MATH CURRICULUM
Eighth Grade
March 2002

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Standard 1: The student uses numerical and computational concepts and procedures in a variety of situations. 

Benchmark 1: NUMBER SENSE
The student demonstrates number sense for real numbers and algebraic expressions in a variety of situations. 

Eighth Grade Knowledge Base Indicators
The student:

1. compares and orders real numbers or algebraic expressions and explains the relative magnitude among them.

2. knows, explains, and uses equivalent representations for the same real numbers, including integers, decimals, ratios, fractions, percents, scientific notation, and numbers with integer exponents.

3. determines the absolute value of real numbers.

Special Note: Remember, real numbers include very large numbers such as one trillion and very small numbers such as one-millionth.

Eighth Grade Application Indicators
The student:

1. uses appropriate representations of real numbers and algebraic expressions to formulate and solve real-world problems.

2. determines reasonableness or magnitude of real number results.

Benchmark 2: NUMBER SYSTEMS AND THEIR PROPERTIES - The student demonstrates an understanding of the real number system, recognizes, applies, and explains its properties, and extends these properties to algebraic expressions. 

Eighth Grade Knowledge Base Indicators
The student:

1. recognizes the need for real numbers.

2. explains and illustrates the relationship among 

3. natural (counting) numbers, whole numbers, integers, rational numbers and irrational numbers (number lines, Venn diagrams).

4. names, uses, and explains the following properties: addition and multiplication properties of equalities and inequalities; commutative, associative, distributive, and substitution properties; identity and inverse properties of addition and multiplication; the zero product property (ab=0, then a=0 or b=0); and names, uses, and explains transitive, reflexive, and symmetric properties.

Eighth Grade Application Indicators
The student:

1. uses properties of the real number system to explain reasoning, and to formulate and solve real-world problems.

2. uses real number properties to perform various computational procedures and explains how they were used.

Benchmark 3: ESTIMATION - The student uses numerical estimation with real numbers in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. uses a variety of computational methods including mental mathematics, paper and pencil, concrete materials, and technological tools such as calculators or computers to estimate real number quantities.

2. estimates quantities using real numbers and simple algebraic expressions, explains the estimation method used such as rounding, special numbers, clustering, truncation, or compatible numbers.

3. knows and explains between which two consecutive integers an irrational number lies.

4. knows and explains why a decimal representation of an irrational number is an approximate value. 

Eighth Grade Application Indicators
The student:

1. uses estimation to check reasonableness of results, and makes predictions in situations involving real numbers and simple algebraic expressions.

2. determines if a problem situation calls for an exact or approximate answer and performs the appropriate computation.

Benchmark 4: COMPUTATION - The student explains and performs computation with real numbers and algebraic expressions in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. uses a variety of computational methods including mental arithmetic (doubles and neighbors), paper and pencil, concrete materials, or technological tools such as calculators or computers.

2. knows, explains and performs arithmetic operations and their inverses on real numbers (addition, subtraction, multiplication, division, square roots of perfect squares, cube roots of perfect cubes, and powers).

3. explains and performs computations using the correct order of operations.

4. finds roots of real numbers using calculators.

5. knows, explains and uses the inverse relationship between whole number exponents and their roots (23=8 and cube root of 8=2).

6. adds algebraic expressions.

7. knows, explains and performs computations involving percent of increase and decrease.

8. finds what percent one number is of another number such as what percent of 80 is 120?

9. finds a number when a percent of the number is given such as 15% of what number is 30?

Eighth Grade Application Indicators
The student:

1. uses arithmetic operations and inverse relationships to formulate and solve real-world problems involving real numbers and algebraic expressions with special emphasis on topics such as percent increase and decrease.

Standard 2: The student uses algebraic concepts and procedures in a variety of situations.

Benchmark 1: PATTERNS - The student recognizes, describes, extends, develops, analyzes and gives the general rule of patterns from a variety of situations involving tenth grade content.

Eighth Grade Knowledge Base Indicators
The student:

1. identifies and continues patterns presented in a variety of formats: numeric, algebraic, visual, oral, written, kinesthetic, pictorial, tabular, graphical, or listing.
2. recognizes patterns with two simultaneous changes.
3. creates a pattern.
   A partial list of patterns to help students meet these indicators could include: real number patterns such as π/4, π/2, 3π/4, π...basic algebraic patterns in one variable such as x, 2-x, 3-x, 4-x, ... or x, x2, x3, ...cyclic patterns such as time, calendars, and tides. Given a table of linear values, graph the ordered pairs and generalize the pattern. 

Eighth Grade Application Indicators
The student:

1. generalizes a pattern using a written description.
2. generalizes numerical and geometric patterns using algebra, and relates the equation, graph and table of values resulting from the generalization. 

Benchmark 2: VARIABLES, EQUATIONS, AND INEQUALITIES - The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. solves linear equations and inequalities in one variable with whole number coefficients and constants with rational number solutions analytically, graphically, and intuitively such as 2x+1=9 or 2x+13<8.
2. finds the number when a percent and a number are given, finds what percent one number is of another number, and finds percent increase and percent decrease.
3. evaluates formulas using substitution.

Eighth Grade Application Indicators
The student:

1. uses symbols, variables, expressions, inequalities, equations and simple systems of equations to represent problem situations which involve variable quantities.
2. formulates and solves problem situations involving percents, variables, expressions, linear equations and inequalities.

Benchmark 3: FUNCTIONS - The student analyzes functions in a variety situations.

Eighth Grade Knowledge Base Indicators
The student:

1. knows and explains the difference between linear and nonlinear relationships.

Eighth Grade Application Indicators
The student:

1. moves between symbolic, numerical, tabular and graphical representations of linear relationships with fluency and accuracy. 
2. uses variables to generalize patterns, and information presented in tables, charts, and graphs.

Benchmark 4: MODELS - The student develops and uses models to represent and justify mathematical relationships found in a variety of situations involving tenth grade knowledge and skills.

Eighth Grade Knowledge Base Indicators
The student:

2. creates mathematical models to show the relationship between two or more things.

The student knows, explains, and uses these specific mathematical models: draws a diagram/picture to model a problem situation, the number line to model the relationship between real numbers and operations on real numbers, the coordinate plane to model ordered pairs and linear equations and inequalities, equations and inequalities to model numerical relationships, graphs, tables or charts to organize, interpret and analyze data.

Eighth Grade Application Indicators
The student:

1. recognizes the same situation can be modeled in more than one way. 
2. determines if a given graphical, algebraic or geometric model is an accurate representation of a specific real-world situation.
3. uses the mathematical modeling process to make inferences about real-world situations.

Special Note: The mathematical modeling process involves:

1. selecting key features and relationships within the real-world situation and representing these concepts in mathematical terms through some sort of mathematical model;
2. performing manipulations and mathematical procedures within the mathematical model;
3. interpreting the results of the manipulations within the mathematical model and using these results to make inferences about the original real-world situation.

Standard 3: The student uses geometric concepts and procedures in a variety of situations.

Benchmark 1: GEOMETRIC FIGURES AND THEIR PROPERTIES - The student recognizes geometric figures, and applies, compares, and justifies the properties of geometric figures in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. recognizes, classifies and discusses properties of all geometric figures listed in previous grade level indicators plus the following figures: point, line, plane.
2. knows angle relationships formed by parallel lines cut by a transversal using appropriate technology (alternate interior, alternate exterior, supplementary, vertical angles, corresponding angles, complementary, consecutive interior).
3. recognizes the following figures and explains their attributes: arc, chord, and semicircle.
4. knows and explains how to use appropriate technology, manipulatives, constructions, or drawings to recognize, classify and compare geometric figures.
5. recognizes or applies properties of corresponding parts of similar and congruent triangles and quadrilaterals.

Eighth Grade Application Indicators
The student:

1. formulates and solves real-world problems by applying properties of geometric figures.

Benchmark 2: MEASUREMENT AND ESTIMATION - The student estimates, measures and uses geometric formulas in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. uses various estimation techniques to determine rational number approximations of surface area, weight, and capacity.
2. uses given measurement formulas for two- and three-dimensional figures including area of a trapezoid, volume of a cylinder and triangular prism, and surface area of a cylinder and triangular prism.
3. applies various measurement techniques, uses appropriate tools; units of measure and degree of accuracy to find accurate real number representations for length, volume, surface area, area, perimeter, weight, capacity, temperature, time and angle measurements.
4. explains how simple measurement formulas involving squares, rectangles, and triangles work.
5. uses ratios and proportions to measure inaccessible objects.
6. performs conversions within each system of measurement.

Eighth Grade Application Indicators
The students:

1. formulates and solves real-world problems by applying measurements and measurement formulas.
2. uses estimation to check reasonableness of measurements and calculations and/or adjusts original measurements based on additional information (estimation from a frame of reference).

Benchmark 3: TRANSFORMATIONAL GEOMETRY - The student recognizes and applies transformation of two- and three-dimensional figures in a variety situations.

Eighth Grade Knowledge Base Indicators
The student:

1. recognizes, explains, and performs single and multiple transformations [rotation, reflection, translation, shrink or magnify (dilate)] to two-dimensional figures.
2. performs reflections of two-dimensional figures in the coordinate plane, over the x- and y-axis.
3. draws three-dimensional shapes as they would appear from a variety of visual perspectives (top, bottom, side, and corner).
4. creates a two-dimensional drawing of a three-dimensional shape.

Eighth Grade Application Indicators
The student:

1. discusses and discerns relationships among combinations of reflections, translations and rotations.
2. describes and draws two-dimensional shapes as they would appear after undergoing two specified transformations.

Benchmark 4: GEOMETRY FROM AN ALGEBRAIC PERSPECTIVE - The student uses an algebraic perspective to analyze the geometry of two- and three-dimensions in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. uses a variety of methods including mental mathematics, paper and pencil, concrete materials, graphing utilities and other technological tools to analyze two-dimensional geometry on the coordinate plane.
2. creates a T-table from a linear equation and plots ordered pairs.
3. lists five coordinate pairs on the graph of a line and states the pattern that exists.
4. determines the length of a side of a figure drawn on a coordinate plane with vertices having the same x-coordinates or y-coordinates.
5. recognizes that ordered pairs that lie on the graph of an equation are solutions to that equation.
6. recognizes that points that do not lie on the graph of an equation are not solutions to that equation.

Eighth Grade Application Indicators
The student:

1. represents, formulates and solves distance and geometry problems using the language and symbols of algebra and the coordinate plane.

Standard 4: The student uses concepts and procedures of data analysis in a variety of situations.

Benchmark 1: PROBABILITY - The student applies probability theory to analyze the validity of arguments, draw conclusions, and make decisions in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. knows and explains the difference between independent and dependent events.
2. identifies situations involving independent or dependent events.
3. calculates the odds of a desired outcome in a simple experiment.

Eighth Grade Application Indicators
The student:

1. conducts experiments and/or simulations, records the results in charts, tables or graphs, and uses the results to draw conclusions and make predictions.
2. compares theoretical (expected) results with empirical (experimental) results.

Benchmark 2: STATISTICS - The student generates, organizes, and interprets real number and other data in a variety of situations.

Eighth Grade Knowledge Base Indicators
The student:

1. rganizes, displays and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized and accurate manner including the use of correct titles, labels, and intervals or categories. Specific formats for data displays include:
   frequency distributions.
   box-and-whiskers plots.
   stem-and-leaf plots.
   histograms.
   scatterplots.
   bar, line, and circle graphs.
   Venn diagrams or other pictorial 
   displays.
   charts and tables.
2. calculates and explains the measures of central tendency (mean, median, mode) for a real number data set.
3. calculates or explains the range, quartiles, and interquartile range for a real number data set.
4. explains the affects of outliers on the mean, median, and range of a real number data set.
5. recognizes valid and invalid sampling techniques.

Eighth Grade Application Indicators
The student:

1. explains advantages and disadvantages of various sampling techniques for gathering data for a given situation (observation, survey, census of total population, random sampling, etc.) 
2. uses data analysis to make accurate inferences, decisions/predictions, and to develop convincing arguments from data displayed in a variety of formats. (see listing of graphical displays)
3. uses changes in scales, intervals or categories to help support a particular interpretation of the data.