is the rate of change and b is the vertical intercept of the graph).

NAEP Statement:
8A5a: Make, validate, and justify conclusions and generalizations about linear relationships.

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


Mathematics (2016)
Grade(s): 8
All Resources: 1
Learning Activities: 1
8 ) Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. [8-EE6]


NAEP Framework
NAEP Statement:
8A4d: Interpret relationships between symbolic linear expressions and graphs of lines by identifying and computing slope and intercepts (e.g., know in y = ax + b, that a is the rate of change and b is the vertical intercept of the graph).

NAEP Statement:
8A5a: Make, validate, and justify conclusions and generalizations about linear relationships.


Analyze and solve linear equations and pairs of simultaneous linear equations.
Mathematics (2016)
Grade(s): 8
All Resources: 1
Learning Activities: 1
9 ) Solve linear equations in one variable. [8-EE7]

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). [8-EE7a]

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions, using the distributive property and collecting like terms. [8-EE7b]


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


Mathematics (2016)
Grade(s): 8
All Resources: 5
Learning Activities: 2
Lesson Plans: 3
10 ) Analyze and solve pairs of simultaneous linear equations. [8-EE8]

a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersections of their graphs because points of intersection satisfy both equations simultaneously. [8-EE8a]

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. [8-EE8b]

Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables. [8-EE8c]

Example: Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.


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:
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).


Functions
Define, evaluate, and compare functions.
Mathematics (2016)
Grade(s): 8
All Resources: 1
Learning Activities: 1
11 ) Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) [8-F1]

Mathematics (2016)
Grade(s): 8
All Resources: 0
12 ) Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [8-F2]

Example: Given a linear function represented by a table of values and linear function represented by an algebraic expression, determine which function has the greater rate of change.


NAEP Framework
NAEP Statement:
4A2a: Translate between the different forms of representations (symbolic, numerical, verbal, or pictorial) of whole number relationships (such as from a written description to an equation or from a function table to a written description).

NAEP Statement:
8A1e: Identify functions as linear or nonlinear or contrast distinguishing properties of functions from tables, graphs, or equations.

NAEP Statement:
8A2a: Translate between different representations of linear expressions using symbols, graphs, tables, diagrams, or written descriptions.


Mathematics (2016)
Grade(s): 8
All Resources: 1
Learning Activities: 1
13 ) Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. [8-F3]

Example: The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4), and (3,9), which are not on a straight line.


NAEP Framework
NAEP Statement:
8A1f: Interpret the meaning of slope or intercepts in linear functions.

NAEP Statement:
8A2b: Analyze or interpret linear relationships expressed in symbols, graphs, tables, diagrams, or written descriptions.


Use functions to model relationships between quantities.
Mathematics (2016)
Grade(s): 8
All Resources: 0
14 ) Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of linear function in terms of the situation it models and in terms of its graph or a table of values. [8-F4]


NAEP Framework
NAEP Statement:
8A1c: Analyze or create patterns, sequences, or linear functions given a rule.


Mathematics (2016)
Grade(s): 8
All Resources: 0
15 ) Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. [8-F5]


NAEP Framework
NAEP Statement:
8A2f: Identify or represent functional relationships in meaningful contexts including proportional, linear, and common nonlinear (e.g., compound interest, bacterial growth) in tables, graphs, words, or symbols.


Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
Mathematics (2016)
Grade(s): 8
All Resources: 0
16 ) Verify experimentally the properties of rotations, reflections, and translations: [8-G1]

a. Lines are taken to lines, and line segments are taken to line segments of the same length. [8-G1a]

b. Angles are taken to angles of the same measure. [8-G1b]

c. Parallel lines are taken to parallel lines. [8-G1c]


NAEP Framework
NAEP Statement:
4G4a: Describe relative positions of points and lines using the geometric ideas of parallelism or perpendicularity.

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

NAEP Statement:
8G3g: Describe or analyze properties and relationships of parallel or intersecting lines.


Mathematics (2016)
Grade(s): 8
All Resources: 0
17 ) Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8-G2]


NAEP Framework
NAEP Statement:
4G2e: Match or draw congruent figures in a given collection.

NAEP Statement:
8G2e: Justify relationships of congruence and similarity and apply these relationships using scaling and proportional reasoning.

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 (2016)
Grade(s): 8
All Resources: 0
18 ) Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. [8-G3]


NAEP Framework
NAEP Statement:
4G2c: Identify the images resulting from flips (reflections), slides (translations), or turns (rotations).

NAEP Statement:
8G2c: Recognize or informally describe the effect of a transformation on two- dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, magnifications, and contractions).

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:
8G4d: Represent geometric figures using rectangular coordinates on a plane.


Mathematics (2016)
Grade(s): 8
All Resources: 0
19 ) Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. [8-G4]


NAEP Framework
NAEP Statement:
4G2c: Identify the images resulting from flips (reflections), slides (translations), or turns (rotations).

NAEP Statement:
8G2e: Justify relationships of congruence and similarity and apply these relationships using scaling and proportional reasoning.

NAEP Statement:
8G2f: For similar figures, identify and use the relationships of conservation of angle and of proportionality of side length and perimeter.

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 (2016)
Grade(s): 8
All Resources: 1
Lesson Plans: 1
20 ) Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. [8-G5]

Example: Arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give argument in terms of transversals why this is so.


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


Understand and apply the Pythagorean Theorem.
Mathematics (2016)
Grade(s): 8
All Resources: 0
21 ) Explain a proof of the Pythagorean Theorem and its converse. [8-G6]

Mathematics (2016)
Grade(s): 8
All Resources: 1
Learning Activities: 1
22 ) Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. [8-G7]


NAEP Framework
NAEP Statement:
8G3d: Use the Pythagorean theorem to solve problems.


Mathematics (2016)
Grade(s): 8
All Resources: 2
Learning Activities: 1
Lesson Plans: 1
23 ) Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. [8-G8]


NAEP Framework
NAEP Statement:
8G1a: Draw or describe a path of shortest length between points to solve problems in context.

NAEP Statement:
8G3d: Use the Pythagorean theorem to solve problems.


Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Mathematics (2016)
Grade(s): 8
All Resources: 0
24 ) Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems. [8-G9]

Statistics and Probability
Investigate patterns of association in bivariate data.
Mathematics (2016)
Grade(s): 8
All Resources: 0
25 ) Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. [8-SP1]


NAEP Framework
NAEP Statement:
8DASP1b: For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, line graphs, scatterplots, circle graphs, and bar graphs).

NAEP Statement:
8DASP2c: Identify outliers and determine their effect on mean, median, mode, or range.


Mathematics (2016)
Grade(s): 8
All Resources: 0
26 ) Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. [8-SP2]


NAEP Framework
NAEP Statement:
8DASP1b: For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, line graphs, scatterplots, circle graphs, and bar graphs).

NAEP Statement:
8DASP2e: Visually choose the line that best fits given a scatterplot and informally explain the meaning of the line. Use the line to make predictions.


Mathematics (2016)
Grade(s): 8
All Resources: 0
27 ) Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. [8-SP3]

Example: In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.


NAEP Framework
NAEP Statement:
8DASP1b: For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, line graphs, scatterplots, circle graphs, and bar graphs).


Mathematics (2016)
Grade(s): 8
All Resources: 0
28 ) Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. [8-SP4]

Example: Collect data from students in your class on whether or not they have a curfew on school nights, and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores'


NAEP Framework
NAEP Statement:
8A1a: Recognize, describe, or extend numerical and geometric patterns using tables, graphs, words, or symbols.

NAEP Statement:
8A1b: Generalize a pattern appearing in a numerical sequence, table, or graph using words or symbols.