KANSAS State Standards

1.1 The student demonstrates number sense for real numbers and simple algebraic expressions in a variety of situations.

1.1.K1 knows, explains, and uses equivalent representations for rational numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; rational number bases with integer exponents; rational numbers written in scientific notation with integer exponents; time; and money.

1.1.K2 compares and orders rational numbers, the irrational number pi, and algebraic expressions, e.g., which expression is greater ??3n or 3n? It depends on the value of n. If n is positive, 3n is greater. If n is negative, -3n is greater. If n is zero, they are equal.

1.1.K3 explains the relative magnitude between rational numbers, the irrational number pi, and algebraic expressions.

1.1.K4 recognizes and describes irrational numbers, e.g., sqare root of 2 is a non-repeating, non-terminating decimal; or pi is a non-terminating decimal.

1.1.K5 knows and explains what happens to the product or quotient when:

1.1.K5.a a positive number is multiplied or divided by a rational number greater than zero and less than one, e.g., if 24 is divided by 1/3, will the answer be larger than 24 or smaller than 24? Explain.

1.1.K5.b a positive number is multiplied or divided by a rational number greater than one,

1.1.K5.c a nonzero real number is multiplied or divided by zero. (For purposes of assessment, an explanation of division by zero will not be expected.)

1.1.K6 explains and determines the absolute value of real numbers.

1.2 The student demonstrates an understanding of the real number system; recognizes, applies, and explains their properties; and extends these properties to algebraic expressions.

1.2.K1 explains and illustrates the relationship between the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] using mathematical models, e.g., number lines or Venn diagrams.

1.2.K2 identifies all the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] to which a given number belongs. (For the purposes of assessment, irrational numbers will not be included.)

1.2.K3 names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects:

1.2.K3.a commutative, associative, distributive, and substitution properties [commutative: a + b = b + a and ab = ba; associative: a + (b + c) = (a + b) + c and a(bc) = (ab)c; distributive: a(b + c) = ab + ac; substitution: if a = 2, then 3a = 3 x 2 = 6];

1.2.K3.b identity properties for addition and multiplication and inverse properties of addition and multiplication (additive identity: a + 0 = a, multiplicative identity: a ?? 1 = a, additive inverse: +5 + -5 = 0, multiplicative inverse: 8 x 1/8 = 1);

1.2.K3.c symmetric property of equality, e.g., 7 + 2 = 9 has the same meaning as 9 = 7 + 2;

1.2.K3.d addition and multiplication properties of equalities, e.g., if a = b, then a + c = b + c;

1.2.K3.e addition property of inequalities, e.g., if a > b, then a + c > b + c;

1.2.K3.f zero product property, e.g., if ab = 0, then a = 0 and/or b = 0.

1.3 The student uses computational estimation with real numbers in a variety of situations.

1.3.K1 estimates real number quantities using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology.

1.3.K2 uses various estimation strategies and explains how they were used to estimate real number quantities and simple algebraic expressions.

1.3.K3 knows and explains why a decimal representation of the irrational number pi is an approximate value.

1.3.K4 knows and explains between which two consecutive integers an irrational number lies.

1.4 The student models, performs, and explains computation with rational numbers, the irrational number pi, and algebraic expressions in a variety of situations.

1.4.K1 computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology.

1.4.K2 performs and explains these computational procedures with rational numbers:

1.4.K2.a addition, subtraction, multiplication, and division of integers;

1.4.K2.b 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);

1.4.K2.c approximation of roots of numbers using calculators;

1.4.K2.d multiplication or division to find:

1.4.K2.d.i a percent of a number, e.g., what is 0.5% of 10?

1.4.K2.d.ii percent of increase and decrease, e.g., if two coins are removed from ten coins, what is the percent of decrease?

1.4.K2.d.iii percent one number is of another number, e.g., what percent of 80 is 120?

1.4.K2.d.iv a number when a percent of the number is given, e.g., 15% of what number is 30?;

1.4.K2.e addition of polynomials, e.g., (3x ?? 5) + (2x + 8).

1.4.K2.f simplifies algebraic expressions in one variable by combining like terms or using the distributive property, e.g., ??3(x ?? 4) is the same as ??3x + 12.

1.4.K3 finds factors and common factors of simple monomial expressions, e.g., given the monomials 10m? n? and 15a?mn?; some common factors would be 5m, 5mn?, and n?.

2.1 The student recognizes, describes, extends, develops, and explains the general rule of a pattern from a variety of situations.

2.1.K1 identifies, states, and continues a pattern presented in various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, graph), verbal (oral description), kinesthetic (action), and written using these attributes:

2.1.K1.a counting numbers including perfect squares, cubes, and factors and multiples with positive rational numbers (number theory).

2.1.K1.b 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), e.g., 1/4, 1/2, 3/4,

2.1.K1.c geometric figures;

2.1.K1.d measurements

2.1.K1.e things related to daily life;

2.1.K1.f variables and simple expressions, e.g., 1 ?? x, 2 ?? x, 3 ?? x, 4 ?? x,...; or x, x squared, x cubed,...

2.1.K2 generates and explains a pattern.

2.1.K3 generates a pattern limited to two operations (addition, subtraction, multiplication, division, exponents) when given the rule for the nth term, e.g., the nth term is n? + 1, find the first 4 terms beginning with n = 1; the terms are 2, 5, 10, and 17.

2.1.K4 states the rule to find the nth term of a pattern using explicit symbolic notation, e.g., given 2, 5, 8, 11, ??; find the rule for the nth term, the rule is 3n ??1.

2.1.K5 describes the pattern when given a table of linear values and plots the ordered pairs on a coordinate plane, e.g., in the table below, the pattern could be described as the x-coordinates are increasing by three, while the y-coordinates are increasing by 6, or the x is doubled and one is added to find the y. x = 2, 5, 8, 11 when y = 5, 11, 17, 23.

2.2 The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.

2.2.K1 identifies independent and dependent variables within a given situation.

2.2.K2 simplifies algebraic expressions in one variable by combining like terms or using the distributive property, e.g., -3(x ?? 4) is the same as -3x + 12.

2.2.K3 solves:

2.2.K3.a one- and two-step linear equations in one variable with rational number coefficients and constants intuitively and/or analytically;

2.2.K3.b one-step linear inequalities in one variable with rational number coefficients and constants intuitively, analytically, and graphically;

2.2.K3.c systems of given linear equations with whole number coefficients and constants graphically.

2.2.K4 knows and describes the mathematical relationship between ratios, proportions, and percents and how to solve for a missing monomial or binomial term in a proportion, e.g., 2/5 = 1/(x+2)

2.2.K5 represents and solves algebraically:

2.2.K5.a the number when a percent and a number are given,

2.2.K5.b what percent one number is of another number,

2.2.K5.c percent of increase or decrease, e.g., the price of a loaf of bread is $2.00. With a coupon, the cost is $1.00. What is the percent of decrease?

2.2.K6 evaluates formulas using substitution.

2.3 The student recognizes, describes, and analyzes, constant, linear, and nonlinear relationships in a variety of situations.

2.3.K1 recognizes and examines constant, linear, and nonlinear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology.

2.3.K2 knows and describes the difference between constant, linear, and nonlinear relationships.

2.3.K3 explains the concepts of slope and x- and y-intercepts of a line.

2.3.K4 recognizes and identifies the graphs of constant and linear functions.

2.3.K5 identifies ordered pairs from a graph, and/or plots ordered pairs using a variety of scales for the x- and y-axis.

2.4 The student generates and uses mathematical models to represent and justify mathematical relationships found in a variety of situations.

2.4.K1 knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships. Mathematical models include:

2.4.K1.a process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations

2.4.K1.b 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;

2.4.K1.c 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;

2.4.K1.d factor trees to model least common multiple, greatest common factor, and prime factorization;

2.4.K1.e equations and inequalities to model numerical relationships;

2.4.K1.f function tables to model numerical and algebraic relationships

2.4.K1.g coordinate planes to model relationships between ordered pairs and linear equations and inequalities;

2.4.K1.h two- and three-dimensional geometric models (geoboards, dot paper, nets, or solids) and real-world objects to model perimeter, area, volume, surface area, and properties of two-and three-dimensional figures;

2.4.K1.i scale drawings to model large and small real-world objects;

2.4.K1.j geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability;

2.4.K1.k frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, and histograms to organize and display data;

2.4.K1.l Venn diagrams to sort data and to show relationships.

3.1 The student recognizes geometric figures and compares their properties in a variety of situations.

3.1.K1 recognizes and compares properties of two- and three-dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology.

3.1.K2 discusses properties of triangles and quadrilaterals related to:

3.1.K2.a sum of the interior angles of any triangle is 180?;

3.1.K2.b sum of the interior angles of any quadrilateral is 360?;

3.1.K2.c parallelograms have opposite sides that are parallel and congruent, opposite angles are congruent;

3.1.K2.d rectangles have angles of 90?, sides may or may not be equal;

3.1.K2.e rhombi have all sides equal in length, angles may or may not be equal;

3.1.K2.f squares have angles of 90?, all sides congruent;

3.1.K2.g trapezoids have one pair of opposite sides parallel and the other pair of opposites sides are not parallel;

3.1.K2.h kites have two distinct pairs of adjacent congruent sides.

3.1.K3 recognizes and describes the rotational symmetries and line symmetries that exist in two-dimensional figures, e.g., draw a picture with a line of symmetry in it. Explain why it is a line of symmetry.

3.1.K4 recognizes and uses properties of corresponding parts of similar and congruent triangles and quadrilaterals to find side or angle measures using standard notation for similarity (~) and congruence (approximately =).

3.1.K5 knows and describes Triangle Inequality Theorem to determine if a triangle exists.

3.1.K6 uses the Pythagorean theorem to:

3.1.K6.a determine if a triangle is a right triangle,

3.1.K6.b find a missing side of a right triangle where the lengths of all three sides are whole numbers.

3.1.K7 recognizes and compares the concepts of a point, line, and plane.

3.1.K8 describes the intersection of plane figures, e.g., two circles could intersect at no point, one point, two points, or all points.

3.1.K9 describes and explains angle relationships:

3.1.K9.a when two lines intersect including vertical and supplementary angles;

3.1.K9.b when formed by parallel lines cut by a transversal including corresponding, alternate interior, and alternate exterior angles.

3.1.K10 recognizes and describes arcs and semicircles as parts of a circle and uses the standard notation for arc and circle.

3.2 The student estimates, measures, and uses geometric formulas in a variety of situations.

3.2.K1 determines and uses rational number approximations (estimations) for length, width, weight, volume, temperature, time, perimeter, area, and surface area using standard and nonstandard units of measure.

3.2.K2 selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, perimeter, area, surface area, and angle measurements.

3.2.K3 converts within the customary system and within the metric system.

3.2.K4 estimates the measure of a concrete object in one system given the measure of that object in another system and the approximate conversion factor, e.g., a mile is about 2.2 kilometers; how far is 2 miles?

3.2.K5 uses given measurement formulas to find:

3.2.K5.a area of parallelograms and trapezoids;

3.2.K5.b surface area of rectangular prisms, triangular prisms, and cylinders;

3.2.K5.c volume of rectangular prisms, triangular prisms, and cylinders;

3.2.K6 recognizes how ratios and proportions can be used to measure inaccessible objects, e.g., using shadows to measure the height of a flagpole.

3.2.K7 calculates rates of change, e.g., speed or population growth.

3.3 The student recognizes and applies transformations on geometric figures in a variety of situations.

3.3.K1 identifies, describes, and performs single and multiple transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a two-dimensional figure.

3.3.K2 describes a reflection of a given two-dimensional figure that moves it from its initial placement (preimage) to its final placement (image) in the coordinate plane over the x- and y-axis.

3.3.K3 draws:

3.3.K3.a three-dimensional figures from a variety of perspectives (top, bottom, sides, corners);

3.3.K3.b a scale drawing of a two-dimensional figure;

3.3.K3.c a two-dimensional drawing of a three-dimensional figure.

3.3.K4 determines where and how an object or a shape can be tessellated using single or multiple transformations

3.4 The student uses an algebraic perspective to examine the geometry of two-dimensional figures in a variety of situations.

3.4.K1 uses the coordinate plane to:

3.4.K1.a list several ordered pairs on the graph of a line and finds the slope of the line;

3.4.K1.b recognize that ordered pairs that lie on the graph of an equation are solutions to that equation;

3.4.K1.c recognize that points that do not lie on the graph of an equation are not solutions to that equation;

3.4.K1.d determine the length of a side of a figure drawn on a coordinate plane with vertices having the same x- or y-coordinates;

3.4.K1.e solve simple systems of linear equations.

3.4.K2 uses a given linear equation with integer coefficients and constants and an integer solution to find the ordered pairs, organizes the ordered pairs using a T-table, and plots the ordered pairs on a coordinate plane.

3.4.K3 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.

4.1 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.

4.1.K1 knows and explains the difference between independent and dependent events in an experiment, simulation, or situation.

4.1.K2 identifies situations with independent or dependent events in an experiment, simulation, or situation, e.g., there are three marbles in a bag. If you draw one marble and give it to your brother, and another marble and give it to your sister, are these independent events or dependent events?

4.1.K3 finds the probability of a compound event composed of two independent events in an experiment, simulation, or situation, e.g., what is the probability of getting two heads, if you toss a dime and a quarter?

4.1.K4 finds the probability of simple and/or compound events using geometric models (spinners or dartboards).

4.1.K5 finds the odds of a desired outcome in an experiment or simulation and expresses the answer as a ratio (2/3 or 2:3 or 2 to 3).

4.1.K6 describes the difference between probability and odds.

4.2 The student collects, organizes, displays, explains, and interprets numerical (rational) and non-numerical data sets in a variety of situations.

4.2.K1 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:

4.2.K1.a frequency tables;

4.2.K1.b bar, line, and circle graphs;

4.2.K1.c Venn diagrams or other pictorial displays;

4.2.K1.d charts and tables;

4.2.K1.e stem-and-leaf plots (single and double);

4.2.K1.f scatter plots;

4.2.K1.g box-and-whiskers plots;

4.2.K1.h histograms.

4.2.K2 recognizes valid and invalid data collection and sampling techniques.

4.2.K3 determines and explains the measures of central tendency (mode, median, mean) for a rational number data set.

4.2.K4 determines and explains the range, quartiles, and interquartile range for a rational number data set.

4.2.K5 explains the effects of outliers on the median, mean, and range of a rational number data set.

4.2.K6 makes a scatter plot and draws a line that approximately represents the data, determines whether a correlation exists, and if that correlation is positive, negative, or that no correlation exists.

Kansas Standards - Grade 8 Math