In this course, students will extend their understanding of the operations and properties of the rational number system to real numbers, including numbers expressed using exponents and roots. Algebraic expressions and relationships are developed as a generalization of the real number system and grow naturally out of work with numerical operations. Facility with expressions involving variables and numerical relationships expressed algebraically and graphically opens the door to using mathematics as a problem-solving tool and is critical to success in more advanced mathematics courses.

A focus of this course is on linear relationships; these are used to introduce students to the concept of a function and its multiple representations. Solving linear equations algebraically and seeing the connection to the graph of the related linear function as well as to a contextual situation from which the equation might have arisen further prepares students for more rigorous mathematical modeling in later courses. It also opens the door to interesting and varied applications of the mathematics students are learning in their classrooms. The inclusion of data analysis in this course extends the algebraic lessons even further into real-life situations. Simple logical arguments that establish facts about geometric figures help to solidify previously studied geometric concepts. Coordinate geometry offers teachers and students opportunities to connect the branches of mathematics through the study of transformations, area of polygons, and slope in the coordinate plane. Upon completion of this two-course sequence, students should be prepared for success on eighth-grade state tests as well as the National Assessment of Educational Progress (NAEP) grade 8 assessment. They should also be prepared to successfully tackle algebra and geometry taught at the high school level.

Appropriate use of technology is expected in all work. In middle school this includes employing technological tools to assist students in creating graphs and data displays, transforming graphs, conceptualizing and analyzing geometric situations, and solving problems. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of this middle school course to concepts students have encountered in earlier grades and courses as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for this course, the second course in a two-year middle school sequence, which is intended to prepare students for a high school–level course in Algebra I or the first course in an Integrated Mathematics sequence. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Real Numbers, Exponents, Roots, and the Pythagorean Theorem
• Number Bases [OPTIONAL ENRICHMENT UNIT]
• Variables and Expressions
• Functions
• Equations and Identities
• Geometric Reasoning, Representation, and Transformations
• Data Analysis

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for Middle School Course 2.

• Whole and Rational Number Operations
• Numeric Relationships
• Ratio/Proportion and Percent
• Similarity and Scaling
• Graphing in the Coordinate Plane
• Geometric Shapes and Measurement
• Data Collection and Organization
• Probability

A. Real Numbers, Exponents, and Roots and the Pythagorean Theorem

Students extend the properties of computation with rational numbers to real number computation, categorize real numbers as either rational or irrational, and locate real numbers on the number line. Powers and roots are studied along with the Pythagorean theorem and its converse—a critical concept in its own right as well as a context in which numbers expressed using powers and roots arise. Students apply this knowledge to solve problems.

Successful students will:

A1 Know and apply the definition of absolute value.

The absolute value is defined by |a| = a if a > 0 and |a| = -a if a < 0.

1. Interpret absolute value as distance from zero.
2. Interpret absolute value of a difference as "distance between" on the number line.

A2 Use the definition of a root of a number to explain the relationship of powers and roots.

If aⁿ = b, for an integer n ≥ 0, then a is said to be an nth root of b. When n is even and b > 0, we identify the unique a > 0 as the principal nth root of b, written .

1. Use and interpret the symbols and ; informally explain why , when .

   By convention, for is used to represent the non-negative square root of a.

2. Estimate square and cube roots and use calculators to find good approximations.
3. Make or refine an estimate for a square root using the fact that if 0 ≤ a < n < b, then ; make or refine an estimate for a cube root using the fact that if a < n < b, then .

A3 Categorize real numbers as either rational or irrational and know that, by definition, these are the only two possibilities; extend the properties of computation with rational numbers to real number computation.

1. Approximately locate any real number on the number line.
2. Apply the definition of irrational number to identify examples and recognize approximations.

   Square roots, cube roots, and nth roots of whole numbers that are not respectively squares, cubes, and nth powers of whole numbers provide the most common examples of irrational numbers. Pi (π) is another commonly cited irrational number.

3. Know that the decimal expansion of a rational number eventually repeats, perhaps ending in repeating zeros; use this to identify the decimal expansion of an irrational number as one that never ends and never repeats.
4. Recognize and use and 3.14 as approximations for the irrational number represented by pi (π).
5. Determine whether the square, cube, and nth roots of integers are integral or irrational when such roots are real numbers.

A4 Interpret and prove the Pythagorean theorem and its converse; apply the Pythagorean theorem and its converse to solve problems.

1. Determine distances between points in the Cartesian coordinate plane and relate the Pythagorean theorem to this process.

B. Number Bases [OPTIONAL ENRICHMENT UNIT]

This should be used as an optional unit of study if time permits. Using their understanding of the base-10 number system, students represent numbers in other bases. Computers and computer graphics have made much more important the knowledge of how to work with different base systems, particularly binary.

Successful students will:

B1 Identify key characteristics of the base-10 number system and adapt them to the binary number base system.

1. Represent and interpret numbers in the binary number system.
2. Apply the concept of base-10 place value to understand representation of numbers in other bases.

   Example: In the base-8 number system, the 5 in the number 57,273 represents the quantity 5 x 8⁴.

3. Convert binary to decimal and vice versa.
4. Encode data and record measurements of information capacity using the binary number base system.

C. Variables and Expressions

In middle school, students work more with symbolic algebra than in the previous grades. Students develop an understanding of the different uses for variables, analyze mathematical situations and structures using algebraic expressions, determine if expressions are equivalent, and identify single-variable expressions as linear or non-linear.

Successful students will:

C1 Interpret and compare the different uses of variables and describe patterns, properties of numbers, formulas, and equations using variables.

While a variable has several distinct uses in mathematics, it is fundamentally just a number we either do not know yet or do not want to specify.

1. Compare the different uses of variables.

   Examples: When a + b = b + a is used to state the commutative property for addition, the variables a and b represent all real numbers; the variable a in the equation 3a - 7 = 8 is a temporary placeholder for the one number, 5, that will make the equation true; the symbols C and r refer to specific attributes of a circle in the formula C = 2πr; the variable m in the slope-intercept form of the line, y = mx + b, serves as a parameter describing the slope of the line.

2. Express patterns, properties, formulas, and equations using and defining variables appropriately for each case.

C2 Analyze and identify characteristics of algebraic expressions; evaluate, interpret, and construct simple algebraic expressions; identify and transform expressions into equivalent expressions; determine whether two algebraic expressions are equivalent.

Two algebraic expressions are equivalent if they yield the same result for every value of the variables in them. Great care must be taken to demonstrate that, in general, a finite number of instances is not sufficient to demonstrate equivalence.

1. Analyze expressions to identify when an expression is the sum of two or more simpler expressions (called terms) or the product of two or more simpler expressions (called factors). Analyze the structure of an algebraic expression and identify the resulting characteristics.
2. Identify single-variable expressions as linear or non-linear.
3. Evaluate a variety of algebraic expressions at specified values of their variables.

   Algebraic expressions to be evaluated include polynomial and rational expressions as well as those involving radicals and absolute value.

4. Write linear and quadratic expressions representing quantities arising from geometric and real-world contexts.
5. Use commutative, associative, and distributive properties of number operations to transform simple expressions into equivalent forms in order to collect like terms or to reveal or emphasize a particular characteristic.
6. Rewrite linear expressions in the form ax + b for constants a and b.
7. Choose different but equivalent expressions for the same quantity that are useful in different contexts.

   Example: p + 0.07p shows the breakdown of the cost of an item into the price p and the tax of 7%, whereas (1.07)p is a useful equivalent form for calculating the total cost.

8. Demonstrate equivalence through algebraic transformations or show that expressions are not equivalent by evaluating them at the same value(s) to get different results.
9. Know that if each expression is set equal to y and the graph of all ordered pairs that satisfy one of these new equations is identical to the graph of all ordered pairs that satisfy the other, then the expressions are equivalent.

D. Functions

Middle school students increase their experience with functional relationships and begin to express and understand them in more formal ways. They distinguish between relations and functions and convert flexibly among the various representations of tables, symbolic rules, verbal descriptions, and graphs. A major focus at this level is on linear functions, recognizing linear situations in context, describing aspects of linear functions such as slope as a constant rate of change, identifying x- and y-intercepts, and relating slope and intercepts to the original context of the problem.

Successful students will:

D1 Determine whether a relationship is or is not a function; represent and interpret functions using graphs, tables, words, and symbols.

In general, a function is a rule that assigns a single element of one set–the output set—to each element of another set—the input set. The set of all possible inputs is called the domain of the function, while the set of all outputs is called the range.

1. Identify the independent (input) and dependent (output) quantities/variables of a function.
2. Make tables of inputs x and outputs f(x) for a variety of rules that take numbers as inputs and produce numbers as outputs.
3. Define functions algebraically, e.g., g(x) = 3 + 2(x - x²).
4. Create the graph of a function f by plotting and connecting a sufficient number of ordered pairs (x, f(x)) in the coordinate plane.
5. Analyze and describe the behavior of a variety of simple functions using tables, graphs, and algebraic expressions.
6. Construct and interpret functions that describe simple problem situations using expressions, graphs, tables, and verbal descriptions and move flexibly among these multiple representations.

D2 Analyze and identify linear functions of one variable; know the definitions of x- and y-intercepts and slope, know how to find them and use them to solve problems.

A function exhibiting a rate of change (slope) that is constant is called a linear function. A constant rate of change means that for any pair of inputs x₁ and x₂, the ratio of the corresponding change in value f(x₂) - f(x₂) to the change in input x₂ - x₁ is constant (i.e., it does not depend on the inputs).

1. Explain why any function defined by a linear algebraic expression has a constant rate of change.
2. Recognize that the graph of a linear function defined for all real numbers is a straight line, identify its constant rate of change, and create the graph.
3. Determine whether the rate of change of a specific function is constant; use this to distinguish between linear and nonlinear functions.
4. Know that a line with a slope equal to zero is horizontal and represents a function, while the slope of a vertical line is undefined and cannot represent a function.

D3 Express a linear function in several different forms for different purposes.

1. Recognize that in the form f(x) = mx + b, m is the slope, or constant rate of change of the graph of f, that b is the y-intercept and that in many applications of linear functions, b defines the initial state of a situation; express a function in this form when this information is given or needed.
2. Recognize that in the form f(x) = m(x - x₀) + y₀, the graph of f(x) passes through the point (x₀, y₀); express a function in this form when this information is given or needed.

D4 Recognize contexts in which linear models are appropriate; determine and interpret linear models that describe linear phenomena; express a linear situation in terms of a linear function f(x) = mx + b and interpret the slope (m) and the y-intercept (b) in terms of the original linear context.

Common examples of linear phenomena include distance traveled over time for objects traveling at constant speed; shipping costs under constant incremental cost per pound; conversion of measurement units (e.g., pounds to kilograms or degrees Celsius to degrees Fahrenheit); cost of gas in relation to gallons used; the height and weight of a stack of identical chairs.

D5 Recognize, graph, and use direct proportional relationships.

A linear function in which f(0) = 0 represents a direct proportional relationship. The linear function f(x) = kx, where k is constant, descries a direct proportional relationship.

1. Show that the graph of a direct proportional relationship is a line that passes through the origin (0, 0) whose slope is the constant of proportionality.
2. Compare and constrast the graphs of x = k, y = k, and y = kx, where k is a constant.

E. Equations and Identities

In this middle school course, students begin the formal study of equations. They solve linear equations and solve and graph linear inequalities in one variable. They graph equations in two variables, relating features of the graphs to the related single-variable equations. Solving systems of two linear equations in two variables graphically and understanding what it means to be a solution of such a system is also included in this unit. Interwoven with the development of these skills, students use linear equations, inequalities, and systems of linear equations to solve problems in context and interpret the solutions and graphical representations in terms of the original problem.

Successful students will:

E1 Distinguish among an equation, an expression, and a function; interpret identities as a special type of equation and identify their key characteristics.

An identity is an equation for which all values of the variables are solutions. Although an identity is a special type of equation, there is a difference in practice between the methods for solving equations that have a small number of solutions and methods for proving identities. For example, (x+2)² = x² + 4x + 4 is an identity which can be proved by using the distributive property, whereas (x+2)² = x² + 3x + 4 is an equation that can be solved by collecting all terms on one side.

1. Know that solving an equation means finding all its solutions and predict the number of solutions that should be expected for various simple equations and identities.
2. Explain why solutions to the equation f(x) = g(x) are the x-values (abscissas) of the set of points in the intersection of the graphs of the functions f(x) and g(x).
3. Recognize that f(x) = 0 is a special case of the equation f(x) = g(x) and solve the equation f(x) = 0 by finding all values of x for which f(x) = 0.

   The solutions to the equation f(x) = 0 are called roots of the equation or zeros of the function. They are the values of x where the graph of the function f crosses the x-axis. In the special case where f(x) equals 0 for all values of x, f(x) = 0 represents a constant function where all elements of the domain are zeros of the function.

4. Use identities to transform expressions.

E2 Solve linear equations and solve and graph the solution of linear inequalities in one variable.

Common problems are those that involve break-even time, time/rate/distance, percentage increase or decrease, ratio and proportion.

1. Solve equations using the facts that equals added to equals are equal and that equals multiplied by equals are equal; more formally, if A = B and C = D, then A + C = B + D and AC = BD; using the fact that a linear expression ax + b is formed using the operations of multiplication by a constant followed by addition, solve an equation ax + b = 0 by reversing these steps.

   Be alert to anomalies caused by dividing by 0 (which is undefined), or by multiplying both sides by 0 (which will produce equality even when things were originally unequal).

2. Graph a linear inequality in one variable and explain why the graph is always a half-line (open or closed); know that the solution set of a linear inequality in one variable is infinite, and contrast this with the solution set of a linear equation in one variable.
3. Explain why, when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality is reversed, but that when all other basic operations involving non-zero numbers are applied to both sides, the direction of the inequality is preserved.

E3 Recognize, represent, and solve problems that can be modeled using linear equations in two variables and interpret the solution(s) in terms of the context of the problem.

1. Rewrite a linear equation in two variables in any of three forms: ax + by = c, ax + by + c = 0, or y = mx + b; select a form depending upon how the equation is to be used.
2. Know that the graph of a linear equation in two variables consists of all points (x, y) in the coordinate plane that satisfy the equation and explain why, when x can be any real number, such graphs are straight lines.
3. Identify the relationship between linear functions in one variable, x maps to f(x) and linear equations in two variables f(x) = y or f(x) - y = 0; explain why the solution to an equation in standard (or polynomial) form (ax + b = 0) will be the point where the graph of f(x) = ax + b crosses the x-axis.
4. Identify the solution of an equation that is in the form f(x) = g(x) and relate the solution to the x-value (abscissa) of the point at which the graphs of the functions f(x) and g(x) intersect.
5. Know that pairs of non-vertical lines have the same slope if and only if they are parallel (or the same line) and slopes that are negative reciprocals if and only if they are perpendicular; apply these relationships to analyze and represent equations.
6. Represent linear relationships using tables, graphs, verbal statements, and symbolic forms; translate among these forms to extract information about the relationship.

E4 Determine the solution to application problems modeled by two linear equations and interpret the solution set in terms of the situation.

1. Determine either through graphical methods or comparing slopes whether a system of two linear equations has one solution, no solutions, or infinitely many solutions, and know that these are the only possibilities.
2. Represent the graphs of two linear equations as two intersecting lines when there is one solution, parallel lines when there is no solution, and the same line when there are infinitely many solutions.
3. Use the graph of two linear equations in two variables to suggest solution(s).

   Since the solution is a set of ordered pairs that satisfy the equations, it follows that these ordered pairs must lie on the graph of each of the equations in the system; the point(s) of intersection of the graphs is (are) the solution(s) to the system of equations.

4. Recognize and solve problems that can be modeled using two linear equations in two variables.

   Examples: Break-even problems, such as those comparing costs of two services

F. Geometric Reasoning, Representation, and Transformations

Middle school students begin to develop the mathematical reasoning needed to verify basic theorems about angles and triangles. Coordinate geometry affords them the opporunity to make valuable connections between algebra concepts and geometry representations, such as slope and distance. Students extend their elementary school experiences with transformations as specific motions in two-dimensions to transformations of figures in the coordinate plane. They describe the characteristics of transformations that preserve distance, relating them to congruence.

Successful students will:

F1 Know and verify basic theorems about angles and triangles.

1. Know the triangle inequality and verify it through measurement.

   In words, the triangle inequality states that any side of a triangle is shorter than the sum of the other two sides; it can also be stated clearly in symbols: If a, b, and c are the lengths of three sides of a triangle, then a < b + c, b < a + c, and c < a + b.

2. Verify that the sum of the measures of the interior angles of a triangle is 180°.
3. Verify that each exterior angle of a triangle is equal to the sum of the opposite interior angles.
4. Show that the sum of the interior angles of an n-sided convex polygon is (n - 2) x 180°.
5. Explain why the sum of exterior angles of a convex polygon is 360°.

F2 Represent and explain the effect of translations, rotations, and reflections of objects in the coordinate plane.

1. Identify certain transformations (translations, rotations, and reflections) of objects in the plane as rigid motions and describe their characteristics; know that they preserve distance in the plane.
2. Demonstrate the meaning and results of the translation, rotation, and reflection of an object through drawings and experiments.
3. Identify corresponding sides and angles between objects and their images after a rigid transformation.
4. Show how any rigid motion of a figure in the plane can be accomplished through a sequence of translations, rotations, and reflections.

F3 Represent and interpret points, lines, and two-dimensional geometric objects in a coordinate plane; calculate the slope of a line in a coordinate plane.

1. Determine the area of polygons in the coordinate plane.
2. Know how the word slope is used in common non-mathematical contexts, give physical examples of slope, and calculate slope for given examples.
3. Find the slopes of physical objects (roads, roofs, ramps, stairs) and express the answers as a decimal, ratio, or percent.
4. Interpret and describe the slope of parallel and perpendicular lines in a coordinate plane.
5. Show that the calculated slope of a line in a coordinate plane is the same no matter which two distinct points on the line one uses to calculate the slope.
6. Use coordinate geometry to determine the perpendicular bisector of a line segment.

G. Data Analysis

Students learn to design a study to answer a question; collect, organize, and summarize data; communicate the results; and make decisions about the findings. Technology is utilized both to analyze and display data. Students expand their repertoire of graphs and statistical measures and begin the use of random sampling in sample surveys. They assess the role of random assignment in experiments. They look critically at data studies and reports for possible sources of bias or misrepresentation. Students are able to use their knowledge of slope to analyze lines of best fit in scatter plots and make predictions from the data, further connecting their algebra and data knowledge.

Successful students will:

G1 Design a plan to collect appropriate data to address a well-defined question.

1. Understand the differing roles of a census, a sample survey, an experiment, and an observational study.
2. Select a design appropriate to the questions posed.
3. Use random sampling in surveys and random assignment in experiments, introducing random sampling as a "fair" way to select an unbiased sample.

G2 Represent both univariate and bivariate quantitative (measurement) data accurately and effectively.

1. Represent univariate data; make use of line plots (dot plots), stem-and-leaf plots, and histograms.
2. Represent bivariate data; make use of scatter plots.
3. Describe the shape, center, and spread of data distributions.

   Example: A scatter plot used to represent bivariate data may have a linear shape; a trend line may pass through the mean of the x and y variables; its spread is shown by the vertical distances between the actual data points and the line.

G3 Summarize, compare, and interpret data sets by using a variety of statistics.

1. Use percentages and proportions (relative frequencies) to summarize univariate categorical data.
2. Use conditional (row or column) percentages and proportions to summarize bivariate categorical data.
3. Use measures of center (mean and median) and measures of spread (percentiles, quartiles, and interquartile range) to summarize univariate quantitative data.
4. Use trend lines (linear approximations or best-fit lines) to summarize bivariate quantitative data.
5. Graphically represent measures of center and spread (variability) for quantitative data.
6. Interpret the slope of a linear trend line in terms of the data being studied.
7. Use box plots to compare key features of data distributions.

G4 Read, interpret, interpolate, and judiciously extrapolate from graphs and tables and communicate the results.

1. State conclusions in terms of the question(s) being investigated.
2. Use appropriate statistical language when reporting on plausible answers that go beyond the data actually observed.
3. Use oral, written, graphic, pictorial and multi-media methods to create and present manuals and reports.

G5 Determine whether a scatter plot suggests a linear trend.

1. Visually determine a line of good fit to estimate the relationship in bivariate data that suggests a linear trend.
2. Identify criteria that might be used to assess how good the fit is.

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

Supporting Resources

Home

Jump to:

A. Real Numbers, Exponents and Roots and the Pythagorean Theorem

B. Number Bases [OPTIONAL ENRICHMENT UNIT]

C. Variables and Expressions

D. Functions

E. Equations and Identities

F. Geometric Reasoning, Representation and Transformations

G. Data Analysis

Tasks Related to This Course

• Archeological Similarity
• Click It
• Common Differences
• Counting Cubes
• Cycling Situations
• Equal Salaries for Equal Work?
• Match That Function
• Neighborhood Park
• Out of the Swimming Pool
• Season Pass
• Talk Is Cheap
• Television and Test Grades
• You're Toast, Dude!