We believe that all students can achieve success in Mathematics which is an integral part of all curriculum areas. Mathematical instruction is a part of everyday routines and should involve real-life situations, current technology, and hands-on experiences. We believe that Mathematical instruction should cultivate creative problem solving and critical thinking skills and should be differentiated to accommodate individual learning styles. Standard 1: Uses a variety of strategies in the problem-solving process Benchmark 1. Understands how to break a complex problem into simpler parts or use a similar problem type to solve a problem Knowledge/skill statements 1. Breaks problems into smaller parts Benchmark 2. Uses a variety of strategies to understand problem-solving situations and processes (e.g., considers different strategies and approaches to a problem, restates problem from various perspectives) Knowledge/skill statements 1. Restates a problem in a number of ways Benchmark 3. Understands that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages Knowledge/skill statements 1. Works backward from a solution 2. Identifies a similar problem type Benchmark 4. Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions Knowledge/skill statements 1. Identifies multiple possible solutions 2. Sets expectations for possible solutions 3. Determines what information is needed to solve problem Benchmark 5. Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms Knowledge/skill statements 1. Translates a verbal problem into a graph 2. Translates a verbal problem into a pictorial representation 3. Translates a verbal problem into a concrete representation Benchmark 6. Generalizes from a pattern of observations made in particular cases, makes conjectures, and provides supporting arguments for these conjectures (i.e., uses inductive reasoning) Knowledge/skill statements 1. Provides supportive arguments for conjectures 2. Generalizes from a pattern of observations 3. Makes conjectures from generalizations Benchmark 7. Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems (i.e., uses informal deductive methods) Knowledge/skill statements 1. Understands the concept of a logical argument 2. Justifies solution methods through logical argument Benchmark 8. Understands the role of written symbols in representing mathematical ideas and the use of precise language in conjunction with the special symbols of mathematics Knowledge/skill statements 1. Understands how written symbols represent mathematical ideas Benchmark 9. Uses a variety of reasoning processes (e.g., reasoning from a counter example, using proportionality) to model and to solve problems Knowledge/skill statements 1. Uses counter examples to model problems 2. Uses proportionality to model problems 3. Uses counter examples to solve problems 4. Uses proportionality to solve problems Standard 2: Understands and applies basic and advanced properties of the concepts of numbers Benchmark 1. Understands the relationships among equivalent number representations (e.g., whole numbers, positive and negative integers, fractions, ratios, decimals, percents, scientific notation, exponentials) and the advantages and disadvantages of each type of representation Knowledge/skill statements 1. Understands the equivalence of decimals and percents 2. Understands the equivalence of whole numbers and fractions 3. Understands the equivalence of fractions and ratios 4. Understands the equivalence of whole numbers and ratios 5. Understands the equivalence of whole numbers and decimals 6. Understands the equivalence of whole numbers and percents 7. Understands the equivalence of whole numbers and scientific notation 8. Understands the equivalence of integers and fractions 9. Understands the equivalence of integers and ratios 10. Understands the equivalence of integers and decimals 11. Understands the equivalence of integers and percents 12. Understands the equivalence of integers and scientific notation 13. Understands the equivalence of integers and exponents 14. Understands the equivalence of fractions and decimals 15. Understands the equivalence of ratios and decimals 16. Understands the equivalence of ratios and percents 17. Understands the equivalence of decimals and scientific notation 18. Understands the equivalence of decimals and exponents 19. Understands the equivalence of scientific notation and exponents 20. Understands the advantages and disadvantages of different number representations Benchmark 2. Understands the characteristics and properties (e.g., order relations, relative magnitude, base-ten place values) of the set of rational numbers and its subsets (e.g., whole numbers, fractions, decimals, integers) Knowledge/skill statements 1. Understands the concept of "less than" with rational numbers 2. Understands the concept of "greater than" with rational numbers 3. Understands the concept of base 10 4. Understands the concept of rational numbers Benchmark 3. Understands the role of positive and negative integers in the number system Knowledge/skill statements 1. Understands the concept of positive integers 2. Understands the concept of negative integers Benchmark 4. Uses number theory concepts (e.g., divisibility and remainders, factors, multiples, prime, relatively prime) to solve problems Knowledge/skill statements 1. Determines whether a problem can be solved by using the concept of divisibility 2. Determines whether a problem can be solved by using the concept of multiples 3. Determines whether a problem can be solved by using the concept of factors 4. Determines whether a problem can be solved by using the concept of prime numbers Benchmark 5. Understands the characteristics and uses of exponents and scientific notation Knowledge/skill statements 1. Understands the concept of exponents 2. Understands the concept of scientific notation Benchmark 6. Understands the structure of numeration systems that are based on numbers other than 10 (e.g., base 60 for telling time and measuring angles, Roman numerals for dates and clock faces) Benchmark 7. Understands the concepts of ratio, proportion, and percent and the relationships among them Knowledge/skill statements 1. Understands the concept of proportion 2. Understands the concept of percent 3. Understands the concept of ratio 4. Understands the relationship between ratio and proportion 5. Understands the relationship between ratio and percent 6. Understands the relationship between proportion and percent Standard 3: Uses basic and advanced procedures while performing the processes of computation Benchmark 1. Adds, subtracts, multiplies, and divides integers, and rational numbers Knowledge/skill statements 1. Adds rational numbers 2. Subtracts rational numbers 3. Multiplies rational numbers 4. Divides rational numbers 5. Adds integers 6. Subtracts integers 7. Multiplies integers 8. Divides integers 9. Understands the concept of integer addition 10. Understands the concept of integer subtraction 11. Understands the concept of integer multiplication 12. Understands the concept of integer division Benchmark 2. Adds and subtracts fractions with unlike denominators; multiples and divides fractions Knowledge/skill statements 1. Adds fractions with unlike denominators 2. Subtracts fractions with unlike denominators 3. Multiplies fractions 4. Divides fractions Benchmark 3. Understands exponentiation of rational numbers and root-extraction (e.g., squares and square roots, cubes and cube roots) Knowledge/skill statements 1. Understands the concept of square roots 2. Understands the concept of roots 3. Understands the concept of cubes 4. Understands the concept of squares Benchmark 4. Selects and uses appropriate computational methods (e.g., mental, paper and pencil, calculator, computer) for a given situation Knowledge/skill statements 1. Knows when paper/pencil is best method for computation 2. Uses calculator for computations 3. Knows when calculator is best method for computation 4. Knows when mental computation is the best method 5. Uses paper/pencil for computation 6. Uses mental computation Benchmark 5. Understands the correct order of operations for performing arithmetic computations Knowledge/skill statements 1. Knows the order of operations Benchmark 6. Uses proportional reasoning to solve mathematical and real-world problems (e.g., involving equivalent fractions, equal ratios, constant rate of change, proportions, percents) Knowledge/skill statements 1. Determines whether a problem can be solved by calculating percents 2. Understands the concept of a constant rate of change 3. Determines whether a problem can be solved using the concept of a constant rate of change 4. Determines whether a problem can be solved using the concept of proportions 5. Determines whether a problem can be solved using the concept of equivalent fractions 6. Understands the concept of equivalent fractions 7. Understands the concept of equal ratios 8. Understands the concept of mathematical proportion 9. Determines whether a problem can be solved using the concept of equal ratios Benchmark 7. Understands the properties of operations with rational numbers (e.g., distributive property, commutative and associative properties of addition and multiplication, inverse properties, identity properties) Knowledge/skill statements 1. Understands the associative property of addition 2. Understands the commutative property of multiplication 3. Understands the commutative property of addition 4. Understands the concept of multiplicative inverse 5. Understands the distributive property of multiplication 6. Understands the associative property of multiplication 7. Understands the concept of additive inverse 8. Understands how the properties of operations relate to rational numbers 9. Understands the concept of additive identity 10. Understands the concept of multiplicative identity Benchmark 8. Knows when an estimate is more appropriate than an exact answer for a variety of problem situations Knowledge/skill statements 1. Understands situations in which an estimate is more appropriate than a calculation Benchmark 9. Understands how different algorithms work for arithmetic computations and operations Knowledge/skill statements 1. Understands how different algorithms work for multiplication 2. Understands how different algorithms work for division 3. Understands how different algorithms work for subtraction 4. Understands how different algorithms work for addition Standard 4: Understands and applies basic and advanced properties of the concepts of measurement Benchmark 1. Understands the basic concept of rate as a measure (e.g., miles per gallon) Knowledge/skill statements 1. Understands the concept of rate 2. Knows types of rates (e.g., miles per gallon) Benchmark 2. Solves problems involving perimeter (circumference) and area of various shapes (e.g., parallelograms, triangles, circles) Knowledge/skill statements 1. Determines whether a problem can be solved by calculating the perimeter 2. Determines whether a problem can be solved by calculating the area 3. Understands the concept of perimeter of a parallelogram 4. Understands the concept of perimeter of a triangle 5. Understands the concept of perimeter (circumference) of a circle 6. Understands the concept of area of a parallelogram 7. Understands the concept of area of a triangle 8. Understands the concept of area of a circle Benchmark 3. Understands the relationships among linear dimensions, area, and volume and the corresponding uses of units, square units, and cubic units of measure Knowledge/skill statements 1. Understands the concept of a square unit 2. Understands the use of square units in measuring area 3. Understands the concept of a cubic unit 4. Understands use of cubic units in measuring volume 5. Understands relationships between length, width, height and volume 6. Understands relationships between length, width, and area Benchmark 4. Solves problems involving units of measurement and converts answers to a larger or smaller unit within the same system (i.e., standard or metric) Knowledge/skill statements 1. Converts from larger to smaller units in the metric system 2. Converts from smaller to larger units in the metric system 3. Converts from smaller to larger units in the standard system 4. Converts from larger to smaller units in the standard system Benchmark 5. Understands the concepts of precision and significant digits as they relate to measurement (e.g., how units indicate precision) Knowledge/skill statements 1. Understands the concept of precision as it relates to measurement 2. Understands how units indicate precision Benchmark 6. Selects and uses appropriate units and tools, depending on degree of accuracy required, to find measurements for real-world problems Knowledge/skill statements 1. Selects an appropriate measurement tool based on needed accuracy 2. Selects an appropriate measurement unit based on needed accuracy Benchmark 7. Understands formulas for finding measures (e.g., area, volume, surface area) Knowledge/skill statements 1. Understands the formula for area 2. Computes area using the formula 3. Understands the formula for volume 4. Computes volume using the formula 5. Understands the formula for circumference 6. Computes circumference using the formula 7. Understands the formula for surface area 8. Computes surface area using the formula Benchmark 8. Selects and uses appropriate estimation techniques (e.g., overestimate, underestimate, range of estimates) to solve real-world problems Knowledge/skill statements 1. Uses a range of estimations effectively 2. Understands concept of overestimation 3. Determines whether a problem can be solved by using overestimation 4. Understands the concept of underestimation 5. Uses underestimation effectively 6. Determines whether a problem can be solved by using underestimation 7. Understands the concept of estimation for measurements Benchmark 9. Understands procedures for basic indirect measurements (e.g., using grids to estimate area of irregular figures) Knowledge/skill statements 1. Understands the concept of indirect measurement 2. Understands the concept of grids 3. Uses grids to estimate area Standard 5: Understands and applies basic and advanced properties of the concepts of geometry Benchmark 1. Uses geometric methods (i.e., an unmarked straightedge and a compass using an algorithm) to complete basic geometric constructions (e.g., perpendicular bisector of a line segment, angle bisector) Knowledge/skill statements 1. Uses a straight edge and compass to construct perpendicular bisectors 2. Understands the concept of geometric constructions 3. Understands the concept of a perpendicular bisector Benchmark 2. Understands the defining properties of three-dimensional figures (e.g., a cube has edges with equal lengths, faces with equal areas and congruent shapes, right angle corners) Knowledge/skill statements 1. Understands the concept of a three-dimensional figure 2. Understands the properties of three-dimensional figures 3. Understands that the properties of a cube include faces that are congruent, faces that have equal areas, edges with equal lengths, and right angle corners Benchmark 3. Understands the defining properties of triangles (e.g., the sum of the measures of two sides of a triangle must be greater than the measure of the third side) Knowledge/skill statements 1. Understands that the sum of two sides of a triangle must be greater than the measure of the third side 2. Understands the properties of triangles Benchmark 4. Understands geometric transformations of figures (e.g., rotations, translations, dilations) Knowledge/skill statements 1. Understands the concept of rotations 2. Understands the concept of transformation 3. Understands the concept of translation 4. Understands the concept of dilations Benchmark 5. Understands the relationships between two- and three-dimensional representations of a figure (e.g., scale drawings, blueprints, planar cross sections) Knowledge/skill statements 1. Understands the relationship between two- and three-dimensional representations of figures Benchmark 6. Understands the mathematical concepts of similarity (e.g., scale, proportion, growth rates) and congruency Knowledge/skill statements 1. Understands the concept of similarity 2. Understands the concept of growth rate 3. Understands the concept of congruence 4. Understands the concept of geometric proportion 5. Understands the concept of growth rates 6. Understands the concept of scale Benchmark 7. Understands the basic concept of the Pythagorean theorem Knowledge/skill statements 1. Understands the concept of the Pythagorean theorem Standard 6: Understands and applies basic and advanced concepts of statistics and data analysis Benchmark 1. Understands basic characteristics of measures of central tendency (i.e., mean, mode, median) Knowledge/skill statements 1. Understands the concept of central tendency 2. Understands the concept of the mean 3. Computes mean 4. Understands the concept of the mode 5. Computes mode 6. Understands the concept of the median 7. Computes median Benchmark 2. Understands basic characteristics of frequency and distribution (e.g., range, varying rates of change, gaps, clusters) Knowledge/skill statements 1. Understands the concept of varying rates of change 2. Understands the concept of range 3. Computes range Benchmark 3. Reads and interprets data in charts, tables, and plots (e.g., stem-and-leaf, box-and-whiskers, scatter) Knowledge/skill statements 1. Reads scatter plots 2. Interprets stem-and-leaf plots 3. Reads stem-and-leaf plots 4. Interprets data in tables Benchmark 4. Uses data and statistical measures for a variety of purposes (e.g., formulating hypotheses, making predictions, testing conjectures) Knowledge/skill statements 1. Test conjectures using data 2. Makes predictions using data 3. Formulates hypotheses using data Benchmark 5. Organizes and displays data using tables, graphs (e.g., line, circle, bar), frequency distributions, and plots (e.g., stem-and-leaf, box-and-whiskers, scatter) Knowledge/skill statements 1. Constructs scatter plots using data 2. Constructs stem-and-leaf plots using data Benchmark 6. Understands faulty arguments, common errors, and misleading presentations of data Knowledge/skill statements 1. Understands faulty arguments from data 2. Understands misleading presentations of data 3. Understands common errors with data Benchmark 7. Understands that the same set of data can be represented using a variety of tables, graphs, and symbols and that different modes of representation often convey different messages (e.g., variation in scale can alter a visual message) Knowledge/skill statements 1. Understands the same data set can be displayed in different ways 2. Understands the influence of scale on a data display 3. Understands that the manner in which data is displayed affects interpretation 4. Understands that data can be represented in tables 5. Understands that data can be represented in graphs 6. Understands that data can be represented using symbols Standard 7: Understands and applies basic and advanced concepts of probability Benchmark 1. Determines probability using mathematical/theoretical models (e.g., table or tree diagram, area model, list, sample space) Knowledge/skill statements 1. Uses tables to determine probability 2. Uses tree diagrams to determine probability 3. Understands the concept of tree diagrams Benchmark 2. Determines probability using simulations or experiments Knowledge/skill statements 1. Understands the concept of a simulation 2. Understands the concept of an experiment 3. Uses simulations to determine probability 4. Uses experiments to determine probability Benchmark 3. Understands how predictions are based on data and probabilities (e.g., the difference between predictions based on theoretical probability and experimental probability) Knowledge/skill statements 1. Understands predictions based on experimental probability 2. Understands predictions based on theoretical probability 3. Contrasts predictions based on theoretical vs experimental probability Benchmark 4. Understands that the measure of certainty in a given situation depends on a number of factors (e.g., amount of data collected, what is known about the situation, how current data are) Knowledge/skill statements 1. Understands that the certainty of conclusion(s) depend upon many factors 2. Understands that the certainty of conclusion(s) depends upon the currency of data 3. Understands that the certainty of conclusion(s) depend upon what is known about a situation 4. Understands that the certainty of conclusion(s) depend upon the amount of data Benchmark 5. Understands the relationship between the numerical expression of a probability (e.g., fraction, percentage, odds) and the events that produce these numbers Knowledge/skill statements 1. Understands the concept of percents representing probability 2. Understands the concept of fractions representing probability 3. Understands the concept of decimals representing probability Standard 8: Understands and applies basic and advanced properties of functions and algebra Benchmark 1. Knows that an expression is a mathematical statement using numbers and symbols to represent relationships and real-world situations (e.g., equations and inequalities with or without variables) Knowledge/skill statements 1. Understands the concept of a mathematical expression 2. Understands the role of numbers in math expressions 3. Understands the role of variables in math expressions 4. Understands the concept of an inequality 5. Understands the role of symbols in math expressions Benchmark 2. Understands that a variable can be used in many ways (e.g., as a placeholder for a specific unknown, such as x + 8 = 13; as a representative of a range of values, such as 4t + 7) Knowledge/skill statements 1. Understands that variables can be used in many ways 2. Understands variables can be placeholders for specific values 3. Understands variables can represent ranges of values Benchmark 3. Understands various representations (e.g., tables, graphs, verbal descriptions, algebraic expressions, Venn diagram) of patterns and functions and the relationships among them Knowledge/skill statements 1. Understands the concept of a Venn diagram 2. Represents patterns using Venn diagrams 3. Understands how Venn diagrams can represent patterns 4. Understands how algebraic expressions can represent patterns 5. Represents patterns using algebraic expressions 6. Understands how verbal descriptions can represent patterns 7. Represents patterns using tables 8. Understands how tables can represent patterns 9. Understands how graphs can represent patterns 10. Represents patterns using graphs 11. Represents patterns using verbal descriptions 12. Understands the relationship between tables and graphs 13. Understands the relationship between tables and verbal descriptions 14. Understands the relationship between tables and algebraic expressions 15. Understands the relationship between tables and Venn diagrams 16. Understands the relationship between graphs and verbal descriptions 17. Understands the relationship between graphs and algebraic expressions 18. Understands the relationship between graphs and Venn diagrams 19. Understands the relationship between verbal descriptions and algebraic expressions 20. Understands the relationship between verbal descriptions and Venn diagrams Benchmark 4. Solves linear equations using concrete, informal, and formal methods (e.g., using properties, graphing ordered pairs, using slope-intercept form) Knowledge/skill statements 1. Understands the concept of a linear equation 2. Solves linear equations using knowledge of properties 3. Solves linear equations by graphing ordered pairs 4. Solves linear equations using slope-intercept form 5. Solves linear equations using concrete methods 6. Solves linear equations using informal methods 7. Solves linear equations using formal methods Benchmark 5. Solves simple inequalities and non-linear equations with rational number solutions, using concrete and informal methods Knowledge/skill statements 1. Solves simple inequalities using informal methods 2. Understands the concept of a non-linear equation 3. Solves simple non-linear equations using informal methods 4. Understands the concept of an inequality 5. Solves inequalities using concrete methods 6. Solves non-linear equations using concrete methods Benchmark 6. Understands special values (e.g., minimum and maximum values, x- and y-intercepts, slope, constant ratio or difference) of patterns, relationships, and functions Knowledge/skill statements 1. Understands x- and y-intercepts 2. Understands the concept of slope 3. Understands constant ratio or difference 4. Understands the concept of maximum 5. Understands the concept of minimum Benchmark 7. Understands basic operations (e.g., combining like terms, expanding, substituting for unknowns) on algebraic expressions Knowledge/skill statements 1. Combines like terms of an algebraic expression 2. Understands combining like terms of an algebraic expression 3. Expands algebraic expressions 4. Understands substituting for unknowns in an algebraic expression 5. Understands expanding algebraic expressions 6. Substitutes for unknowns in an algebraic expression 7. Understands operations on algebraic expressions Benchmark 8. Uses the rectangular coordinate system to model and to solve problems Knowledge/skill statements 1. Determines whether a problem can be modeled by using rectangular coordinates 2. Determines whether a problem can be solved by using rectangular coordinates Benchmark 9. Solves simple systems of equations graphically Knowledge/skill statements 1. Solves simple systems of equations graphically 2. Understands the concept of systems of equations Benchmark 10. Understands the properties of arithmetic and geometric sequences (i.e., linear and exponential patterns) Knowledge/skill statements 1. Understands the properties of arithmetic sequences 2. Understands the concept of a linear sequence 3. Understands the concept of an exponential sequence Standard 9: Understands the general nature and uses of mathematics Benchmark 1. Understands that mathematics has been helpful in practical ways for many centuries Knowledge/skill statements 1. Understands that mathematics has helped humankind for centuries Benchmark 2. Understands that mathematicians often represent real things using abstract ideas like numbers or lines; they then work with these abstractions to learn about the things they represent Knowledge/skill statements 1. Understands that mathematicians often represent real things using abstract ideas 2. Understands that mathematicians work with abstract symbols to learn about the concept those symbols represent |
6-8 Mathematics |