Standard 1:  Number and Computation

The student uses numerical and computational cncepts and procedures in a variety of situations.

Seventh Grade

Benchmark 1
The student demonstrates number sense for rational numbers, the irrational number pi, and simple algebraic expressions in one variable in a variety of situations. Knowledge Base Indicators
  1. The student knows, explains, and uses equivalent representations for rational numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; integer bases with whole number exponents; positive rational numbers written in scientific notation with positive integer exponents; time; and money.
  2. The student compares and orders rational numbers and the irrational number pi.
  3. The student explains the relative magnitude between rational numbers and between rational numbers and the irrational number pi.
  4. The student knows and explains what happens to the product or quotient when:
    1. a whole number is multiplied or divided by a rational number greater than zero and less than one,
    2. a whole number is multiplied or divided by a rational number greater than one,
    3. a rational number (excluding zero) is multiplied or divided by zero.
  5. The student explains and determines the absolute value of rational numbers.
Application Indicators
  1. The student generates and/or solves real-world problems using:
    1. equivalent representations of rational numbers and simple algebraic expressions.
    2. fraction and decimal approximations of the irrational number pi.
  2. The student determines whether or not solutions to real-world problems using rational numbers, the irrational number pic, and simple algebraic expressions are reasonable.
 Benchmark 2
The student demonstrates an understanding of the rational number system and the irrational number; recognizes, uses, and describes their properties; and extends these properties to algebraic expressions in one variable. Knowledge Base Indicators
  1. The student knows and explains the relationships between natural (counting) numbers, whole numbers, integers, and rational numbers using mathematical models.
  2. The student classifies a given rational number as a member of various subsets of the rational number system.
  3. The student names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects:
    1. commutative properties of addition and multiplication (changing the order of the numbers does not change the solution);
    2. associative properties of addition and multiplication (changing the grouping of the numbers does not change the solution);
    3. distributive property [distributing multiplication or division over addition or subtraction];
    4. substitution property (one name of a number can be substituted for another name of the same number).
  4. The student uses and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects:
    1. identify properties for addition and multiplication (additive identity - zero added to any number is equal to that number; multiplicative identity - one multiplied by any number is equal to that number);
    2. symmetric property of equality;
    3. zero property of multiplication (any number multiplied by zero is zero);
    4. addition and multiplication properties of equality (adding/multiplying the same number to each side of an equation results in an equivalent equation);
    5. additive and multiplicative inverse properties. (Every number has a value known as its additive inverse and when the original number is added to that additive inverse, the answer is zero).
  5. The student recognizes that the irrational number pi can be represented by approximate rational values.
Application Indicators
  1. The student generates and/or solves real-world problems with rational numbers and the irrational number pi using the concepts of these properties to explain reasoning;
    1. commutative and associative properties of addition and multiplication.
    2. distributive property.
    3. substitution property.
    4. symmetric property of equality
    5. additive and multiplicative identities.
    6. zero property of multiplication.
    7. addition and multiplication properties of equality.
    8. additive and multiplicative inverse properties.
  2. The student analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals, or the irrational number pi and its rational approximations in solving a given real-world problem.
 Benchmark 3
The student uses computational estimation with rational numbers and the irrational number pi in a variety of situations. Knowledge Base Indicators
  1. The student estimates quantities with combinations of rational numbers and/or the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology.
  2. The student uses various estimation strategies and explains how they were used to estimate rational number quantities and the irrational number pi.
  3. The student recognizes and explains the difference between an exact and approximate answer.
  4. The student determines the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result.
  5. The student knows and explains why the fraction (22/7) or decimal (3.14) representation of the irrational number pi is an approximate value.
Application Indicators
  1. The student adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference).
  2. The student estimates to check whether or not the result of a real-world problem using rational numbers, the irrational number pi, and/or simple algebraic expressions is reasonable and makes predictions based on the information.
  3. The student determines a reasonable range for the estimation of a quantity given a real-world problem and explains the reasonableness of the range.
  4. The student determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology.
 Benchmark 4
The student models, performs, and explains computation with rational numbers, the irrational number pi, and first-degree algebraic expressions in one variable in a variety of situations. Knowledge Base Indicators
  1. The student computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology.
  2. The student performs and explains these computational procedures:
    1. adds and subtracts decimals from ten millions place though hundred thousandths place;
    2. multiplies and divides a four-digit number by a two-digit number using numbers from thousands place through thousandths place;
    3. multiplies and divides using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each;
    4. adds, subtracts, multiplies, and divides fractions and expresses answers in simplest form;
    5. adds, subtracts, multiplies, and divides integers;
    6. uses order of operations (evaluates within grouping symbols, evaluates powers to the second or third power, multiplies or divides in order from left to right, then adds or subtracts in order from left to right) using whole numbers;
    7. simplifies positive rational numbers raised to positive whole number powers;
    8. combines like terms of a first degree algebraic expression.
  3. The student recognizes, describes, and uses different ways to express computational procedures.
  4. The student finds prime factors, greatest common factor, multiples, and the least common multiple.
  5. The student finds percentages of rational numbers.
Application Indicators
  1. The student generates and/or solves one- and two-step real-world problems using these computational procedures and mathematical concepts:
    1. addition, subtraction, multiplication, and division of rational numbers with a special emphasis on fractions and expressing answers in simplest form.
    2. addition, subtraction, multiplication, and division of rational numbers with a special emphasis on integers.
    3. first degree algebraic expressions in one variable.
    4. percentages of rational numbers.
    5. approximation of the irrational number pi.

Standard 2:  Algebra

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

Seventh Grade

Benchmark 1
The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations. Knowledge Base Indicators
  1. The student identifies, states, and continues a pattern presented in various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes:
    1. counting numbers including perfect squares, cubes, and factors and multiples;
    2. positive rational numbers including arithmetic and geometric sequences (arithmetic: sequence of numbers in which the difference of two consecutive numbers is the same, geometric: a sequence of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number);
    3. geometric figures;
    4. measurements;
    5. things related to daily life.
  2. The student generates a pattern.
  3. The student extends a pattern when given a rule of one or two simultaneous changes (addition, subtraction, multiplication, division) between consecutive terms.
  4. The student states the rule to find the nth term of a pattern with one operational change (addition or subtraction) between consecutive terms.
Application Indicators
  1. The student generalizes a pattern by giving the nth term using symbolic notation.
  2. The student recognizes the same general pattern presented in different representations.
 Benchmark 2
The student uses variables, symbols, rational numbers, and simple algebraic expressions in one variable to solve linear equations and inequalities in a variety of situations. Knowledge Base Indicators
  1. The student knows and explains that a vaiable can represent a single quantity that changes.
  2. The student knows, explains, and uses equivalent representations for the same simple algebraic expressions.
  3. The student shows and explains how changes in one variable affects other variables.
  4. The student explains the difference between an equation and an expression.
  5. The student solves:
    1. one-step linear equations in one variable with positive rational coefficients and solutions;
    2. two-step linear equations in one variable with counting number coefficients and constants and positive rational solutions;
    3. one-step linear inequalities with counting numbers and one variable.
  6. The student explains and uses the equality and inequality symbols (= ≠ ≤ < > ≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) to represent mathematical relationships with rational numbers.
  7. The student knows the mathematical relationship between ratios, proportions, and percents and how to solve for a missing tern in a proportion with positive rational number solutions and monomials.
  8. The student evaluates simple algebraic expressions using positive rational numbers.
Application Indicators
  1. The student represents real-world problems using variables and symbols to write linear expressions, one- or two-step equations.
  2. The student solves real-world problems with one-or two-step linear equations in one variable with whole number coefficients and constants and positive rational solutions intuitively and analytically.
  3. The student generates real-world problems that represent one- or two-step linear equations.
  4. The student explains the mathematical reasoning that was used to solve a real-world problem using a one- or two-step linear equation.
 Benchmark 3
The student recognizes, describes, and analyzes constant and linear relationships in a variety of situations. Knowledge Base Indicators
  1. The student recognizes constant and linear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology.
  2. The student finds the values and determines the rule through two operations using a function table (input/output machine, T-table).
  3. The student demonstrates mathematical relationships using ordered pairs in all four quadrants of a coordinate plane.
  4. The student describes and/or gives examples of mathematical relationships that remain constant.
Application Indicators
  1. The student represents a variety of constant and linear relationships using written or oral descriptions of the rule, tables, graphs, and when possible, symbolic notation.
  2. The student interprets, describes, and analyzes the mathematical relationships of numerical, tabular, and graphical respresentations, including translations between the representations.
 Benchmark 4
The student generates and uses mathematical models to represent and justify mathematical relationships found in a variety of situations. Knowledge Base Indicators
  1. The student knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships.  Mathematical models include:
    1. process models (concrete objects, pictures, diagrams, number division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations.
    2. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures;
    3. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities;
    4. factor trees to find least common multiple and greatest common factor and to model prime factorization;
    5. equations and inequalities to model numerical relationships;
    6. function tables to model numerical and algebraic relationships;
    7. coordinate planes to model relationships between ordered pairs and linear equations;
    8. two- and three-dimensional geometric models (geoboards, dot paper, nets or solids) to model perimeter, area, volume, and surface area, and properties of two- and three-dimensional;
    9. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability;
    10. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single stem-and-leaf plots, scatter plots, and box-and-whisker plots to organize and display data;
    11. Venn diagrams to sort data and show relationships.
Application Indicators
  1. The student recognizes that various mathematical models can be used to represent the same problem situation. Mathematical models include:
    1. process models (concrete objects, pictures, diagrams, flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations;
    2. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations;
    3. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities;
    4. factor trees to find least common multiple and greatest common factor and to model prime factorization.
    5. equations and inequalities to model numerical relationships.
    6. function tables to model numerical and algebraic relationships;
    7. coordinate planes to model relationships between ordered pairs and linear equations;
    8. two- and three-dimensional geometric models (geoboards, dot paper, nets or solids) to model perimeter, area, volume, and surface area, and properties of two-and three-dimensional models;
    9. scale drawings to model large and small real-world objects;
    10. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability;
    11. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single stem-and-leaf plots, scatter plots, and box-and-whisker plots to describe, interpret, and analyze data;
    12. Venn diagrams to sort data and show relationships.
  2. The student selects a mathematical model and justifies why some mathematical models are more accurate than other mathematical models in certain situations.
  3. The student uses the mathematical modeling process to make inferences about real-world situations when the mathematical model used to represent the situation is given.

Standard 3:  Geometry

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

Seventh Grade

Benchmark 1
The student recognizes geometric figures and compares their properties in a variety of situations. Knowledge Base Indicators
  1. The student recognizes and compares properties of two- and three-dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology.
  2. The student classifies regular and irregular polygons having through ten sides as convex or conclave.
  3. The student identifies angle and side properties of triangles and quadrilaterals:
    1. sum of the interior angles of any triangle is 180°,
    2. sum of the interior angles of any quadrilateral is 360°,
    3. parallelograms have opposite sides that are parallel and congruent;
    4. rectangles have angles of 90°, opposite sides are congruent;
    5. rhombi have all sides the same length, opposite angles are congruent;
    6. squares have angles of 90°, all sides congruent;
    7. trapezoids have one pair of opposite sides parallel and the other pair of opposite sides are not parallel.
  4. The student identifies and describes:
    1. the altitude and base of a rectangular prism and triangular prism,
    2. the radius and diameter of a cylinder.
  5. The student identifies corresponding parts of similar and congruent triangles and quadrilaterals.
  6. The student uses symbols for right angles within a figure, parallel, perpendicular, and trangle to describe geometric figures.
  7. The student classifies triangles as:
    1. scalene, isosceles, or equilateral;
    2. right, acute, obtuse, or equiangular.
  8. The student determines if a triangle can be constructed given sides of three different lengths.
  9. The student generates a pattern for the sum of angles for 3-, 4-, 5-, ... n-sides polygons.
  10. The student describes the relationship between the diameter and the circumference of a circle.
Application Indicators
  1. The student solves real-world problems by applying the properties of:
    1. plane figures (regular and irregular polygons through 10 sides, circles, and semicircles) and the line(s) of symmetry.
    2. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases.
  2. The student decomposes geometric figures made from:
    1. regular and irregular polygons through 10 sides, circles, and semicircles;
    2. nets (two-dimensional shapes that can be folded into three-dimensional figures).
    3. prisms, pyramids, cylinders, cones, spheres, and hemispheres.
  3. The student composes geometric figures made from:
    1. regular and irregular polygons through 10 sides, circles, and semicircles;
    2. nets (two-dimensional shapes that can be folded into three-dimensional figures);
    3. prisms, pyramids, cylinders, cones, spheres, and hemispheres.
 Benchmark 2
The student estimates, measures, and uses measurement formulas ina variety of situations. Knowledge Base Indicators
  1. The student determines and uses rational number approximations (estimations) for length, width, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure.
  2. The student selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area, and angle measurements.
  3. The student converts within the customary system and within the metric system.
  4. The student knows and uses perimeter and area formulas for circles, squares, rectangles, triangles, and parallelograms.
  5. The student finds perimeter and area of two-dimensional composite figures of circles, squares, rectangles, and triangles.
  6. The student uses given measurement formulas to find:
    1. surface area of cubes,
    2. volume of rectangular prisms.
  7. The student finds surface area of rectangular prisms using concrete objects.
  8. The student uses appropriate units to describe rate as a unit of measure.
  9. The student finds missing angle measurements in triangles and quadrilaterals.
Application Indicators
  1. The student solves real-world problems by:
    1. converting within the customary and metric systems.
    2. finding perimeter and area of circles, squares, rectangles, triangles, and parallelograms.
    3. finding perimeter and area of two-dimensional composite figures of squares, rectangles, and triangles.
    4. using appropriate units to describe rate as a unit of measure.
    5. finding missing angle measurements in triangles and quadrilaterals.
    6. applying various measurement techniques (selecting and using measurement tools, units of measure, and level of precision) to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area appropriate to a given situation.
  2. The student estimates to check whether or not measurements or calculations for length, width, weight, volume, temperature, time, perimeter, and area in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference).
 Benchmark 3
The student recognizes and performs transformations on two- and three- dimensional geometric figures in a variety of situations. Knowledge Base Indicators
  1. The student identifies, describes, and performs single and multiple transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a two-dimensional figure.
  2. The student identifies three-dimensional figures from various perspectives (top, bottom, sides, corners).
  3. The student draws three-dimensional figures from various perspectives (top, bottom, sides, corners).
  4. The student generates a tessellation.
Application Indicators
  1. The student describes the impact of transformations [reflection, rotation, translation, reduction (contraction/shrinking, enlargement (magnification/growing)] on the perimeter and area of squares and rectangles.
  2. The student investigates congruency and similarity of geometric figures using transformations.
  3. The student determines the actual dimensions and/or measurements of a two-dimensional figure represented in a scale drawing.
 Benchmark 4
The student relates geometric concepts to a number line and a coordinate plane in a variety of situations. Knowledge Base Indicators
  1. The student finds the distance between the points on a number line by computing the absolute value of their difference.
  2. The student uses all four quadrants of a coordinate plane to :
    1. identify in which quadrant or on which axis a point lies when given the coordinates of a point,
    2. plot points,
    3. identify points,
    4. list through five ordered pairs of a given line.
  3. The student uses a given linear equation with whole number coefficients and constants and a whole number solution to find the ordered pairs, organize the ordered pairs using a T-table, and plot the ordered pairs on the coordinate plane.
  4. The student examines characteristics of two-dimensional figures on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology.
Application Indicators
  1. The student represents and/or generates real-world problems using a coordinate plane to find:
    1. perimeter of squares and rectangles.
    2. circumference (perimeter) of circles.
    3. area of circles, parallelograms, triangles, squares, and rectangles.

Standard 4:  Data

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

Seventh Grade

Benchmark 1
The student applies the concepts of probability to draw conclusions, generate convincing arguments, and make predictions and decisions including the use of concrete objects in a variety of situations. Knowledge Base Indicators
  1. The student finds the probability of a compound event composed of two independent events in an experiment or simulation.
  2. The student explains and gives examples of simple or compound events in an experiment of simulation having probability of zero or one.
  3. The student uses a fraction, decimal, and percent to represent the probability of:
    1. a simple event in an experiment or simulation;
    2. a compound event composed of two independent events in an experiment or simulation.
  4. The student finds the probability of a simple event in an experiment or simulation using geometric models.
Application Indicators
  1. The student conducts an experiment or simulation with a compound event composed of two independent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions and make predictions about future events.
  2. The student analyzes the results of an experiment or simulation of a compound event composed of two independent events to draw conclusions, generate convincing arguments, and make predictions and decisions in a variety of real-world situations.
  3. The student compares results of theoretical (expected) probability with empirical (experimental) probability in an experiment or situation with a compound event composed of two simple independent events and understands that the larger the sample size, the greater the likelihood that the experimental results will equal the theoretical probability.
  4. The student makes predictions bases on the theoretical probability of a simple event in an experiment or simulation.
 Benchmark 2
The student collects, organizes, displays, and explains numerical (rational numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency. Knowledge Base Indicators
  1. The student organizes, displays, and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays:
    1. frequency tables;
    2. bar, line, and circle graphs;
    3. Venn diagrams or other pictorial displays;
    4. charts and tables;
    5. stem-and-leaf plots (single);
    6. scatter plots;
    7. box-and-whiskers plots.
  2. The student selects and justifies the choice of data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, or purposeful sampling) in a given situation.
  3. The student conducts experiments with sampling and describes the results.
  4. The student determines the measures of central tendency (mode, median, mean) for a rational number data set.
  5. The student identifies and determines the range and the quartiles of a rational number data set.
  6. The student identifies potential outliers within a set of data by inspection rather than formal calculation.
Application Indicators
  1. The student uses data analysis (mean, median, mode, range) of a rational number data set to make reasonable inferences and predictions, to analyze decisions, and to develop convincing arguments from these data displays:
    1. frequency tables;
    2. bar, line, and circle graphs;
    3. Venn diagrams or other pictorial displays;
    4. charts and tables;
    5. stem-and-leaf plots (single);
    6. scatter plots;
    7. box-and-whiskers plots.
  2. The student explains advantages and disadvantages of various data displays for a given data set.
  3. The student recognizes and explains:
    1. misleading representations of data;
    2. the effects of scale or interval changes on graphs of data sets.
  4. The student determines and explains the advantages and disadvantages of using each measure of central tendency and the range to describe a data set.